The most outlying direction

Denote the PC score, i.e. the projection of the jth point onto theith sample eigenvector, by ˆ

y_{i j}. Suppose a hypothetical scenario that an outlier signalU_i0 is consistently estimated by one of the sample eigenvectors, say ˆU_i0. Then, the outliers that involveU_i0 are expected to stick out in the direction ˆU_i0 in that the ˆy_i0_jcorresponding to those outliers are much larger than the others. That is, the normalized PC scores, defined as

ˆ

v_{i j} = _qyˆi j

ˆ λi

can be used to detect outliers by choosing cases that have substantially large ˆv_i0_j. As we discussed in Section 2.5, however, the PCA subspace consistency does not guarantee that outlier signals are estimated individually. Therefore, the PC scores corresponding to the singleKsample eigenvectors might be unsuitable for quantifying the outlyingness and less powerful for detecting outliers.

Then, what is a more informative direction for outliers than the individual PC directions? The type of structure of interest here isoutlyingand there is no guarantee that the large-variance PC directions from PCA will find such features. To find such outlying structure, we employ the projection depth function in (3.2.1) over the low-dimensional space, with the goal of detecting outliers based on the outlying structure. Using the same notation as in (3.2.1), denote a normalized projection score of a pointX with respect toXon a directionhbys_h(X|X), i.e.

s_h(X|X) =h

T_X−µ(F hT_X)

σ(F_hT_X)

such thatkhk=1.

Thes_h(X|X)reflects the outlyingness ofX over the directionhwith respect to the datasetX. Scaling a projection scorehTX by the spread of{hT_X

j}1≤j≤nallows us to compare the outlyingness ofX

over the directionhwith the other direction vectors,h0∈span(X1,· · ·,Xn). Therefore, we define the

most outlying directionh∗of a data pointX as a direction that yields the largests_h(X|X)over all directionshas follows:

Definition 3.5.1. The MOD (Most Outlying Direction) of a pointX∈_Rd_{, denoted byh}∗_{(X|X), is}

defined as a unit vectorh∈span(X₁,· · ·,X_n)that maximizess_h(X|X), i.e.

h∗(X|X) =argsup_h|sh(X|X)| subject toh∈span(X1,· · ·,Xn)andkhk=1.

From the definition of a projection depth function, an MOD is the direction that achieves the supremum in a projection depth function. That is, the directionh∗(X|X)provides the most projection outlyingness of the pointX over all possible directions. In contrast to PC directions that describe overall structures relevant to the majority of data points, the most outlying direction describes individual structure of each data point that makes the point the most distinguished. Accordingly, the

−2 −1 0 1 2 3 4 −4 −2 0 2 4 ● −4 −2 0 2 4 −4 −2 0 2 4 X0[2, ] ●

Figure 3.11: In the left (right) figure, the outlier is indicated by a red (blue) circle and the other normal points are indicated by black. The red (blue) line illustrates the MOD of the red (blue) outlier.

MODs of outliers may be used to recover hidden outlier directions, and further the visualization of the MOD often provides a useful interpretation as to why the RNA-seq outliers are abnormal as will be seen in Section 3.6. Such interpretation is informative in that users can understand the biological meaning of the abnormalities.

Figure 3.11 illustrates the MODs of two outliers with(µ,σ)=(Median, MAD). In the left figure, an outlier is indicated by a red circle with normal observations indicated by black. The MOD of the red circle is indicated by a red line. It is obviously the direction that describes the unique structure of the red outlier in the sense that the other points have little variation in the red direction. Here, the MOD is very close to the red point since the red point only goes along the vertical axis and the others rarely do. In the right panel, an outlier is indicated by a blue circle and its MOD is illustrated by a blue line. This MOD also provides good insight about the abnormality of the blue outlier compared to the other points. In this case, the MOD is not close to the blue outlier because the MOD here is balancing the projected value of the blue point with the spread of projected values of the other points. Thus, the MOD of an outlier point can be thought of as a compromise between the outlier and a direction of minimal projected variation.

The projection depth function can be rewritten based on a MOD as follows: o(X|X) = sup khk=1 hTX−µ(F_hT_X) σ(F_hT_X) (3.5.2) = sup khk=1 |s_h(X|X)|=|s_h∗(X|X)| (3.5.3)

where h∗ =h∗(X|X). Then, the o(X|X) is understood as the maximum value over all possible one-dimensional outlyingness and we call this value theprojection outlyingness(PO) orprojection outlyingness score. As a special case, ifh∗(X_j|X) =Uˆ_i0 for a data pointX_j with(µ,σ)=(Mean, SD), theno(X_j|X)becomes the absolute value of the normalized PC score|vˆ_i0_j|in (3.5.1). Hence, a PO score can be seen as a generalized version of the normalized PC score, allowing projection onto other directions (MODs) and other location and scale measures rather than the PC directions and (Mean, SD). In contrast to PCA that obtains min(d,n)orthogonal PC directions by sequentially maximizing the sample variances, Scissor obtainsnMODs each of which maximizes the projection outlyingness of each data point. Note that thenMODs do not need to be orthogonal to each other and besides, if several outliers are involved in similar outlying structure, then the corresponding MODs tend to be close to each other, constructing fairly small angles.

ThenMODs yield thenPO scores, allowing appropriate comparison of the depth of outlying- ness across data points. For example, if a point has a higher PO score, then it is more likely to be an outlier. Based on a cutoff value, Scissor declares a data point as an outlier when its PO score is so large that it cannot be justified by randomness as will be seen later.

Although any univariate location and scale measures can be used forµ(·)andσ(·), the Med and MAD have been widely employed due to their robustness and simplicity. Scissor also uses

(µ,σ)=(Med, MAD) by default. However, for the purpose of exploring theoretical properties of the MODs and PO scores, the projection depth function involved with (Med, MAD) is very tricky to deal with because it not differentiable with respect toh. On the other hand, the projection depth function based on (Mean, SD) may not be a reliable measure of outlyingness because the mean and SD easily break down at extreme points. Nevertheless, the function with (Mean, SD) is much easier

to deal with, making the theoretical investigation much simpler. Therefore, we first investigate some interesting properties of the MODs and PO scores based on (Mean, SD) and then propose the main procedure based on (Med, MAD).