Self-similarity of high dimensional random walk process

A high dimensional random walk process (RWP) is the trajectory of a vector variable whose components change independently and randomly step by step, in discrete time space, for a small constant percentage. High dimensional RWP can be used as the starting point to investigate changing complex systems. For instance, as discussed in this blog, the stocks price history of 500 stocks can be considered as a 500 dimensional RWP.

Self-similarity means that an object is, from certain point of view, similar to parts of itself. A high dimensional RWP is self-similar in the sense that each sub-section of it follow the same statistical constraints. Thus, the randomness is a property shared by RWP and its sub-sections. We can generate series of 1000 data points of a 500-dimensional RWP with the following VisuMap's JavaScript code:

var n = 1000;
var dim = 500;
var rwp = New.NumberTable(n, dim);
var m = rwp.Matrix;
for( var col=0; col < dim; col++) {
    m[0][col] = 1.0;
    for(var t=1; t < n; t++) {
        m[t][col] = m[t-1][col] * (1 + 0.01*(Math.random() - 0.5) );
    }
}
rwp.ShowValueDiagram();

How can we visualize the self-similar randomness in high dimensional RWP? A simple way is to use a dimensionality reduction method to map the high dimensional trajectory to low dimensional space, then plot it out on paper or screen. The following picture for instance shows two CCA (curvilinear component analysis) maps of the 1000 data points mapped to the 3 dimensional space:

We can see the randomness of the trajectory in above picture, but the self-similarity is hardly apparent. This is because, as intrinsically to the human perception, we normally only good at recognizing similarity between geometric patterns, but not similarity between random patterns.

Fortunately, when we use principal component analysis (PCA) to project the trajectory to low dimensional space, we get much easier recognizable patterns. For instance, the following picture shows the projections of our sample RWP to some major principal components (ie. eigenvectors):

The PCA projection to the major principal components apparently filtered out the randomness of the data, like a low pass filter suppresses high frequency noises. Now, we can select three principal components, say the second, third and forth components as our projection axes; then plot the PCA projection of the data (or sub-sections of it) to these three axes. As be illustrated in the following picture, we can see that they are all geometrically similar to each other albeit with different densities.

Notice that in above picture, the projection axes are the second, third and forth principal components of the selected data, not those of the complete dataset. The following video clip shows what we have discussed above in a more intuitively way:

One practical use of studying RWP is to find non-randomness ( that is information ) in seemly random data. Using the PCA technique we can show random data as easy recognizable patterns which enable us to detect deviations from those patterns, so that we can quickly find potential information in apparently random data.