Pattern Recognition and Machine Learning

2.3. The Gaussian Distribution 107

Figure 2.18 The von Mises distribution can be derived by considering
a two-dimensional Gaussian of the form (2.173), whose
density contours are shown in blue and conditioning on
the unit circle shown in red.

x 1

x 2

p(x)

r=1

to one, but it must also be periodic. Thusp(θ)must satisfy the three conditions

p(θ)  0 (2.170)

∫ 2 π

0

p(θ)dθ =1 (2.171)

p(θ+2π)=p(θ). (2.172)

From (2.172), it follows thatp(θ+M 2 π)=p(θ)for any integerM.

We can easily obtain a Gaussian-like distribution that satisfies these three prop-

erties as follows. Consider a Gaussian distribution over two variablesx=(x 1 ,x 2 )

having meanμ=(μ 1 ,μ 2 )and a covariance matrixΣ=σ^2 IwhereIis the 2 × 2

identity matrix, so that

p(x 1 ,x 2 )=

1

2 πσ^2

exp

{

−

(x 1 −μ 1 )^2 +(x 2 −μ 2 )^2

2 σ^2

}

. (2.173)

The contours of constantp(x)are circles, as illustrated in Figure 2.18. Now suppose

we consider the value of this distribution along a circle of fixed radius. Then by con-

struction this distribution will be periodic, although it will not be normalized. We can

determine the form of this distribution by transforming from Cartesian coordinates

(x 1 ,x 2 )to polar coordinates(r, θ)so that

x 1 =rcosθ, x 2 =rsinθ. (2.174)

We also map the meanμinto polar coordinates by writing

μ 1 =r 0 cosθ 0 ,μ 2 =r 0 sinθ 0. (2.175)

Next we substitute these transformations into the two-dimensional Gaussian distribu-

tion (2.173), and then condition on the unit circler=1, noting that we are interested

only in the dependence onθ. Focussing on the exponent in the Gaussian distribution

we have

−

1

2 σ^2

{

(rcosθ−r 0 cosθ 0 )^2 +(rsinθ−r 0 sinθ 0 )^2

}

= −

1

2 σ^2

{

1+r^20 − 2 r 0 cosθcosθ 0 − 2 r 0 sinθsinθ 0

}

=

r 0

σ^2

cos(θ−θ 0 )+const (2.176)