Encoding Analysis and Representational Similarity

The methods in this paper were first developed in the context of fMRI data analysis, and our examples will come from this domain. A simple way to apply the analyses to fMRI data is to use as activity estimates ( ${\widehat{\mathbf{U}}}^{\left(m\right)}$ ) the regression coefficients, or “beta”-weights, from a first-level time series analysis [36 (link), 37 (link)]. The time-series model accounts for the hemodynamic lag and the temporal autocorrelation of the noise. The activity estimates usually express the difference in activity during a condition relative to rest. Activity estimates commonly co-vary together across fMRI imaging runs, because all activity estimates within a partition are measured relative to the same resting baseline. This positive correlation can be reduced by subtracting, within each partition, the mean activity pattern (across conditions) from each activity pattern. This makes the mean of each measurement channel (across condition) zero and thus centers the ensemble of points in activity-pattern space that is centered on the origin.
Rather than using the concatenated activity estimates from different partitions, encoding analysis and PCM can also be applied directly to time series data. As a universal notation that encompasses both situations, we can use a standard linear mixed model [38 (link)]:
$\mathbf{Y}=\mathbf{\text{ZU}}+\mathbf{\text{XB}}+ϵ,$
where Y is an N × P matrix of all activity measurements, Z the N × K design matrix, which relates the activity measurements to the K experimental conditions, and X is a second design matrix for nuisance variables. U is the K × P matrix of activity patterns (the random effects), B are the regression coefficients for these nuisance variables (the fixed effects), and E is the matrix of measurement errors. If the data Y are the concatenated activity estimates, the nuisance variables typically only model the mean pattern for each run. If Y consists of time-series data, the nuisance variables typically capture additional effects such as time-series drifts and residual head-motion-related artifacts.