##### 106 2. PROBABILITY DISTRIBUTIONS

`Figure 2.17 Illustration of the representation of val-`

uesθnof a periodic variable as two-

dimensional vectorsxnliving on the unit

circle. Also shown is the averagexof

those vectors.

`x 1`

`x 2`

`x 1`

`x 2`

`x 3`

x 4

`x ̄`

̄r

θ ̄

`instead to give`

`x=`

##### 1

##### N

`∑N`

`n=1`

`xn (2.167)`

`and then find the corresponding angleθof this average. Clearly, this definition will`

ensure that the location of the mean is independent of the origin of the angular coor-

dinate. Note thatxwill typically lie inside the unit circle. The Cartesian coordinates

of the observations are given byxn=(cosθn,sinθn), and we can write the Carte-

sian coordinates of the sample mean in the formx=(rcosθ,rsinθ). Substituting

into (2.167) and equating thex 1 andx 2 components then gives

`rcosθ=`

##### 1

##### N

`∑N`

`n=1`

`cosθn, rsinθ=`

##### 1

##### N

`∑N`

`n=1`

`sinθn. (2.168)`

`Taking the ratio, and using the identitytanθ=sinθ/cosθ, we can solve forθto`

give

`θ= tan−^1`

`{∑`

∑nsinθn

ncosθn

`}`

. (2.169)

`Shortly, we shall see how this result arises naturally as the maximum likelihood`

estimator for an appropriately defined distribution over a periodic variable.

We now consider a periodic generalization of the Gaussian called thevon Mises

distribution. Here we shall limit our attention to univariate distributions, although

periodic distributions can also be found over hyperspheres of arbitrary dimension.

For an extensive discussion of periodic distributions, see Mardia and Jupp (2000).

By convention, we will consider distributionsp(θ)that have period 2 π.Any

probability densityp(θ)defined overθmust not only be nonnegative and integrate