Comparing representational geometries using whitened unbiased-distance-matrix similarity — part 6/6

Cleaned-text code points [78232, 85328). Bibliography omitted here; source.md includes it.

S i , j = 1 M 2 { M ( Ξ i , j M − 1 + Ξ i , j ) + M ( M − 1 ) ( ( M − 2 ) ​ Ξ i , j ( M − 1 ) 2 + 2 ​ Ξ i , j M − 1 ) } = 1 M 2 ​ Ξ i , j ​ { M M − 1 + M + M ⁡ ( M − 2 ) M − 1 + 2 ​ M } = 4 M ​ Ξ i , j , N i , j = { M ​ Ξ i , j ​ Ξ i , j M − 1 + M ⁡ ( M − 1 ) ​ Ξ i , j ​ Ξ i , j ( M − 1 ) 2 } = 2 ​ Ξ i , j ​ Ξ i , j M ⁡ ( M − 1 ) , \begin{split}{S}_{i,j}=\ &\frac{1}{M^{2}}\Big\{M\big(\frac{{\Xi}_{i,j}}{M-1}+{\Xi}_{i,j}\big)\\ &+M(M-1)\Big(\frac{(M-2){\Xi}_{i,j}}{(M-1)^{2}}+\frac{2{\Xi}_{i,j}}{M-1}\Big)\,\,\Big\}\\ =\ &\frac{1}{M^{2}}\,{\Xi}_{i,j}\Big\{\frac{M}{M-1}+M+\frac{M(M-2)}{M-1}+2M\Big\}\\ =\ &\frac{4}{M}\,{\Xi}_{i,j},\\ {N}_{i,j}=\ &\Big\{\frac{M{\Xi}_{i,j}{\Xi}_{i,j}}{M-1}+\frac{M(M-1){\Xi}_{i,j}{\Xi}_{i,j}}{(M-1)^{2}}\Big\}\\ =\ &\frac{2\,{\Xi}_{i,j}{\Xi}_{i,j}}{M(M-1)},\end{split} (50)

and

Cov ⁡ ( d ~ i , d ~ j ) = 1 P 2 ​ ( 4 ​ P ​ 𝜹 i ​ 𝚺 P ​ 𝜹 j T M ​ Ξ i , j + 2 ​ t ​ r ​ ( 𝚺 P ​ 𝚺 P ) M ⁡ ( M − 1 ) ​ Ξ i , j ​ Ξ i , j ) . {\rm Cov}(\tilde{d}_{i},\,\tilde{d}_{j})=\frac{1}{P^{2}}\Big(4\frac{P\bm{\delta}_{i}\bm{\Sigma}_{P}\bm{\delta}_{j}^{T}}{M}\Xi_{i,j}+\frac{2\rm tr\big(\bm{\Sigma}_{P}\bm{\Sigma}_{P}\big)}{M(M-1)}\Xi_{i,j}\Xi_{i,j}\Big). (51)

Finally, on writing the desired complete covariance matrix using element-by-element multiplication (Hadamard product), we obtain the result given in Eq. 29 .

### 6.2 Unbalanced designs

In unbalanced designs, the noise covariance 𝚺 K \bm{\Sigma}_{K} is different across different partitions - i.e. each partition has their own covariance matrix 𝚺 K m \bm{\Sigma}_{K}^{m} . In the calculation of the distances, this ideally should be taken into account. To simplify this problem we here assume that all signals are zero, as we did for the derivation of the WUC.

Given the zero mean assumption and independent runs, the product of patterns from one pair of partitions, is uncorrelated to the product of two patterns from any other pair of partitions (Eq. 36 ). Thus, for both biased and unbiased distance estimates, the optimal pooling of the estimates from the pairs is their precision weighted average.

The distance estimate from single pair of partitions 𝐝 ^ m , n = diag ⁡ ( 𝐂 ​ 𝐁 ^ m ​ 𝐁 ^ n T ​ 𝐂 ) / P \hat{\mathbf{d}}_{m,n}=\mathrm{diag}(\mathbf{C}\hat{\mathbf{B}}_{m}\hat{\mathbf{B}}_{n}^{T}\mathbf{C})/P has the following expected value ( E \mathrm{E} ) and covariance ( Var \mathrm{Var} ):

If m ≠ n m\neq n : E ⁡ ( 𝐝 ^ m , n ) = 0 \mathrm{E}(\hat{\mathbf{d}}_{m,n})=0 (52)

Var ⁡ ( 𝐝 ^ m , n ) = tr ⁡ ( 𝚺 𝑷 ​ 𝚺 𝑷 ) ​ 𝚵 m ∘ 𝚵 n = tr ⁡ ( 𝚺 𝑷 ​ 𝚺 𝑷 ) ​ ( 𝐂 ​ 𝚺 k m ​ 𝐂 T ) ∘ ( 𝐂 ​ 𝚺 k n ​ 𝐂 T ) \mathrm{Var}(\hat{\mathbf{d}}_{m,n})=\mathrm{tr}(\bm{\Sigma_{P}}\bm{\Sigma_{P}})\bm{\Xi}^{m}\circ\bm{\Xi}^{n}=\mathrm{tr}(\bm{\Sigma_{P}}\bm{\Sigma_{P}})(\mathbf{C}\bm{\Sigma}_{k}^{m}\mathbf{C}^{T})\circ(\mathbf{C}\bm{\Sigma}_{k}^{n}\mathbf{C}^{T}) (53)

If m = n m=n : E ⁡ ( 𝐝 ^ m , m ) = tr ⁡ ( 𝚺 𝑷 ​ 𝚺 𝑷 ) ​ diag ​ ( 𝐂 ​ 𝚺 k m ​ 𝐂 T ) \mathrm{E}(\hat{\mathbf{d}}_{m,m})=\mathrm{tr}(\bm{\Sigma_{P}}\bm{\Sigma_{P}})\mathrm{diag}(\mathbf{C}\bm{\Sigma}_{k}^{m}\mathbf{C}^{T}) (54) Var ⁡ ( 𝐝 ^ m , m ) = tr ⁡ ( 𝚺 𝑷 ​ 𝚺 𝑷 ) ​ 𝚵 m ∘ 𝚵 n = tr ⁡ ( 𝚺 P ​ 𝚺 P ) ​ ( 𝐂 ​ 𝚺 k m ​ 𝐂 T ) ∘ ( 𝐂 ​ 𝚺 k m ​ 𝐂 T ) \mathrm{Var}(\hat{\mathbf{d}}_{m,m})=\mathrm{tr}(\bm{\Sigma_{P}}\bm{\Sigma_{P}})\bm{\Xi}^{m}\circ\bm{\Xi}^{n}=\mathrm{tr}(\bm{\Sigma}_{P}\bm{\Sigma}_{P})(\mathbf{C}\bm{\Sigma}_{k}^{m}\mathbf{C}^{T})\circ(\mathbf{C}\bm{\Sigma}_{k}^{m}\mathbf{C}^{T}) (55)

Using these formulas we derive an optimal unbiased estimate using precision weighting: 𝐝 ~ = ( ∑ m ∑ n ≠ m Var − 1 ​ ( 𝐝 ^ m , n ) ) − 1 ​ ∑ m ∑ n ≠ m Var − 1 ​ ( 𝐝 ^ m , n ) ​ 𝐝 ^ m , n \tilde{\mathbf{d}}=\left(\sum_{m}\sum_{n\neq m}\mathrm{Var}^{-1}(\hat{\mathbf{d}}_{m,n})\right)^{-1}\sum_{m}\sum_{n\neq m}\mathrm{Var}^{-1}(\hat{\mathbf{d}}_{m,n})\hat{\mathbf{d}}_{m,n} (56)

The covariance matrix of this combined estimate is then:

( ∑ m ∑ n ≠ m Var − 1 ​ ( 𝐝 ^ m , n ) ) − 1 \left(\sum_{m}\sum_{n\neq m}\mathrm{Var}^{-1}(\hat{\mathbf{d}}^{m,n})\right)^{-1} (57)

### 6.3 Simulations with spatially correlated noise

To study the influence of spatial noise correlations, we repeated the simulations for Experiment 2 (see Fig. 6), using noise that was spatially correlated across voxels, Σ P ≠ 𝐈 \Sigma_{P}\neq\mathbf{I} . To parametrically vary the level of spatial co-dependence, we generated the data on a 6 × 6 × 6 6\times 6\times 6 cube of isotropic voxels. The covariance between the noise term of the two voxels e i e_{i} and e j e_{j} was determined by

cov ( e i , e j ) = exp ( − d i , j 2 / s 2 ) \mathrm{cov}(e_{i},e_{j})=\mathrm{exp}(-d_{i,j}^{2}/s^{2})

with d i , j d_{i,j} being the distance between two voxels (in voxel widths), and s 2 s^{2} the variance of the Gaussian covariance kernel, which ranged from 0 0 (independent noise) to 5 5 (highly correlated noise). The "effective number" of voxels, tr ​ ( 𝚺 P ) 2 / tr ⁡ ( 𝚺 P ​ 𝚺 P ) \mathrm{tr}(\bm{\Sigma}_{P})^{2}/\mathrm{tr}(\bm{\Sigma}_{P}\bm{\Sigma}_{P}) , was 216 216 for s 2 = 0 s^{2}=0 and 15 15 for s 2 = 5 s^{2}=5 . The signal patterns ( 𝐁 \mathbf{B} ) for the 31 conditions were generated to be spatially independent, with a row-covariance as specified by either of the 2 candidate models.

As expected, the accuracy of model selection decreased as the spatial correlation in the noise increased. Nonetheless, the advantage of the whitened RDM Pearson correlation and whitened unbiased cosine similarity over their traditional non-whitened cousins remained stable .

Interestingly, PCM was superior to the whitened RDM correlation only for spatially independent, or for nearly independent, data. For a high degree of spatial covariance ( s 2 = 5 s^{2}=5 ), the model selection accuracy was higher for whitened Pearson correlation (64.3%) than for PCM (62.4%). Thus, the likelihood-based test implemented in PCM is only better for model comparison for data where the residual covariances are limited to a size that typically remain after multivariate prewhitening of fMRI data [ 31 ] . For highly correlated raw fMRI data (non- or uni-variately prewhitented), RSA-based methods can perform slightly superior.

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