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1
GPS error and its effects on movement analysis
Peter Ranacher 1, Richard Brunauer2, Stefan Christiaan Van der
Spek3, and Siegfried Reich 2
1 Department of Geoinformatics - Z GIS, University of Salzburg,
Schillerstrae 30,5020 Salzburg, Austria2 Salzburg Research
Forschungsgesellschaft mbH, Jakob Haringer Strae 5/3, 5020Salzburg,
Austria3 Delft University of Technology, Faculty of Architecture,
Department of Urbanism,Julianalaan 134, 2628BL Delft, The
Netherlands E-Mail: [email protected]
Abstract
Global Navigation Satellite Systems (GNSS), such as the Global
Positioning System (GPS), areamong the most important sensors in
movement analysis. GPS data loggers are widely used torecord the
movement trajectories of vehicles, animals or human beings.
However, these trajecto-ries are inevitably affected by GPS
measurement error, which influences conclusion drawn aboutthe
behavior of the moving objects. In this paper we investigate GPS
measurement error anddiscuss its influence on movement parameters
such as speed, direction or distance. We identifythree
characteristic properties of GPS measurement error: it follows
temporal (1) and spatial (2)autocorrelation and causes a systematic
overestimation of distances (3). Based on our findingswe give
recommendations on how to collect movement data in order to
minimize the influence oferror. We claim that these recommendations
are essential for designing an appropriate samplingstrategy for
collecting movement data by means of a GPS.
Introduction
1 Introduction
Global Navigation Satellite Systems (GNSSs) such as the Global
Positioning System (GPS) have become essential sensors for
collecting the movement of objects in geographical space.
Inmovement ecology, GPS tracking is used to unveil the migratory
paths of birds [1], elephants [2]or whales [3], in urban studies it
helps detecting traffic flows [4] or human activity patterns
incities [5], in transportation research GPS allows monitoring of
intelligent vehicles [6] or mappingof transportation networks [7],
to name but a few examples of application.
Movement recorded by a GPS is commonly stored in form of a
trajectory. A trajectory is anordered sequence of spatio-temporal
positions: = < (P1, t1), ..., (Pn, tn) >, with t1 < ...
< tn [8].The expression (P , t) indicates that the moving object
was at a position P at time t. In order torepresent the continuity
of movement, consecutive positions (Pi, ti) and (Pj , tj) along the
trajectoryare connected by an interpolation function [9].
However, although satellite navigation provides global
positioning at an unprecedented accu-racy, GPS trajectories remain
affected by errors. There are two types of error inherent in any
kindof movement data, measurement error and interpolation error
[10], and these errors inevitably alsoaffect trajectories recorded
by a GPS.
Measurement error refers to the impossibility of determining the
true position (P , t) of anobject due to the limitations of the
measurement system. In the case of satellite navigation,it reflects
the spatial uncertainty associated with each position estimate.
-
2 Interpolation error refers to the limitations on interpolation
representing the true motionbetween consecutive positions (Pi, ti)
and (Pj , tj). This error is influenced by the temporalsampling
rate at which a GPS unit records positions.
Measurement and interpolation errors cause the movement recorded
by a GPS receiver to differfrom the true movement, which needs to
be taken into account in order to achieve meaningfulresults from
GPS data. In this paper we focus on GPS measurement error. We show
that it issubject to both spatial as-well as temporal
auto-correlation and we demonstrate how this affectsthe calculation
of movement parameters. Movement parameters are physical quantities
of move-ment such as the distance travelled by an object, its
speed, or direction [11]. Moreover, we identifya systematic GPS
error that results in GPS trajectories systematically
overestimating distances.We demonstrate this error using real-world
data and provide a mathematical explanation. Basedon our results we
give recommendations on how to record movement to minimize the
influence ofGPS measurement error when calculating movement
parameters.
Section 2 introduces relevant work from previously published
literature. Section 3 discusses theinfluence of measurement error
on movement parameters. Section 4 describes the experiment
andpresents our experimental results, Section 5 concludes the paper
and gives recommendations onhow to collect movement data with a GPS
in order to minimize the effects of error on
movementparameters.
2 Related work
Since GPS data have become a common component of scientific
analyses its quality parametershave received considerable
attention. These parameters include the accuracy of the signal, as
wellas its availability, continuity, integrity, reliability,
coverage, and update rate [12]. The accuracyof GPS data (i.e. the
expected conformance of a position provided by a GPS receiver to
the trueposition, or the anticipated measurement error) is clearly
of utmost importance. Measurementerror and its causes, influencing
factors, and scale have been extensively discussed in
publishedliterature: measurement error has been shown to vary over
time [13] and to be location-dependent.Shadowing effects, for
example due to canopy cover, have a significant influence on its
magnitude[14]. Measurement error is both random, i.e. caused by
external influences, and systematic,i.e. caused by the systems
limitations [15]. As such it is a result of several influencing
factors.According to Langley [16], these include:
Propagation delay: atmospheric variations can affect the speed
of the GPS signal and hencethe time that it takes to reach the
receiver;
drift in the GPS clock: a drift in the on-board clocks of the
different GPS satellites causesthem to run asynchronously with
respect to each other and to a reference clock;
Ephemeris error: imprecise satellite data and incorrect physical
models affect estimations ofthe true orbital position of each GPS
satellite [17];
Hardware error: the GPS receiver, being as fault-prone as any
other measurement instrument,produces an error when processing the
GPS signal;
Multipath propagation: infrastructure close to the receiver can
reflect the GPS signal andthus prolong its travel time from the
satellite to the receiver;
-
3 Satellite geometry: an unfavourable geometric constellation of
the satellites reduces the ac-curacy of positioning results.
A detailed overview of current GPS accuracy is provided in the
quarterly report of the FederalAviation Administration [18]. A good
introduction to the GPS in general, and to its error sourcesand
quality parameters in particular, has been provided by
Hofmann-Wellenhof et al. [12].
The above-mentioned research has mainly focused on describing
and understanding GPS mea-surement error. In addition to this
filtering and smoothing approaches have been proposed forrecording
movement data, in order to reduce the influence of errors on
movement trajectories. Aconcise summary of these approaches can be
found in Parent et al. [15]. Jun et al. [19] testedsmoothing
methods that best preserve the true travelled distance, speed, and
acceleration. Theauthors found that Kalman filtering resulted in
the least difference between the true movementand its
representation.
Our work is based on the published literature discussed above
but differs in its objectives, whichwere to explain the influence
of GPS measurement error (regardless of its causes) on the
calculationof movement parameters from GPS trajectories and give
recommendations for recording these.
3 Movement parameters and measurement error
A GPS measurement consists of a spatial component (i.e. latitude
, longitude ) and a temporalcomponent (i.e. a time stamp t). The
GPS uses the World Geodetic System 1984 (WGS84) asa spatial
coordinate reference system (CRS). WGS84 is a global CRS and
latitude and longitudeare angular measures that uniquely identify a
position on the earths surface, approximated bythe WGS84 reference
ellipsoid. For reasons of simplicity it is preferable to transform
the GPSmeasurements to a Cartesian CRS such as the Universal
Transversal Mercator (UTM) CRS. Atransformation from a spheroid
(WGS84) to a Cartesian plane (UTM) leads to a distortion ofthe
original trajectories [12]. For vehicle, pedestrian, or animal
movements consecutive positionsalong a trajectory are usually
sampled in intervals ranging from seconds to minutes. Thus,
thesepositions are very close together in space so that the
distortion is insignificant for most practicalapplications. Hence,
for all the following consideration we assume that the movement is
recordedand all movement parameters are calculated in a UTM
CRS.
3.1 Movement parameters
In navigation, a moving objects behaviour at a particular
instant in time is most commonlydescribed by a state vector. A
state vector consists of the objects current position, its
instantaneousvelocity and acceleration and the time [12]. Such
detailed information on the nature of movement isoften not
available for trajectories recorded using a GPS receiver. If
distance, speed or accelerationare to be retrieved from , they
inevitably have to be derived from consecutive
spatio-temporalpositions along the trajectory. Table 1 summarizes
all movement parameters analysed in thisarticle and describes how
they are calculated from consecutive spatio-temporal positions.
In Table 1 the true movement parameters (left) are calculated
from spatio-temporal positions(Pi, ti), (Pj , tj), (Pk, tk). These
are free of measurement error, i.e. they are calculated
frompositions that the moving object has attained along its path.
The movement parameters measuredby a GPS (right) are all affected
by GPS measurement error. They are calculated from GPSpositions
(Pmi , ti), (P
mj , tj), (P
mk , tk). In Table 1 and in the the remainder of this paper,
a
variable followed by the superscript m indicates a movement
parameter (or position) affected by
-
4Table 1: Movement parameters and their definitions
Movement parametertrue measured by a GPS
Variable Definition Variable Definition
Distance 1 d di,j = d(Pi,Pj) dm dmi,j = d(P
mi ,P
mj )
Speed 2 v vi,j =di,jt
vm vmi,j =dmi,jt
Acceleration 2 a ai,j,k =vj,kvi,j
tam ami,j,k =
vmj,kvmi,j
t
Direction 3 i,j = (Pi,Pj) m mi,j = (P
mi ,P
mj )
Turning angle 3 i,j,k = j,k i,j m mi,j,k = mj,k mi,j1d(A,B) is
the Euclidean distance between A and B, 2 t = tj ti = constant, for
all i, j
3 (A,B) is the angle between the UTM x-axis and the vector
AB
measurement error, whereas the same variable with no superscript
denotes the true parameter (orposition).
The following Section describes the spatial component of GPS
measurement error in moredetail. Strictly speaking, also the
temporal component of a GPS measurement is affected by error,in
practice, however, this error can be neglected [20].
3.2 GPS measurement error
Very generally, a spatial position in UTM is a two-dimensional
coordinate
P =
x
y
, (1)
where x is the metric distance of the position from a reference
point in eastern direction andy in northern direction. If a moving
object is recorded at position P by means of a GPS, theposition
estimate Pm = (xm, ym) is affected by measurement error. The
relationship between thetrue position and its estimate is very
trivially
Pm = P + , (2)
where = (x, y) is the horizontal measurement error expressed as
a vector in the horizontalplane. We adopt the convention used by
Codling et al. [21] to denote random variables with upper
case letters (e.g.,E ) and their numerical values with lower
case letters (e.g., ).
Let pdfgps be the probability density function of horizontal GPS
measurement errorE at
position P . For reasons of generality, it is desirable to treat
pdfgps as being location-independent[22], however, in reality it
varies considerably with the location of P [18]. Moreover, the
pdfgpsis often assumed to have a bivariate normal distribution and
to be independent in both the x-and y-direction [23, 24]. However,
Chin [25] puts forward convincing arguments that GPS error isvery
likely not independent in both the x- and y-direction, but rather
follows an elliptical errordistribution. The shape of this ellipse
is different for different locations. Hence, and also for reasonsof
generality, it preferable to assume pdfgps follows an arbitrary
bivariate distribution.
There are several quality measures to describe the observed
distribution ofE , the most common
being the 95 % radius (R95), which is defined as the radius of
the smallest circle that encompasses
-
595 % of all position estimates [25]. This circle is centred on
the true position of the pdfgps . Theofficial GPS Performance
Analysis Report for the Federal Aviation Administration [18] states
thatthe current set-up of the GPS allows receivers to measure a
spatial position with an average R95 ofslightly over 3 meters. The
values in the report were, however, obtained from reference
stations thatwere equipped with high quality receivers and had
unobstructed views of the sky. It is reasonableto assume that the
accuracy would be reduced in other recording environments, as
measurementerror depends to a considerable extent on the receiver,
as well as on the geographic location [16,18].This assumption is
supported by published literature on GPS accuracy in forests [26]
and on urbanroad networks [27], as well as on the accuracies of
different GPS receivers [28, 29].
3.3 The influence of time and space on GPS measurement error
According to Menditto et al. [30] the accuracy of a measurement
system combines two qualityparameters: its trueness and precision.
The trueness is the proximity of the mean of a set ofmeasurements
to the true value, while the precision is the proximity of the
measurements to oneanother. In this section we discuss the
influence of time and space on the trueness and precisionof GPS
measurements and, consequently, on the calculation of movement
parameters.
We have so far considered the distribution of horizontal GPS
measurement error independentof space and time. The pdfgps
describes how a GPS position estimate P
m deviates from its trueposition P . The pdfgps has very high
trueness (i.e. the expectation value of position estimates
iscentred on the true position), but low precision (i.e. the
measurements are widely spread). Theerror appears to be random.
These assumptions are supported by empirical GPS measurements(see,
Figure 1 a).
We will now consider the distribution of GPS measurement error
with time. We would arguethat GPS measurements are not independent
of time [24], but that measurements taken at the samelocation and
at similar times will have a similar error due to similar
atmospheric conditions and asimilar satellite constellation: the
error appears to be more systematic. We therefore define
pdftemp
as the distribution ofE at position P during the time interval t
= tj ti; t is assumed to be
short (around several minutes). The pdftemp therefore describes
how any two position estimates,Pm at ti and P
m at tj , will deviate from their (identical) true position P
given that both arerecorded during a short time interval t. If t is
short, the influence of the satellite constellationand the
atmosphere should be very similar and we can therefore expect the
error to be less dispersed,yielding a high precision. However, in
contrast to the pdfgps position estimates will most likely notby
distributed around P and hence the trueness of pdftemp is expected
to be low.
We carried out a simple experiment to visualize the difference
between pdfgps and pdftemp.This involved placing a GPS receiver at
a known position P and recording the deviation in eachposition
estimate Pm. For the experiment we used a QSTARZ:BT-Q1000X GPS unit
1 unit withAssisted GPS activated to record about 720 positions
over a period of about six hours, at asampling rate of 1/30 Hz. The
resulting point cloud generated an R95 of about 3 m (Figure 1a).
The distribution was centred on the true position (high trueness)
but was widely spread (lowprecision). If only those measurements
are displayed that were recorded within a certain timeinterval the
resulting point cloud appears less dispersed, but is not equally
spread around the trueposition. The positional measurements are
therefore very precise but not very true. Figure 1 bshows only
those measurements that were taken within periods covering 5
minutes before and aftert1, t2, t3.
When tracking a moving object, the time interval between any two
consecutive position esti-mates is generally short. However, these
measurements do not relate to a static true position butto one that
dynamically changes over a short time interval and within a small
distance. We would
1for specifications, please refer to:
http://www.qstarz.com/Products/GPS/20Products/BT-Q1000.html
-
6R95 3m
Pm around t2
s-neighbourhood of PR95 3m
R95 3m
P
Pm
P
P
Q
Pm
Qm
Pm around t1
Pm around t3
Figure 1: pdfgps (a), pdftemp (b) and pdfmove (c)
argue that consecutive GPS measurements are not independent of
space and time: measurementstaken at similar times and in a similar
location will have similar error vectors. We therefore in-troduce
pdfmove to be the distribution of
E in the s-neighbourhood of P during the short time
interval t = tj ti. Here, the s-neighbourhood of P comprises all
points whose distance to Pis less than s, where s is assumed to be
small (usually only a few meters). The pdfmove describeshow any two
position estimates, Pm at ti and Q
m at tj , will deviate from their respective truepositions, P
and Q, assuming that both measurements are taken during a short
time interval tand that Q is within the s-neighbourhood of P .
Figure 1 c shows a (theoretical) example of twoposition estimates
at P and Q, where Q is in the s-neighbourhood of P .
R95
Pm
P
Qm
Q
R95
dm
d
P
Q
dm
Pm
Qm
P
Q
Figure 2: pdfmove and its effects on calculating dm
The pdfmove is relevant for calculating movement parameters from
GPS position estimates.This is illustrated in Figure 2, where an
object moves the true distance d between P and Q (blackline). If
the measurement error P at P is similar to
Q at Q (red solid lines), dm (red dashed
line) is similar to d; if the measurement errors are not similar
(blue solid lines), dm (blue dashed
-
7lines) and d are different. Both P andQ are drawn from pdfmove
. Hence, the difference between
dm and d depends solely on pdfmove at the time and the vicinity
of recording. A similar point canbe made for the relationship
between true and measured direction. If P and
Q are similar thetrue direction is similar to the measured
direction m.
The characteristics of pdfmove are largely unknown and,
moreover, not empirically determinable.They depend on a plethora of
factors including the type and quality of the GPS receiver used,
thesatellite constellation, the atmospheric conditions, the
location and time of recording the positionestimates and the
temporal and spatial proximity of the true positions. Despite these
sources ofuncertainty, we propose that the pdfmove has three
characteristic properties:
Property 1 : GPS measurment error is affected by temporal
autocorrelation. As the time intervalt between two consecutive
position estimates increases, we would expect the pdfmove tobecome
more widely dispersed and the precision of the measurements to
decrease on average.Note that high precision does not imply high
accuracy and highly precise measurements canbe very inaccurate due
to low trueness.
Property 2 : GPS measurment error is affected by spatial
autocorrelation. As the true distanced between two position
estimates increases we would expect the pdfmove to become
morewidely dispersed and the precision of the measurements to
decrease on average .
Property 3 : The pdfmove causes a systematic error in movement
parameters. Distances recordedby a GPS receiver are, on average,
larger than the true distances travelled by the movingobject.
Properties 1 and 2 are evident from the observations described
above, Property 3 is discussedin detail in section 3.4. In section
4 all three properties are supported by experimental results
withreal-world GPS data.
3.4 Overestimation of distance through the GPS
In this section we discuss the reason why measurement error
causes the distance recorded by aGPS receiver to be on average
greater than the true distance travelled by a moving object.In
subsection 3.4.1 we provide a mathematical explanation for
measurement error resulting inan overestimation of distance, in
subsection 3.4.2 we calculate the overestimation for
differentmagnitudes of measurement error. In Section 4 we
illustrate this property, together with the othercharacteristics of
pdfmove, in an experiment.
3.4.1 Geometric explanation
Consider two true positions, P and Q, these being positions the
moving object passes through onits path. The distance d = d(P ,Q)
is the true distance that the moving object travels between Pand Q.
Consider now that a GPS receiver measures the position of the
moving object at P and a
short t later at Q. These position estimates Pm and Qm are
affected by measurement errorE .
This error propagates to measured distance dm = d(Pm,Qm). As a
consequence of measurementerror we conjecture that
E(dm) > d,
where E(dm) is the expected value of measured distance.
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8Theorem 1 For two arbitrary consecutive GPS positions and if
interpolation error can be ignored,the expected measured distance
E(dm) along a GPS trajectory is always larger than the true
distanced travelled by the moving object, due to the influence of
measurement error.
This inevitable overestimation of distance OD = E(dm) d referred
to in Theorem 1 is boundto occur in GPS data and can be explained
geometrically.
Let us assume that measurement errorE for both Pm and Qm is
drawn from pdfmove . Again,
for reasons of generality, we assume that pdfmove does not
necessarily follow a bivariate normaldistribution, and that its
components are not necessarily independent of each other. The first
andsecond moments of the pdfmove are very generally defined by the
expected error and thecovariance matrix :
=
E(Ex)E(Ey)
and =
2x x y
y x 2y
.
This implies that the expected error for pdfmove is the same at
P as it is at Q, which isreasonable as both points are close
together in both space and time. The variance associated withthe
difference vector Pm Qm is now simply V = 2 . The situation is,
however, complicatedby the fact that this relationship is only
valid if
E at Qm is independent of Pm. It is important
to note that, at a certain time and location both measurements,
Pm and Qm, will be similar, butthis is due to the constellation of
the GPS satellites and the characteristics of the environment
anddoes not imply that one measurement affects the other. Hence,
the relationship V = 2 holds.
Now d2 = (P Q)2 = (P Q)T I2 (P Q) is the square of the true
distance anddm
2
= (Pm Qm)2 = (Pm Qm)T I2 (Pm Qm) is the square of the measured
distance,where I2 is the identity matrix with dimensions of 2
2:
I2 =
1 00 1
.
According to Johnson et al. [31] the expected value of the
quadratic form for dm is:
E(dm2
) = Trace(I V ) + E(Pm Qm)T I E(Pm Qm).In this equation, Trace(I
V ) = 22x + 22y, and therefore is always positive. If E(Pm) = P
andE(Qm) = Q, i.e. if the expected positions are the true
positions, then
E(dm2
) = 22x + 22
y + (P Q)T I (P Q)= 22x + 2
2
y + d2. (3)
It therefore follows that E(dm) > d. However, this is only
the case if we consider the measurementerror to be equally
distributed around the true position (i.e. the measurements are
expected tohave a high trueness) and this cannot be guaranteed for
pdfmove . If the expected positions differfrom the true positions,
E(Pm) 6= P and E(Qm) 6= Q, then
E(dm2
) = 22x + 22
y + d2,
where d2 is the square of the distance between E(Pm) and E(Qm).
An overestimation of distancecan now still be expected if
P E(Pm) = Q E(Qm) = .In this case the expected values of the
position estimates deviate by the same vector fromtheir true
positions. This is assumed by the definition of pdfmove . It
therefore follows thatd(E(Pm), E(Qm)) = d(P ,Q) and d2 = d2, and
Equation 3 therefore holds true.
-
93.4.2 The characteristics of overestimation
From Equation 3 it follows that the difference between E(dm) and
d is related to the variance ofmeasurement error of each position
estimate:
2(2x + 2
y) = E(dm
2
) d2.If we assume that 2 = 2x =
2y , i.e. that the error is the same in x- and y- directions,
then
=
E(dm2) d2
2(4)
yields a quality measure for pdfmove since describes the
dispersal of the measurements andtherefore allows conclusions to be
drawn on their precision. Note that 2 is assumed to be thesame in
both x and y purely for convenience, since the precision would
otherwise be the squareroot of the variance in x and y and Equation
4 would change slightly.
a b
over
estim
atio
n o
f dista
nce
OD
[m
]
over
estim
atio
n o
f dista
nce
OD
[m
]
Figure 3: Overestimation of distance for different values of d
(a) and (b)
The expected overestimation of distance OD = E(dm) d generally
increases as the precisionof measurements decreases (Note: when the
precision of the measurements decreases increases!).Figure 3 a
shows the value of OD as increases, with d assumed to be constant.
For a truedistance d = 5m (yellow line) a precision of = 1m yields
an overestimation of 1m, a precisionof = 5m an overestimation of
2m. In contrast, the overestimation of distance decreases as
theseparation between the true positions d increases. Figure 3 b
shows the value of OD as the truedistance d increases, with assumed
to be constant. For a precision of = 2m (blue line) a truedistance
d = 5m yields an overestimation of 2m, a true distance d = 10m an
overestimation of 1m.
In the following section we show the overestimation of distance
in real-world GPS trajectoriesand use it to discuss the precision
of the measurements.
4 Experimental evaluation of measurement error
4.1 Experimental setup
In the following experiment we attempt to show that the
characteristics concerning pdfmove dis-cussed in Section 3.3 agree
with observations obtained from real-world GPS data. In the
experiment
-
10
we performed distance measurements using a GPS data logger along
a course with well-establishedreference distances. In this way we
were assured of obtaining a true distance d not affected bynoise.
We calculated the average measured distance dm and from this derive
OD = dm d and =
dm2d2
2. OD and are estimators for OD and .
The reference course was located in an empty parking lot to
avoid shadowing and multi-patheffects. We staked out a square with
sides that were 10 m long and had markers at one metreintervals. A
square was used in order to allow distance measurements to be
collected in all fourcardinal directions (approximately). The
distance between the markers was used as a referencedistance,
henceforth simply referred to as the true distance d. All positions
were recorded usinga QSTARZ:BT-Q1000X GPS unit 2 with Assisted GPS
activated. We considered that it wassufficient to use only a single
GPS unit as the aim of the experiment was not to investigate
thequality of the particular GPS unit, but the suggestion that
measurement error follows a certainlogic and yields results that
are independent of the device used. The GPS unit was treated as
ablack box, i.e. the algorithm to calculate the position estimates
from the raw GPS signal wasnot known.
The GPS measurements were obtained by placing the GPS receiver
on each reference markerin turn and recording the position, moving
around the square until the position of all markershad been
recorded. Positions were only recorded at the reference markers,
and only when therecording button was pushed manually. Two
consecutive position estimates were taken promptly(within 35
seconds) in order to ensure similar satellite and atmospheric
conditions. A full circuitaround the square took between 2 and 3
minutes (approximately) and resulted in 40 positionsbeing recorded.
A total of 25 circuits around the square were completed, without
any breaks. Thisresulted in 1000 GPS positions being collected in
approximately one hour.
In pre-processing the recorded positions were connected to form
trajectories and the distancesdm along these trajectories were
compared to the true distances d between the markers. Distancewas
calculated according to Table 1.
4.2 Results
The results support all three properties raised in Section 3.3,
namely that GPS measurement erroris affected by (1) temporal and
(2) spatial autocorrelation and causes a (3) systematical
overesti-mation of distances. In this Section we explain our
results in detail.
Figure 4 shows OD (black dots) for different true distances d.
The data therefore support theproposition in Property 3 , that the
pdfmove causes distance measurements by GPS to overestimatethe true
distance.
In contrast to the theoretical findings, overestimation of
distance tended to increase as thetrue distance d increased. We
argue that this was due to a decrease in the precision of the
GPSmeasurements induced by their increasing difference in location.
The position estimates becamemore dispersed, and the of the GPS
measurement error increased (black crosses in Figure 4);These
results therefore support the proposition in Property 2 that
pdfmove becomes more dispersedas d between two position estimates
increases.
Figure 5 shows the histograms of distance difference dm d for d
= 1 m (a), and for d = 5 m(b), and their fit to a Gaussian
distribution. There is no indication in either histogram that
theoverestimation of distance is due to positive outliers. On the
contrary, both follow a Gaussiandistribution N (d, 2d) rather well
and outliers are almost non-existent. Note that d and 2d inFigure 5
refer to the values of the fitted Gaussian distribution and not to
the empirically derived
2for specifications, please refer to:
http://www.qstarz.com/Products/GPS/20Products/BT-Q1000.html
-
11
over
estim
atio
n o
f dista
nce
OD
[m
]^
prec
isio
n
[m
]^
OD^
^
Figure 4: Overestimation of distance measured with a GPS with
increasing true distance betweentwo position estimates
frequency.
a b
Figure 5: Difference between measured and true distance for d =
1 m (a), d = 5 m (b)
In order to illustrate the suggested influence of time on the
pdfmove we also calculated the dis-tance between non-consecutive
position estimates around the square. One example is the
distancebetween two position estimates taken at two consecutive
markers, where the latter measurementwas collected one circuit
later. The true distance between the markers remains the same, i.e.
1 m,but the measurements are taken within a longer time interval.
Figure 6 shows the relationship be-tween the time interval between
consecutive measurements and OD (black dots) as well as
(blackcrosses) for a true distance d = 1 m. Both OD and increase
with longer time intervals. Thesharpest increase occurs between
measurements that were taken promptly and those taken afterabout 2
1
2minutes. After 40 minutes the curve levels out, the dispersal
of the position estimates
-
12
no longer increases. Up until that time, the data support the
proposition in Property 1 that thepdfmove becomes increasingly
dispersed as the t between two position estimates increases.
over
estim
atio
n o
f dista
nce
OD
[m
]^
prec
isio
n
[m
]^
OD^
^
Figure 6: Overestimation of distance measured with a GPS with
increasing time interval betweentwo position estimates (d = 1
m)
5 Conclusion and Discussion
In this paper we have evaluated the effects of measurement error
on GPS trajectories. First,we showed how movement parameters are
calculated from GPS measurements. Then, we took acloser look at GPS
measurement error and the influence of space and time. We suggested
thatthe pdfmove has three characteristic properties and
demonstrated these using real-world GPS data.Property 1 is that, in
general, the closer two GPS position estimates are in time the more
similarare their measurement error vectors; Property 2 is that, in
general, the closer two GPS positionestimates are in space along a
trajectory, the more similar are their measurement error
vectors;and Property 3 is that measurement error causes a
systematic overestimation of distance (i.e. ifinterpolation error
is ignored then the distances recorded by a GPS receiver are on
average largerthan the true distances travelled by the moving
object). In addition to an empirical experiment,we also provided a
mathematical explanation for the third property.In our analysis we
treated the GPS unit as a black box. This raises the legitimate
question,whether our results were produced by a smoothing algorithm
rather than the behaviour of theGPS. Let us assume that a smoothing
algorithm was used in the GPS unit. In simplified form,the current
position estimate is then calculated from the last position
estimate, the current GPSmeasurement and a movement model. For
movement with constant speed and direction, smoothingyields
trajectories that represent the true movement very accurately.
However, sudden changes inmovement, i.e. a sharp turn, are not
followed by the trajectory. The current measurement implies asharp
turn, however, the movement model does not. Thus, the sharp turn
becomes more elongated,the overestimation of distance increases.
This should be visible in our data. However, we did notfind any
support for an increase in the overestimation of distance after a
sharp turn. It was thesame for all parts along the square, no
matter if the movement was recorded immediately afterturning the
corner or not.
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13
Our findings in Section 4 have implications for the way movement
is collected when using aGPS. We now synthesize our results and
give recommendations for recording the movement pa-rameters
distance, speed, acceleration, direction and turning angle. Our
recommendations focuson the temporal sampling rate of recording,
filtering or smoothing are not addressed. An appro-priate temporal
sampling strategy for GPS trajectories is always tailored to the
characteristicsof the moving object and must take interpolation
error into consideration. Hence, before givingrecommendations, we
briefly discuss the role of interpolation error when recording
movement.
5.1 The role of interpolation error when recording movement
Interpolation error refers to the limitations on interpolation
representing a moving objects truebehaviour, i.e., it is the
difference between the objects true and interpolated movement.
Interpo-lation error strongly affects the calculation of movement
parameters, which is briefly discussed inthis paragraph. All
considerations relate to linear interpolation, as it is the easiest
and most com-mon type of interpolation [9]. Linear interpolation
assumes that between two recordings an objectmoves in a straight
line and at a uniform speed; any change of speed and direction is
assumed tooccur abruptly and only at one of the position
recordings.
Firstly, interpolation error causes the interpolated spatial
path to differ from the true pathwith respect to its direction and
distance. Interpolation always follows a straight line between
twopositions, hence interpolated distances are always less than or
equal to the true distance. This sys-tematic underestimation of
distances also affects speed. Interpolation error tends to
underestimatespeed.
Secondly, interpolation error concerns the objects
spatio-temporal progression along its path.The interpolated speed
and direction are average values over the time period between two
record-ings. They do not provide any information on instantaneous
speed and direction. An objectmoving at a variable speed and
another moving at a uniform speed can have the same averagespeed
between two recordings, but only the latter of the two will be
captured appropriately by aGPS trajectory.
Interpolation error can be controlled by the temporal sampling
rate at which movement isrecorded: the shorter the time interval
between two recordings, the smaller the interpolation error.The
choice of an appropriately short temporal sampling rate depends to
a large degree on themoving object, its speed and turning angle and
the tendency to change these. In general, anobject that tends to
move uniformly with infrequent changes in direction requires less
frequentsampling compared to an object that moves non-uniformly and
frequently changes its direction.
For each movement parameter we now give sampling
recommendations. We consider boththe theoretical and experimental
findings about measurement error as well as interpolation error.It
is, however, beyond the scope of this paper to fully reveal the
complex interaction betweenmeasurement and interpolation error when
recording GPS trajectories.
5.2 Sampling recommendations to reduce the influence of
measurement
error
(Cumulative) distance:
Distance is calculated from two consecutive position estimates.
For high sampling rates theseare close together in both space and
time and are therefore likely to have similar error vectors(see
Figure 6). In general, this spatio-temporal autocorrelation yields
a measured distance dm
that is similar to the true distance d (see Section 4). However,
measurement error also causes asystematic overestimation of d (see
Section 3.4). This effect has been explained mathematically
insubsection 3.4.1, and demonstrated using real-world trajectory
data in subsections 4.
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14
In our experiment the moving object travels along a straight
line between successive measure-ment points. For such a scenario
interpolation error can be ignored. For other scenarios, suchas a
car moving in a street network or a pedestrian walking along a
winding road, the effect ofinterpolation error can counterbalance
or even outweigh the effect of measurement error. This
haspreviously been noted in the trajectories of fishing vessels
[32]: for high sampling rates the distancetravelled by the vessel
was overestimated due to measurement error, while for lower
sampling ratesit was underestimated due to the increasing influence
of interpolation error.
Hence, a temporal sampling rate must balance out these opposing
effects of measurement andinterpolation error to properly represent
the distances a moving object has travelled.
Speed and acceleration:
In order to ensure that the measured speed vm and acceleration
am are similar to the true speed vand the true acceleration a we
require reliable distance measurements dm. Due to
spatio-temporalautocorrelation of measurement error we can achieve
a reliable dm with very high sampling rates,i.e. position estimates
that are close together in space and time (see, Figure 6). However,
highsampling rates result in a systematic overestimation of speed.
In contrast to distance, speed is notcumulative and therefore these
slight systematic errors are also not cumulative.
In conclusion, high sampling rates are advisable for reducing
both measurement and interpo-lation error when recording speed and
acceleration.
Direction and turning angle:
Similar to distance, the measured direction m and turning angle
m are more similar to the truedirection and turning angle when they
are calculated from positions that are close together inspace and
time. Therefore, high sampling rates reduce the effects of
measurement and interpolationerror.
5.3 Discussion
GPS trajectories have been widely used for describing and
analyzing movement [3335]. However,the influence of error on these
trajectories has only been addressed briefly if at all in
thesestudies, which is rather surprising. Moreover, the choice of
the sampling strategy has not alwaysbeen explained. We claim that
an appropriate sampling strategy for collecting movement data
bymeans of a GPS is crucial and that it must consider the following
aspects:
The sampling strategy must reflect the aims of movement
analysis, i.e. which information isneeded for the analysis and at
which level of detail?
It must respond to the characteristics of the moving object
under observation, i.e. it mustconsider the expected interpolation
error.
It must address the influence of measurement error when
collecting the movement data.In this paper we reveal three
fundamental properties of GPS measurement error and we discuss
their implications when calculating movement parameters. These
properties are independent of thetype of movement under
consideration. Hence, our findings are generally applicable when
analysingmovement by means of a GPS. We claim that they provide an
important basis for designing anappropriate sampling strategy for
collecting movement data by means of a GPS.
Our findings have implications not only for recording movement
but also for simulations. Laubeand Purves [36] performed a
simulation to reveal the complex interaction between measurement
er-ror and interpolation error and their effects on recording the
movement parameters speed, turning
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15
angle and sinuosity. Their Monte Carlo approach assumed GPS
errors to scatter entirely ran-domly between each two consecutive
positions. Our empirical analysis shows that this assumptionis not
realistic, since measurement error is not purely random, but
affected by spatio-temporalautocorrelation, and, thus, tends to be
similar for similar positions in space and time.
Acknowledgments
This research was funded by the Austrian Science Fund (FWF)
through the Doctoral CollegeGIScience at the University of Salzburg
(DK W 1237-N23). We thank Arne Bathke and WolfgangTrutschnig from
the Department of Mathematics of the University of Salzburg for
their invaluablehelp on quadratic forms.
References
1. Higuchi H, Pierre JP (2005) Satellite tracking and avian
conservation in asia. Landscapeand Ecological Engineering 1:
3342.
2. Douglas-Hamilton I, Krink T, Vollrath F (2005) Movements and
corridors of african ele-phants in relation to protected areas.
Naturwissenschaften 92: 158-163.
3. Elwen SH, Best PB (2004) Environmental factors influencing
the distribution of southernright whales (eubalaena australis) on
the south coast of south africa ii: Within bay distri-bution.
Marine Mammal Science 20: 583-601.
4. Zheng Y, Liu Y, Yuan J, Xie X (2011) Urban computing with
taxicabs. In: Proceedings ofthe 13th international conference on
Ubiquitous computing. ACM, pp. 89-98.
5. Van der Spek S, Van Schaick J, De Bois P, De Haan R (2009)
Sensing human activity: Gpstracking. Sensors 9: 3033-3055.
6. Zito R, dEste G, Taylor MA (1995) Global positioning systems
in the time domain: Howuseful a tool for intelligent
vehicle-highway systems? Transportation Research Part C:Emerging
Technologies 3: 193209.
7. Mintsis G, Basbas S, Papaioannou P, Taxiltaris C, Tziavos I
(2004) Applications of gpstechnology in the land transportation
system. European Journal of Operational Research152: 399-409.
8. Guting R, Schneider M (2005)Moving objects databases. San
Francisco: Morgan Kaufmann.
9. Macedo J, Vangenot C, Othman W, Pelekis N, Frentzos E, et al.
(2008) Trajectory DataModels, Berlin: Springer, book section 5. pp.
123 - 150.
10. Schneider M (1999) Uncertainty management for spatial data
in databases: Fuzzy spatialdata types, Berlin: Springer. pp.
330-351. doi:10.1007/3-540-48482-5\ 20.
11. Dodge S, Weibel R, Lautenschutz AK (2008) Towards a taxonomy
of movement patterns.Information visualization 7: 13.
12. Hofmann-Wellenhof B, Legat K, Wieser M (2003) Navigation:
Principles of positioning andguidance. Wien: Springer Verlag, 427
pp.
13. Olynik M (2002) Temporal characteristics of GPS error
sources and their impact on relativepositioning. Report, University
of Calgary.
-
16
14. DEon RG, Serrouya R, Smith G, Kochanny CO (2002) GPS
radiotelemetry error and biasin mountainous terrain. Wildlife
Society Bulletin : 430-439.
15. Parent C, Spaccapietra S, Renso C, Andrienko G, Andrienko N,
et al. (2013) Semantictrajectories modeling and analysis. ACM
Computing Surveys 45: 1-32.
16. Langley RB (1997) The GPS error budget. GPS world 8:
51-56.
17. Colombo OL (1986) Ephemeris errors of GPS satellites.
Bulletin geodesique 60: 64-84.
18. William J Hughes Technical Center & NSTB/WAAS T&E
Team (2013) Global PositioningSystem (GPS) Standard Positioning
Service (SPS) Performance Analysis Report . Report,Federal Aviation
Administration.
19. Jun J, Guensler R, Ogle JH (2006) Smoothing methods to
minimize impact of Global Po-sitioning System random error on
travel distance, speed, and acceleration profile
estimates.Transportation Research Record: Journal of the
Transportation Research Board 1972: 141-150.
20. Mumford P (2003) Relative timing characteristics of the one
pulse per second (1PPS) outputpulse of three GPS receivers. In:
Proceedings of the 6th International Symposium on
SatelliteNavigation Technology Including Mobile Positioning &
Location Services (SatNav 2003). pp.22-25.
21. Codling EA, Plank MJ, Benhamou S (2008) Random walk models
in biology. Journal of theRoyal Society Interface 5: 813-834.
22. Buchin K, Sijben S, Arseneau TJM, Willems EP (2012)
Detecting movement patterns usingbrownian bridges. In: Proceedings
of the 20th International Conference on Advances inGeographic
Information Systems. ACM, pp. 119128.
23. Jerde CL, Visscher DR (2005) GPS measurement error
influences on movement model pa-rameterization. Ecological
Applications 15: 806-810.
24. Bos, MS, Fernandes R, Williams S, Bastos L (2008) Fast error
analysis of continuous GPSobservations. Journal of Geodesy 82:
157-166.
25. Chin GY (1987) Two-dimensional Measures of Accuracy in
Navigational Systems. ReportDOT-TSC-RSPA-87-1, U.S. Department of
Transportation.
26. Sigrist P, Coppin P, Hermy M (1999) Impact of forest canopy
on quality and accuracy ofGPS measurements. International Journal
of Remote Sensing 20: 3595-3610.
27. Modsching M, Kramer R, ten Hagen K (2006) Field trial on GPS
Accuracy in a mediumsize city: The influence of built-up. In: 3rd
Workshop on Positioning, Navigation andCommunication. pp.
209-218.
28. Wing MG, Eklund A, Kellogg LD (2005) Consumer-grade global
positioning system (GPS)accuracy and reliability. Journal of
Forestry 103: 169-173.
29. Zandbergen PA (2009) Accuracy of iPhone locations: A
comparison of assisted GPS, WiFiand cellular positioning.
Transactions in GIS 13: 5-25.
30. Menditto A, Patriarca M, Magnusson B (2007) Understanding
the meaning of accuracy,trueness and precision. Accreditation and
Quality Assurance 12: 45-47.
-
17
31. Johnson NL, Kotz S, Balakrishnan N (1994) Continuous
Univariate Distributions: Vol.: 1,volume 1. New York: John Wiley
and Sons.
32. Palmer MC (2008) Calculation of distance traveled by fishing
vessels using GPS positionaldata: A theoretical evaluation of the
sources of error. Fisheries Research 89: 57-64.
33. Zheng Y, Li Q, Chen Y, Xie X, Ma WY (2008) Understanding
mobility based on GPS data.In: Proceedings of the 10th
international conference on Ubiquitous computing. ACM,
pp.312-321.
34. Giannotti F, Nanni M, Pedreschi D, Pinelli F, Renso C, et
al. (2011) Unveiling the complexityof human mobility by querying
and mining massive trajectory data. The VLDB Journal -The
International Journal on Very Large Data Bases 20: 695-719.
35. Andrienko G, Andrienko N, Hurter C, Rinzivillo S, Wrobel S
(2011) From movement tracksthrough events to places: Extracting and
characterizing significant places from mobility data.In: 2011 IEEE
Conference on Visual Analytics Science and Technology (VAST). IEEE,
pp.161170.
36. Laube P, Purves RS (2011) How fast is a cow? Cross-Scale
Analysis of Movement Data.Transactions in GIS 15: 401-418.
1 Introduction2 Related work3 Movement parameters and
measurement error3.1 Movement parameters3.2 GPS measurement
error3.3 The influence of time and space on GPS measurement
error3.4 Overestimation of distance through the GPS3.4.1 Geometric
explanation3.4.2 The characteristics of overestimation
4 Experimental evaluation of measurement error 4.1 Experimental
setup4.2 Results
5 Conclusion and Discussion5.1 The role of interpolation error
when recording movement5.2 Sampling recommendations to reduce the
influence of measurement error5.3 Discussion