Biometric Conference 2009, Taupo1 Maryann Pirie PhD candidate Department of Statistics and School of Environment University of Auckland.

Biometric Conference 2009, Taupo 1

Maryann PiriePhD candidate

Department of Statistics and School of EnvironmentUniversity of Auckland

OverviewKey questionData setsDeveloping the methodsConclusions from simulationsApplication to tree-ring

dataset

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To investigate a possible failure of the uniformitarianism principle in the use of kauri

ring-widths to investigate past climates

Contains rings from the inner of the core, formed when tree was smaller

Contains rings from the outer of cores, formed when tree was larger

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The key question:

Data

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MethodsFor a given core we have a series of ring

widths, wijt t = 1, … , T

We may have several cores from the same tree, j = 1, … , Ci – typically Ci = 2

We have many trees, i = 1, … , L

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Method-issue

Assemble series into an array W is an array with elements wijt :

Problem: not all series are the same length

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Method-issue

Tree 1 Tree i

7

wijt = width tree, core, index

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Method-issueAssume times Tij are all equal, We have two matrices of time series, X,YWhere X,

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For each time we average

To give:

And, for a similar matrix of time series for Y

Statistical QuestionHow do we formalise

the difference between the two series;

and ? This will be termed the

concordance

These are not stationary series

We do not want to use correlation coefficients

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Method-ideafor the common period m=nProduce bootstrapped

replicates of:

For each time, t sort the averaged bootstrapped time series,

Count the number of bootstrap replicates that overlap at each time, t

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0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

Picture illustrating bxt, byt and R

Index

valu

e

*

*

*

*

*

*

######

*

*

*

*

*

*

#

#

#

#

#

#

Bxt = 3R = 6

Byt = 6R = 6

Bxt = 5R = 6

Byt = 5R = 6

Concordance, PThe concordance at time, t, can be defined

as:

t lies between 0 and 1The overall concordance of how similar the

two time seriesCombines concordances for all (common)

times.

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Simulated Results – Time series generated from normally distributed white noise

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Average time series for matrix X (black) and Y (red), ~N(0,1)

Time

x.m

aste

r

1860 1880 1900 1920 1940

-0.2

-0.1

0.0

0.1

0.2

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5 10 15 20

-0

.4-0

.20

.00

.20

.4

sorted bootstrapped replicated for X (black), and Y (red), for the first 20 series

series index

va

lue

Simulated Results – Time series generated from normally distributed white noise

Simulated Results – Time series generated from normally distributed white noise

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Histogram of prec$x.over.y

prec$x.over.y

Fre

qu

en

cy

0 20 40 60 80 100

02

04

0

Histogram of prec$y.over.x

prec$y.over.x

Fre

qu

en

cy

0 20 40 60 80 100

01

03

0

Simulated Results – Time series generated from normally distributed white noise

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Normally distributed time series - Differences in level

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Normally distributed time series - Difference in scale

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Correlated time series - Differences in level

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Other design issuesSensitivity to sample sizeRagged arrays

Adjust the overlap counts bxt, byt to proportions

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A case study: Tree ring analysis using kauri from Northern New Zealand

Two subsets: small = 0-20cm from pith, large = 20-200cm from pith

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Concordance indices

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ConclusionConcordance indices are able to identify

periods of similarity/dissimilarity between two matrices of time series

The Concordance tends to zero when there is little or no overlap between matrices of time series

There was a difference detected between the subsets ‘small’ and ‘large’ for Huapai

Suggesting failure of uniformitarianism principle

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Thank youQuestions/comments

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