For each observation Si, label its eigen-decompositions as
F−1(Si) =
2(p−1)p! [
j=1
{ Di(j), Ui(j)},
where versions (D1(j), U1(j)), . . . , (Dn(j), Un(j)) form an i.i.d. sample from a continuous distri- bution F with location parameter (D(j), U(j)) satisfying assumptions 1-3 from the previ- ous section. To simplify the notation of the set of location parameters, let (D(1), U(1)) = (D∗, U∗). By assumption 2 from the previous section, (D(j), U(j)) = (GjD∗GTj, U∗GTj) for some Gj ∈ G(p) for j = 1, . . . , 2(p−1)p!. Then we will denote the set of location parameters as [(D∗, U∗)] ∈ (Diag+(p) × SO(p))/G(p), which is the orbit of (D∗, U∗) under the group action of G(p) on Diag+(p) × SO(p) (as defined in subsection 3.2.1).
Let k(p) = 2(p−1)p!, and suppose that (D10, U10), . . . , (D0n, Un0) is a sample of observed eigen-decompositions as specified in assumption 3 from the previous section. Each observed eigen-decomposition has the mixture representation
(D0i, Ui0) = k(p) X j=1 (ZjD (j) i , ZjU (j) i )
where ¯Z = (Z1, . . . , Zk(p)) follows a multinomial distribution with the constraint Pk(p)
j=1Zi = 1 and probabilities ¯π = (1/k(p), . . . , 1/k(p)). This representation leads to the observed data log-likelihood from Equation 5.2, which will be difficult to optimize in practice.
If we were able to observe the version indicator variable zij for each observed eigen- decomposition (Di0, Ui0), then we could form the complete data log-likelihood
`C((D∗, U∗), Γ) = n X i=1 k(p) X j=1 zijlog[f ((GTjD 0 iGj, Ui0Gj); (D∗, U∗), Γ)] − n log(k(p))
and estimate the parameters using standard maximum likelihood estimation. The EM algo- rithm will address our missing data problem by iteratively imputing values for the unobserv- able version indicator variables and then performing maximum likelihood estimation using the complete data log-likelihood with imputed values plugged in for the version indicator variables. The steps are:
1. Choose initial estimates ( ˆD∗(0), ˆU∗(0)) and ˆΓ(0) for the parameters (D∗, U∗) and Γ and set tolerance ε > 0.
2. Expectation Step: Compute ˆ zij(k) = f ((G T jD0iGj, Ui0Gj); ( ˆD∗(k), ˆU∗(k)), ˆΓ(k)) Pk(p) j=1f ((GTjD 0 iGj, Ui0Gj); ( ˆD∗(k), ˆU∗(k)), ˆΓ(k)) 3. Maximization Step: Compute
(( ˆD∗(k+1), ˆU∗(k+1)), ˆΓ(k+1)) = argmax n X i=1 k(p) X j=1 ˆ zij(k)log[f ((GTjDi0Gj, Ui0Gj); (D∗, U∗), Γ)]
4. If |`(( ˆD∗(k+1), ˆU∗(k+1)), ˆΓ(k+1)) − `(( ˆD∗(k), ˆU∗(k)), ˆΓ(k))| > ε, then return to Step 2. Oth- erwise, set the final parameter estimates equal to ˆΦ(k+1).
We illustrate how to perform parameter estimation via the EM algorithm when the generating eigen-decompositions follow distribution F2 with p = 2 from Section 5.1 with density function f2((D, U ); (D∗, U∗), Γ) = γ1 2πN (γ2)|D| exp −γ1 2d 2 D+(2)(D, D∗) − γ2 2d 2 SO(2)(U, U ∗ )
where |D| = det(D) and N (γ2) = p2π/γ2erf (πpγ2/2). Let S1, . . . , Sn be a random sample of SPD matrices arising from eigen-composition of the distribution F2and let (D0i, Ui0) denote an observed eigen-decomposition of Si randomly drawn from the uniform distribution on F−1(S
i). We begin with initial guesses ( ˆD∗(0), ˆU∗(0)) and ˆΓ(0) for the parameters (D∗, U∗) and Γ.
Performing the expectation step after k iterations simply requires computing ˆ zij(k)= f2((G T jD0iGj, Ui0Gj); ( ˆD∗(k), ˆU∗(k)), ˆΓ(k)) P4 j=1f2((GTjDi0Gj, Ui0Gj); ( ˆD∗(k), ˆU∗(k)), ˆΓ(k)) for i = 1, . . . , n and j = 1, . . . , 4.
Performing the maximization for the k + 1 iteration requires solving ˆ D∗(k+1) = argmin D∈Diag+(2) n X i=1 4 X j=1 zˆ(k) ij n d2D+(2)(GTjDi0Gj, D) (5.3) ˆ U∗(k+1) = argmin U ∈SO(2) n X i=1 4 X j=1 zˆ(k) ij n d2SO(2)(Ui0Gj, U ) (5.4)
to obtain estimates for the location parameters. It can be shown that ˆ D∗(k+1) = n X i=1 4 X j=1 zˆ(k) ij n GTjD0iGj
and that solving for ˆU∗(k+1) requires locating U ∈ SO(2) that satisfy n X i=1 4 X j=1 zˆ(k) ij n Log(Ui0GjUT) = 0.
After substituting ( ˆD∗(k+1), ˆU∗(k+1)) for (D∗, U∗) into the approximate complete data log-likelihood, we can solve the equations
ˆ γ1(k+1) = argmax γ1 n log(γ1) −γ1 2 n X i=1 4 X j=1 zˆ(k) ij n d2D+(2)(GTjD0iGj, ˆD∗(k+1)) (5.5) ˆ γ2(k+1) = argmax γ2 −n log(N (γ2)) − γ2 2 n X i=1 4 X j=1 zˆ(k) ij n d2SO(2)(Ui0Gj, ˆU∗(k+1)) (5.6)
to obtain estimates for the concentration parameters. After differentiatingEquation 5.5and setting the derivative equal to 0, we have that
ˆ γ1(k+1) = 1 2 n X i=1 4 X j=1 zˆ(k) ij n d2D+(2)(GTjDi0Gj, ˆD∗(k+1)) −1 .
From differentiating Equation 5.6 and setting the derivative equal to 0, we get that ˆγ2(k+1) should satisfy V (ˆγ2(k+1)) = −N 0(ˆγ(k+1) 2 ) N (ˆγ2(k+1)) = n X i=1 4 X j=1 zˆ(k) ij n d2SO(2)(Ui0Gj, ˆU∗(k+1)).
It is shown in [12] that V is a one-to-one function, so ˆγ2(k+1)is unique and can be approximated with a numerical procedure such as Newton’s method.
To demonstrate the performance of this estimation method, we performed simulation experiments drawing 500 samples of size n = 50, 100, and 500 from the distribution F2 with p = 2 and parameters γ1 = 2, γ2 = 5, D∗ = 10 0 0 2 , and U∗ = cos(π/6) − sin(π/6) sin(π/6) cos(π/6) .
Table 6: Empirical Mean Squared Error (MSE) from 500 Simulations Parameter n = 50 n = 100 n = 500
(D∗, U∗) 1.6042 0.8928 0.1756
γ1 0.1068 0.0493 0.0089
γ2 1.7845 0.8069 0.1237
We computed the empirical mean squared error from estimating (D∗, U∗) using the formula
M SE((D∗, U∗)) = 1 500 500 X t=1 kF ( ˆD∗t, ˆUt∗) − F (D∗, U∗)k2F
where ( ˆDt∗, ˆUt∗) is the sample estimate for (D∗, U∗) from sample t. We can see in Table 6
that the empirical MSE for each parameter decreases as n increases, suggesting that the EM estimation procedure is consistent for this example.
5.4 DISCUSSION
It would be interesting to establish conditions on the distribution of the generating eigen- decompositions which would guarantee that the EM algorithm would find a global maximizer of the likelihood function and would provide consistent estimates. Perhaps more interesting, the mixture likelihood-based estimation framework proposed in this chapter can easily be extended to create likelihood ratio tests for single and multi-sample inference. For example, to test the null hypothesis that (D∗, U∗) = (D(0), U(0)), Γ = Γ0, one could use the test statistic
Λ = 2[`(( ˆD∗, ˆU∗), ˆΓ) − `((D0, U0), Γ0)],
where `(( ˆD∗, ˆU∗), ˆΓ) is the observed log-likelihood evaluated at the EM parameter estimates and `((D0, U0), Γ0) is the observed log-likelihood evaluated at the null hypothesis param- eter values. Under the null hypothesis, we conjecture that Λ will asymptotically follow a chi-square distribution with degrees of freedom equal to the number of free parameters in D∗, U∗, and Γ.
Given data from two independent samples, it might be of interest to test the null hy- pothesis that the two samples have the same location parameters (i.e. (D∗1, U1∗) = (D∗2, U2∗)) while assuming that they have the same concentration parameters. For this test, we could use the test statistic
Λ = 2[`(( ˆD1∗, ˆU1∗), ˆΓpooled) + `(( ˆD2∗, ˆU ∗
2), ˆΓpooled) − `(( ˜D∗, ˜U∗), ˜Γ)],
where ( ˆD∗1, ˆU1∗) and ( ˆD2∗, ˆU2∗) are the EM location estimates for samples 1 and 2 under the alternative hypothesis, ˆΓpooled contains the pooled concentration parameter estimates under the alternative hypothesis, and ( ˜D∗, ˜U∗) and ˜Γ are the EM estimates for the location and concentration parameters under the null hypothesis after combining both samples into one sample. We conjecture that, under the null hypothesis, this test statistic will asymptoti- cally follow a chi-square distribution with degrees of freedom equal to the number of free parameters in D∗ and U∗.
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