CHAPTER 4 : Unsupervised machine learning for the analysis of heterogeneity in
4.4 Addressing the challenge: The approach
4.4.0.1. Overview and notations
For the remainder of this manuscript we use the following standard set of notations. The dataset containsm1appropriately processed brain images from group 1, andm2such images from group 2. We wish to analyze heterogeneity in group 2 with respect to group 1. Without loss of generality, we assume a vector representation of three dimensional images. Such a representation may be generated by concatenating voxel intensity values into a single long vector of dimensionality equal to the total number of image voxels (d). Keeping with this notation, we index the vectors associated with images of the first group withjasx1j ∈R1×d wherej∈ {1,· · ·m1}. Vectors associated with group 2 images are denoted asx2i ∈R1×dand
indexed by i ∈ {1,· · ·m2}. One may stack all the vectors associated with the first group into a matrix denoted by X1 ∈ Rm1×d and vectors from group 2 into a matrix denoted by X2 ∈ Rm2×d. In addition to imaging data, most clinical studies collect a large set of ancillary clinical variables as well. Examples of such variables may be age, sex, ethnicity, scanner types, and even the concomitant presence or absence of another disease. CHAMP uses a sub set of these ancillary variables that are a) distributed independently of the group labeling under study and b)have a substantial influence on brain anatomy and function. A prime example of such a variable might be ‘age’ in an ‘age-matched’ imaging study. Knowledge of a subject’s age gives us no additional information about it’s group label in such a study. Yet, it’s influence on brain anatomy is undeniable. We call such a variable a ‘meta-variable’ and matching subjects between groups using these ‘meta-variables’ is the basis of the CHAMP approach. We denote the collection meta-variables associated with x1j in the vector vj1 and meta-variables associated with x2i with v2i. These meta-variable vectors may be assumed to be multi-dimensional, but their dimensionality is assumed to be much lower than the sample size of the study.
4.4.1. The difference representation matrix
As stated earlier, we aim to identify heterogeneous patterns inX2that are not concomitantly present inX1, and are associated with the differences between groups 1 and 2. We operate under the assumption that the majority of such variation is driven by the group difference itself. Here, we propose to encapsulate this variation into a difference representation matrix, denoted by D ∈ Rm2×d. The subsequent text presents a procedure to construct such a matrix alongside the intuition behind this procedure.
Assume p(x1) and p(x2) to be probability distributions associated with groups 1 and 2. Recall that we chose meta-variable distribution to be independent of group labeling. Thus, if we assume p(v) to denote the distribution from which meta-variables are sampled, then we may write: p(x1) = Z v∈V p(x1|v)p(v)dv (4.1) and p(x2) = Z v∈V p(x2|v)p(v)dv (4.2)
where V indicates the domain of v. In the expressions above we may be able to model
p(v) using data. However, it is nearly impossible to estimate the distributions p(x1|v) and p(x2|v) from sample sizes typically available in neuroimaging studies. This is because imaging data are high dimensional in nature. Yet, we use these conditional distributions to elucidate certain concepts that are critical in the design of our method.
Consider, x2i ∼ p(x2) to be a subject that was observed in group 2. Define x1j(i) ∼ p(x1) to be the image that would have been observed for subject i if, contrary to fact, subject
i had been observed in group 1. That is, we would have observed x1j(i) had this subject been spared of the pathological process that drove subjects from group 1 to group 2. In the
outcome Rubin (1974) Holland (1986) Rubin (2005).
Under certain assumptions, it is possible to obtain a reasonable estimate ofx1j(i) using the observed data from group 1. First, as previously mentioned, we must assume thatv arises from common distributionp(v) for both groups. Second, we must assume most or all of the variation in the images beyond that caused by the disease itself is captured by v. When these assumptions hold, x1
j(i) can be estimated using data from patient group 1. Next, we outline our proposed approach for estimatingx1j(i) for each patient.
In this work, we propose to modelx1j(i)as the mean of the conditional distributionp(x1|v= v2i). The intuition behind using such an estimation is that meta-variables like age and sex provide a fairly low dimensional measure that captures a large proportion of variation in normal brain anatomy. Like the size of the cartoon in figure 1, the variation captured by meta-variables is assumed to be common across the populations being compared. Thus, the mean ofp(x1|v=v2i) provides a reasonable estimate of how the truex1j(i) might look like. Formally, one may write:
x1j(i) =
Z
x1∈X 1
x1p(x1|v=vi2)dx1, (4.3)
where the integration only uses the subset ofx1 that has meta-variable values which match vi2. This definition of x1j(i) is based on a distribution that is conditioned on the meta-data rather than the full distribution of the imaging data themselves.
Now in real data, sample size constraints severely limit the total number of samples with v=vi2. So we estimate x1j(i) using the mean ofr samples chosen from group 1 which have meta-variable values closest tov2i in a Euclidean sense. If we let l ∈Si index this set ofr
samples for a particularx2i, then we may approximate x1j(i) as:
ˆ x1j(i)= 1 r X l∈Si x1l. (4.4)
The expression on the right of the above expression represents our best estimate of x1j(i). Using our estimate ofx1j(i), we may finally compute:
di=x2i −ˆx1j(i). (4.5)
This vector represents our best guess of the component of x2i that arises from phenom- ena driving the group difference. Thus, the distributional assumptions made above ul- timately lead to a construction of di that is based on nearest ‘group 1’ neighbors of x2i
where,‘nearness’ is defined using meta-data rather than the imaging itself. This contrasts the traditional machine learning perspective, where feature vectors themselves are used to define ‘nearness’. The intuition behind this meta-variable centric approach may be exem- plified by considering the following scenario. Suppose, we were analyzing imaging data from a study comparing autistics to normal controls. If a six year old male autistic child did not have autism, his brain should look like a six year old normal male child. It would be incorrect to assume that a 6 year old autistic male brain would look like to a 4 year old normal female brain if autism never manifested. The invalidity of such an assumption would hold in spite of any observed similarity in terms of brain structure/imaging.
This idea is presented visually in figures 41 and 46 and captured in the theory presented here. Thus, we use meta-variable based neighborhoods to compute the vectors di for all
i∈ {1,· · · , m2}and stack them row-wise to construct the matrixD. We theorize that this matrix exposes variation associated with group differences.
Heterogeneity visualization and mapping
The study of heterogeneity using the difference vectors di is the search for clusters of self
similar difference vectors. A re-ordered cross-correlations matrix-based approach may be used to visualize similarity between the rows of D. While it is the similarity between the
Figure 41: Group differences are most likely expressed as changes in voxel intensity in case of imaging data. Thus, if group 2 was essentially generated by subtracting or adding a fixed number to the intensities in group 1, neighborhoods based on Euclidean distances would not generate appropriate difference maps. In the illustrations above the orange ellipses indicate Euclidean distance based neighbors, whereas the black arrows indicate the neighbors we want to find.
the extremely high similarity of every row with itself. With this in mind, we compute the hollow matrix F∈Rm2×m2 with elements given by:
Fpq = (1−δpq)Kpq, (4.6)
where the matrixKis defined as: K=DDT andδ
pq is the Kronecker delta symbol. Using
the rows ofF as features in a k-means algorithm we define cluster memberships that define sub groups of group 2. In order to understand how each of these subgroups differs from group 1, we run voxelwise univariate analysis between each subgroup and group 1. We visualize the resulting p-value maps by overlaying them on a template brain.