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Eigendecomposition

Eigendecompositions are used in the following algorithms within TableCurve 2D. They are the principal methodology in the:

Eigendecompositions are also intrinsically a part of all TableCurve 2D procedures that include SVD (singular value decomposition) methods:

Processing the eigenmodes (eigenvectors and eigenvalues/singular values) of a data series is an important tool in signal analysis. Unlike Fourier decompositions, which partition signals based on harmonic frequency using parametric sines and cosines, an eigendecomposition partitions by signal strength using adaptive non-parametric basis functions. Signal components can thus be separated by differences in power.

Nomenclature

The identification, isolation, and reconstruction of signal components via eigendecomposition is known by a variety of names. "Singular Spectrum Analysis", "Principal Component Analysis", and "Eigenfiltering" are common. TableCurve 2D exclusively uses the "Eigendecomposition" designation since it represents a more precise description of the numeric method.

Reference

An excellent reference for eigendecomposition can be found in:

  • J. B. Elsner and A. A. Tsonis, "Singular Spectrum Analysis", Plenum Press, 1996.

Eigenvalues and Singular Values

Although the terms eigenvectors and singular vectors are often used interchangeably, the singular values are the square roots of the eigenvalues. In terms of visually selecting eigenmodes for signal-noise separation, or for component isolation, either the eigenvalues or their square root, the singular values, can be used. For signal-noise thresholding, TableCurve 2D plots the singular values directly. The Eigendecomposition Denoising and Eigendecomposition Filtering procedures plot the eigenvalue (the square of the respective singular value) divided by the order of the decomposition.

Lagged Covariance or Data Matrix

An eigendecomposition can be achieved in any number of ways. The first step is always the creation of a matrix that uses lagged copies of subsets of the data series. This can be a straightforward data or trajectory matrix, such as the forward prediction (Fwd), backward prediction (Bwd), or forward-backward (FB) prediction matrices used in autoregressive modeling. These data matrices are usually rectangular, and SVD is used to extract the eigenvectors and singular values. If the eigendecomposition is used only for reconstruction and does not involve the least-squares computation of parametric model coefficients, there is no difference between using a Fwd or Bwd data matrix. A FB prediction matrix will contain twice the number of rows as a Fwd or Bwd matrix, and will require a greater processing time. This approach is used in the AR Modeling and Prediction procedure.

Another option is to use one of several methods to construct a covariance matrix from lagged copies of the data. This is a square matrix whose eigenvectors and singular values can be computed using SVD. A covariance-matrix procedure that enforces Toeplitz symmetry (all of the elements along each diagonal are equal), is reported to be of value for short data records. This analysis imposes a stationary assumption. Typically, non-Toeplitz matrices will better map the variance in a data series. The covariance matrix approach is used in the Eigendecomposition Denoising and Eigendecomposition Filtering options.

Eigendecomposition Order

The order of the eigendecomposition is the number of data elements from the data set that is extracted for each segment or subset of the overall series, not the number of segments. For rectangular data matrices, the order or "embedding dimension" specifies the number of columns in the matrix and the number of segments specifies the rows. For a covariance matrix, both the column and row count will be equal to the order.

It is essential that the order of the eigendecomposition be high enough to offer good signal-noise separation. The order must be sufficient to fully partition the noise components and prevent them from corrupting the signal-being eigenmodes. In general, the higher the order (data length permitting), the more complete the partitioning as more eigenmodes are available to capture the random noise.

Denoising

An inverse FFT can reconstruct a filtered version of the original data by zeroing a subset of the frequency channels. Similarly an eigendecomposition can reconstruct a filtered data set by discarding one or more eigenmodes. The most obvious eigendecomposition-based filtering is for the removal of random noise from a deterministic signal. Noise removal via eigendecomposition is the sole purpose of the Eigendecomposition Denoising procedure. Unlike Fourier low-pass filtering, where some noise passes in the low frequency channels, and time-domain smoothing, which can introduce spurious spectral features, eigendecomposition procedures can theoretically isolate and discard virtually all of the noise corrupting a given signal, across all frequencies.

Isolating Signal or Noise Components

The Eigendecomposition Filtering and Reconstruction option can be used to isolate individual signal components based on signal strength. The first eigenmode captures the greatest measure of the variance in the data; the least eigenmode usually captures only the least significant noise oscillations. In general, eigendecompositions cannot isolate a single sinusoid amongst a number of sinusoids when signal strengths are approximately comparable.

The first eigenmode will capture the predominant data trend in the signal, the second the next predominant, and so forth. It does not matter if the component captured is sinusoidal, a square wave, a sawtooth, or an anharmonic pattern. Further, the signal may be a slowly varying low frequency anharmonic oscillation, or a high frequency sinusoid. The eigenmodes are said to be adaptive, because they capture, in an eigenvalue-ordered sequence, the variance of the data in a non-parametric manner.

Two eigenmodes are need to capture an oscillatory trend. A pair of eigenmodes with nearly identical eigenvalues usually signifies a specific harmonic or anharmonic oscillation in the signal. One strength of eigendecomposition is the isolation of secondary signal components containing very little power. Such secondary components may not be visible in Fourier spectra because of spectral leakage and resolution considerations. Similarly these components may be invisible in AR and ARMA models because an autoregressive model will emphasize the primary components. The same is true for parametric sinusoid models because the total variance associated with the components of greater power can often overwhelm a least-squares fit (and even a robust fit) that seeks to also include secondary components of far lesser power. Eigendecomposition offers the means to isolate the high and low power components for separate spectral analysis and fitting.

With eigendecomposition, it is possible to isolate backgrounds by reconstructing only the noise eigenmodes. This can be done to confirm the presence or absence of white (normally distributed) noise, or to isolate a red noise background for subsequent modeling by one of TableCurve 2D’s spectral procedures. It is also possible to reconstruct only the initial noise components, those judged to contain no meaningful signal, and to check for oscillatory trends in the time domain, or for significant spectral content in the frequency domain.