A Path gives points on a sphere

Suppose we have a matrix $latex X$ with $latex N$ rows and $latex D$ columns. Here we consider each row as a time point and each each column as a feature which could be expression or normalized expression of a given gene or a principal component. We therefore have a path consisting of $latex N$ points in $latex D$ dimensional space. In our experience, biological trajectories are capable of changing direction, so we may wish to query the data for directionality starting at the $latex k$th time point. Let $latex x_i$ be the $latex i$th row of $latex X$. We will consider the matrix $latex X'$ starting with the $latex k$th row of $latex X$. We view the successive points from $latex x_k$ and consider the successive vectors $latex y_1=x_{k+1} - x_k$, $latex y_2=x_{k+2} - x_k$, …, $latex y_m=x_{N} - x_k$. (Thus $latex m = N - k - 1$.) Taking $$latex p_i = { y_i \over || y_i ||}$$ gives $latex m$ points on the unit sphere which represent the directionality of the path $latex x_k, \dots, x_N$.

How close are these points on the sphere?

We would like to measure how compactly a set of points $latex p_1,\dots,p_m$ sit on the sphere. To to this we would like to find a center for these points and a circle (more generally as $latex N-2$ dimensional sphere) which measures their collective separation from that center. The measure of their collective distance from their center could be their median, their mean or their maximum distance from this center. The center is chosen to minimize whichever summary statistic we’re using. Distance in this context is spherical distance, i.e., the angular distance between points as viewed from the center of the sphere. That is, we take $latex \mu$ to be one of the functions median, mean or max. The center is then $$latex C = {\mathrm{argmin}}_x \mu(d(x,p_1),d(x,p_2),\dots,d(x,p_m))$$ and $$latex R_\mu = \mu(d(x,p_1),d(x,p_2),\dots,d(x,p_m))$$ In this way, we get a map $$latex X' \mapsto R_\mu$$ and we take $latex R_\mu$ as our measure of the clustering of the points $latex p_i$ and therefore of the directionality of $latex X'$.

Testing for statistical significance

We would like to test whether this clustering is statistically significant. To ask whether the points $latex p_i$ are close is to ask, close compared to what? We do this by producing $K$ randomized paths $latex Y_1, \dots, Y_K$ modeled on $latex X'$. For each of these we compute $latex R_{\mu,i}$. A p-value for the directionality of $latex X'$ comes from the ranking of $latex R_\mu$ among $latex R_{\mu,1}, \dots, R_{\mu,K}$. If it is less than all but 5% of these, you have detected a p-value of p<0.05.

Randomization

This package offers several methods for producing a set of randomized paths based on a path $latex X$. They are roughly divided into two methods. One produces a random path by taking random steps in the ambient space of $latex X$ and one produces random paths by permuting the entries in $latex X$.

• Random steps. Here we produce a path starting at the same point as $latex X$ and having the same number steps.
  
• There is an option to require the steps to have the same length as those of $latex X$. This is done by including ‘preserveLengths’ in the randomization parameter. If this is omitted all steps will have length 1.

library(TrajectoryGeometry)
randomizationParams = c('bySteps','preserveLengths')
Y = generateRandomPaths(path=straight_path,
                        randomizationParams=randomizationParams,
                        N=10)

• Negative values in the randomized paths may be reasonable in the case where one is working with PCA data and can expect negative values. However, in the case of non-negative gene expression data, the randomized paths should also be non-negative and this can be accomplished by including the value ‘nonNegative’

randomizationParams = c('bySteps','preserveLengths','nonNegative')
Y = generateRandomPaths(path=straight_path,
                        randomizationParams=randomizationParams,
                        N=10)

• Permutation. Here we produce the randomized paths by permuting the entries of $latex X$. This can be done either by permuting the entries of $latex X$ freely or permuting values within each column spearately. The former may be appropriate in the case of PCA data or in the case of normalized gene expression data where the expression levels are fairly uniform between genes. However, in the case where different genes have significantly different expression levels, it seems appropriate to restrict to permutations which preserve columns. These operations are carried out as above using either of the following:

randomizationParams = c('byPermutation','permuteAsMatrix')
randomizationParams = c('byPermutation','permuteWithinColumns')