A Collaborative Approach to Time Series Decomposition

I consider the problem of decomposing a time series into its constituent components. In general, this is underdetermined, and assumptions must be made. Previous attempts include applying nonnegative matrix factorization to the spectrograms of isolated sources to learn a basis for each. A new time series composed of a sum of these sources is then projected onto these bases. For this project, I attempt to generalize this approach first by showing that a bag of segments is a better representation of a time series than the spectrogram, and second by introducing a more natural collaborative approach. 1. Background When learning from time series it is common to compute the spectrogram matrix and treat each column as a feature vector. Each feature then approximately represents the energy in the frequency of the time series at that time. The spectrogram is computed by breaking the time series into equal size segments and layering the element-wise magnitudes of the discrete Fourier transforms (DFT) into the columns of a matrix1. An example of the spectrogram of a violin is shown in Figure 1. One problem in machine learning where spectrograms are used is in single source separation. The spectrogram is factored using nonnegative matrix factorization (Virtanen, 2007; Schmidt & Olsson, 2006; Smaragdis, 2004). This factored representation is a weighted average (nonnegative sum) of positive basis vectors, i.e. no complicated positivenegative cancellations. 2. Bag of Segments vs. Spectrogram Consider a time series x ∈ R . Let X be a matrix whose columns are length d segments of x (bag of segments) In the general formulation, the segments are allowed to overlap and a window function (e.g. Gaussian) can be applied to each. CS229T Project Report Figure 1. Spectrogram of a violin; time on horizontal axis, frequency on vertical X def =  x[1] x[d+ 1] x[T − d+ 1] .. .. · · · .. x[d] x[2d] x[T ]  (1) Let F ∈ Cn×n be the unitary DFT matrix (F ∗F = I): F = 1 √ n  1 1 1 · · · 1 1 ω ω · · · ωn−1 1 ω ω · · · ω2(n−1) .. .. .. .. 1 ωn−1 ω2(n−1) · · · ω(n−1)2  (2) where ω = exp ( − 2π √ −1 n ) . The short-time-Fouriertransform (STFT) of x is given by: STFT(x) = FX (3) Note that the STFT is invertible. F ∗STFT(x) = F ∗FX = X and x = vector(X). The spectrogram is the elementwise magnitude of the STFT. Spectrogram(x)t,f = |STFT(x)|t,f (4) Computing the spectrogram destroys information since the phases are lost. Let’s take a step back and consider the A Collaborative Approach to Time Series Decomposition STFT which contains all the information of the original time series. The columns of this matrix (i.e. the features) lie in a Hilbert space H defined by the feature mapping φ : u 7→ Fu and kernel k(x, y) = 〈φ(x), φ(y)〉 = x>F ∗Fy = 〈x, y〉. Since the Fourier transform is unitary, the kernel of this space is equivalent to the kernel of segment space. Therefore, using the columns of the STFT as feature vectors to learn is in some sense no better than learning from the segments. More concretely, consider doing PCA on the STFT columns. This is equivalent to kernel PCA on the segments with feature mapping φ : u 7→ Fu. First define the empirical covariance matrix