Data reduction is a step to reduce the size of the subject's functional data. Principal Components Analysis (PCA) is used as a technique to reduce the dimensions. A single subject is reduced from 53*63*34* 220 to 53*63*34* 50, where 220 refers to the time points or scans.
A maximum of three data reduction steps or groups is allowed. First group contains the individual data-sets and each data-set is reduced in time dimension and put in another group where the data-sets are concatenated into groups and put through another data reduction step. Table 1 shows how groups are formed depending upon the number of data-sets. Table 2 shows the procedure involved in the data-reduction steps. In addition to the tables, figure shows how the data-reduction step is performed.
Table 1 : Shows how groups are formed depending upon the number of data-sets
Number of Data-Sets (n) | Number of groups or reduction steps |
n = 1 | 1 |
n < 4 | 2 |
n >= 4 | User specified number (2 or 3, See Setup ICA) |
Table 2: Shows how many sub-groups can be formed in a group
Number of groups or reduction steps | Procedure |
1 | Only one sub-group can be formed from one data-set. This data-set is reduced in time dimension (scans) to PC1 (See, Setup ICA). |
2 | First group contains the number of data-sets. These data-sets are reduced in time dimension to first data-reduction number and put in second group. In the second group, the number of sub-groups (say m) is determined by ratio of number of data-sets divided by 4. The reduced data-sets from first group are placed in m groups and concatenated in each sub-group. Each concatenated sub-group is reduced to a second data-reduction number (PC2, See Setup ICA). |
3 | First step involves all the steps for two data-reduction steps. Third group contains only one sub-group. In this sub-group, the reduced sub-groups from data-reduction step 2 are concatenated. The concatenated data-set is reduced to a third data-reduction number (PC3, See Setup ICA). |
Figure 1: Three data-sets reduced in time dimension using two data-reduction steps
Calculation involved in data-reduction step:
The example here discussed will be for one data-set. Let us say if the data is a matrix of dimensions (V by I) where V is volume and I is images or time points. Covariance matrix in time dimension is calculated from the observed data and eigen values are computed from the covariance matrix. The eigen values are arranged in decreasing order. Only the eigen vectors that correspond to non-zero and first m (Reduction parameter PC1) components are taken into account. These components are orthogonal to each other and then passed to the whitening step. Whitening matrix is computed by solving the square root of diagonal matrix of eigen values and transpose of the eigen vectors. This matrix is then multiplied to the data to make the variances equal for all the components. Thus the components are orthogonal and the variances are equal.
Note: The whitening matrix, de-whitening matrix (pseudo inverse of whitening matrix) are stored for future use during the back reconstruction as they contain information regarding the reduced set of the concatenated groups. These are stored as .mat files with suffixes as '_pca_r*' where * refers to the reduction step count.