arX

iv:1

504.

0450

4v1

[phy

sics.i

ns-d

et] 1

7 Apr

2015

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: peter.ranacher@sbg.ac.at

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.

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.

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.

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

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