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Series espacio-temporales, NDVI, MODIS, HMM, Componentes cuasiperiódicos, Fenología, Detección de zonas incendiadas
Miguel Antonio García Ferrández
Departamento de Matemática Aplicada
Análisis de series de datos de
teledetección de índices de vegetación
Memoria presentada por el licenciado en Ciencias
Matemáticas Miguel Antonio García Ferrández para optar al
grado de Doctor por la Universidad de Alicante
Alicante, 2015
Francisco Rodríguez Mateo, Profesor Titular de la Universidad de
Alicante,
HACE CONSTAR:
Que el trabajo descrito en la presente memoria, titulado: “Análisis de
series de datos de teledetección de índices de vegetación”, ha sido
realizado bajo su dirección por Miguel Antonio García Ferrández en la
Universidad de Alicante y reúne todos los requisitos necesarios para su
aprobación como Tesis Doctoral.
Alicante, diciembre de 2014
Dr. Francisco Rodríguez Mateo
It’s not the same to solve for a hundred, as to solve for a thousand
E.W. Dijkstra, in
Structured Programming
O.-J. Dahl, E.W. Dijkstra, C.A.R. Hoare
90% of the scheduled time solves 90% of the problem;
the remaining 10% of the problem takes another 90% of the time
Murphys Laws of Computer Science
La diferencia entre manjar y bazofia son tres minutos de cocción
Primera Ley de García, de la Cocina con Pasta
i
Índice
Agradecimientos iii
Capítulo 1
Objetivos y resumen de la tesis
1
Capítulo 2
Modelos ocultos de Markov para la determinación de parámetros
fenológicos
7
Capítulo 3
Ajuste de funciones lineales a trozos a series temporales de índices de
vegetación
23
Capítulo 4
Análisis tiempo-frecuencia de series de datos de índices de vegetación
mediante componentes cuasi-periódicos
49
Capítulo 5
Detección de áreas afectadas por incendios forestales
67
Conclusiones generales
103
Anexo
Conceptos básicos sobre modelos ocultos de Markov
109
iii
Agradecimientos
El trabajo desarrollado en esta tesis se inició en el marco del proyecto “Análisis
espacio-temporal de datos de índices de vegetación derivados del sistema de
teledetección MODIS. Aplicación a la determinación de parámetros fenológicos y al
seguimiento de la regeneración post-incendio en la Comunidad Valenciana”
(GVPRE/2008/310), financiado por la Generalitat Valenciana.
El grupo de investigación de “Análisis de datos y modelización de procesos en Biología
y Geociencias”, en cuyo seno se ha realizado el trabajo, ha recibido financiación de la
Universidad de Alicante a través de las sucesivas convocatorias anuales de ayudas a
grupos de investigación (VIGROB-162).
Algunos de los trabajos presentados en esta memoria se han desarrollado en
colaboración con investigadores de otros grupos de investigación, de la Universidad de
Alicante y de otras universidades y centros de investigación. En los apartados de
Agradecimientos de los capítulos correspondientes se especifican los proyectos en
cuyo marco se han llevado a cabo estas colaboraciones y las correspondientes
entidades financiadoras.
Capítulo 1:
Objetivos y estructura de la tesis
Objetivos y estructura de la tesis
3
El análisis de la dinámica de la vegetación a partir de datos de teledetección constituye
actualmente una herramienta esencial en agricultura, ecología y otras ciencias
ambientales. Desde el año 2000 se dispone de los datos de observación por satélite de
distintas variables de ecosistemas terrestres proporcionados por los sensores MODIS
(Moderate Resolution Imaging Spectroradiometers). Entre los productos derivados, se
encuentran índices de vegetación como los denominados NDVI (Normalized Difference
Vegetation Index) y EVI (Enhanced Vegetation Index), disponibles cada 16 días con una
resolución de 250mx250m, que pueden ser utilizados, entre otras aplicaciones, para
analizar la evolución de cultivos, realizar cartografías de tipos funcionales de
vegetación o estudiar diversos parámetros fenológicos.
El desarrollo de métodos efectivos de análisis de este tipo de series de datos espacio-
temporales representa una cuestión clave en las distintas aplicaciones y constituye el
objetivo general de este trabajo de tesis. Entre las aplicaciones que pueden derivarse,
se plantea la utilización de los métodos desarrollados en dos aspectos específicos de
interés para el estudio del medio ambiente en la Comunidad Valenciana, como son la
determinación de parámetros fenológicos de distintos tipos de vegetación y el análisis
de la dinámica de la regeneración de la vegetación después de incendios forestales.
Los modelos ocultos de Markov (HMMs), que ya habían sido utilizados previamente
por el grupo de “Análisis de datos y modelización de procesos en Biología y
Geociencias” de la Universidad de Alicante en el contexto del análisis del patrón
espacial de la vegetación, ofrecen la posibilidad de incorporar en la estructura del
modelo utilizado información a priori sobre la dinámica del tipo de vegetación que se
quiere modelizar, permitiendo así el desarrollo de modelos específicos para distintos
tipos de vegetación y que estén adaptados a las condiciones ambientales locales.
En el Capítulo 2 se explora la utilización de los HMMs para la determinación de
parámetros fenológicos, mediante su aplicación en dos zonas contrastadas de la
Cominidad Valenciana. El objetivo es poder definir, de forma más precisa que con los
ajustes generales de tipo polinómico usuales, indicadores fenológicos como las fechas
de inicio y fin de los periodos de aumento y disminución de biomasa fotosintética,
permitiendo con ello la aplicación de este tipo de análisis para estudiar de las posibles
modificaciones en estos parámetros en relación con covariables de interés (p. ej.,
datos de precipitación, temperatura o altitud) y las posibles tendencias que puedan
estar relacionadas con fenómenos de cambio global.
Capítulo 1
4
El ajuste a una serie de datos de índices de vegetación como el NDVI de modelos de
tipo HMMs supone la existencia de distintos estados de la vegetación, en distintos
periodos de tiempo de la serie, caracterizados, cada uno de ellos, por una cierta
distribución de probabilidad que determina la variación en los valores observados de
NDVI a lo largo del tiempo. Un modelo de referencia simple, que correspondería
esencialmente a considerar los valores medios de incremento o disminución del NDVI,
consistiría en un modelo continuo lineal a trozos, de modo que según el estado de la
vegetación se tendría una variación media constante del NDVI, con cambios de
pendiente entre segmentos contiguos al cambiar de un estado al siguiente.
En el Capítulo 3 se considera el problema del ajuste de modelos lineales a trozos a
series de datos con un alto número de puntos y con la posibilidad de un alto número, a
priori desconocido, de puntos de cambio. El problema se aborda mediante un
algoritmo de tipo iterativo, que puede ser adaptado en función del tipo de información
previa que se tenga sobre el sistema analizado o de la complejidad del modelo que se
quiera ajustar.
En el estudio de la dinámica de la vegetación es preciso separar las posibles
tendencias, debidas por ejemplo a la regeneración natural tras un incendio forestal o
tras un periodo de estrés hídrico, de las oscilaciones estacionales, que pueden no estar
definidas de forma precisa. Una característica que es preciso tener en cuenta de los
datos de teledetección como los suministrados por MODIS es el enorme volumen de
datos de los que puede disponerse para los análisis, pues las series temporales están
repetidas para cada pixel, en general con una alta correlación entre píxeles próximos.
Sólo si se dispone de métodos especialmente eficientes para tratar de forma conjunta
con un número elevado de datos es posible aprovechar esta redundancia espacial,
pues de otra forma, incluso para los análisis en teoría más simples, se termina abocado
al análisis de series exclusivamente temporales utilizando valores correspondientes a
promedios espaciales.
En el Capítulo 4 se presenta un modelo para series de datos que incluyen
componentes seculares y componentes cíclicas no constantes, denominadas
componentes cuasi-periódicas. El objetivo es obtener estimaciones más ajustadas a la
realidad que las proporcionadas por los modelos de análisis espectral con
componentes periódicas constantes, permitiendo con ello el análisis de las relaciones
entre las variaciones en los parámetros que definen las variaciones estacionales y las
covariables o factores ecológicos de interés.
Objetivos y estructura de la tesis
5
Los incendios forestales constituyen una de las perturbaciones naturales comunes en
los ecosistemas mediterráneos y uno de los problemas ambientales principales en
zonas como el sureste español, donde la frecuencia de incendios se ha incrementado
notablemente en el último medio siglo, debido fundamentalmente a factores
relacionados con la actividad humana, como el aumento de combustibles debido al
abandono de zonas agrícolas o el incremento de igniciones provocadas o debidas a
descuidos o actividades agrícolas o recreativas.
Los métodos de análisis de la dinámica de la vegetación, como los mencionados
anteriormente, pueden ser aplicados a zonas incendiadas, permitiendo estudiar la
dinámica de la regeneración post-incendio en relación con distintos factores
ambientales. Aunque desde hace algunos años existen en lugares como la Comunidad
Valenciana registros oficiales de zonas incendiadas, el análisis extensivo de las
condiciones de regeneración requiere la identificación de zonas incendiadas a partir de
los propios datos de teledetcción, para lo que se han desarrollado distintos tipos de
algoritmos y métodos.
En el Capítulo 5 se presenta un método en dos fases para la detección de áreas
incendiadas, que puede ser aplicado de forma eficiente en zonas extensas. En primer
lugar se explica el funcionamiento del método con un ejemplo detallado y a
continuación se analizan sus propiedades mediante su aplicación en una amplia zona
de la Comunidad Valenciana y la comparación entre las zonas incendiadas detectadas
con el método y las registradas en la base de datos de incendios de la Dirección
general de Prevención, Extinción de Incendios y Emergencias de la Generalitat
Valenciana. El objetivo es disponer de un método eficiente, que pueda ser utilizado de
forma automática o semiautomática en áreas amplias con distintos tipos de
vegetación, de modo que se facilite el estudio de los factores ambientales que afectan
a la regeneración vegetal tras los incendios forestales.
En el capítulo final del trabajo se presenta un resumen de conclusiones generales, en
donde se recapitulan los distintos resultados obtenidos en los distintos apartados. Por
último, se incluye un apéndice en donde se exponen los conceptos básicos sobre
HMMs, con el fin de facilitar la comprensión del trabajo presentado en el Capítulo 2.
Los trabajos recogidos en los Capítulos 2 a 5 han sido presentados en distintos
congresos internacionales y se han traducido en distintas publicaciones, como se
indica a continuación.
Capítulo 1
6
García, M.A.; Moutahir, H.; Bautista, S. y Rodríguez, F. Determination of phenological
parameters from MODIS derived NDVI data using hidden Markov models. En:
Proceedings of SPIE, Volume 9229. D.G. Hadjimitsis, K. Themistocleous, S.
Michaelides, G. Papadavid, eds. SPIE Press - The International Society for
Optical Engineering, 2014, Article number 92291K.
García, M.A. y Rodríguez, F. An Iterative Algorithm for Automatic Fitting of Continuous
Piecewise Linear Models. WSEAS Transactions on Signal Processing, 4(8): 474
– 483, 2008.
García, M.A. y Rodríguez, F. HANDFIT: An Algorithm for Automatic Fitting of
Continuous Piecewise Regression, with Application to Feature Extraction from
Remote Sensing Time Series Data. En: New Aspects of Signal Processing,
Computational Geometry and Artificial Vision. Mastorakis, N.E.; Demiralp, M.;
Mladenov, V.; Bojkovic, Z. (eds.), WSEAS Press, 2008, pp. 28-33.
García, M.A. y Rodríguez, F. Analysis of MODIS NDVI time series using quasi-periodic
components. En: Proceedings of SPIE, Volume 8795. D.G. Hadjimitsis, K.
Themistocleous, S. Michaelides, G. Papadavid, eds. SPIE Press - The
International Society for Optical Engineering, 2013, pp. 879523-1 - 879523-8.
García, M.A.; Alloza, J.A.; Bautista, S. y Rodríguez, F. Detection and analysis of burnt
areas from MODIS derived NDVI time series data. En: Proceedings of SPIE,
Volume 8795. D.G. Hadjimitsis, K. Themistocleous, S. Michaelides, G.
Papadavid, eds. SPIE Press - The International Society for Optical Engineering,
2013, pp. 879521-1 - 879521-9.
García, M.A.; Alloza, J.A.; Mayor, A.G.; Bautista, S. y Rodríguez, F. Detection and
mapping of burnt areas from time series of MODIS derived NDVI data in a
Mediterranean region. Central European Journal of Geosciences, 6(1): 112-
120, 2014.
Capítulo 2:
Modelos ocultos de Markov para la determinación
de parámetros fenológicos
OBJETIVOS
En este capítulo se explora la utilización de los HMMs para la determinación de
parámetros fenológicos, mediante su aplicación en dos zonas contrastadas de la
Cominidad Valenciana. El objetivo es poder definir, de forma más precisa que con los
ajustes generales de tipo polinómico usuales, indicadores fenológicos como las fechas
de inicio y fin de los periodos de aumento y disminución de biomasa fotosintética,
permitiendo con ello la aplicación de este tipo de análisis para estudiar de las posibles
modificaciones en estos parámetros en relación con covariables de interés (p. ej.,
datos de precipitación, temperatura o altitud) y las posibles tendencias que puedan
estar relacionadas con fenómenos de cambio global.
RESUMEN
Las características fenológicas de la vegetación son elementos fundamentals para
comprender la respuesta de la vegetación en distintos scenarios de cambio climático,
así como indicadores de procesos activos de aridez. La determinación de los
parámetros fenológicos para distintos tipos de vegetación en áreas grandes puede
ayudar a evaluar los impactos actuales o futuros del cambio climático en los
ecosistemas, especialmente en los más vulnerables. Los datos de teledetección, como
los proporcionados por MODIS, han sido usados para extraer características
fenológicas de series de datos de índices de vegetación, normalmente mediante el uso
de técnicas de suavizado y ajuste de modelos polinomiales. En el trabajo presentado
en este capítulo se utilizan modelos ocultos de Markov (HMMs, por sus iniciales en
inglés) para determinar parámetros fenológicos a partir de series de datos de NDVI
procedentes de MODIS en una región semiárida mediterránea. Se aplican diferentes
tipos de HMMs en areas seleccionadas con comunidades vegetales bien definidas y se
discute el potencial de los HMMs para el análisis fenológico automático a gran escala.
Modelos ocultos de Markov
11
INTRODUCTION
The determination of the phenological parameters of vegetation communities,
characterising the vegetation dynamics, at different scales, from local and regional to
global scopes, constitutes a key topic in understanding the functioning of terrestrial
ecosystems. Describing and analysing phenological characteristics for different types
of vegetation in large areas help evaluate current impacts of climate change and other
perturbations in ecosystems, as well as predict vegetation responses in different
climate change scenarios1,2.
Remote sensing from different space-borne sensors such as AVHRR, Landsat or MODIS
has been used to describe land cover phenology, by extracting phenological
characteristic from time series data of vegetation indices such as the normalized
difference vegetation index (NDVI)3-8.
A wide variety of methods have been used to analyze remote sensed NDVI time series
data in order to extract phenological metrics that describe the phenological traits of
the vegetation dynamics. Most frequently, some type of smoothing method is first
applied to reduce the usually high level of noise present in the raw data, and then
different strategies based on polynomial fittings or spectral analyses are used to
characterize seasonal changes, thus providing phenological parameters as the onset of
the growing season9-13.
Hidden Markov models (HMMs) are a modelling technique for sequential data analysis,
originally developped in the field of automatic speech recognition, where they
constitute a basic and widespread tool14. These models have pervaded other scientific
and technical fields, as climatology15 or bioinformatics16-17. The application of HMMs to
phenological analysis of remote sensed vegetation indices was proposed two decades
ago18, but alternative methods of analysis were preferred, and there was almost no
other example of application until very recently19.
The objective of this work was to apply different types of HMMs to determine
phenological parameters from MODIS derived NDVI time series data in a semiarid
Mediterranean region, analysing their potentials for automatic phenological analysis in
different types of vegetation.
Capítulo 2
12
METHODS AND RESULTS
The tutorial by Rabiner14 presents a general introduction to HMMs and the algorithms
to analyse them. Further details can be found in the monographs by Cappé et al.20 and
MacDonald and Zucchini21. In the next subsection we briefly present the main aspects
of the particular HMMs used in this work.
Elements and structure of the HMMs applied to NDVI data for phenological analysis
Given a sequence of observations, tY , which may consists in a time series of NDVI
values for a given pixel, we assume the existence of a corresponding sequence of
hidden states, tS , which may represent the different phenological states of the
vegetation (Fig. 1).
Figure 1. Scheme of the structure of a first order HMM. Temporal sequence of
hidden states (St) and corresponding observed states (Yt). Lack of a direct
arrow connecting two states implies their conditional independence.
For instance, assuming a simple phenological model with three states (Fig.2, left and
centre), representing growing, S+ , declining, S− , or stationary periods, 0S , with,
respectively, general positive, negative or null changes in NDVI values, there is not a
perfect correspondence between the unobserved states of the vegetation and the
observed NDVI values at particular moments. Thus, the sequence of hidden states has
to be inferred from the sequence of observations, or emissions, as they are usually
denoted.
Modelos ocultos de Markov
13
Figure 2. Topologies of the HMMs used in this work. Three completely
connected states (left), three states with forbidden transitions or blocking
(centre), and four states with blocking (right).
To carry out this inference, it is needed a plausible model describing the transitions
between (hidden) states and the probabilities of the emissions given the states. We
assumed first order HMMs, which means that the transitions between states depend
only on the current state, and not on the past history. In the case of continuous
emissions, as is the case with NDVI values or changes, appropriate density functions
have to be selected, which we assumed to be normal. The so called topology of the
HMM is then complete by specifying the number of states and the forbidden
transitions, if any, between them.
The different HMM topologies used in this work are presented in Fig. 2. The first
model, denoted 3S, (Fig. 2, left) allows for all possible transitions between its three
states, while the second one, 3SB, (Fig. 2, centre) only allows for transitions between
different states from stationary, or baseline, to upward, then to downward, and then
back to baseline. The last model, 4SB, (Fig. 2, right) includes two different stationary
states, connecting the upward and downward states.
Once the topology has been selected, the parameters of the model can be fitted using
an expectation maximization algorithm, and then the sequence of hidden states can be
inferred using a dynamic programming algorithm known as Viterbi algorithm14,20,21.
NDVI data and vegetation classes
MODIS data were downloaded from the NASA website (currently accessed through the
Reverb data gateway, http://reverb.echo.nasa.gov/reverb/). We used the NDVI 16-
Capítulo 2
14
days composite band from a time series of MOD13Q1 MODIS/Terra product at 250m
resolution (tile h17v05), from February 2000 to December 2012.
We selected a large area in the centre of the Valencia province, in the Valencia region
in Eastern Spain, and automatically classified the pixels in the area in clusters according
to the similarities in their distributions of NDVI changes (Fig. 3).
Figure 3. Cluster analysis based on similarities of NDVI changes in an area in
the centre of the Valencia province, Eastern Spain. Different colours represent
different clusters. Dark blue at right corresponds to the Mediterranean Sea.
Phenological analysis of NDVI time series data using HMMs
For the analysis carried out in this work, two different zones were selected. The first
one, composed of three clusters with rather similar dynamics, comprised a zone of rice
crops located in the Albufera area. Two of these clusters are displayed in Fig. 4, the
third one corresponding to the border of the zone, with a slightly less well-defined
dynamics. The second zone in this study corresponds to natural vegetation with a
much more diffused dynamics and much larger variations (Fig. 5).
Modelos ocultos de Markov
15
The zone with rice crops depicts a highly coherent vegetation dynamics (Fig. 4), and
hence it is expected that any reasonable method should provide sufficiently good
results.
Figure 4. Top: Two selected clusters (white) from Fig. 3 corresponding to rice
crops in the Albufera area. Bottom: For each of the selected clusters, mean
NDVI (x104) and percentiles time series values, reconstituted from their
distributions of NDVI changes.
Figure 5. Selected clusters (white) corresponding to natural vegetation in the
Valencia province (left), and mean (red) and reconstituted from the fitting of
the four states HMM (blue) NDVI (x104) time series values (right).
Capítulo 2
16
Table 1. Estimated values of the parameters for HMM models 3S and 3SB
fitted to changes in NDVI (x104) for rice crops in the Albufera area.
Model Transition probabilities Emissions mean and
standard deviation From: To:
3S 0S S+ S− µ̂ σ̂
0S 0.8809 0.0545 0.0646 1.2388 127.2428
S+ 0.1618 0.8363 0.0019 576.5619 345.7381
S− 0.1387 0.0063 0.8550 -447.7662 231.3246
3SB 0S − S+ S− µ̂ σ̂
0S 0.9021 0.0979 0 -4.9107 117.6908
S+ 0 0.8420 0.1580 439.5495 383.9313
S− 0.1360 0 0.8640 -368.2144 271.1641
Though, it also gives the opportunity to compare different methods, as the different
HMM models here considered, which may shed light on fine differences between
them. We will focus the analysis on the onset of the growing season, but other
phenological parameters are also readily obtained from the sequence of inferred
hidden states.
The fitted values of the parameters for the different models considered, 3S, 3SB and
4SB are presented in Tables 1 and 2. For each hidden state, the emissions, i.e., the
changes in NDVI values, are modelled as a normal distribution with the corresponding
mean and standard deviation.
Modelos ocultos de Markov
17
Table 2. Estimated values of the parameters for HMM model 4SB fitted to
changes in NDVI (x104) for rice crops in the Albufera area (RC) and a zone
with natural vegetation in the Valencia province (NV).
Transition probabilities Emissions mean and standard
deviation From: To:
RC 0S − S+ 0S + S− µ̂ σ̂
0S − 0.8948 0.1052 0 0 -16.0189 118.8479
S+ 0 0.8168 0.1832 0 562.3302 352.8487
0S + 0 0 0.8023 0.1977 42.3842 133.4238
S− 0.1365 0 0 0.8635 -424.4757 240.7890
NV 0S − S+ 0S + S− µ̂ σ̂
0S − 0.7323 0.2677 0 0 -6.1832 51.0317
S+ 0 0.7039 0.2961 0 143.9279 99.8844
0S + 0 0 0.1310 0.8690 3.1719 51.6534
S− 0.3190 0 0 0.6810 -143.8033 99.4484
The values for models 3S and 3SB for the zone of rice crops are presented in Table 1.
The values of the fitted parameters for model 4SB for both zones, rice crops and
natural vegetation, is presented in Table 2. Compared with the corresponding values
for the rice crops, the parameters in the zone with natural vegetation reflect much less
Capítulo 2
18
defined states, with higher probabilities of changes between different states, and
much smaller absolute values of changes in the upward and downward states.
Figure 6. Start of season (SOS), as MODIS time sections, estimated using HMM
models 3S (green) and 4SB (blue), for rice crops in the Albufera area for the
years 2000-20012.
Comparison of the estimated dates for the onset of the growing period obtained
applying models 3S and 4SB are showed in Fig. 6. Although both estimations are highly
correlated in each pixel, with a global correlation coefficient of 0.89ρ = , the more
complex model 4SB coherently and significantly provide earlier detections of the start
of the growing season.
CONCLUSIONS
In this work, different HMM models were applied in two types of vegetations in a
Mediterranean area to define phenological characteristics from remote sensed NDVI
MODIS derived time series data.
HMMs can be efficiently applied with large sets of data, and they can provide
consistent results when homogeneous sets of pixels, in terms of their vegetation
dynamics, are modelled. This condition can be fulfilled by either previous classification
Modelos ocultos de Markov
19
from land cover and vegetation mapping or by automatic clustering of pixels based on
the distributions of NDVI changes.
The estimated parameters of the HMMs, transition probabilities and means and
standard deviations of emissions, reflect the strengths of the different phenological
states, and could be used to compare the dynamics of vegetation communities
affected by perturbations or experimental treatments.
The use of HMMs allows the incorporation of previous knowledge of the system in its
modelling, by appropriately selecting the number of states or the blocked transitions,
thus providing more specific models for different vegetation communities, that may
result in a better determination of the parameters of interest.
Once a suitable HMM model is selected, the two steps involved in the estimation
process, estimation of parameters and inference of the sequence of hidden states, are
carried out at different levels of data processing. For the estimation of the parameters,
a whole set of pixels, and also their whole time series values, can be used, thus
providing highly consistent estimates. However, the inference of the hidden states
proceeds at the pixel level, and it can also be carried out in a yearly basis, thus allowing
for individual pixel analysis in relation with particular environmental factors.
More complex types of HMMs than those used in this work could also be applied, for
instance to include specific modelling of the duration of the different phases of the
seasons by using higher order or semi-Markov models, and they could also result in
more accurate estimations. Work in progress includes the comparison of the
estimations provided by different HMMs with the phenological parameters obtained
with other well-established smoothing and curve fitting methods.
ACKNOWLEDGEMENTS
This work was supported by the research projects FEEDBACK (CGL2011-30515- C02-
01), funded by the Spanish Ministry of Innovation and Science, and CASCADE
(GA283068), funded by the EC 7FP.
REFERENCES
[1] Cleland, E. E., Chuine, I., Menzel, A., Mooney, H. A. and Schwartz, M. D.,
“Shifting plant phenology in response to global change,” Trends in Ecology &
Evolution 22, 357-365 (2007).
Capítulo 2
20
[2] Galford, G.L., Mustard, J.F., Melillo, J., Gendrin, A., Cerri, C.C. and Cerri, C.E.P.,
“Wavelet analysis of MODIS time series to detect expansion and intensification
of row-crop agriculture in Brazil,” Remote Sensing of the Environment 112, 576-
587 (2008).
[3] Moulin, S., Kergoat, L., Viovy, N. and Dedieu, G., “Global-scale assessment of
vegetation phenology using NOAA/AVHRR satellite measurements,” Journal of
Climate 10, 1154-1170 (1997).
[4] Zhang, X., Friedl, M.A., Schaaf, C.B., Strahler, A.H., Hodges, J.C.F., Gao, F., Reed,
B.C. and Huete, A., “Monitoring vegetation phenology using MODIS,” Remote
Sensing of the Environment 84, 471-475 (2003).
[5] Fontana1, F., Rixen, C., Jonas, T., Aberegg1, G. and Wunderle1, S., “Alpine
grassland phenology as seen in AVHRR, VEGETATION, and MODIS NDVI time
series - a comparison with in situ measurements,” Sensors 8, 2833-2853 (2008).
[6] Fisher, J. I. and Mustard, J. F., “Cross-scalar satellite phenology from ground,
Landsat, and MODIS data,” Remote Sensing of the Environment 109, 261-273
(2007).
[7] Ganguly, S., Friedl, M. A., Tan, B., Zhang, X. and Verma, M. “Land surface
phenology from MODIS: Characterization of the Collection 5 global land cover
dynamics product,” Remote Sensing of Environment 114, 1805-1816 (2010).
[8] Hmimina, G., Dufrêne, E., Pontailler, J.-Y., Delpierre, N., Aubinet M., Caquet, B.,
de Grandcourt, A., Burban, B., Flechard, C., Granier, A., Gross, P., Heinesch, B.,
Longdoz, B., Moureaux, C., Ourcival, J.-M., Rambal, S., Saint André, L. and
Soudani, K., “Evaluation of the potential of MODIS satellite data to predict
vegetation phenology in different biomes: An investigation using ground-based
NDVI measurements,” Remote Sensing of Environment 132, 145-158 (2013).
[9] Jönsson, P. and Eklundh, L., “Seasonality extraction and noise removal by
function fitting to time-series of satellite sensor data,” IEEE Transactions on
Geoscience and Remote Sensing 40, 1824-1832 (2002).
[10] Jönsson, P. and Eklundh, L., “TIMESAT– a program for analyzing time-series of
satellite sensor data,” Computers & Geosciences 30, 833-845 (2004).
Modelos ocultos de Markov
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[11] Bradley, B. A., Jacob, R. W., Hermance, J. F. and Mustard, J. F., “A curve fitting
procedure to derive inter-annual phenologies from time series of noisy satellite
NDVI data,” Remote Sensing of Environment 106, 137-145 (2007).
[12] Moody, A. and Johnson, D., “Land-surface phenologies from AVHRR using the
discrete Fourier transform,” Remote Sensing of Environment 75, 305-323 (2001).
[13] Wagenseil, H. and Samimi, C., “Assessing spatio-temporal variations in plant
phenology using Fourier analysis on ndvi time series: results from a dry savannah
environment in Namibia,” International Journal of Remote Sensing 27(16), 3455-
3471 (2006).
[14] Rabiner, L.R., “A tutorial on hidden Markov models and selected applications in
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Discrete-Valued Time Series], Chapman and Hall, London, (1997).
Capítulo 3:
Ajuste de funciones lineales a trozos a series
temporales de índices de vegetación
OBJETIVOS
En este capítulo se considera el problema del ajuste de modelos lineales a trozos a
series de datos con un alto número de puntos y con la posibilidad de un alto número, a
priori desconocido, de puntos de cambio. El problema se aborda mediante un
algoritmo de tipo iterativo, que puede ser adaptado en función del tipo de información
previa que se tenga sobre el sistema analizado o de la complejidad del modelo que se
quiera ajustar.
RESUMEN
Los modelos lineales continuos a trozos representan herramientas útiles para extraer
las características básicas sobre los patrones de variación en series de datos complejas.
En el trabajo presentado en este capítulo se desarrolla un algoritmo iterative para el
ajuste de modelos de regresión lineal continua a trozos con estimación automática de
los puntos de cambio. El algoritmo necesita partir de unos valores inicializales en
cuanto al número y posición de los puntos de cambio, que se pueden obtener
mediante diferentes métodos, y a continuación procede mediante un ajuste iterativo
similar a los desplazamientos del método de Newton para la obtención de raíces de
funciones. El algoritmo se puede aplicar a altos volúmenes de datos, con muy rápida
convergencia en la mayoría de los casos, permitiendo la simplificación del modelo en
cuanto a la reducción del número de puntos de cambio al identificar puntos
suficientemente próximos. Se presentan ejemplos de aplicaciones para la extracción
de características fenológicas a partir de series de datos de índices de vegetación por
teledetección.
Ajuste de funciones lineales a trozos
27
INTRODUCTION
Remote sensing of vegetation dynamics, soil properties, and other ecosystem variables
and indicators constitutes a key tool in ecology, agriculture and environmental studies
at several temporal and spatial scales1,2. Many different techniques can be used to
analyze this kind of data3-5, and the development of efficient methods to identify
patterns and extract features from remote sensing derived spatio-temporal data series
is a key point in the applications6.
Time series of vegetation indices, such as the normalized difference vegetation index
(NDVI), are derived products from data of Earth observing systems like the Moderate
Resolution Imaging Spectroradiometers (MODIS)7 on the Terra platform. MODIS
derived NDVI data are available from the year 2000, every 16 days for a global grid of
pixels with a maximum resolution of 250 m, and they are just an example, as
paradigmatic as it could be, of many different fields where huge amounts of time
series data are produced and need to be analyzed with efficient methods capable of
extracting their main features, some of which may be readily noticeable to a human
observer.
Figure 1. NDVI time series values (x 104) for an area of semiarid vegetation in southeast
Spain. Time values (abscissas) are number of days, starting from 01/01/2000.
A common characteristic in NDVI time series is the presence of different regions with
increasing and decreasing trends, which correspond to periods of growth and decline
of vegetation. This pattern is well-defined in Figure 1, which shows data -four years
period- for a set of contiguous pixels in a semiarid area of the Valencia region
Capítulo 3
28
(southeast Spain), although it may not be so clear in the individual curves, i.e., in the
data series for each pixel.
A simplified model of the functional dependence suggested by this type of data is
shown in Figure 2, a continuous polygonal which characterizes the sequence of growth
rate changes, providing the position of the change-points and the slopes of the linear
segments.
Figure 2. NDVI time series values from Figure 1 with a continuous piecewise linear
model fitted to the data.
The problem of fitting a continuous piecewise linear model to a series of data is
referred to as piecewise8 or segmented9 regression, linear regression with multiple
structural changes10 or regimes, or in the case of two segments as broken-line or two-
phase11,12 regression. In the classical statistical framework, this problem has been
tackled as a particular case of nonlinear regression13, or with specific approaches
aimed at minimizing the sum of squares of errors, yielding least squares estimates of
the parameters, or maximum likelihood estimates in the case of independent
identically distributed normal errors14-16 or under particular hypothesis on the error
structure9,17 .
When the number and positions of the change-points are known, the estimation of the
model is straightforward. Segmented linear models with change-point estimation
without the continuity requirement are special cases of model trees18-19, where
induction methods such as Quinlan's M520 are well-designed for predictive
Ajuste de funciones lineales a trozos
29
performance with many regressors. The restrictions imposed by the continuity
condition, and the discontinuities in the derivative implied by a polygonal model, make
the estimation of the model by minimizing some form of risk function much more
difficult. Some authors use approximate smooth models to avoid these problems21,22,
while the more direct algorithms are mainly based on grid search16 or some form of
greedy exploration of the possible change-points15. Other computationally intensive
approaches include bayesian23 and fuzzy methods24.
Piecewise linear models are usually approximations to complex real phenomena, that
allow to extract the basic features of the data, and so find application in many
different fields, as in economy10, ecology25 or cancer research26. The objective of this
work is to efficiently fit a continuous polygonal model to large datasets, with a
computational approach that does not intend to yield a global optimum for some
measure of adjustment, but to capture in an objective manner the main trends of the
data, providing estimates for the relevant parameters in the problem considered.
DESCRIPTION OF THE METHOD
The method proposed, which we denote with the acronym HANDFIT, standing for
Hinges Adjustment by Newton-like Displacements FIT, consists of two phases. First, an
initial guess about the number and positions of the change-points or hinges is made,
for which various alternatives suited for different particular problem are considered.
Then, an iterative process to displace these hinges, analogous to Newton method for
root finding, is applied. Several parameters of the algorithm can be adjusted so that
the type of features that are of main interest be extracted, although a completely
automated functioning is also possible.
Iterative Adjustment of the Hinges
Assume we have an initial estimation of the abscissas of the increasing sequence of
change-points, { } 1i i nx
= K
, in a fixed domain with endpoints 0a x= and 1nb x += . The
iterative algorithm proceeds in two steps. Firstly, for each interval [ ]1,i ix x + a line
segment is adjusted to the data by ordinary least squares, although any other fitting
method could be used as well (Figure 3).
Capítulo 3
30
Figure 3. A section of the NDVI data from Figure 1, showing a step of the fitting
process, where for the current selection of change-points (indicated by vertical lines) a
line is fitted in each segment (bars) and their intersections obtained as new change-
points (crossed circles).
Secondly, the intersection points between consecutive segments are computed, so
that their abscissas define the new change-points, and the process is repeated until
convergence is achieved (Figure 4). The stopping criterion defining convergence is
simply that a fixed threshold for the displacements of the hinges not be exceeded, and
for sound choices of the threshold it is usually reached after only a few iterations.
Figure 4. Continuous piecewise linear model fitted to the data from Figure 3, given by
the final convergent solution of the algorithm.
Ajuste de funciones lineales a trozos
31
The second step in the algorithm is essentially a Newton iteration, as the new points
are the intersection of affine varieties, which in the simplest Newton method, the
Newton-Raphson algorithm to compute zeros of real functions, are the tangent line
and the X-axis. Newton-like algorithms are in most cases very fast, but it is well-known
that these type of methods may produce abrupt jumps, which in our problem could
yield non-admissible values when the ordering of the hinges is not preserved (Figure
5).
Figure 5. Example of data where non-admissible values would be produced at an
iteration step.
To tackle this eventuality, if ˆix is to be the abscissa of the new i-hinge, the relative
increments,
1 1
ˆ ˆˆ ˆ, ; , ;i i i i
i i i ii i i i
x x x xx x x x
x x x x+ +
− −> <− −
are always transformed using a sigmoid function (Figure 6), with a dampening
coefficient that can be adjusted to successfully avoid any problems. This cautionary
safeguard has the cost of increasing the number of iterations, but it results in a more
robust algorithm, with still a fast speed of convergence for most data. It should be
clear, however, that convergence can not be guaranteed for any arbitrary dataset, as
Capítulo 3
32
no sound continuous linear model can be expected to fit a data series resulting from a
process with intrinsic discontinuities.
Figure 6. Example of sigmoid function used to correct for possible jumps during the
iteration process.
Depending on the data and the number and positions of the starting points, two
consecutive hinges might get close enough to consider that they should be identified,
and this is what the algorithm does when a proximity threshold is crossed, which can
also be automatically defined in terms of the minimum number of different data points
in a segment for the least square adjustment be considered sound. Thus, the algorithm
can automatically correct to a certain extent an excess of hinges in the initial set, and
hence results in models that are not of much higher complexity than that suggested by
the data.
Initial Selection of the Change-Points
Different strategies can be applied to select the initial set of change-points, according
to the type of data and the particular features of interest, although the iterative step
leads from many distinct reasonable elections of the starting points to the same final
convergent solution, as will be discussed in the next section.
Ajuste de funciones lineales a trozos
33
For the data in Figure 1, where the pattern is essentially a sequence of periods with
alternate growing and decay behavior, a plausible election would be those points
where the mean slope of the curves changes in sign, i.e., where it passes trough zero.
Figure 7. Up: Local slopes for the NDVI data series in Figure 3, showing their medians
and the gaussian kernel used to filter them. Down: Smoothed curve of the medians. The
values of the abscissas corresponding to zeros of the filtered medians are displayed,
and the change-points of the final solution indicated by vertical lines.
In Figure 7 (up), the cloud of slopes for 120 similar curves is displayed, showing their
medians as robust estimations of the slopes. Although there are many points where
the curve of the medians changes in sign, in the three sections marked in the graph,
which correspond to the segments of a final convergent fitted model, the second one
consists essentially in positive values, whereas in the other two the values of the
medians are mostly negative. Considering all the zeros in this curve as initial change-
Capítulo 3
34
points would produce a model of very high complexity, despite the limited reduction in
the number of hinges that the iteration step of the algorithm is capable of perform. A
more reasonable election is obtained filtering the medians using a gaussian kernel
filter, as the one shown in the same figure, and working with the curve of the
smoothed medians (Figure 7, down). The zeros of the filtered median give values close
to the final iterated solution obtained from a subjective selection of the starting points,
and very similar results are obtained if the data are averaged and the smoothed slope
of the mean curve is used instead.
Consider, however, the NDVI data presented in Figure 8, corresponding to rice crops,
also in the Valencia region. Here the dynamics of the vegetation is more complex, as
besides the clearly defined evolution of the crop, there are also other periods that
correspond to phases of harvesting, growing of natural vegetation and preparation of
the fields for the new season. All the pixels show similar and synchronized behaviors,
since they are subjected to the same labors at specific moments in time, and so the
curves are much more better defined than those in Figure 1, allowing for a more
detailed description than just the incresing/decreasing pattern.
Figure 8. NDVI time series values ($\times 10000$) for an area of rice crops in
southeast Spain. Time values (abscissas) are number of days, starting from
01/01/2000.
Ajuste de funciones lineales a trozos
35
In Figure 9, a simple model that represents fairly well the periods of growth and decay
of the crops is fitted to the data. In Figure 10, a different model of higher level of
complexity, in terms of the number of change-points considered, is presented. This last
model reflects better than the previous one the transitions between the periods of
growth and decay, as well as the phases between consecutive cropping seasons. Both
models are final equilibrium solutions of the iterative algorithm for different elections
of the number and positions of the initial change-points. The rationale for deciding
between these two models depends on the kind of features that we are interested in,
either the basic characterization of the crop dynamics, as in Figure 9, or a more
detailed description of the whole vegetation dynamics as in Figure 10. Hence, this
decision must be set by the analyst according to the objectives of the study, although it
can be incorporated into the algorithm, either in an explicit or implicit way, trough the
the method used for the selection of the initial points and with the setting of the
different parameters modulating the outcome of the algorithm, as window sizes of the
smoothing filters or thresholds levels.
Figure 9. NDVI time series values from Figure 8, with a continuous piecewise linear
model fitted to the data. Note the variations in the slopes of contiguous segments,
which do not restrict to a positive/negative sequence.
Capítulo 3
36
Figure 10. NDVI time series values from Figure 8, with a continuous piecewise linear
models of higher complexity than that of Figure 9 fitted to the data.
In any case, in this more general context the previously discussed method relying on
the change in sign of the slopes is clearly inappropriate, and the selection of the
starting points could instead be based on the distribution of the curvatures (Figure 11).
In Figure 11 (up), the cloud of curvatures for the data in Figure 8 is shown, where it has
been obtained using a moving window of suitable amplitude, to compute the
curvatures for each curve and abscissa fitting a second degree polynomial to the points
inside the window. The medians of the curvatures have been smoothed with a
gaussian kernel, as shown in Figure 11 (down), and the zeros of the derivative of the
filtered curvature function, corresponding to the most extreme values above some
threshold, have been selected (Figure 12).
Ajuste de funciones lineales a trozos
37
Figure 11. Up: Local curvatures for the data in Figure 8. Down: Medians of the
distribution of local curvatures (dots), and continuous smoothed median after filtering
with a gaussian kernel.
Capítulo 3
38
Figure 12. Positions of the ten most extreme values of the local curvatures for the NDVI
data in Figure 8.
For equally spaced data, curvatures can be computed in an very efficient way using a
Savitzki-Golay type method27 to adjust the quadratic polynomials for each position of
the moving window. In case that for the specific data quadratic polynomials were not
flexible enough to detect the zones of interest, polynomials of higher degree could be
used, and the computational effort would be comparable if a Savitzki-Golay strategy
could be employed, i.e., if abscissas were uniformly spaced.
A robust and computationally efficient alternative, to avoid computing curvatures
through fitting of second or higher degree polynomials, is to consider the angles
between lines adjusted to contiguous sets of points (Figure 13). Using a double-sized
moving window, two lines are fitted to the points at the left and the right of each
abscissa, and the angles between these lines are computed (Figure 13 up). Then, they
can be post-processed as in the previous method (Figure 13 down), and the most
extreme values selected with some threshold criteria (Figure 14).
Ajuste de funciones lineales a trozos
39
Figure 13. Up: Local angles for the data in Figure 8. Down: Medians of the distribution
of local angles (dots), and continuous smoothed median after filtering with a gaussian
kernel.
Capítulo 3
40
Figure 14. Positions of the ten most extreme values of the local angles for the NDVI
data in Figure 8.
Sensitivity to initial conditions
Although a variety of methods can be used to determine the starting set of hinges, we
note that many different selections of the initial points lead to the same final result,
which is one of a very restricted set, that of the fixed points for the Newton-like
iteration algorithm. To illustrate this behavior, for the data presented in Figure 3,
where two change-points seem to provide a sound model, an exhaustive search on the
initial positions of the two change-points was performed.
For each position of the two change-points, when a simple piecewise linear model is
fitted, i.e., when each segment is optimally fitted to their data by ordinary least
squares without requiring the continuity condition, a measure of the magnitude of the
jumps across contiguous segments gives an idea of the regions that can sustain a
continuous model (Figure 15), the zeros of this function being the points sought by the
algorithm. If we apply the iterative algorithm from any pair of these initial points, a
convergent solution is eventually reached, and the set of the positions of the change-
points in the possible final solutions is shown in Figure 16. The regions defined by the
Ajuste de funciones lineales a trozos
41
set of initial points that result in the same final solution are presented in Figure 17. The
larger central region in this figure, comprising more than half of the possible elections
for the initial change-points, result in the solution shown in Figure 4, which intuitively
provides a sound model for the data. In fact, this is also the two change-points model
with the minimum global error of fit (Figure 18).
Figure 15. Index of discontinuity (root mean squares of jumps between contiguous
segments) as a function of the positions of the two change-points when fitting a
piecewise linear model to the data in Figure 3.
Capítulo 3
42
Figure 16. Abscissas of the two change-points in the set of the final iterated solutions
for the data in Figure 3, resulting from an exhaustive search of the initial positions for
the change-points.
Figure 17. For the data in Figure 3, regions of initial positions for the change-points
that result in the same final solution.
Ajuste de funciones lineales a trozos
43
Figure 18. Global errors of fit for the models corresponding to the different final
solutions in Figure 17. The values displayed correspond to the models with the two
lowest global errors.
A different question that can be raised is the sensitivity of the algorithm to small
variations in the original data. In real applications, data are measured with a certain
degree of error, and any feature extraction algorithm should not give much different
outcomes for close data inputs. To explore the robustness of the algorithm to random
perturbations of the data, we selected a section of two curves from data in Figure 8
and added gaussian noise to the positions of the data points. The final positions of the
change-points given by the algorithm were consistent, determining models exhibiting
similar behaviors (Figure 19).
Capítulo 3
44
Figure 19. Sensitivity of the final positions of the hinges to random perturbation of the
data. Two sections of data series from Figure 8 were perturbed, adding independent
gaussian noise to the x and y coordinates of the points, with standard deviations as
indicated in the figure, producing 50 replications. For each cluster of the final change-
points, elipsoids determined by three standard deviations of their distributions are
shown.
DISCUSSION
The algorithms presented in this work provide a computationally efficient method to
fit continuous piecewise linear models to data series when the number of points is
high and many change-points have to be considered, and can be an alternative to
methods based on exhaustive or grid searches aimed at minimizing some global risk
function.
Our objective in fitting these kinds of models is to extract the main features, in terms
of different growth regimes, present in the data, and in this context it is clear that
some kind of a priori information, either explicit or implicit, has to be used to define
what a trait of interest is. Consider, for instance, the data presented in Figure 8. As
shown in Figure 9 and Figure 10, models of different levels of complexity can be fitted,
depending wether the interest lies essentially in the succession of grothw/decline
seasons or a more detailed description is sought.
Ajuste de funciones lineales a trozos
45
Our primary envisaged application for these type of models is in the analysis of remote
sensing vegetation data, as exemplified along the paper, although there are many
other fields where continuous piecewise linear models are sound models and can
provide basic description of the patterns of growth exhibited by the data, and where
efficient algorithms are needed to cope with high volumes of data. However, it should
be kept in mind that no algorithm for continuous piecewise regression can be
successful when the data does not reflect the continuity properties needed in these
models (Figure 20).
Figure 20. Example of simulated data produced by a discontinuous model.
There are several options and parameters in the algorithms that can be set to fine tune
the method, and obtain the type of model more adequate for the data in
consideration. Besides the different options for the selection of the initial points, the
size of the moving windows used to compute local curvatures or angles, the shape of
the smoothing kernels and the values of the thresholds to select the most extreme
values determine the number and positions of the initial change-points. Nevertheless,
any sound choice for these parameters would lead to very similar sets of starting
points, as exemplified by comparing Figure 12 and Figure 14. Moreover, as discussed in
the previous section, the final set of change-points are the points of equilibrium of the
iteration process, and so there is no need for an intensive effort to optimally
Capítulo 3
46
determine the positions of the initial points, as most of them will usually lead to the
same final solution.
Finally, it should be clear that the method can be run in a completely automated way.
The choice between different models with the same number of change-points can be
based on the global error of fit, while some suitable model selection criteria28 taking
into account the number of parameters of the model, such as AIC29 or BIC30, can be
employed to select between models with different number of change-points.
ACKNOWLEDGEMENTS
This work has been partly funded by grants from Valencia Regional Government
(GVPRE/2008/310, Conselleria de Educación, Generalitat Valenciana) and University of
Alicante (VIGROB-162).
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[23] B.P. Carlin, A.~E. Gelfand and A.F.M. Smith, Hierarchical Bayesian analysis of
change point problems, Appl. Statist. 41, 1992, pp. 389--405.
[24] J.-R. Yu, G.-H. Tzeng and H.-L. Li, General fuzzy piecewise regression analysis with
automatic change-point detection, Fuzzy Sets and Systems 119, 2001, p. 247--
257.
[25] J.D. Toms and M.L. Lesperance, Piecewise regression: A tool for identifying
ecological thresholds, Ecology 84, 2003, pp. 2034--2041.
[26] B. Yu, M.J. Barrett, H.-J. Kim and E.J. Feuer, Estimating joinpoints in continuous
time scale for multiple change-point models, Comput. Statist. Data Anal. 51,
2007, pp. 2420--2427.
[27] A. Savitzky and M.J.E. Golay, Smoothing and Differentiation of Data by Simplified
Least Squares Procedures, Anal. Chem. 36, 1964, pp. 1627--1639.
[28] K.P. Burham and D.R. Anderson, Model Selection and Inference. A Practical
Information-Theoretic Approach}, Springer-Verlag, New York, 1998.
[29] H. Akaike, Information theory and an extension of the maximum likelihood
principle, in: B.N. Petrov and F. Csáki (eds.), 2nd International Symposium on
Information Theory, Akadémia Kiado, Budapest, 1973, pp. 267--281.
[30] G. Schwarz, Estimating the dimension of a model, Ann. Statist. 6, 1978, pp. 461–
464.
Capítulo 4:
Análisis tiempo-frecuencia de series de datos de
índices de vegetación mediante componentes cuasi-
periódicos
OBJETIVOS
En este capítulo se presenta un modelo para series de datos que incluyen
componentes seculares y componentes cíclicas no constantes, denominadas
componentes cuasi-periódicas. El objetivo es obtener estimaciones más ajustadas a la
realidad que las proporcionadas por los modelos de análisis espectral con
componentes periódicas constantes, permitiendo con ello el análisis de las relaciones
entre las variaciones en los parámetros que definen las variaciones estacionales y las
covariables o factores ecológicos de interés.
RESUMEN
Las series de datos temporales de NDVI muestran usualmente comportamientos
cíclicos debidos a las características fenológicas de la vegetación. Se han utilizado de
forma efectiva diferentes formas de análisis de Fourier para describir datos de NDVI
procedentes de teledetección, que permiten el ajuste simultáneo de componentes
seculares y cíclicos. No obstate, pueden existir variaciones en las frecuencias y/o en las
fases de los componentes cíclicos, que no son recogidas en los análisis espectrales
típicos. En el trabajo presentado en este capítulo, se analizan series de datos
temporales de NDVI procedentes de MODIS considerando components seculars y
cuasi-periódicos, que se ajustan a los datos mediante un algoritmo de suavizado
basado en análisis espectral de tipo tiempo-frecuencia. El algoritmo funciona
especialmente en el caso de datos equiespaciados, permitiendo el análisis de
variaciones temporales en las frecuencias y fases de los componentes cíclicos. Se
presentan ejemplos de aplicaciones en diferentes tipos de vegetación y condiciones
em la Comunidad Valenciana.
Análisis tiempo-frecuencia con componentes cuasi-periódicos
53
INTRODUCTION
The description and analysis of vegetation dynamics at regional and global scales
constitutes a topic of major importance to understand the functioning of terrestrial
ecosystems and their interactions with factors such as climate change1 or human
driven natural ecosystems degradation2. Remote sensing has been used for the past
three decades to extract phenological characteristic by analyzing time series data of
vegetation indices such as the normalized difference vegetation index (NDVI), derived
from several space-borne sensors such as AVHRR, Landsat or MODIS3-6. Its capacity to
properly characterize the phenology of the vegetation has been assessed with ground
data7.
The use of remotely sensed time series of vegetation indices has produced global
descriptions of land cover dynamics8, and has provided a valuable tool for landscape
analysis, as in the classification of vegetation types9, vegetation ecology, as in the
analysis of the relations between vegetation dynamics and climatic factors10,11, and the
study of vegetation recovery after disturbances such as wildfires12,13.
Different methods have been used to analyzed NDVI time series data, to reduce the
noise present in the original data by applying some smoothing method, extract
phenological metrics that describe the phenological traits of the vegetation, or to best
characterize the temporal behavior of the NDVI data corresponding to the underlying
vegetation dynamics14-17. Seasonal characteristics of the vegetation produce
oscillations with approximate annual and semiannual periods in NDVI data, and hence
different types of harmonic, Fourier or spectral analysis have been effectively used to
extract the cyclic components of NDVI time series18-22.
In Mediterranean landscapes, as in the Valencia region in Southeast Spain, it is normal
to find a high interspersion of different types of vegetations, even at reduced scales,
showing a wide variability in their phenological dynamics (Fig. 1). Also, high regional
and local variations in climatic factors, including the presence of droughts or extended
dry seasons, may produce alterations in the typical seasonal oscillations, that may
affect not only their amplitudes but also the onset and duration of the growing season.
The objective of this work was to devise, and efficiently implement, a method to
analyze MODIS derived NDVI biweekly time series data that explicitly consider the
presence of variations in the parameters defining the cyclic components which model
Capítulo 4
54
the oscillations of the data, to account for variations in the data that are not the result
of inter-years trends, and hence facilitate the analysis of the relations between the
variations in the seasonal characteristics of the vegetation and different environmental
factors.
Figure 1. Cluster analysis of the annual dynamics of the vegetation, as derived
from similarities between NDVI time series, for an area in the Valencia region.
Map of dynamics classes (left) and characteristic NDVI annual curves for the
six classes considered (right, colors refer to the corresponding class in the
map).
METHODS AND RESULTS
NDVI time series usually exhibit behaviors that may result from the combination of
trends, only apparent when several years are analyzed, and intra-year oscillations with
a certain amount of variability in their amplitudes and phases. To better study this type
of temporal variations, a specific model will be proposed, and then appropriate
algorithms to efficiently analyze high volumes of data will be developed.
Formulation of the model with quasi-periodic components
Consider as a model for NDVI time series a time dependent function ( )Y t expressed
in the form
Análisis tiempo-frecuencia con componentes cuasi-periódicos
55
1
( ) ( ) ( ) sin( ( ) ( )) ( ),n
k k kk
Y t s t A t t t t tω θ ε=
= + ⋅ ⋅ − +∑ (1)
that is, the superposition of a slowly varying function ( )s t , what will be called trend or
secular term, and a sum of almost periodic functions, corresponding to the terms
( ) sin( ( ) ( ))k k kA t t t tω θ⋅ ⋅ − , which will also be called cyclic, periodic, oscillating or
sinusoidal components.
Here, ( )k tω and ( )k tθ are almost constant functions, which will be called,
respectively, the quasi-frequencies and quasi-phases of the periodic signal, or simply
frequencies and phases if there is no confusion. Frequencies ( )k tω are ordered from
lower to higher values, so that, for NDVI data from areas including vegetation types
with usual annual and semiannual phenological cycles, when time is measured in years
the first two frequencies will be 1( ) 2tω π≅ and 2( ) 4tω π≅ , and higher
frequencies can also be considered to better fit some data.
The functions ( )kA t are the amplitudes of the corresponding cyclic components, and
it will be assumed that they vary much more slowly than the mean value of the lower
frequency 1( )tω , a condition that will also be assumed to be satisfied by the functions
( )k tω , ( )k tθ , and ( )s t .
The model includes the error term ( )tε , incorporating measurement errors, random
variations, or any other source of deviations from the deterministic part of the model,
which is the model to be fitted to the data. Once a model is fitted, it can be used to
replace the original data values for different purposes, producing a smoothing of the
NDVI time series.
Consider the NDVI time series shown in Figure 2, corresponding to four years data for
an individual pixel. A classical linear harmonic analysis with polynomial trend, i.e., a
model of the type
0 1
( ) sin(2 / ) ( ),g n
jj k k
j k
H t c t a k t T tπ φ ε= =
= ⋅ + ⋅ ⋅ ⋅ − +∑ ∑ (2)
Capítulo 4
56
is not able to capture variations in frequencies or phases, so that the shape of the
estimated cyclic component is necessarily constant, as shown in Figure 2, where a
model with two periodic terms, corresponding to the annual and semiannual
frequencies, has been fitted (Fig. 2, left), and the resulting cyclic component extracted
(Fig. 2, right).
Figure 2. Left: NDVI (x104) time series (time in days from January 2000) for an
area in Millares (Valencia province) and a model with a quadratic trend and a
two harmonic terms fitted to the data. A linear trend is also shown for
reference. Right: Cyclic component resulting from adding the two harmonic
terms of 1-year and ½-year periods.
Fitting a model with quasi-periodic components produces the results shown in Figure
3, where, as in the previous model (Figure 2), two basic frequencies have been
considered in the fitted model (Fig. 3, left), resulting in a cyclic component exhibiting
variations similar to those present in the data (Fig. 3, right).
Análisis tiempo-frecuencia con componentes cuasi-periódicos
57
Figure 3. NDVI (x104) data of Figure 2 and a fitted model with a quadratic
trend and two quasi-periodic components (left). The cyclic part of the fitted
model is shown on the right subfigure.
Figure 4. Harmonic terms with approximate annual (left) and semiannual
(right) periods corresponding to the fitted model presented in Figure 3.
The individual quasi-periodic components of the cyclic part of the fitted model (Fig. 3,
right) are shown in Figure 4, where clear variations in both annual (Fig. 4, left) and
semiannual (Fig. 4, right) oscillations are present. These variations can be decomposed
in changes in the amplitudes of the quasi-periodic components (Fig. 5, left), and
changes in their corresponding phases (Fig. 5, right).
Capítulo 4
58
Figure 5. For the model fitted in Figure 3, with the two cyclic components
shown in Figure 4, amplitudes (left) and phases (right) of the approximate
annual (blue) and semiannual (green) quasi-periodic terms. Phases are
periodic, and the origin of phases in the right subfigure is arbitrary.
The root mean square errors (RMS) of the fitted models, 325.7 for the classical
harmonic and 270.9 for the quasi-periodic, showed, as expected, better results for the
more flexible model. Also, since the variation in the cyclic part may affect the
estimation of the trend, a stronger quadratic effect in the secular component was
present in the classical harmonic model.
To better compare the fitting strengths of the models considered above, a model with
only one annual quasi-periodic component was fitted to the data, obtaining still a good
adjustment, with an RMS of 292.0 (Fig. 6). As will be explained next, the greater
flexibility and fitting capacity of the method can be achieved with a computational cost
not much higher than the classical harmonic analysis.
Análisis tiempo-frecuencia con componentes cuasi-periódicos
59
Figure 6. Comparison between the fitted model of Figure 3 (top), which
included two variable harmonic components, and a model fitted to the same
data with only one annual quasi-periodic term (bottom).
Basic algorithm
The basis of the method is the use of a time-frequency, or local Fourier, analysis. First,
to extract the cyclic component, for each data point ( )Y t a local Fourier analysis is
carried out in a window centered in t . A few periodic components with frequencies
multiple of a basic frequency 1ω are considered, the size of the moving window being
one or two periods of the basic wave with frequency 1ω . The Fourier components,
with frequencies 1ω , 12ω , and so on, are obtained using appropriate envelopes for
spectral analysis, but any other method to obtain the spectral components, like the
FFT, can also be used. Then, the cyclic part of the signal ( )Y t is estimated, as
Capítulo 4
60
represented in Figure 7, and only the values of these cyclic components at time t are
to be saved (red dots in Fig. 7, bottom).
Figure 7. Top: Data included in the window around the point to be analyzed
(red dot, abscissa marked by a vertical line). Bottom: First two harmonic terms
in the local Fourier analysis and their values at the abscissa of the point shown
in the left subfigure (red dots). Composite wave (red line) is shown for
comparison with the raw data in the left subfigure.
As a result of the first part of the algorithm, quasi-periodic functions ( )kV t , with
approximate frequencies multiple of 1ω , are obtained. Next, a basis of functions
including the ( )kV t is completed with an appropriate basis of functions for the secular
component, e.g., powers of t if a polynomial trend is to be fitted, obtaining a matrix of
basis functions
Análisis tiempo-frecuencia con componentes cuasi-periódicos
61
21 2
cyclic basis secular basis
... ... ... ... ... ... ...
( ) ( ) ... 1 ...
... ... ... ... ... ... ...
V t V t t t
=
X. (3)
Finally, a simple linear model Y b= ⋅X is fitted to the data vector Y by ordinary
least squares. For sound fitting models, i.e., if the selected frequencies are appropriate
for the data being analyzed, the coefficients for the cyclic terms in this linear model are
close to unity. In any case, the final cyclic components are those obtained in this last
step, with only differences in amplitudes, if any, with the previously obtained ( )kV t .
An efficient algorithm for equispaced data
The first step of the basic algorithm described above is in general computationally very
costly, since a Fourier analysis has to be carried out for each data point, including all
surrounding points inside the moving window, resulting in an algorithm of complexity
( )O N W× , where N is the number of data points and W the number of points in
the window. However, for equispaced data, as is the case for nominal dates in NDVI
biweekly time series, an efficient algorithm, of complexity ( )O N , can be devised.
Let ( )Y t be measured at the equispaced points { }0 0 0, , 2 , ...t t t h t h∈ + + . For each
frequency kω , create the unitary complex vector
( ) ( ) ( )0 0 0exp ( ) , exp ( ) , exp ( 2 ) , ... ,k k ki t i t h i t hτ ω ω ω= − ⋅ − ⋅ + − ⋅ +
and compute the pointwise product
[ ]0 0 1 0 2 0( ), ( ), ( 2 ), ... .Y Y t Y t h Y t hτ τ τ τ⋅ = ⋅ ⋅ + ⋅ +
Then, apply a moving window average to Yτ ⋅ , with an appropriate window size and
repeat an adequate number of times. As a result, a smoothed vector ( )k w Yα τ= ⋅ is
obtained, where the function w denotes the smoothing operator on the points inside
the window. Finally, the values of the cyclic component associated with the frequency
kω are obtained from the real part of the pointwise product of kα and τ ,
Capítulo 4
62
( )( ) 2 Re .k kV t α τ= ⋅ ⋅
Also, the variable amplitudes and phases of the quasi-periodic component are
obtained, respectively, as the absolute value of 2 kα⋅ and the argument of kα . In this
algorithm, the key point is the application of the moving average, and for equispaced
data it can be computed in ( )O N time. Thus, the algorithm can be efficiently applied
to large sets of data (Fig. 8).
Figure 8. Average values (x104) of the amplitudes of the annual (left) and
semiannual (right) quasi-periodic components for models fitted to NDVI time
series in an area comprising most of the Valencia province. The area (center-
right) with very high values of both amplitudes corresponds to a zone of rice
crops.
CONCLUSIONS
NDVI time series data, as exemplified by maps of annual dynamics in the Valencia
region (Fig. 1), show complex behaviors, including oscillations that may not be properly
analyzed by using classical spectral analysis with constant values of the parameters
defining the fitted models.
A general model, including secular and cyclic components, was proposed to take into
account temporal variations in the oscillations of NDVI time series, by considering time
dependent parameters defining the characteristics of the cyclic part of the models.
Thus, frequencies, phases and amplitudes of the periodic functions included in the
Análisis tiempo-frecuencia con componentes cuasi-periódicos
63
model were allowed to slowly vary in time, producing quasi-periodic components that
better represented the behavior of the real data.
Models with quasi-periodic components were fitted to some examples of NDVI time
series data using a time-frequency analysis approach, where in order to extract the
cyclic component of the model a local Fourier analysis has to be carried out for each
point in the data series and including all points in a moving window of appropriate size.
The type of time-frequency analysis needed to fit models with quasi-periodic
components would imply in general a high computational cost. For equispaced time
series data, as when nominal dates are consider for biweekly NDVI data, an efficient
algorithm was proposed, facilitating the application of the method to large volumes of
data.
The values of the parameters defining the quasi-periodic components, and their
variations in time, could provide a useful tool to investigate relations with different
types of vegetation, external factors such as climate or land use changes, as well as
their interactions. Work in progress includes the application of the method in different
types of vegetations and conditions in extensive areas of the Valencia region, to assess
the potential of the method to help analyze different problems in landscape analysis
and vegetation ecology.
ACKNOWLEDGEMENTS
This work was supported by the research projects FEEDBACK (CGL2011-30515- C02-
01), funded by the Spanish Ministry of Innovation and Science, and GVPRE/2008/310,
funded by the Valencia Regional Government (Generalitat Valenciana).
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Capítulo 4
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[5] Fontana1, F., Rixen, C., Jonas, T., Aberegg1, G. and Wunderle1, S., “Alpine
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[7] Hmimina, G., Dufrêne, E., Pontailler, J.-Y., Delpierre, N., Aubinet M., Caquet, B.,
de Grandcourt, A., Burban, B., Flechard, C., Granier, A., Gross, P., Heinesch, B.,
Longdoz, B., Moureaux, C., Ourcival, J.-M., Rambal, S., Saint André, L. and
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[8] Ganguly, S., Friedl, M. A., Tan, B., Zhang, X. and Verma, M. “Land surface
phenology from MODIS: Characterization of the Collection 5 global land cover
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[9] Geerken, R., Zaitchik, B. and Evans, J. P., “Classifying rangeland vegetation type
and fractional cover of semi-arid and arid vegetation covers from NDVI time-
series,” International Journal of Remote Sensing 26, 5535-5554 (2005).
[10] Li, Z. and Kafatos, M., “Interannual variability of vegetation in the United States
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[11] Sarkar, S. and Kafatos, M., “Interannual variability of vegetation over the Indian
sub-continent and its relation to the different meteorological parameters,”
Remote Sensing of Environment 90, 268-280 (2004).
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65
[12] van Leeuwen, W. J. D., “Monitoring the effects of forest restoration treatments
on post-fire vegetation recovery with MODIS multitemporal data,” Sensors 8,
2017-2042 (2008).
[13] van Leeuwen, W.J.D., Casady, G., Neary, D., Bautista, S., Alloza, J.A., Carmel, Y.,
Wittenberg, L., Malkinson, D. and Orr, B., “Monitoring post-wildfire vegetation
response with remotely sensed time-series data in Spain, USA and Israel,”
International Journal of Wildland Fire 19, 75-93 (2010).
[14] Chen, J., Jönsson, P., Tamura, M., Gu, Z., Matsushita, B. and Eklundh, L., “A
simple method for reconstructing a high-quality NDVI time series data set based
on the Savitzky–Golay filter,” Remote Sensing of Environment 91, 332-344
(2004).
[15] Jönsson, P. and Eklundh, L., “Seasonality extraction and noise removal by
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Geoscience and Remote Sensing 40, 1824-1832 (2002).
[16] Jönsson, P. and Eklundh, L., “TIMESAT– a program for analyzing time-series of
satellite sensor data,” Computers & Geosciences 30, 833-845 (2004).
[17] Bradley, B. A., Jacob, R. W., Hermance, J. F. and Mustard, J. F., “A curve fitting
procedure to derive inter-annual phenologies from time series of noisy satellite
NDVI data,” Remote Sensing of Environment 106, 137-145 (2007).
[18] Moody, A. and Johnson, D., “Land-surface phenologies from AVHRR using the
discrete Fourier transform,” Remote Sensing of Environment 75, 305-323 (2001).
[19] Jakubauskas, M. E., Legates, D. R. and Kastens, J. H., “Harmonic analysis of time
series AVHRR NDVI data,” Photogrammetric Engineering and Remote Sensing 67,
461-470 (2001).
[20] Wagenseil, H. and Samimi, C., “Assessing spatio-temporal variations in plant
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[21] Verbesselt, J., Hyndman, R., Newnham, G. and Culvenor, D., “Detecting trend and
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66
[22] Verbesselt, J., Hyndman, R., Zeileis, A. and Culvenor, D., “Phenological change
detection while accounting for abrupt and gradual trends in satellite image time
series,” Remote Sensing of Environment, 114, 2970-2980 (2010).
Capítulo 5:
Detección de áreas afectadas por incendios
forestales
OBJETIVOS
En este capítulo se presenta un método en dos fases para la detección de áreas
incendiadas, que puede ser aplicado de forma eficiente en zonas extensas. En la
primera parte del capítulo se explica el funcionamiento del método con un ejemplo
detallado y a continuación, en una segunda parte, se analizan sus propiedades
mediante su aplicación en una amplia zona de la Comunidad Valenciana y la
comparación entre las zonas incendiadas detectadas con el método y las registradas en
la base de datos de incendios de la Dirección general de Prevención, Extinción de
Incendios y Emergencias de la Generalitat Valenciana. El objetivo es disponer de un
método eficiente, que pueda ser utilizado de forma automática o semiautomática en
áreas amplias con distintos tipos de vegetación, de modo que se facilite el estudio de
los factores ambientales que afectan a la regeneración vegetal tras los incendios
forestales.
RESUMEN
En la primera parte de este capítulo se presenta un método en dos fases para la
detección automática de áreas quemadas a partir de series de datos quincenales de
índices de vegetación procedentes de MODIS. Para cada pixel en el area, se ajustan
modelos de diversa complejidad a las subseries de datos anterior y posterior al punto
considerado. Las discrepancias o saltos entre ambos modelos que exceden un cierto
umbral se utilizan para definir pixels semilla, a partir de los cuales se extienden y
definen los grupos de pixels correspondientes al area quemada potencial. Se construye
un algoritmo computacionalmente eficiente mediante una aproximación de tipo filtro
digital.
Detección de áreas quemadas
71
INTRODUCTION
Fire constitutes a key factor in the functioning and shaping of Mediterranean
ecosystems1, with fire regimes affected by anthropogenic and climatic factors2.
Accurate mappings of burnt areas as a result of wildfires are essential to analyze the
spatiotemporal distribution of wildfires and its relation with different environmental
factors, as well as to monitor the recovery of vegetation and the effect of restoration
treatments3-8.
Fire scars can be confidently mapped from visual comparison of pre- and post-fire high
resolution aerial photographs or satellite imagery of the zone that includes the
wildfire. Remote sensing from MODIS and other space-borne sensors has been
increasingly used in the last decades for automatic or semiautomatic detection of
active or past wildfires, usually from daily records of a suitable combination of
reflectance bands, allowing the analysis of wildfires distribution at regional and global
scales9-18.
Some methods to detect fire scars from remotely sensed time series data incorporate
two different phases, first detecting possible abrupt changes in the temporal data for
some pixels, usually by jointly considering several spectral indices, and then using
some algorithm for region growing to finally delimitate the perimeter of the fire
scar19,20. The objective of the present work was to test some simple two-phase
algorithms and variations for automatic or semiautomatic detection of burnt areas
from MODIS 250m biweekly NDVI time series data for the Valencia region, Southeast
Spain.
METHODS AND RESULTS
The first step for detecting fire scars from NDVI time series poses the general problem
of identifying change points or jumps in a noisy time series, corresponding to times,
pre- and post-fire, where the value of the NDVI drops as a consequence of vegetation
burning (Fig. 1). This apparently simple problem needs to be tackled taking into
account the high noise to signal ratio characteristic of NDVI series for individual pixels,
and in a computationally efficient way which would allow its practical application to a
very high number of pixels. Once this problem is solved, an initial set of pixels
Capítulo 5
72
exhibiting sufficiently high jumps can be obtained, acting as seeds for the second
phase of the algorithm, where the potential burnt area is delimited.
Figure 1. NDVI (x104) time series for an area in Bixquert Valley (Valencia
province) where a wildfire occurred in June 2005 (vertical line), and different
models fitted to pre- and post-fire data.
Change points detection in NDVI time series
Consider as a sample illustration the data shown in Fig. 1, corresponding to NDVI
values for an area in the Valencia region burned in 2005. Different models, such as
polynomials of different degrees, or models including constant or variable cyclic terms,
can be fitted to pre- and post-fire data, and discrepancies between the parameters of
these models can provide a measure of the magnitude of the jump, expectedly more
robust than just the change between the last pre-fire and the first post-fire values.
For an individual pixel in a region, where the presence of a wildfire is unknown, all the
points in the series are considered potential change points, and they are consecutively
analyzed outside some minimal border zones. Then, separate models can be fitted to
the data left and right to each potential change point, and discrepancies between
models computed. When the maximum value of these discrepancies exceeds some
appropriate threshold, a potential fire affecting that pixel, corresponding to the
extreme discrepancy date, is considered.
Detección de áreas quemadas
73
The application of this procedure to a very high number of pixels would be
computationally very costly, but an efficient algorithm can be devised for equispaced
data, as is the case for nominal dates in NDVI biweekly series, and for a wide family of
fitting models, including combinations of polynomials and trigonometric terms. In this
setting, discrepancies between parameters of the fitted models, or between their
predicted values, are linear functions of the data, and they can be computed with an
appropriate digital filter, as the convolution of the vector of data with a suitable vector
of weights, in a similar way to the classical Savitzky-Golay smoothing method21.
This process is related to wavelet analysis, and it is equivalent to it for certain fitting
models and particular wavelet base functions. Thus, for example, if a constant model is
fitted to n points at each side of the change point, the difference between both models
is obtained by applying a Haar wavelet with amplitude n+n (Fig. 2).
Figure 2. Haar filter output (green) and traveling Haar pulse (red, see text) for
the data in Figure 1.
The detection accuracy of the method using fitting models with polynomials of varying
degrees, with and without harmonic terms, was evaluated for the period 2000-2012 in
most of the Valencia region, where a record of wildfires with approximate perimeters
from field surveys was available. In most cases, best results were obtained with a
simple constant model of 1-year amplitude, reflecting the dominance of an annual
phenological cycle of the vegetation, more complex models being affected by the high
noise level of MODIS data at the individual pixel level.
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In some areas where a clear biannual cycle is present, this simple approach would
produced false positives, but this effect can be corrected by considering both the
difference in 1-year NDVI averages and the difference between consecutive points, by
simply multiplying them, what we called a traveling Haar pulse (Fig. 2).
Burnt areas delimitation
The steps of the algorithm to map fire scars will be illustrated with its application to an
area in Simat de la Valldigna (Valencia province) affected by a wildfire in September
2000, close to a zone that was burnt in July 2005. For each pixel in the area, the value
of the detected maximum drop in the series of NDVI values is shown in Fig. 3, and the
map of nominal dates corresponding to those drops is presented in Fig. 4.
Figure 3. Maximum drops in NDVI (x104) for each pixel in an area near Simat
de la Valldigna (Valencia province), and approximate perimeters of September
2000 (center) and July 2005 (right-down corner) wildfires.
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Figure 4. Dates for the maximum drops shown in Figure 3.
Figure 5. Smoothing of the maximum drops shown in Figure 3.
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Next, in order to reduce the noise in the data, a smoothed version of the map of
maximum drops is computed (Fig. 5), using a Gaussian kernel with a short amplitude
(Fig. 6).
Figure 6. Kernel applied to the data in Figure 3 to obtain the smoothed map of
maximum drops shown in Figure 5.
From the map of smoothed maximum drops (Fig. 5), a few NDVI curves corresponding
to individual pixels in the proximity of the local maxima in the map, are selected (Fig.
7), as the curves with maximum real, non smoothed, change in NDVI values in a very
reduced neighborhood of the local maxima. Then, for each curve, data at both sides of
the putative change point are modeled using polynomial and harmonic terms, fitted to
the data by ordinary least squares. As a heuristic rule, derived from wide previous
experience, the degree of the polynomial in time is increased in one unit for each two
years of data in the partial data series.
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Figure 7. Selected NDVI (x104) data series corresponding to pixels with
detected maximum drops in the proximities of the local maxima in the map of
smoothed maximum drops shown in Figure 5, and models fitted to the partial
series of pre- and post-change data.
Next, the selected curves are ranked by the magnitude of their maximum NDVI change
(Fig. 8), and only those with drops exceeding a certain threshold are retained. For the
set of spatial and temporal data analyzed, a value of 0.13 in the detected maximum
drop in NDVI was selected as the threshold. The selection of the threshold value can be
based on the distribution of detected maximum drops, or it may be derived from
cross-validation from a set of independently mapped fire scars. The location of the
four finally selected curves, on the contour map of maximum drops, is shown in Fig. 9.
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Figure 8. Maximum drops in NDVI (x104) values for the set of curves in Figure 7
ranked by their magnitude. Drops in actual values (blue) and in the smoothed
series (green). A threshold with a value of 1300 is marked (horizontal line).
Figure 9. Contour plot for the map of smoothed maximum drops shown in
Figure 5, with the positions of the selected curves (1 to 4) presented in Figure 7
whose maximum changes in NDVI exceed the threshold shown in Figure 8.
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The objective of the selection process is to obtain a small set of pixels that will act as
seeds in an extension clustering algorithm, to define spatially contiguous clusters
through some measure of similarity between each pixel in the region and those used
as seeds. A simple effective approach consists of clustering spatially contiguous pixels
with detected maximum drops in NDVI values corresponding to dates similar to those
of the seed curves (Fig.10).
Figure 10. Clusters of contiguous pixels defined by similarity of their dates of
maximum drops in NDVI with those of the pixels used as seeds (see Figure 9).
Pixels belonging to the same cluster defined by the date of the putative wildfire show,
as expected, similar behaviors in their temporal series, or curves, of NDVI values, with
a high correlation between the NDVI curve of each pixel and the curve of the seed pixel
used to obtain the cluster (Fig. 11).
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Figure 11. Correlations between NDVI data series of pixels in each cluster
shown in Figure 10 and NDVI series of the pixel used as seed to define the
cluster. Different colors identify different clusters and magnitudes of the
correlations are given by color intensities (grey scale).
CONCLUSIONS
In the Mediterranean Valencia region, for the temporal period 2000-2012 analyzed,
burnt areas as a consequence of wildfires were efficiently detected from 250 m
resolution MODIS derived NDVI biweekly data series. A two-phase algorithm, first
identifying seed pixels with maximum NDVI drops above some threshold and then
clustering contiguous pixels with an extension algorithm provided good results.
Detection of abrupt changes in NDVI as a consequence of vegetation burning was best
achieved by considering jumps between 1-year averages for partial series before and
after each possible change-point, or by using the product of the jumps in 1-year
averages and the drops in NDVI values of two consecutive points. No improvement
was achieved by fitting more complex models to the partial series, as the high
variability of MODIS data for individual pixels degraded the accuracy of the method.
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Delimitation of fire scars was based on propagating clusters from the seed pixels to
spatially connected pixels with similarities between their dates of detected maximum
drops, which would correspond to the date of the wildfire. Pixels in the same cluster
presented high correlation between their NDVI data series, and the maps of the
detected burnt areas showed a good agreement with the approximate perimeters
resulting from field surveys.
The algorithms of the method can be efficiently implemented, allowing its application
to large regions (Fig. 12). Work in progress include carrying out detailed comparisons
of the accuracy of the method with a curated database of wildfire perimeters and
other remote sensing fire mapping products and algorithms, exploring variations in the
automatic selection of thresholds in NDVI jumps to facilitate its use in regions with
different types of vegetation, and optimizing the modifications needed to best map
areas with fire recurrence.
Figure 12. Map of detected maximum drops in NDVI values for most part of
the Valencia province.
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ACKNOWLEDGEMENTS
This work was supported by the research projects FEEDBACK (CGL2011-30515-C02-01),
funded by the Spanish Ministry of Innovation and Science, and GVPRE/2008/310,
funded by the Valencia Regional Government (Generalitat Valenciana).
REFERENCES
[1] Pausas, J. G. and Vallejo, V. R., “The role of fire in European Mediterranean
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European Mediterranean basin], Springer-Verlag, 3-16 (1999).
[2] Pausas, J. G. and Fernandez-Muñoz, S., “Fire regime changes in the Western
Mediterranean Basin: From fuel-limited to drought-driven fire regime,” Climatic
Change 110, 215-226 (2012).
[3] Levin, N. and Heimowitz, A., “Mapping spatial and temporal patterns of
Mediterranean wildfires from MODIS,” Remote Sensing of Environment 126, 12-
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[4] Díaz-Delgado, R. and Pons, X., “Spatial patterns of forest fires in Catalonia (NE of
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[5] Wittenberg, L., Malkinson, D., Beeri, O., Halutzy, A. and Tesler, N., “Spatial and
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[6] Gouveia, C., DaCamara, C. C. and Trigo, R. M., “Post-fire vegetation recovery in
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[7] van Leeuwen, W.J.D., Casady, G., Neary, D., Bautista, S., Alloza, J.A., Carmel, Y.,
Wittenberg, L., Malkinson, D. and Orr, B., “Monitoring post-wildfire vegetation
response with remotely sensed time-series data in Spain, USA and Israel,”
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[8] van Leeuwen, W. J. D., “Monitoring the effects of forest restoration treatments
on post-fire vegetation recovery with MODIS multitemporal data,” Sensors 8,
2017-2042 (2008).
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[9] Barbosa, P. M., Grégoire, J. -M. and Pereira, J. M. C., “An algorithm for extracting
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[10] Chuvieco, E., Ventura, G., Martín, M. P. and Gómez, I., “Assessment of
multitemporal compositing techniques of MODIS and AVHRR images for burned
land mapping,” Remote Sensing of Environment 94, 450-462 (2005).
[11] Chuvieco, E., Martín, M. P. and Palacios, A., “Assessment of different spectral
indices in the red—Near-infrared spectral domain for burned land
discrimination,” International Journal of Remote Sensing 23, 5103-5110 (2002).
[12] Fraser, R. H., Li, Z. and Cihlar, J., “Hotspot and NDVI differencing synergy
(HANDS): A new technique for burned area mapping,” Remote Sensing of
Environment 74, 362-376 (2000).
[13] Giglio, L., Descloitres, J., Justice, C. O. and Kaufman, Y. J., “An enhanced
contextual fire detection algorithm for MODIS,” Remote Sensing of Environment
87, 273-382 (2003).
[14] Giglio, L., van der Werf, G. R., Randerson, J. T., Collatz, G. J. and Kasibhatla, P.,
“Global estimation of burned area using MODIS active fire observations,”
Atmospheric Chemistry and Physics 6, 957-974 (2006).
[15] Roy, D. P., Jin, Y., Lewis, P. E. and Justice, C. O., “Prototyping a global algorithm
for systematic fire-affected area mapping using MODIS time series data,”
Remote Sensing of Environment 97, 137-162 (2005).
[16] Davies, D. K., Ilavajhala, S., Wong, M. M. and Justice, C. O., “Fire information for
resource management system: Archiving and distributing MODIS active fire
data,” IEEE Transactions on Geoscience and Remote Sensing 47, 72-79 (2009).
[17] Roy, D. P., Boschetti, L., Justice, C. O. and Ju, J., “The Collection 5 MODIS burned
area product—global evaluation by comparison with the MODIS active fire
product,” Remote Sensing of Environment 112, 3690-3707 (2008).
[18] Boschetti, L., Roy, D., Barbosa, P., Boca, R. and Justice, C., “A MODIS assessment
of the summer 2007 extent burned in Greece,” International Journal of Remote
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[19] Bastarrika, A., Chuvieco, E. and Martín, M.P., “Mapping burned areas from
Landsat TM/ETM+ data with a two-phase algorithm: Balancing omission and
commission errors,” Remote Sensing of Environment 115, 1003-1012 (2011).
[20] Stroppiana, D., Bordogna, G., Carrara, P., Boschetti, M., Boschetti, L. and Brivio,
P. A., “A method for extracting burned areas from Landsat TM/ETM+ images by
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ISPRS Journal of Photogrammetry and Remote Sensing 69, 88-102 (2012).
[21] Savitzky, A. and Golay, M.J.E., “Smoothing and differentiation of data by
simplified least squares procedures,” Analytical Chemistry 36, 1627-1639 (1964).
RESUMEN
En esta segunda parte del capítulo se aplica el método en dos fases para la detección
de áreas quemadas, presentado en la primera parte del capítulo, a datos de NDVI
procedentes de MODIS, con una resolución espacial de 250m y una resolución
temporal quincenal, en área extensa de la Comunidad Valenciana. Los resultados se
comparan con los registros de incendios disponibles en la base de datos de incendios
de la Dirección general de Prevención, Extinción de Incendios y Emergencias de la
Generalitat Valenciana. Los mapas de incendios detectados mostraron un alto acuerdo
con los perímetros registrados en la base de datos usada como referencia, como se
muestra con diversas medidas e indices de precision y porcentajes de errors, con un
comportamiento similar o mejor que los proporcionados por otros métodos
disponibles en la literatura, incluso de mayor complejidad y dificultad de aplicación en
zonas con gran extensión.
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INTRODUCTION
Wildfires are common and natural disturbances in Mediterranean regions worldwide,
where they largely contribute to shape the structure and functioning of flammable
ecosystems [1]. Thus, many Mediterranean species have acquired adaptive
mechanisms to persist and regenerate after wildfires, facilitating the autosuccession of
the plant community and contributing to accelerate the recovery of the vegetation
cover [1, 2]. However, depending on fire attributes such as severity, frequency or size,
wildfires may promote critical changes in Mediterranean ecosystems and landscapes.
For example, in the Mediterranean Basin, frequent wildfires contribute to maintain the
dominance of shrubs in areas where the climate and soil conditions would be suitable
for a forest ecotype [3, 4, 5]. Severe and/or frequent wildfires may also promote
structural changes towards more fire-prone vegetation [6, 7]. Furthermore, the
synergistic effect of wildfire and post-fire extreme climatic events, such as droughts or
torrential rainfall, may result in long windows of disturbance, and challenge the
resilience of Mediterranean ecosystem [8, 9].
During the twentieth century, the fire regime in the Mediterranean Basin has shifted
to larger and more frequent wildfires [6, 10]. For example, in eastern Spain, there was
a major shift around the early 1970s, resulting in double annual fire frequency and four
times larger average fire size for the post-1970s period as compare with the previous
100 years. [11]. The main driving factor for this change was the rapid increase in fuel
amount and continuity due to the combined effect of agricultural land abandonment
and extensive reforestation programs [6, 11]. The rapid pace of these changes in fire
regimes explains the growing concern about the future spatiotemporal dynamics and
the associated ecological and socio-economic impacts of wildfires in the
Mediterranean Basin, particularly under a climate change context that could imply a
further increase in fire risk and frequency in the Mediterranean Basin [12, 13].
Accurate mappings of burnt areas as a result of wildfires are essential to analyze the
spatiotemporal distribution of wildfires and its relation with different environmental
factors, as well as to monitor the recovery of vegetation and the effect of management
and restoration treatments. Fire scars can be confidently mapped from visual
comparison of pre- and post-fire high resolution aerial photographs or satellite
imagery of the zone that includes the wildfire; the resulting burnt area maps are
widely used to investigate the spatiotemporal patterns of wildfire impacts [14-19].
Remote sensing from MODIS and other space-borne sensors has been increasingly
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used in the last decades for automatic or semiautomatic detection of active or past
wildfires, usually from daily records of a suitable combination of reflectance bands,
and with varying results in terms of accuracies and omission and commission errors
[14, 20-30].
The objective of the present work was to develop and test some simple algorithms and
variations for automatic or semiautomatic detection of burnt areas from MODIS 250m
biweekly NDVI time series data for a Mediterranean region. We evaluated the
detection and mapping method on a target area located in the Valencia region, East
Spain, which is a good model for most of the western Mediterranean Basin [11].
Methods
Data
MODIS data were downloaded from the NASA website (currently accessed through the
Reverb data gateway, http://reverb.echo.nasa.gov/reverb/). We used the NDVI 16-
days composite band from a time series of MOD13Q1 MODIS/Terra product at 250m
resolution (tile h17v05), starting in February 2000.
A database of fire event records for the time period 2000-2005 was provided by the
Fire Prevention, Extinction and Emergencies Office of the Valencia Regional
Government (Dirección general de Prevención, Extinción de Incendios y Emergencias,
Conselleria de Gobernación y Justicia, Generalitat Valenciana). For each fire event, the
fields in the database included information on date; Municipality where the wildfire
started, which gives name to the fire; forest and total affected area, and cartography
of the fire perimeter, obtained from IRS or SPOT images and field surveys.
Burnt areas detection and mapping
We used a two-phase algorithm to detect and map burnt areas, first detecting a subset
of seed pixels showing significant drops, as described below, in the series of NDVI data,
and then delimitating the fire scar for each potential wildfire using an extension
algorithm from the seed pixels. The main steps of the detection and mapping method
used in this work are as follows (see [31] for a description of the steps of the algorithm
illustrated with examples):
For each individual pixel in the area, all the points in the time series of NDVI data were
considered as potential change points, i.e., potential dates of wildfire occurrence. We
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fitted separate models to the data previous and posterior to the change point
(hereafter pre and post models, and computed the discrepancies between the models.
We tested different types of models, such as polynomials of different degrees and
models including cyclic terms. Discrepancies between the parameters of the pre and
post models were used as measure of the magnitude of the jump in NDVI values at the
potential change-point. We devised an efficient algorithm for equispaced data, as is
the case for nominal dates in NDVI biweekly series, valid for a wide family of fitting
models, for which discrepancies between parameters of the fitted models are linear
functions of the data, so that they can be computed with an appropriate digital filter,
as the convolution of the vector of data with a suitable vector of weights, in a similar
way to the classical Savitzky-Golay smoothing method [32]. As a result of a wide set of
trials, we finally discarded complex models and selected a combination of two simple
models: a simple constant model of 1-year amplitude, to account for the dominance
of an annual phenological cycle of the vegetation, and the difference between two
consecutive points. Discrepancies between pre and post models were calculated using
the product of two Haar wavelets of amplitude n+n and 1+1, what we called a
travelling Haar pulse.
For each pixel in the area, we computed the value of the detected maximum drop in
the series of NDVI values; obtained a map of smoothed maximum drops using a
Gaussian kernel of short amplitude, and selected a few NDVI curves, corresponding to
individual pixels in the proximity of the local maxima in the map, as the curves with
maximum real, non smoothed, change in NDVI values in a very reduced
neighbourhood of the local maxima. The selected curves were ranked by the
magnitude of their maximum NDVI change, and only those with drops exceeding a
certain threshold, which in this work was set to a value of 0.13, were retained as seed
pixel for the second phase of the method.
From the set of seed pixels, we apply an extension clustering algorithm, to define
spatially contiguous clusters through some measure of similarity between each pixel in
the region and those used as seeds. A simple effective approach consisted of clustering
spatially contiguous pixels with detected maximum drops in NDVI values
corresponding to dates similar to those of the seed curves, or by using a combination
of dates closeness and curves similarity, measured as the correlation between their
temporal series, or curves, of NDVI values.
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Evaluation of the method
We evaluated the detection and mapping method against the Regional database of fire
event records of the Valencia region. We compared the burnt areas detected by our
method with the registered wildfire perimeters and burnt area, for the time period
2000-2005, in a region of interest (ROI) in the Valencia region, located on the
Mediterranean east coast of the Iberian Peninsula (Figure 1).
Figure 1: Location of the Region of Interest (ROI) in the Valencia region, East
Spain.
The ROI is a large area of 10.178 Km2, including the central part, and 44% of the total
area, of the Valencia region. For the period 2000-2005, and considering a minimum fire
size of either 15 ha or 90 ha, total burnt area in the ROI was more than 62% of total
burnt area for the whole Valencia region (Table 1).
We compared the maps of detected and registered burnt areas using the GIS system
ESRI© ArcGis 9.0 (Environmental System Research Institute Inc., California). From this
comparison, we estimated the confusion matrices for the sets of detected and
registered fires with areas larger than 15ha and 90ha, and computed overall
accuracies, omission and commission errors, and several indices of agreement [33, 34].
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Table 1: Number of fires and total and forest burnt areas in the ROI and the whole
Valencia region, for the time period 2000-2005, for wildfires with affected areas
exceeding 15ha and 90ha.
ROI Valencia region
Burnt area
>15ha
Burnt area
>90ha
Burnt area
>15ha
Burnt area
>90ha
Number of fires 52 18 77 31
Total burnt area (ha) 11130 10011 17862 16189
Forest area (%) 21 20 26 25
RESULTS
The application of the method to the ROI, using the combination of local and 1-year
average drops to define seed pixels, yielded the results shown in Figure 2 (only
detected fire scars with areas larger than 15ha are mapped).
Table 2 shows a set of accuracy measures estimated from the confusion matrices that
resulted from the comparisons between detected fire scars and the reference
perimeters registered in the official, regional database. Values of overall accuracy and
the various indices of agreements were very high, particularly for the subset of
medium-to-large (> 90 ha) detected burnt areas as compared with the whole set (> 15
ha) of detected scars. Omission errors were very low, less than 4% for fire scars larger
than 90 ha, while commission errors were moderately low.
The accuracy of the method for the assessment of burned surface is illustrated in
Figure 3. Either for large wildfires, such as the Chiva wildfire, or small ones, such as the
Vall de Ebo wildfire, the agreement between registered perimeters and detected fire
scars is very high.
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Figure 2. Detected fire scars (> 15 ha) from MODIS in the ROI for the period 2000-2005.
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Table 2: Accuracy measures for detected burnt areas, larger than 15ha and larger than
90ha, as compared with the Regional database of fire event records used as reference.
Detected burnt
areas >15ha
Detected burnt
areas >90ha
Overall accuracy (%) 99.52 99.75
Quantity disagreement (%) 0.23 0.19
Allocation disagreement (%) 0.25 0.07
Omission error (%) 11.55 3.43
Commission error (%) 26.93 18.76
Standard Kappa index of agreement (kstandard) 0.798 0.881
Kappa for no information (kno) 0.990 0.995
Kappa for allocation (kallocation) 0.883 0.965
Kappa for quantity (kquantity) 0.995 0.996
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Figure 3. Registered perimeters (Regional database of fire event records) and detected
burnt areas from MODIS (this study) for the largest (Chiva wildfire, upper panel) and
smallest (Vall de Ebo wildfire, bottom panel) burnt area (> 90 ha) recorded in the ROI
for the period 2000-2005.
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Table 3: Main characteristics of the registered (Regional database of fire event records)
and detected (this study) burnt areas for wildfires with affected area larger than 90ha
recorded in the ROI for the period 2000-2005.
Registered burnt area (R) Detected burnt area (D)
Date Wildfire Total (ha) Forest (%) Total (ha) D-R overlap (%)†
16/09/2000 Chiva 2164.46 18.4 2508.41 99.09
28/08/2003 Buñol 1701.64 8.0 1974.55 98.37
03/09/2000 Simat de Valldigna 1259.13 25.5 1381.84 97.59
12/07/2005 Simat de Valldigna 641.95 12.1 792.55 97.92
12/08/2004 Serra 624.39 44.7 674.61 95.92
27/08/2000 Planes 558.42 16.2 639.17 97.74
09/08/2001 Vall de Gallinera 457.11 4.5 586.60 95.41
31/07/2000 Alcalalí 413.45 44.9 444.31 96.55
22/06/2005 Xativa 400.00 33.3 418.88 92.85
31/01/2003 Eslida 391.37 56.7 614.89 91.69
25/06/2001 Villalonga 275.05 0.4 293.44 96.30
24/01/2005 Vall de Almonacid 249.92 30.0 264.01 93.81
21/08/2000 Jérica 208.71 60.0 196.30 78.89
03/08/2000 Requena 190.76 50.0 214.87 91.86
29/08/2001 Chiva 174.30 3.4 199.84 95.34
11/10/2002 Beniganim 105.32 10.3 121.15 90.81
15/02/2005 Llaurí 99.13 9.1 115.15 90.13
27/08/2000 Vall de Ebo 96.16 8.4 98.29 89.40
†Overlapping of registered and detected burnt areas as percentage of the registered
areas
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Table 3 compares the registered and detected burnt areas for each individual wildfire
(>90 ha) in the ROI for the period 2000-2005. Overlapping areas ranged from78.89 to
99.09 % (average value: 93.9 ± 1.1) of the reference registered area, with larger
overlapping fractions for larger wildfires. Except for one case, Jérica wildfire, the
detected fire scar was slightly larger than the reference registered area.
DISCUSSION
A wide range of Mediterranean burnt areas, ranging in size between 15 ha and 2500
ha, and including contrasting fractions of forest and non-forest (shrublands,
grasslands, croplands) can be effectively detected and mapped with a simple two-
phase algorithm for automatic detection of burnt areas from MODIS 250m biweekly
NDVI time series data. In terms of overall accuracies, omission and commission errors
and indices of agreement, the results obtained with our method were at least as good
as previous studies that used a variety of automatic or semiautomatic fire scars
detection methods from remote sensing in the Mediterranean Basin and other regions
(see, e.g., [14, 29, 35, 36, 37, 38] and references therein).
Some methods to detect fire scars from remotely sensed time series data incorporate,
as is the case with our method, two different phases: first detecting possible abrupt
changes in the temporal data for some pixels, usually by jointly considering several
spectral indices, and then using some algorithm for region growing to finally delimitate
the perimeter of the fire scar. This two-phase approach is a way of balancing or
compensating omission and commission errors, by setting strict thresholds or
conditions for a pixel to be selected as a seed for the second phase, thus lowering
commission errors, and then using, explicitly or implicitly, lower requirements to
incorporate additional pixels in the extension phase, thus reducing omission errors [35,
36, 37].
In our method, detection of abrupt changes in NDVI as a consequence of vegetation
burning was best achieved by using the product of the jumps in 1-year averages and
the drops in NDVI values of two consecutive points, with no clear improvement by
fitting more complex models to the pre- and post- partial data series. Local drops,
between to consecutive values, are normally used in different detection methods,
although the use of 1-year averages has also been incorporated in some algorithms
[39]. We found a clear improvement when using the combination of both differences,
in comparison with the use of each of them separately. We think that both local drops
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in NDVI and substantial drops in pre- and post- 1-year averages can be produced by
many factors different of fire, as local fluctuations in NDVI measurements, droughts
and other climatic variations, agriculture and forest management, among others, but
their combination is likely to be the result of a wildfire.
In the second phase, delimitation of fire scars was based on propagating clusters from
the seed pixels to spatially connected pixels with similarities between their dates of
detected maximum drops and the correlations between their NDVI data series. The
maps of the detected burnt areas showed a good agreement with the perimeters
recorded in the database of fire events records used as reference, at least for medium-
to-large fires, with affected area exceeding 90ha. For these set of fires, the average
percentage of burnt area detected by our method was very high (≈ 94%), with an
average percentage of overestimation of 14.0 ± 3.1, which was mainly due to the low
spatial resolution of the MODIS data. We think that our method can equally be applied
to higher resolution data, as Landsat images, from which the expected results would
improve. The method could also be used in a semiautomatic way, to analyse a window
area where a wildfire is known to have occurred, in order to automatically delimitate
its perimeter, and also it could be fine tuned by adjusting to the local characteristics of
the area the value of the threshold to select the seeds, the amplitude of the filter used
to compute the map of smoothed maximum drops, and the relative weights of the
similarities in nominal dates of fire and correlations between NDVI curves in the
clustering algorithm used to extent the seeds.
Although the evaluation of the algorithm was carried out in a relatively large area, it
was applied with a fixed threshold value, which should be adapted to the particular
region to be analysed. As in other methods, the selection of the threshold value can be
based on the distribution of detected maximum drops, or it may be derived from
cross-validation from a set of independently mapped fire scars.
As the objective of the work was to test the ability of using MODIS-derived NDVI and
simple methods to detect and map burnt areas, no other spectral band or indices were
used, no systematic procedure to optimize the parameters of the method was
performed, and no previous filtering of the evaluated area to discard agricultural or
non-forest areas was applied, leaving room for further improvements in the accuracy
of the method.
Capítulo 5
98
CONCLUSIONS
In a large area in the Mediterranean Valencia region, for the temporal period analyzed,
burnt areas as a consequence of wildfires were efficiently detected from 250 m
resolution MODIS derived NDVI 16-days time series data. A two-phase algorithm, first
identifying seed pixels with maximum NDVI drops above some threshold, combining
local and 1-year average drops, and then clustering contiguous pixels with an
extension algorithm provided good results. The maps of the detected burnt areas
larger than 90ha showed a good agreement with the perimeters registered in the
database of fire records used as reference.
The algorithms of the method can be efficiently implemented, allowing its application
to large regions, thus providing a useful tool for assessing ecological and
environmental factors of wildfire patterns and impacts. Further work could extend the
applicability of the method, by including the automatic selection of thresholds in NDVI
jumps to facilitate its use in regions with different types of vegetation, and optimizing
the modifications needed to best map areas with fire recurrence.
ACKNOWLEDGEMENTS
This work was supported by the research projects FEEDBACK (CGL2011-30515- C02-
01), funded by the Spanish Ministry of Innovation and Science, CASCADE (GA283068),
funded by European Commission under the Seventh Framework Program, and
GVPRE/2008/310, funded by the Valencia Regional Government (Generalitat
Valenciana).
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Conclusiones generales
Conclusiones generales
105
Como se ha puesto de manifiesto en el Capítulo 2, los modelos ocultos de Markov
(HMMs) pueden ser utilizados para caracterizar las propiedades fenológicas de la
vegetación a partir de datos de índices de vegetación por teledetección.
Los HMMs pueden ser aplicados de forma eficiente a grandes conjuntos de datos y
pueden proporcionar resultados consistentes cuando se consideran conjuntos de
píxeles con dinámicas de vegetación similares, lo que puede conseguirse utilizando una
clasificación externa o mediante un agrupamiento automático previo en función de las
distribuciones de cambios en los valores de NDVI.
Los parámetros estimados mediante el ajuste de HMMs, probabilidades de transición y
medias y varianzas de las emisiones, reflejan las propiedades de los diferentes estados
fenológicos de la vegetación, pudiendo ser utilizados para comparar las dinámicas de
comunidades vegetales afectadas por perturbaciones naturales o tratamientos
experimentales.
La utilización de HMMs permite la incorporación del conocimiento previo del sistema
en la modelización del mismo, mediante la selección del número de estados o de las
transiciones permitidas, proporcionando modelos específicamente ajustados a
distintas comundades, que podrían permitir una major determinación de los
parámetros fenológicos de interés.
Una vez seleccionada la topología de un HMM, la estimación de sus parámetros se
puede realizar a partir de todo el conjunto de píxeles y valores temporales, lo que
produce estimaciones altamente consistentes. Sin embargo, la inferencia de los
estados ocultos se realiza al nivel de píxeles individuales, pudiendo llevarse a cabo en
distintos periodos temporales, permitiendo analizar relaciones con factores
ambientales locales.
Se pueden considerar distintas extensiones al análisis presentado en el Capítulo 2,
incluyendo la utilización de tipos de HMMs más complejos y su comparación con otros
métodos de ajuste de curvas usuales en el análisis fenológico. En particular, podrían
utilizarse modelos de HMMs de orden superior o semi-Markovianos, que permitirían
modelizar de forma más específica la duración de las diferentes fases estacionales.
Los modelos lineales a trozos constituyen uno de los tipos de modelos más simples que
pueden considerarse para extraer las tendencias generales y los puntos de cambio en
series largas de datos con cambios de comportamientos. Los algoritmos presentados
en el Capítulo 3 proporcionan un método computacionalmente eficiente para ajustar
Conclusiones generales
106
modelos continuos lineales a trozos a series de datos con un alto número de puntos y
en los que es preciso considerar un alto número de puntos de cambio. Estos
algoritmos pueden ser una alternativa eficiente a los métodos basados en la
optimización de funciones de riesgo mediante búsquedas exhaustivas.
El ajuste de funciones a trozos para estimar tendencias y periodos de crecimiento y
decrecimiento en series de datos, como las correspondientes a series temporales de
índices de vegetación por teledetección, implica la asunción explícita o implícita de de
cierta información sobre el sistema, que permite seleccionar el nivel de complejidad
del modelo a ajustar o el rango esperable de los parámetros que lo definen.
Los algoritmos presentados en el Capítulo 3 incluyen diferentes opciones y parámetros
que permiten adaptarlos al tipo de modelo más adecuado en función de los datos
considerados. No obstante, los resultados presentados muestran que no es preciso
realizar una optimización de los valores iniciales o los parámetros de los algoritmos,
pues un amplio rango de valores razonables generalmente proporciona ajustes finales
similares.
Una extension simple de los algoritmos de ajuste de modelos lineales a trozos
presentados en el Capítulo 3 consistiría en su implementación en forma
completamente automática, utilizando medidas globales de ajuste para comparar
modelos con el mismo número de puntos de cambio y algún tipo de índice de
selección de modelos para decidir entre modelos con distinto número de parámetros.
Los datos de series temporales de índices de vegetación pueden mostrar dinámicas
complejas, como se muestra en el Capítulo 4 con datos de NDVI procedentes de
MODIS en zonas de la Comunidad Valenciana. Estas dinámicas incluyen oscilaciones
que no siempre pueden ser adecuadamente analizadas utilizando modelos clásicos de
análisis espectral con valores constantes de los parámetros.
El modelo de ajuste con componentes cuasi-periódicos presentado en el Capítulo 4,
que incluye un componente secular y componentes cíclicos con parámetros variables
en el tiempo, de modo que las fases, amplitudes y frecuencias de las funciones
periódicas del modelo pueden variar de forma lenta a lo largo del tiempo, proporciona
una mayor flexibilidad y una mejor representación del comportamiento de los datos
reales
El ajuste de modelos con componentes cuasi-periódicos puede realizarse mediante un
análisis de tipo tiempo-frecuencia, en el que se realiza un análisis de Fourier local en
Conclusiones generales
107
cada punto de la serie de datos, utilizando una ventana móvil de amplitud adecuada. El
alto costo computacional de este tipo de análisis pude reducirse notablemente en el
caso de datos equiespaciados, como los correspondientes a fechas nominales en datos
de NDVI, mediante el algoritmo eficiente propuesto en el Capítulo 4, facilitando su
aplicación a grandes volúmenes de datos.
Los valores de los parámetros que definen los componentes cuasi-periódicos, y su
variación en el tiempo, pueden proporcionar una herramienta útil para investigar las
relaciones de la dinámica de la vegetación en función de la composición de la
comunidad vegetal, de factores externos como cambios en los usos del suelo o en
variables climáticas, así como de sus interacciones. La aplicación extensiva del método
en distintas zonas, como se prevé realizar en áreas de la Comunidad Valenciana,
permitiría evaluar el potencial de este tipo de modelos como herramienta de análisis
en ecología vegetal y del paisaje.
La detección de zonas afectadas por incendios forestales, en las que aparecen cambios
abruptos en las series temporales de datos de NDVI, puede llevarse a cabo de forma
efectiva mediante el algoritmo en dos fases presentado en el Capítulo 5.
La identificación de los píxeles semilla del algoritmo puede realizarse de forma simple
combinando las caídas locales, entre un punto temporal y el siguiente, en los valores
de NDVI con las caídas en los promedios anuales previos y posteriores a cada punto,
sin que se obtengan mejores resultados al utilizar modelos de ajuste más complejos. La
delimitación de las áreas quemadas puede llevarse a cabo mediante un algoritmo de
extensión a partir de los píxeles semilla hacia píxeles contiguos con dinámicas de NDVI
similares.
La comparación de las áreas quemadas detectadas con las zonas registradas en una
amplia zona de la Comunidad Valenciana, presentada en el Capítulo 5, mostró una alta
precisión del método, especialmente en el caso de los incendios de mayor superficie.
Los algoritmos del método de detección de áreas quemadas propuesto en el Capítulo 5
pueden ser implementados de forma eficiente, permitiendo su aplicación en áreas
extensas.
Posibles extensiones del método que faciliten su aplicación automática en áreas
diversas incluirían la selección automática de los valores umbrales en las caídas de
NDVI para incorporar un píxel como semilla potencial del algoritmo y optimizar las
modificaciones necesarias para la detección de zonas con recurrencia de incendios. La
Conclusiones generales
108
aplicación automática del método en regiones amplias, con diferentes tipos de
vegetación, puede proporcionar una herramienta útil para el análisis del efecto de
distintos factores ecológicos y ambientales en la distribución e impacto de los
incendios forestales.
El conjunto de herramientas de análisis desarrolladas en este trabajo puede ser de
utilidad en distintos aspectos del análisis del paisaje y la ecología vegetal. Su aplicación
efectiva a problemas reales de interés ecológico, en algunos casos ya en proceso en
colaboración con otros grupos de investigación, mostrará la potencialidad de los
distintos métodos propuestos.
Anexo
Conceptos básicos sobre modelos ocultos de Markov
Análisis espacial mediante modelos ocultos de Markov
Francisco Rodríguez1, Miguel A. García1 y Susana Bautista2
1Departamento de Matemática Aplicada y 2 Departamento de Ecología, Universidad de
Alicante, Ctra. San Vicente del Raspeig s/n, 03690 San Vicente del Raspeig, Alicante,
España.
Extracto del Capítulo 7 del libro: Introducción al Análisis Espacial de Datos en Ecología
y Ciencias Ambientales: Métodos y Aplicaciones. Maestre, F.T.; Escudero, A.; Bonet, A.
(eds.). Servicio de publicaciones de la Universidad Rey Juan Carlos de Madrid, 2008.
ÍNDICE
Resumen
7.1. Introducción
7.2. Elementos y algoritmos básicos de los modelos ocultos de Markov
7.3. Generalizaciones y variaciones del modelo básico
7.4. Casos prácticos
7.4.1 Análisis de transectos de vegetación con datos de presencia-ausencia
7.4.2 Análisis de transectos de vegetación con datos cuantitativos
7.5. Consideraciones finales
7.6. Revisión de software
7.6.1. Herramientas en el entorno MATLAB
7.6.1.1. Funciones básicas en MATLAB
7.6.1.2. Toolbox HMM y conjunto de funciones H2M
7.6.2. Herramientas para el entorno R
7.6.3. Otros programas
7.7. Páginas web de interés
Conceptos básicos sobre modelos ocultos de Markov
113
7.1. Introducción
Existe una gran variedad de métodos disponibles para el análisis de patrones
espaciales, como se muestra en los diversos capítulos de este libro o en la bibliografía
específica (p. ej. Dale 1999, Fortin y Dale 2005), con diverso grado de relación entre
ellos (Dale et al. 2002). Algunas de las técnicas en uso fueron desarrolladas
originalmente en otros campos, como es el caso del análisis espectral (Ripley 1978,
Renshaw y Ford 1984) o más recientemente el empleo de wavelets (Dale y Mah 1998),
provenientes ambas del campo de la teoría de la señal y que ya se encuentran
incorporadas en el conjunto de herramientas estándares para realizar análisis espacial
en ecología.
Los modelos ocultos de Markov (que escribiremos, en adelante, HMM, tanto en
singular como en plural, utilizando las iniciales del término en inglés, Hidden Markov
Models, con las que se denotan usualmente), son una técnica de modelización de
datos secuenciales desarrollada y aplicada originalmente en el campo del
reconocimiento automático del habla (Rabiner 1989), donde actualmente es una
herramienta casi imprescindible, pues la mayor parte de las aplicaciones en este
campo incluyen algún tipo de HMM en su estructura. Los HMM han encontrado
aplicación, más recientemente, en disciplinas muy diversas, como el análisis de imagen
(Aas et al. 1999) o la psicología (Visser et al. 2002), destacando su uso creciente en el
análisis de electroencefalogramas y otras señales biológicas (p. ej. Penny y Roberts
1998, Novák et al. 2004) y, muy especialmente, en bioinformática, donde los HMM
están ya bien establecidos como una de las técnicas básicas (p. ej. Baldi et al. 1994,
Baldi y Brunak 1998, Durbin et al. 1998) y donde se están utilizando distintas variantes
de HMM y generalizaciones (p. ej. Winters-Hilt 2006).
A pesar del enorme incremento en las publicaciones sobre HMM y aplicaciones en los
últimos quince años (véanse Cappé 2001a, para una recopilación bibliográfica de la
pasada década, y la Figura 7.1), la utilización de HMM en áreas de interés en ecología,
aunque en aumento, ha sido muy escasa hasta la fecha. En Viovy y Saint (1994) se
aplican los HMM para el estudio de la dinámica temporal de la vegetación a partir de
datos de teledetección. En Tucker y Anand (2005) se discute la utilidad de los HMM, en
comparación con la modelización mediante cadenas de Markov clásicas, para detectar
dinámicas ecológicas complejas. En Guilford et al. (2004) se utilizan HMM para analizar
datos de navegación de aves. En Franke et al. (2004) se analizan mediante HMM los
estados de comportamiento del caribú, mientras que en Franke et al. (2006) se trata
Anexo
114
de predecir los lugares de caza del lobo a partir de datos de localización por GPS. En
Ver Hoef y Cressie (1997) se utilizan HMM para modelizar transectos de vegetación en
pastizales, con el objetivo de definir los bordes, o puntos de cambio, entre las zonas
con y sin vegetación. Una aplicación similar, con el objetivo de analizar patrones
complejos en transectos de vegetación, se lleva a cabo en Rodríguez y Bautista (2001),
trabajo en el que se basa parcialmente la revisión que se presenta en Rodríguez y
Bautista (2006).
En la siguiente sección se exponen los aspectos esenciales de los HMM, explicando los
conceptos básicos, pero sin desarrollar con todo detalle las cuestiones matemáticas y
computacionales. Para el lector interesado, una buena recomendación para empezar a
profundizar en el tema es la exposición clásica de Rabiner (1989), así como la revisión
de Bengio (1999), en la que se incluyen distintas extensiones de los modelos básicos.
Entre los libros específicos sobre el tema, posiblemente el de un nivel más asequible
sea MacDonald y Zucchini (1997), mientras que en Elliot et al. (1995) y Cappé et al.
(2005) se presentan diversas generalizaciones y aspectos más avanzados. También
puede encontrarse un tratamiento de los aspectos básicos de los HMM en manuales
sobre aprendizaje automático (p. ej. Sierra 2006).
7.2. Elementos y algoritmos básicos de los modelos ocultos de Markov
Para explicar los conceptos básicos de los HMM, conviene empezar por los modelos
más simples, los denominados HMM discretos de primer orden. Asimismo, es
conveniente recordar en qué consiste una cadena o modelo de Markov, un concepto
bien conocido en ecología, por sus aplicaciones en dinámica de poblaciones (p. ej.
Caswell 2001) o en la modelización de la dinámica de la sucesión (p. ej. Waggoner y
Stephens 1970, Wootton 2001).
Un modelo de Markov consiste esencialmente en un conjunto de K estados, en los que
puede encontrarse el sistema en cada momento, y ciertas probabilidades de transición
o de paso de cada estado a todos los demás, incluyendo la transición al propio estado
de partida. Los estados pueden representar cualquier situación de interés del sistema
que se está modelizando, como las posibles distintas fases en la sucesión de un bosque
o los cuatro nucleótidos que se pueden encontrar en una cadena de ADN, pero una vez
decidido cómo ordenamos los K estados posibles podemos referirnos a ellos por sus
índices y hablar del estado i, donde i puede tomar los valores 1…K. Una realización del
modelo es una cadena de n estados, S1… Sn, en los que se encuentra el sistema en n
Conceptos básicos sobre modelos ocultos de Markov
115
momentos sucesivos, es decir, una sucesión {St} de longitud n, en donde cada St puede
tomar uno de los valores 1…K. La propiedad esencial de una cadena de Markov de
orden r es que la probabilidad de que St tome uno de los K valores posibles no
depende de lo que haya ocurrido antes de los r momentos anteriores; en particular, en
una cadena de Markov de primer orden, que es el caso que consideraremos en
adelante, la evolución futura del sistema sólo depende del estado actual y no de cómo
se haya llegado a él, es decir, es independiente de la historia del sistema. Cuando las
probabilidades de transición son constantes a lo largo del tiempo se dice que la cadena
de Markov es homogénea. En ese caso las probabilidades de transición pueden darse
en forma de una matriz T=(Tij) de tamaño K×K (K filas y K columnas), donde el
elemento Tij es la probabilidad de pasar del estado i al estado j, es decir, la
probabilidad condicionada de que sea St+1=j dado que St=i. Si las frecuencias con las
que aparecen los distintos estados permanecen constantes, se dice que la cadena es
estacionaria; las frecuencias de estados en una cadena estacionaria están
determinadas por las probabilidades de transición (la propiedad matemática que
define estas probabilidades es que constituyen un vector propio de T correspondiente
al valor propio 1). Una forma de representar un modelo de Markov es mostrar los
distintos estados posibles y las probabilidades de transición entre ellos (Fig. 7.2); un
esquema alternativo, para indicar cómo evoluciona el sistema y destacar la
independencia condicional respecto de la historia previa, se presenta en la Figura 7.3.
En un HMM existe una cadena de Markov, pero corresponde a un proceso oculto, no
observable. Estos estados ocultos corresponden a propiedades del sistema, reales o
ideales, que no podemos observar directamente, pero que se corresponden, a través
de un modelo probabilístico, con un conjunto de manifestaciones que sí pueden ser
observadas, lo que en los HMM discretos se denominan los símbolos del sistema, de
modo que para cada estado oculto existe una cierta probabilidad de que se observe
uno de estos símbolos, denominada probabilidad de emisión del correspondiente
símbolo.
Precisando estas ideas para el caso de un HMM discreto de primer orden, un HMM de
este tipo queda definido por los siguientes cuatro elementos esenciales: un conjunto
de K estados, un conjunto de D símbolos, una matriz T=(Tij), de tamaño K×K, de
probabilidades de transición y una matriz E=(Eip), de tamaño K×D, de probabilidades de
emisión. Los datos observables consisten en una cadena de n símbolos, es decir, una
sucesión {Yt} de longitud n, donde el valor de cada Yt puede ser uno de los D símbolos,
para t = 1,…n. Existe una sucesión correspondiente oculta, no observable, de estados
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116
{St}, donde cada St puede tomar uno de los K valores que constituyen el conjunto de
estados. De igual modo que en las cadenas de Markov, siempre que no se produzca
ambigüedad podremos identificar las etiquetas de los símbolos o estados y sus índices,
de modo que diremos que St toma los valores 1…K y que Yt toma los valores 1…D.
Cuando el sistema se encuentra en el estado i, tiene una probabilidad Tij de pasar al
estado j, incluyendo la posibilidad de pasar al mismo estado, es decir, de que en el
momento St+1 el sistema permanezca en el mismo estado en el que se encontraba en el
instante St, lo que ocurrirá con probabilidad Tii; asimismo, existe una cierta
probabilidad Eip de emitir el símbolo p, que sólo depende del estado en el que se
encuentra el sistema. Estas probabilidades de transición, Tij, y de emisión, Eip, son
independientes de cuál haya sido la historia del sistema hasta llegar al estado actual,
es decir, de cuáles hayan sido las sucesiones de estados (ocultos) y de observaciones.
Por tanto, podemos decir que existen dos relaciones de independencia condicional
para las sucesiones de estados y de observaciones. Dado un cierto estado St, se tiene
que Yt es independiente del resto de observaciones, mientras que dado St-1 se tiene
que St es independiente de todos los estados anteriores S1 … St-2. Esta última relación
nos dice que la sucesión de estados ocultos constituye una cadena de Markov de
primer orden.
Una representación gráfica de relaciones de probabilidad condicionadas, como las
anteriores, entre un conjunto de variables aleatorias se denomina red bayesiana (Pearl
1988; Heckerman 1996). La Figura 7.4 muestra una red bayesiana correspondiente a
un HMM de primer orden, representada mediante un grafo acíclico dirigido, donde los
nodos del grafo representan las variables aleatorias correspondientes a los estados del
sistema en momentos sucesivos (círculos para los estados ocultos y cuadrados para las
observaciones) y donde la ausencia de una flecha entre dos variables nos indica su
independencia condicional.
Aunque para caracterizar un HMM necesitamos conocer todos sus elementos, la
estructura básica, lo que se denomina la topología, de un HMM queda definida por el
número de estados ocultos, el número de símbolos y las transiciones de estados y
emisiones de símbolos no permitidas (para las que las correspondientes
probabilidades de transición o emisión se asume que son cero). Estos son los
elementos básicos que deben tenerse en cuenta para modelizar un cierto sistema,
pues quiénes sean las etiquetas correspondientes a los estados ocultos y a los símbolos
no afecta al modelo, aunque es esencial para su interpretación, mientras que los
valores concretos de los parámetros del HMM, esto es, los valores de las
Conceptos básicos sobre modelos ocultos de Markov
117
probabilidades de transición y de emisión que pueden tomar valores no nulos, serán
normalmente estimados a partir de los datos experimentales, ajustando de esta forma
a los datos el modelo cuya estructura se ha propuesto previamente.
Conociendo la topología de un HMM y los valores de sus parámetros (probabilidades
de transición y emisión) se podría simular su comportamiento, es decir, podríamos
obtener secuencias aleatorias de estados ocultos y observaciones generadas por el
modelo. Para ello, sin embargo, necesitamos conocer un último elemento, las
frecuencias iniciales o probabilidades de iniciar la cadena en cada uno de los K estados
posibles, es decir, un vector con elementos πi, para i=1..K, donde πi=P(S1=i),
probabilidad de que la sucesión de estados comience en el estado i. Si suponemos que
la sucesión de estados ocultos es estacionaria, entonces las probabilidades de
encontrar los distintos estados permanecen constantes a lo largo del tiempo y las
frecuencias iniciales serán iguales a estas probabilidades que, como se dijo antes,
quedan perfectamente determinadas a partir de la matriz de probabilidades de
transición. Esta suposición de estacionariedad será razonable, por ejemplo, cuando la
secuencia de observaciones que intentamos modelizar sea un segmento aleatorio
dentro de una serie más larga, posiblemente infinita. Por el contrario, en algunas de las
aplicaciones más extendidas de los HMM, como en reconocimiento de habla o en
bioinformática, la identificación de una cierta frase, en una sucesión de sonidos, o de
una cierta estructura, en la secuencia de una proteína, requerirá disponer de un
modelo para el estado inicial de la secuencia a identificar, o su determinación, de la
forma más precisa posible, a partir de las observaciones.
En todo caso, la mayor utilidad de los HMM como herramientas de análisis se basa en
la posibilidad de estimar un modelo a partir de una serie de datos, que suponemos que
son el resultado observable de una serie de estados no directamente accesibles y en
los que estamos interesados, bien sea por tener un cierto significado para el problema
abordado o porque de esta forma se obtiene un modelo con una mayor capacidad de
predicción.
Una vez que hemos definido una cierta topología, existen dos problemas básicos en el
análisis de HMM, conocidos como el problema del aprendizaje y el problema de la
inferencia. Dada una secuencia de observaciones, el problema del aprendizaje consiste
en estimar los parámetros del modelo, es decir, las probabilidades de transición y
emisión. En realidad podría plantearse un problema de aprendizaje más general, en el
que se incluyese la selección de la topología, pero, aunque se han propuesto técnicas
para abordar este problema general (Heckerman 1996), se trata de un tema más
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118
complejo, sobre el que comentaremos algunos aspectos más adelante. Dada una
secuencia de observaciones, y una vez que los parámetros han sido estimados, o bien
si son previamente conocidos, el problema de la inferencia consiste en obtener la
correspondiente sucesión de estados ocultos.
El problema del aprendizaje puede ser resuelto mediante el algoritmo denominado EM
(Dempster et al. 1977), o de maximización de la esperanza (expectation maximisation),
que proporciona los valores de los parámetros que maximizan (el logaritmo de) la
verosimilitud de las observaciones en función de los parámetros; usualmente se utiliza
el denominado algoritmo de Baum-Welch (Baum et al. 1970), que es una versión
particular, eficiente desde el punto de vista computacional, del algoritmo EM. El
problema de la inferencia puede ser resuelto obteniendo la sucesión de estados
ocultos más probable mediante un algoritmo de programación dinámica conocido
como algoritmo de Viterbi (Viterbi 1967), que es un caso especial de algoritmos de
inferencia aplicables a modelos gráficos más generales desarrollados por Pearl (1988) y
otros autores (p. ej. Smyth 1997, Smyth et al. 1997).
Veamos con un ejemplo los distintos conceptos expuestos en este apartado. La
estructura del HMM que vamos a considerar se muestra en la Figura 7.5, donde se
indica que el modelo consta de dos estados ocultos y dos símbolos, con las
probabilidades de transición y emisión que allí se muestran. Este modelo es similar a
uno de los que vamos a considerar más adelante para modelizar transectos con datos
de presencia o ausencia de vegetación; en ese caso, los estados ocultos
corresponderán a zonas densas y ralas de vegetación (manchas y claros) mientras que
los símbolos corresponderán a que en el punto concreto del transecto haya habido o
no contacto con vegetación. Utilizando el modelo indicado en la Figura 7.5, se ha
simulado una cadena de estados ocultos, y la correspondiente secuencia de símbolos,
de longitud 50 (Figura 7.6). Para obtener el estado inicial de la secuencia de estados
ocultos, se han considerado probabilidades de inicio correspondientes a las
frecuencias estacionarias, π = (π1, π2) = (0,5, 0,5), que se han obtenido a partir de la
matriz de (probabilidades de) transición como el vector propio correspondiente al
valor propio 1, normalizado de modo que la suma de sus componentes sea la unidad.
Supongamos ahora, como es habitual en las aplicaciones, que no conocemos ni los
parámetros del modelo ni la sucesión de estados ocultos, sino únicamente la secuencia
de símbolos. Dados ciertos valores de los parámetros, es decir, dado un modelo (M), se
puede calcular la probabilidad de cualquier sucesión de estados ocultos, P(estados |
M), pues sólo hay que multiplicar la probabilidad de inicio por las probabilidades de
Conceptos básicos sobre modelos ocultos de Markov
119
transición correspondientes; asimismo, dada una cierta sucesión de estados ocultos, la
probabilidad de que se emita una secuencia de observaciones puede calcularse,
P(observaciones | estados), pues sólo hay que multiplicar las correspondientes
probabilidades de emisión; por tanto, la probabilidad de que el modelo emita la
secuencia de observaciones de la Figura 7.6 se obtendrá sumando, para todas las
sucesiones de estados ocultos posibles, el producto de estas dos probabilidades,
P(observaciones | estados) x P(estados | M), y esta probabilidad nos indicará la
verosimilitud de los datos observados como función de los parámetros del modelo. Los
valores de los parámetros que hacen máxima la probabilidad de los datos observados
son las estimaciones de máxima verosimilitud, que se obtienen aplicando el algoritmo
de Baum-Welch. Una vez que se han obtenido estimaciones de los parámetros,
podemos calcular la probabilidad a posteriori de cada una de las secuencias posibles
de estados ocultos dados los datos observados; la estimación de la secuencia de
estados ocultos que proporciona el algoritmo de Viterbi es precisamente la secuencia
que hace máxima esta probabilidad. Una explicación detallada y muy clara del
funcionamiento de estos algoritmos puede verse en Rabiner (1989). En la Figura 7.6 se
muestran también los valores estimados de los parámetros del modelo y la secuencia
estimada de estados ocultos. Como puede observarse, existen discrepancias, en algún
caso apreciables, entre los valores reales y estimados de los parámetros, debido al
tamaño reducido de la secuencia de observaciones utilizada para ajustar el modelo.
7.3. Generalizaciones y variaciones del modelo básico
Una característica esencial de los HMM es que el conjunto de estados ocultos es
discreto, aunque algunos autores relajan el término para incluir modelos estocásticos
más generales (p. ej. Elliot et al. 1995). Sin embargo, los datos observables pueden
corresponder a una variable aleatoria cualquiera que siga una cierta distribución cuyos
parámetros dependan del estado en el que se encuentra el sistema. El caso que hemos
considerado anteriormente, con un número finito de símbolos que se emiten con
ciertas probabilidades, dependiendo del estado oculto, es sólo el caso más simple que
podemos encontrar.
Supongamos, por ejemplo, que se dispone un transecto con unidades de muestreo en
las que se cuenta el número de individuos de una especie y se trata de identificar
zonas de alta y baja densidad. El modelo que podemos considerar es que las
observaciones siguen una distribución de Poisson, pero que el parámetro de esta
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120
distribución, es decir, el número medio de individuos por unidad de muestreo, es
mayor o menor según que nos encontremos en una zona de alta o baja densidad. De
esta forma, consideraríamos un HMM con dos estados ocultos (alta y baja densidad) y
emisiones discretas, pero no finitas, determinadas por una función de probabilidad con
un parámetro que depende del estado oculto.
También podríamos considerar situaciones en las que las observaciones correspondan
a una variable aleatoria continua, como ocurriría, por ejemplo, al considerar transectos
de cuadrados de vegetación en los que la variable de interés fuese la biomasa o la
cobertura. En estos casos, en lugar de los símbolos y probabilidades de emisión que
encontramos en los HMM discretos, lo que se tiene es, para cada estado oculto, una
función de densidad que determina la distribución de probabilidad de las
observaciones. Podríamos, por ejemplo, suponer que la biomasa en cada unidad de
muestreo sigue una distribución normal, pero que los parámetros de la distribución,
media y/o varianza, dependen del estado oculto del sistema. En la sección siguiente se
presentará un ejemplo de modelización de transectos de vegetación con datos de
cobertura, donde se mostrará cómo se aplican los HMM con observaciones continuas.
Existen diversas extensiones al modelo básico, discreto o continuo, considerado
anteriormente, que añaden aún mayor flexibilidad a la capacidad de modelización de
los HMM. Entre ellas, podemos destacar los modelos ocultos semimarkovianos, en los
que un estado puede emitir una cadena de símbolos cuya longitud viene determinada
por una cierta distribución de probabilidad, y que han sido utilizados para analizar la
variación espacial en datos de precipitación (Sansom 1999, Sansom y Thompson 2003).
También podrían ser de interés para el análisis de patrones espaciales los HMM
jerárquicos (Fine et al. 1998) y los HMM factoriales (Ghahramani y Jordan 1997), en los
que se asume una cierta estructura en el conjunto o en la sucesión de estados ocultos,
lo que los hace adecuados para la modelización de sistemas con patrones a distintas
escalas.
Los HMM son herramientas de modelización de datos secuenciales, debido a que se
basan en que los estados ocultos constituyen una cadena de Markov. En dos o más
dimensiones, el instrumento análogo a las cadenas de Markov es lo que se denomina
campos aleatorios de Markov (MRF o Markov random fields), en los que la realización
del modelo, en lugar de venir dado por una secuencia de estados del sistema en
momentos sucesivos, consiste en un conjunto de estados del sistema con ciertas
relaciones de contigüidad, como, por ejemplo, los estados en distintas posiciones
espaciales. La propiedad análoga a la condición de que en una cadena de Markov de
Conceptos básicos sobre modelos ocultos de Markov
121
orden r la probabilidad de que un estado tenga un cierto valor sólo depende de los r
estados anteriores, y no de los demás, consiste en un MRF en que la probabilidad de
que un estado tenga un cierto valor sólo depende de los elementos vecinos, según la
relación de contigüidad y el tipo de MRF considerado, y es independiente del resto de
elementos (véase, por ejemplo, Rue y Held 2005). Existen también los
correspondientes modelos con estados ocultos, denominados HMRF (Kunsch et al.
1995), que permiten la modelización de, por ejemplo, sucesiones bidimensionales de
datos con una lógica similar a la de los HMM, aunque con una mayor complejidad que
en el caso unidimensional.
7.5. Consideraciones finales
El aspecto que consideramos más destacable en la aplicación de HMM al análisis de
patrones espaciales es la utilización de un modelo explícito de la estructura del patrón,
a través de la selección de la topología del HMM. De esta forma, el conocimiento que
se tenga a priori sobre el sistema, o que se derive de otro tipo de análisis descriptivos
previos, puede incorporarse en el proceso de modelización, permitiendo un estudio
más profundo y detallado.
En las áreas en las que los HMM están más extendidos, se utilizan, en general, con un
enfoque de aprendizaje automático, en problemas en los que se dispone de una
abundante base de datos (denominados de entrenamiento) con los que es posible
seleccionar y estimar el HMM más apropiado (a veces, con un gran número de estados
ocultos), obteniéndose modelos con una alta capacidad de predicción, pues el
problema típico es clasificar o identificar nuevos datos. Aunque una situación similar
puede darse en algunos problemas de interés en ecología (p. ej. modelización de series
de precipitaciones o de datos de teledetección), es más habitual que se disponga de un
conjunto reducido de datos y que el enfoque sea más de tipo estadístico, en el sentido
de que se desee poder obtener intervalos de confianza para las estimaciones de los
parámetros y poder contrastar tanto los valores de ajuste de un cierto modelo como
distintos modelos alternativos. Aunque estas cuestiones no solían ser tratadas con
detalle en las publicaciones tradicionales sobre HMM, es un tema que está recibiendo
mayor atención en los últimos tiempos, lo que sin duda contribuirá a la extensión del
uso de HMM en problemas de ecología y de otras disciplinas con necesidades
similares.
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122
Además de la posibilidad de usar bootstrapping, se tiene que, dada una cierta
topología, las estimaciones de máxima verosimilitud de los parámetros (obtenidas con
el algoritmo EM) son asintóticamente normales (Bickel et al. 1998) y es posible
obtener intervalos de confianza (Visser et al. 2000) y realizar tests de razón de
verosimilitudes para contrastar sus valores (Giudici et al. 2000). Sin embargo, estos
tests no sirven para decidir entre modelos con diferente número de estados; aunque
se pueden utilizar criterios de selección de modelos para elegir el modelo más
apropiado entre un conjunto de modelos candidatos (Visser et al. 2002, MacDonald y
Zucchini 1997), la fundamentación teórica de este enfoque no está completamente
justificada. No obstante, existen diversas aproximaciones al problema (p. ej.
Stinchcombe y White 1998) e investigaciones en marcha, que deben proporcionar a
corto plazo métodos alternativos plenamente contrastados.
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128
0
20
40
60
80
100
120
140
1983 1986 1989 1992 1995 1998 2001 2004 2007
Nº p
ublic
acio
nes
SC
I
Figura 7.1. Distribución por año del número de artículos en revistas científicas
internacionales incluidas en el “Science Citation Index” (SCI), desde 1983, que incluyen
el término “hidden Markov” en el título (1047 publicaciones en total). Fecha de
búsqueda: 18/04/07.
Figura 7.2. Representación de una cadena de Markov, indicando los estados del
sistema y las probabilidades de transición entre ellos.
Conceptos básicos sobre modelos ocultos de Markov
129
StSt-1 St+1
Figura 7.3. Representación de una cadena de Markov de primer orden, indicando los
estados del sistema en momentos sucesivos. Las flechas directas entre estados indican
las relaciones de (in)dependencia condicional. Para una cadena de primer orden, la
probabilidad de que St tome un cierto valor sólo depende del valor del estado en el
momento inmediatamente anterior, St-1, y no de los valores de los estados en
momentos previos.
StSt-1 St+1
Yt Yt+1 Yt-1
Figura 7.4. Representación de un HMM de primer orden en forma de red bayesiana.
Anexo
130
0,9 0,1T0,3 0,7
= 0,8 0,2E0,1 0,9
=
1 2
Figura 7.5. Esquema de los estados ocultos y transiciones permitidas en un ejemplo de
HMM discreto de primer orden. El modelo consta de dos estados, con las
probabilidades de transición indicadas en la matriz T, y dos símbolos, con las
probabilidades de emisión mostradas en la matriz E. Aunque en el esquema no se han
representado las emisiones, el número de símbolos queda indicado por el número de
columnas de la matriz E.
Conceptos básicos sobre modelos ocultos de Markov
131
Estados ocultos:
Observaciones:
Estados ocultos inferidos con el algoritmo de Viterbi:
Parámetros estimados:
est
0,96 0,04T
0,07 0,93
=
est
0,89 0,11E
0,27 0,73
=
Figura 7.6. Sucesiones de estados ocultos y observaciones (de longitud 50) simuladas
con el modelo de la Figura 7.5, parámetros estimados a partir de la secuencia de
observaciones utilizando el algoritmo de Baum-Welch y sucesión de estados ocultos
inferida utilizando el algoritmo de Viterbi.
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