circular data

Advanced Review

Circular dataAlan Lee∗

The special nature of circular data means that conventional methods suitable forthe analysis of linear data do not apply. In this article, we survey a range ofmethods that have been developed over the last 50 years to handle the specialcharacteristics of data consisting of angular measurements. We discuss summarystatistics and graphical methods, methods for the analysis of single and multiplesamples of circular data, circular correlation, regression methods, and time series.We discuss the standard probability models on which these analyses are based,and give several examples of the application of these methods. A reasonablycomprehensive bibliography is provided. © 2010 John Wiley & Sons, Inc. WIREs Comp Stat2010 2 477–486

Observations consisting of directions or angles,while not as common as continuous measure-

ments on a ratio scale, are nevertheless to be foundacross many areas of science, including ecology, earthsciences, environmental science, and medicine. Exam-ples of such data are the angular movements of ananimal relative to a food source or other attractor,wind directions, diurnal measurements of admissiontimes to an intensive care unit, and departure direc-tions of birds after release.

Circular data arise whenever directions are mea-sured, and are usually expressed as angles relative tosome fixed reference point, such as Due North. Timedata measured on a 24 h clock may also be convertedto angular measurements, with 0:00 correspondingto 0◦ and 24:00 to 360◦. Angles can be measuredin degrees, or, as we shall do for the most part, inradians.

In this article, we offer a brief survey ofthe statistical methods that can be used to handlesuch data. It is immediately apparent that statisticalmethods that are used for ordinary numerical data arequite inappropriate for circular data. For example,suppose we have angular measurements 1◦, 0◦, 359◦.The arithmetic mean of these three numbers is 120◦,but it is clear that a more sensible result is 0◦. Overthe past 50 years, a reasonably comprehensive suite ofspecial techniques has been developed that respect thefact that the circle (where angular measurements live)is very different topologically from the line (whereordinary numeric measurements live). We now have

∗Correspondence to: [email protected]

Department of Statistics, University of Auckland, Auckland,New Zealand

DOI: 10.1002/wics.98

data-analytic techniques that parallel the standardmethods for the analysis of single and multiplesamples of angular data, and for handling problemsof regression and correlation. Paralleling the usualprobability models such as the normal for linear data,we have corresponding models for distributions onthe circle and the torus.

LOCATION, DISPERSION ANDCIRCULAR MOMENTS

Suppose we have a sample θ1, . . . , θn which we canvisualize as a set of unit vectors. Sensible measuresof angular location and dispersion can be derived byconsidering the resultant of these n unit vectors. Thedirection of the resultant, θ say, can be taken as thecircular mean direction, analogous to the mean of asample of linear data. Moreover, the length of theresultant provides a measure of dispersion of the data.At one extreme, if all the angles are identical, the unitvectors are coincident and the length of the resultantis n. At the other, if the data are uniformly dispersedaround the circle, the length of the resultant is zero.A convenient measure of angular dispersion is 1 − R,where R is the mean resultant length (the length of theresultant divided by n.)

Higher moments can also be defined. The pthsample trigonometric moments are complex numbersdefined by

mp = Cp + iSp, (1)

where

Cp = 1n

n∑i=1

cos pθ i, Sp = 1n

n∑i=1

sin pθ i.

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Other summary measures, such as the circularvariance 1 − R, analogues of skewness and kurtosisand other measures of location such as the circularmedian can also be defined. See Fisher 1 pp. 34, 35 fordetails. The large-sample joint distribution of thesestatistics is derived in Ref 2.

There are also specialized methods of datadisplay for samples of circular data, such as rosediagrams, circular histograms, circular dotplots andnonparametric density estimates based on kernelsmoothing. These data displays are discussed in Refs1,3,4. Density estimation is considered in Refs 5–8.

CIRCULAR DISTRIBUTIONSThere are several distributions for angular data,corresponding to the usual distributions for lineardata. Before describing these, we must define theconcept of the circular density and the trigonometricmoments of a distribution. Circular distributions areusually described in terms of a circular density, whichis a function f (θ) defined for each angle θ (which neednot be confined to [−π , π ]) satisfying the conditions

1. f (θ ) ≥ 0 for all angles θ ,

2. f is periodic with period 2π ,

3.∫ θ0+2π

θ0f (θ )dθ = 1 for all θ0.

Any such function describes a probability distributionon the circle. If θ is a circular random variable withdensity f , and θ1 and θ2 are fixed angles with 0 ≤θ1 ≤ θ ≤ θ2 ≤ 2π or −π ≤ θ1 ≤ θ ≤ θ2 ≤ π , then

Pr[θ1 ≤ θ ≤ θ2] =∫ θ2

θ1

f (θ) dθ. (2)

The trigonometric moments of the distributionare the complex numbers

μp = E(eipθ ) = αp + iβp, p = 0, ±1, ±2, . . . , (3)

where αp = ∫ 2π

0 cos pθf (θ)dθ and βp = ∫ 2π

0 sin pθf (θ)dθ . The first trigonometric moment, expressed inpolar coordinates, is

μ1 = E(eiθ ) = ρeiμ, (4)

where ρ is the resultant length and μ the meandirection. The circular variance is defined as 1 − ρ, andthe circular dispersion as δ2 = (1 − ρ2)/2ρ2 where ρ2is the modulus of the complex number μ2. For asample of size n, the quantity δ/

√n is called the

circular standard error of the mean direction, and asample version of this plays a role in inference for asingle sample of data, as will be seen in the followingtext.

We can also interpret the trigonometric momentsin terms of characteristic functions: The periodicityof the circular density (which implies that θ

and θ + 2π have the same distribution) meansthat the characteristic function φ(t) = E(eitθ ) onlymakes sense for the integer values of t. As fordistributions on the line the characteristic functiondetermines the distribution. It follows that circulardistributions are determined by their trigonometricmoments.

Univariate Distributions

Two Standard DistributionsStandard circular distributions include the uniform,with density f (θ) = 1

2π, which has no mean direction,

and the von Mises, with density

f (θ) = 12πI0(κ)

eκ cos(θ−μ), (5)

where μ is the mean direction, and I0 is a modifiedBessel function of order zero. The parameter κ is aconcentration parameter: for κ = 0, the distributionreduces to the uniform, and for κ → ∞, thedistribution converges to a point mass at μ. Theresultant length is given by ρ = I1(κ)

I0(κ) where I1(κ) isa modified Bessel function of order one. The vonMises is often regarded as the circular analogue ofthe normal distribution on the line. Several propertiesof the normal have natural analogues for the vonMises. For example, as we shall see, the maximumlikelihood estimate (MLE) of the mean direction isthe sample mean direction, and the von Mises is thecircular distribution having maximum entropy for afixed mean direction and circular variance.

The von Mises distribution is unimodal. Tomodel multimodal data, mixtures of von Misesdistributions have been widely used. See Refs 9–15for details.

Wrapped DistributionsIf X is a linear random variable, with density g, thenwe may define the corresponding ‘wrapped’ circularversion of X by

θ = X(mod 2π ). (6)

478 © 2010 John Wiley & Sons, Inc. Volume 2, Ju ly /August 2010

WIREs Computational Statistics Circular data

The variable θ has density

f (θ) =∞∑

k=−∞g(θ + 2πk), (7)

obtained by ‘wrapping’ g around the circle. Two of themost commonly used wrapped distributions are thewrapped normal and the wrapped Cauchy. Using thetheory of theta functions, the density of the wrappednormal can be expressed as:

f (θ) = 12π

⎛⎝1 + 2

∞∑p=1

ρp2cos(θ − μ)

⎞⎠ , (8)

where ρ is the resultant length, given by ρ = e−σ2/2

where σ 2 is the variance of the wrapping normal. Themean direction is μ.

Suppose Y is a (linear) Cauchy random variable,with median μ and scale parameter σ , and density

g(y) = 12π

σ

σ 2 + (y − μ)2 . (9)

The density of the corresponding wrapped Cauchycan be expressed as:

f (θ) = 12π

1 − ρ2

1 + ρ2 − 2ρ cos(θ − μ)(10)

where again μ and ρ are the mean direction andresultant length, with ρ = e−σ .

Another elegant derivation of the wrappedCauchy, given in Ref 16, can be made using Mobiustransformations in the complex plane. Define acomplex random variable Z by

Z = 1 + iY1 − iY

. (11)

This transformation is a special case of the generalMobius transformation, and as it maps onto theunit circle, Z can be regarded as a circular randomvariable θ = arg(Z). Its density is given by Eq. (10),and μ and ρ are again the mean direction andresultant length. However, the relationship betweenρ and the parameters of the (linear) Cauchy isdifferent: let z0 = μ + iσ and ψ = (1 + iz0)/(1 − iz0).Then ρ = |ψ | if |ψ | < 1 and 1/|ψ | otherwise.

The wrapped Cauchy has some very nice invari-ance properties which make it a useful distributionfor regression modeling, as we explain in the sec-tion on Regression Models below. For example, writeZ ∼ C∗(z0) if Z is derived from Y through the Mobius

transformation as described above. Then if w is a fixedcomplex number with |w| = 1, wZ ∼ C∗(wz0). More-over, if Z ∼ C∗(z0), and W = (Z + w)/(1 + wZ), thenW ∼ C∗ ((w + z0)/(1 + wz0)) , so that the wrappedCauchy is invariant under these types of Mobiustransformations. See Ref 16 for details.

Other wrapped distributions are the wrappedt,17 the wrapped skew-normal,18–19 and the wrappedstable distributions.20 See also Ref 21 for thedescription of a family of distributions that includesthe wrapped Cauchy, as well as the von Mises and thecardioid.

Projection MethodsAnother general method for constructing angulardistributions is to project univariate or bivariate linearvariables onto the circle. For example, the image ofa linear variable under a stereographic projectionis a circular variable. This amounts to definingθ = 2 arctan(X), where X is linear. Another commonmethod is to project a bivariate linear random vectorY = (Y1, Y2) onto the unit circle. This results in acircular variable θ , where cos θ = Y1/

√(Y2

1 + Y22),

sin θ = Y2/√

(Y21 + Y2

2). If Y is bivariate normal, thisresults in the offset normal distribution (see Ref 3,p. 46).

Bivariate DistributionsA simple way of creating bivariate circular distribu-tions on the torus is by wrapping bivariate distribu-tions on the plane: If g(x, y) is a bivariate density onthe plane, the wrapped version of g is

f (θ , ψ) =∞∑

k=−∞

∞∑l=−∞

g(θ + 2kπ , ψ + 2lπ ). (12)

A simple calculation shows that the marginal densitiesof f are the wrapped versions of the marginal densitiesof g.

Another general method for constructing bivari-ate densities was given in Ref 22. If f1, f2 and f3 aredensities on the circle, then

f (θ , ψ) = 2πf3(2π{F1(θ) + F2(ψ)})f1(θ)f2(ψ) (13)

and

f (θ , ψ) = 2πf3(2π{F1(θ) − F2(ψ)})f1(θ)f2(ψ) (14)

are both bivariate distributions on the torus withmarginal distributions f1 and f2. The functions F1 and



F2 are given by

F1(θ) =∫ θ

0f1(θ ′) dθ ′ and F2(ψ) =

∫ ψ

0f2(ψ ′) dψ ′,

(15)

and are the circular distribution functions correspond-ing to f1 and f2.

The use of projection methods to constructbivariate distributions having specified marginaldistributions is described in Ref 23. For example,if U and V are bivariate random vectors such thatthe elements of U are uncorrelated standard normalsand similarly for V, but (U, V) has a multivariatenormal distribution, then the projections of U and Vonto the unit circle generate two-correlated uniformdistributions.

Finally, a bivariate analogue of the von Misesdistribution has been considered by several authors.Consider a distribution on the torus having density

f (θ , ψ) = C exp(κ1aT1 U + κ2aT

2 V + UTAV) (16)

where U = (cos θ , sin θ), V(cos ψ , sin ψ), and A is a2 × 2 matrix. In general, this distribution does nothave von Mises margins. Special cases have beenstudied in Refs 24–26.

INFERENCE FOR CIRCULAR DATA

Inference for a Single SampleSuppose we have a sample θ1 . . . , θn of angulardata, assumed to be generated from some circulardistribution. How can we be sure our distributionalassumption is reasonable? If it is, how can we estimatethe model parameters? How can we estimate theunderlying mean direction, if indeed it exists, (theunderlying model might be the uniform distribution).How can we construct tests and confidence intervalsfor the mean direction?

Turning first to questions of goodness of fit, wenote that the classical methods used to test uniformityof linear data can be used (with suitable modificationsto make them invariant to the choice of referencedirection) for testing uniformity of circular data. Notethat the uniformity of the underlying distribution maybe of interest in its own right, e.g., when astronomerswish to test if stars are uniformly distributed on thecelestial horizon. Tests for uniformity of linear dataare usually based on either the χ2 test, tests based onthe empirical distribution or tests based on spacingsbetween observations. Circular versions of these testshave been developed in Refs 27–29 in the case of

empirical distribution function (EDF)-based tests, andRefs 30,31 in the case of tests based on arc lengths.

The linear tests based on EDFs are not verypowerful against multimodal alternatives, and this istrue for the circular versions as well. Smooth testsof uniformity which have better power against thistype of alternative are discussed in Ref 32. Parametrictests of uniformity can be performed by, e.g., testingif κ = 0 in the von Mises distribution. As κ = 0 (orequivalently ρ = 0) implies uniformity for the vonMises, a natural test is based on the sample meanresultant length R̄. This is the Raleigh test.

Having satisfied ourselves that the mean direc-tion exists, we can proceed to estimate it with aconfidence interval. For large samples, we can use theinterval

θ̄ ± arcsin(z(α/2)σ̂ ) (17)

where z(α/2) is the usual normal quantile and σ̂

is the estimated circular standard error defined inthe discussion above of circular distributions. Thisinterval, introduced in Ref 33, is based on a seriesof asymptotic approximations and is recommendedwhen the sample size exceeds 25. A bootstrapapproach when n ≤ 25 is recommended in Ref 1.Bootstrap approaches to setting confidence intervalsbased on asymptotically pivotal statistics are discussedin Refs 34,35, with the solution proposed in the secondpaper being superior.

If we are prepared to make distributionalassumptions, we can employ conventional likelihoodmethods. For the von Mises, inference is straightfor-ward: the MLE of the mean direction is the samplemean direction, and the MLE of κ is obtained bysolving the equation I1(κ)/I0(κ) = R, where R isthe sample mean resultant length. See Ref 3, p. 85for details. An approximate confidence interval isdescribed in Ref 36. Tests for the mean directionbased on large-sample theory are discussed in Refs 1,p. 93 and 3, p. 122. A goodness of fit test for the vonMises is provided in Ref 37.

For likelihood inference for the wrappeddistributions, see the references above in the discussionof wrapped distributions.

Inference for Several SamplesAn approximate test for comparing several von Misesmean directions was introduced in Ref 38. Supposeni and Ri, i = 1, 2, . . . , k represent the sample size andresultant length of k samples, from each of k differentvon Mises populations, assumed to have the sameconcentration parameter, and let n and R be the



size and resultant length of the combined sample.The quantities ni − Ri represent the concentrationof the individual samples, and n − R represents theconcentration of the combined sample. If the meandirections coincide, the identity

2κ(n − R) = 2κ∑

i

(ni − Ri) + 2K( ∑

i

Ri − R)

(18)

has an approximate χ2 decomposition

χ2n−1 = χ2

p−1 + χ2n−p (19)

from which an approximate analysis of variancecan be constructed. The approximation improves asthe concentration parameter increases, but correctionfactors are available; see Ref 39. If the assumption ofequal dispersions is untenable, see Ref 40. A test basedon the likelihood ratio test for this problem is givenby Ref 41. For a bootstrap approach, see Ref 42.

More Complicated DesignsSeveral authors (see Refs 41,43,44) have attemptedto extend these results for the several sample case tocover multifactorial designs, but the lack of a suitablenotion of interaction has hampered these efforts, withno compelling breakdown of tables of mean directionsinto main effects and interactions being available.

REGRESSION AND CORRELATION

Circular CorrelationMeasuring association between two circular variables(or a circular and a linear variable) is less straight-forward than between two linear variables, largelybecause there is no natural relationship between them.In the linear–linear case, a linear relationship providesa natural standard against which to measure asso-ciation. No such natural relationship exists wherecircular variables are involved. Consequently, manydifferent correlation coefficients for angular data havebeen proposed, corresponding to different types ofrelationship.

A measure of association between two variablesshould have the following properties:

1. It should have a particular value (by conventionzero) when the variables are independent.

2. It should lie between two particular values (byconvention ±1).

3. It should take these particular values when thevariables have some specified relationship, suchas a linear relationship for linear variables.

We treat the cases of circular–linear and circular–circular association separately, because different typesof relationship are appropriate in these two situations.

Circular–linear AssociationThe joint distribution of a circular and a linear variableis supported by an infinite cylinder, and a possibleform of relationship between a linear variable y and acircular variable θ is

y = a + f (θ − μ) (20)

where f is a continuous function with f (0) = 0. If fis a bounded periodic function with period 2π , thenf defines a closed curve around the cylinder. In thespecial case when f is a multiple of the sine function,the relationship becomes

y = a + b sin θ + c cos θ (21)

and an obvious measure of correlation in thiscase, proposed in Ref 45, is the ordinary multiplecorrelation between y and the variables (cos θ , sin θ).However, this has the disadvantage that it isnot signed. An alternative based on the canonicalcorrelation between y and (cos θ , sin θ) is consideredin Ref 24. In Ref 46, more general forms ofthe relationship (requiring only that g have onemaximum and one minimum in [0, 2π]) are discussed.A correlation coefficient to measure this type ofassociation is introduced, as are distribution-free testsof independence.

Another form of f is an unbounded periodicfunction that is monotone on (−π , π ), such as thefunction tan( θ

2 ). Although in Ref 47 a regression modelbased on this type of relationship is introduced, nocorresponding correlation coefficient seems to havebeen proposed. This regression model is discussedfurther in the following text.

Circular–circular AssociationConsider a one-to-one mapping taking a circularvariable θ onto another circular variable ψ , withthe property that as θ moves around the circle ina clockwise direction, so does ψ . In Ref 48, this istermed positive toroidal association; correspondinglyif ψ moves in an anticlockwise direction as θ movesclockwise, we have negative toroidal association. Twosigned correlation coefficients to measure this typeof association, together with the associated tests of



independence are also presented. A generalization ofone of these tests to higher dimensions is given inRef 49.

A much simpler form of relationship betweenangular variables can be expressed in terms ofreflections and rotations: in Ref 50, relationships ofthe form

ψ = θ + α(mod 2π) (22)

and

ψ = −θ + α(mod 2π) (23)

are considered, together with a signed correlationcoefficient

ρ = E[sin(θ1 − θ2) sin(ψ1 − ψ2)]

{E[sin2(θ1 − θ2)]E[sin2(ψ1 − ψ2)]}1/2(24)

where θ1 and θ2 are independently distributed as θ ,and ψ1 and ψ2 are independently distributed as ψ .The correlation ρ takes the value zero when θ andψ are independent, the value 1 if and only if Eq.(22) holds, and the value 1 if and only if Eq. (23)holds. There is also an associated large-sample test ofindependence.

Other authors have obtained correlations byapplying multivariate methods to the unit vectorsU = (cos θ , sin θ) and V = (cos ψ , sin ψ). In Ref 24,an alternative correlation is proposed based on themaximum canonical correlation between the vectorsU and V, i.e., the largest singular value of the matrix −1

11 12 −122 21, where

Cov(

UV

)=

( 11 12

21 22

). (25)

This correlation is not signed, but takes the value1 if and only if the relationships (22) or (23) hold.A correlation based on on the trace of −1

11 12 −122 21

is studied in Ref 51. Using the maximum singular valueof the matrix 12 is suggested in Ref 25.

Tests for IndependenceIf we are prepared to assume a particular form forthe bivariate distribution of (θ , ψ) which includesindependence as a special case, we can test the inde-pendence of the variables by testing the parameters ofthe distribution assume the appropriate values cor-responding to independence. For example, in thewrapped bivariate normal, if the correlation of theunderlying bivariate normal is zero, then the wrappedbivariate normal has independent margins.

If we do not wish to assume a parametricform the joint distribution, we can use a nonpara-metric test based on circular ranks.52 Other non-parametric tests include those in Ref 53 based onEDFs, and Ref 54, based on weighted degenerateU-statistics.

Regression Models for a Circular ResponseRegression models have been proposed for the threecases

1. Linear response, angular covariates;

2. Angular response, linear covariates;

3. Angular response, angular covariate.

Case 1 is straightforward with the angular covariatesentering as trigonometric polynomials, and will notbe discussed further. Case 2 has been considered inRef 47, where a model of the form

θ = β0 + g(βTx) + ε (mod 2π ), (26)

is proposed. As discussed earlier, g is a monotonelink function mapping the real line to (−π , π), withg(0) = 0, and ε has a von Mises distribution withmean direction 0. In this reference, a procedure forestimating the parameters using maximum likelihoodis proposed, but the likelihood is difficult to work withfor more than a few covariates.

An alternative approach is to use the offsetnormal distribution, allowing the mean vector (μ1, μ2)of the bivariate normal to be a linear function of thecovariates. See Ref 55 for parameter estimation usingmaximum likelihood. This method seems to havebetter numerical properties than the method usingmonotone link functions.

Case 3 is studied in Ref 56, using the relationship

ψ = μψ + g(ωg−1(θ − μθ )) + ε (mod 2π ) (27)

with g(x) = 2 arctan x, where ε has a von Mises distri-bution. Parameter estimation and certain degeneratecases are discussed. Note that the regression relation-ship ψ = μψ + g(ωg−1(θ − μθ )) is a special case of aMobius transformation.

A regression model also involving Mobiustransformations, but based on the circular Cauchydistribution, has recently been proposed in Ref 57.Consider a circular variate U (regarded as a complexnumber with modulus 1) having a C∗(φ) distribution,where 0 < φ < 1. Their regression model, for an



angular covariate θ , is given by

Z = β0eiθ + β

βeiθ + 1U, (28)

where β0 and β are complex numbers with|β0| = 1 and |β| �= 1. By the properties of thecircular Cauchy distribution, the angular variableψ = arg Z has a circular Cauchy distribution, withparameter β0

eiθ+β

βeiθ+1φ. The parameter φ represents the

strength of the regression relationship. As φ → 1, thevariables are related without error, whereas φ → 0,the response is uniformly distributed independentlyof the covariates. A bivariate distribution forθ , ψ in which the conditional distribution of ψ

given θ is the regression function above is alsodescribed in Ref 57. Estimation via maximumlikelihood is straightforward. A test of independencecorresponding to this bivariate distribution can beobtained by testing β = 1.

Finally, a tree-based regression model, witha data-driven choice of regression function, isconsidered in Ref 58.

Circular Time SeriesThe regression models above can be adapted toanalyze time series of angles. For example, inRef 59 an autoregressive model is obtained bywrapping a linear AR(1) model around the circle,and applied it to the analysis of wind directions.Four types of time series model are studied in Ref60. The first is obtained by wrapping standard linearARMA processes, and the second by projecting abivariate linear process onto the unit circle. Twofurther models are based on the regression modelsdiscussed above. One is a linked process of theform

θ t = μ + g(Xt) (29)

where Xt is a linear process, and the second one ofthe form

θ t = μ + g(ω1g−1(θ t−1 − μ))

+ · · · + ωp g−1(θ t−p − μ)) + εt (30)

where the εt’s are independent von Mises variables.Model identification and fitting issues are alsodiscussed. Other recent approaches include the onebased on generalized estimating equations,61 andanother on hidden Markov models.62

APPLICATIONS

The techniques described above have been applied toseveral fields of scientific endeavor, most notably inmeteorology, ecology, and medicine. In meteorology,the von Mises distribution has been used as a modelfor the analysis of wind directions. See Ref 59 fortime series of wind directions, see Ref 63 for theestimation of the mean direction of winds, and thecalculation of correlations between wind directionand wave heights. Mixtures of von Mises distributionsare used to model wind speeds and wind directionsin Refs 64,65. Angular-linear correlation is used toexamine the relationship between wind directions andpollution concentrations in Ref 66.

In ecology, angular measurements arise whenthe directional movement of animals in response tostimuli is considered. Projected normal regression isused to study the orientation of sandhoppers in Ref67. Circular regression is applied to the same problemin Ref 68. Circular time series methods are applied tothe tracks of bark beetles in Ref 69. A multifactorialanalysis of intertidal snail movements is presented inRefs 43 and 44. Circular regression is used to studythe influence of a landscape feature on the directionof animal movement in Ref 70.

Diurnal or seasonal data can also be analyzedby these methods, by identifying the position in thecycle with an angle. Thus, in Ref 71, von Misesdistributions are fitted and density estimation is usedto study crime incident statistics, identifying time ofday with an angle. Circular regression is used to studythe relationship between the seasonality of diseaseonset and other covariates in Ref 72. Mixtures of vonMises distributions are applied to model the seasonalpattern of sudden infant death syndrome in Ref 73.

OTHER TOPICS

Bayesian Approaches to Circular DataSeveral of the techniques described above haveBayesian counterparts. For the one-sample problem, aBayesian analysis based on the von Mises distributionis described in Ref 74. In Ref 75, the same groundis covered from an empirical Bayes perspective.One and two sample problems are treated inRef 76. A hierarchical Bayesian treatment of thecircular–linear regression model is given in Ref 77.

Robustness and OutliersFor directional data, the problem of outliers is notas acute as is the case for linear data, as angles areconfined to the interval [0, 2π]. However, individual



points can still exert an undue influence, particularlyon measures of concentration. See Ref 78, Ch11, fora discussion. Several tests to detect outliers have beenproposed. In Ref 79, a test based on the effect on theresultant length of a sample when a single observationis deleted is presented. Another approach is to fit amixture model, and test for the homogeneity of themixture. See Ref 4, Ch 10 for a discussion. Finally,Collett79 and SenGupta80 approach the outlier prob-lem though slippage tests, where all observations butone have a common distribution, but the distributionof the remaining observation has a different meandirection.

Software for Circular Data AnalysisThe book by Jammalamadaka and SenGupta4 con-tains examples illustrating the use of an R packageCircStats, written by Ulrick Lund, that will per-form many simple calculations (such as summarystatistics and some simple tests) on circular data. Itwill also generate random angles from the standarddistributions, plot circular data and fit the standarddistributions to data. The package circular byClaudio Agostinelli and Ulrick Lund is an updatedversion with more functionality.

CONCLUSION

In this article, we have given a brief survey of statis-tical methods for the analysis of circular data. As wehave pointed out, the standard methods used for lin-ear data are not appropriate for the analysis of angles.However, over the last 50 years, a comprehensiveset of methods for circular data analysis have beendeveloped, covering summary statistics, probabilitydistributions, one- and multi-sample problems, corre-lation, regression, and time series. There is an activeand expanding literature, and statistical software isavailable for the implementation of the methods,which are finding application in many fields of sci-entific endeavor.

The reader wanting to know more about circulardata should consult one of the excellent texts avail-able. The first comprehensive treatment was publishedby Mardia,81 followed by Batschelet,82 Fisher,1 Mar-dia and Jupp,3, and Jammalamadaka and SenGupta.4

The texts by Batschelet and Fisher are more appliedbooks oriented toward the working scientist wantingto apply the methods, whereas the others are more the-oretical. The book by Mardia and Jupp is an updatedversion of Ref 81.

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