• Posted by Konstantin 16.01.2012 No Comments

    This post presumes you are familiar with PCA.

    Consider the following experiment. First we generate a random vector (signal) as a sequence of random 5-element repeats. That is, something like

    (0.5, 0.5, 0.5, 0.5, 0.5,   0.9, 0.9, 0,9, 0.9, 0,9,   0.2, 0.2, 0.2, 0.2, 0.2,   ... etc ... )

    In R we could generate it like that:

    num_steps = 50
    step_length = 5;
    initial_vector = c();
    for (i in 1:num_steps) {
      initial_vector = c(initial_vector, rep(runif(1), step_length));
    }

    Here's a visual depiction of a possible resulting vector:

    Initial random vector

    plot(initial_vector), zoomed in

    Next, we shall create a dataset, where each element will be a randomly shifted copy of this vector:

    library(magic) # Necessary for the shift() function
    dataset = c()
    for (i in 1:1000) {
      shift_by = floor(runif(1)*num_steps*step_length) # Pick a random shift
      new_instance = shift(initial_vector, shift_by)   # Generate a shifted instance
      dataset = rbind(dataset, new_instance);          # Append to data
    }

    Finally, let's apply Principal Component Analysis to this dataset:

    pca = prcomp(dataset)

    Question - how do the top principal components look like? Guess first, then read below for the correct answer.

    Read more...

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