Least-squares spectral analysis

Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the most used spectral method in science, generally boosts long-periodic noise in long gapped records; LSSA mitigates such problems.

LSSA is also known as the Vaníček method after Petr Vaníček, and as the Lomb method (or the Lomb periodogram) and the Lomb–Scargle method (or Lomb–Scargle periodogram), based on the contributions of Nicholas R. Lomb and, independently, Jeffrey D. Scargle. Closely related methods have been developed by Michael Korenberg and by Scott Chen and David Donoho.

Historical background

The close connections between Fourier analysis, the periodogram, and least-squares fitting of sinusoids have long been known. Most developments, however, are restricted to complete data sets of equally spaced samples. In 1963, J. F. M. Barning of Mathematisch Centrum, Amsterdam, handled unequally spaced data by similar techniques, including both a periodogram analysis equivalent to what is now referred to the Lomb method, and least-squares fitting of selected frequencies of sinusoids determined from such periodograms, connected by a procedure that is now known as matching pursuit with post-backfitting or orthogonal matching pursuit.

Petr Vaníček, a Canadian geodesist of the University of New Brunswick, also proposed the matching-pursuit approach, which he called "successive spectral analysis" and the result a "least-squares periodogram", with equally and unequally spaced data, in 1969. He generalized this method to account for systematic components beyond a simple mean, such as a "predicted linear (quadratic, exponential, ...) secular trend of unknown magnitude", and applied it to a variety of samples, in 1971.

The Vaníček method was then simplified in 1976 by Nicholas R. Lomb of the University of Sydney, who pointed out its close connection to periodogram analysis. The definition of a periodogram of unequally spaced data was subsequently further modified and analyzed by Jeffrey D. Scargle of NASA Ames Research Center, who showed that with minor changes it could be made identical to Lomb's least-squares formula for fitting individual sinusoid frequencies.

Scargle states that his paper "does not introduce a new detection technique, but instead studies the reliability and efficiency of detection with the most commonly used technique, the periodogram, in the case where the observation times are unevenly spaced," and further points out in reference to least-squares fitting of sinusoids compared to periodogram analysis, that his paper "establishes, apparently for the first time, that (with the proposed modifications) these two methods are exactly equivalent."

Press summarizes the development this way:

A completely different method of spectral analysis for unevenly sampled data, one that mitigates these difficulties and has some other very desirable properties, was developed by Lomb, based in part on earlier work by Barning and Vanicek, and additionally elaborated by Scargle.

Michael Korenberg of Queen's University in 1989 developed the "fast orthogonal search" method of more quickly finding a near-optimal decomposition of spectra or other problems, similar to the technique that later became known as orthogonal matching pursuit. In 1994, Scott Chen and David Donoho of Stanford University have developed the "basis pursuit" method using minimization of the L1 norm of coefficients to cast the problem as a linear programming problem, for which efficient solutions are available.

The Vaníček method

In the Vaníček method, a discrete data set is approximated by a weighted sum of sinusoids of progressively determined frequencies, using a standard linear regression, or least-squares fit. The frequencies are chosen using a method similar to Barning's, but going further in optimizing the choice of each successive new frequency by picking the frequency that minimizes the residual after least-squares fitting (equivalent to the fitting technique now known as matching pursuit with pre-backfitting). The number of sinusoids must be less than or equal to the number of data samples (counting sines and cosines of the same frequency as separate sinusoids).

A data vector Φ is represented as a weighted sum of sinusoidal basis functions, tabulated in a matrix A by evaluating each function at the sample times, with weight vector x:

$\phi \approx \textbf{A}x$

where the weight vector x is chosen to minimize the sum of squared errors in approximating Φ. The solution for x is closed-form, using standard linear regression:

$x = (\textbf{A}^{\mathrm{T}}\textbf{A})^{-1}\textbf{A}^{\mathrm{T}}\phi.$

Here the matrix A can be based on any set of functions that are mutually independent (not necessarily orthogonal) when evaluated at the sample times; for spectral analysis, the functions used are typically sines and cosines evenly distributed over the frequency range of interest. If too many frequencies are chosen in a too-narrow frequency range, the functions will not be sufficiently independent, the matrix will be badly conditioned, and the resulting spectrum will not be meaningful.

When the basis functions in A are orthogonal (that is, not correlated, meaning the columns have zero pair-wise dot products), the matrix A^TA is a diagonal matrix; when the columns all have the same power (sum of squares of elements), then that matrix is an identity matrix times a constant, so the inversion is trivial. The latter is the case when the sample times are equally spaced and the sinusoids are chosen to be sines and cosines equally spaced in pairs on the frequency interval 0 to a half cycle per sample (spaced by 1/N cycle per sample, omitting the sine phases at 0 and maximum frequency where they are identically zero). This particular case is known as the discrete Fourier transform, slightly rewritten in terms of real data and coefficients.

$x = \textbf{A}^{\mathrm{T}}\phi$     (DFT case for N equally spaced samples and frequencies, within a scalar factor)

Lomb proposed using this simplification in general, except for pair-wise correlations between sine and cosine bases of the same frequency, since the correlations between pairs of sinusoids are often small, at least when they are not too closely spaced. This is essentially the traditional periodogram formulation, but now adopted for use with unevenly spaced samples. The vector x is a good estimate of an underlying spectrum, but since correlations are ignored, Ax is no longer a good approximation to the signal, and the method is no longer a least-squares method – yet it has continued to be referred to as such.

The Lomb–Scargle periodogram

Rather than just taking dot products of the data with sine and cosine waveforms directly, Scargle modified the standard periodogram formula to first find a time delay τ such that this pair of sinusoids would be mutually orthogonal at sample times t_j, and also adjusted for the potentially unequal powers of these two basis functions, to obtain a better estimate of the power at a frequency, which made his modified periodogram method exactly equivalent to Lomb's least-squares method. The time delay τ is defined by the formula

$\tan{2 \omega \tau} = \frac{\sum_j \sin 2 \omega t_j}{\sum_j \cos 2 \omega t_j}.$

The periodogram at frequency ω is then estimated as:

$P_x(\omega) = \frac{1}{2} \left( \frac { \left[ \sum_j X_j \cos \omega ( t_j - \tau ) \right] ^ 2} { \sum_j \cos^2 \omega ( t_j - \tau ) } + \frac {\left[ \sum_j X_j \sin \omega ( t_j - \tau ) \right] ^ 2} { \sum_j \sin^2 \omega ( t_j - \tau ) } \right)$

which Scargle reports then has the same statistical distribution as the periodogram in the evenly sampled case.

At any individual frequency ω, this method gives the same power as does a least-squares fit to sinusoids of that frequency, of the form

$\phi(t) \approx A \sin \omega t + B \cos \omega t.$

Korenberg's "fast orthogonal search" method

Michael Korenberg of Queen's University in Kingston, Ontario, developed a method for choosing a sparse set of components from an over-complete set, such as sinusoidal components for spectral analysis, called fast orthogonal search (FOS). Mathematically, FOS uses a slightly modified Cholesky decomposition in a mean-square error reduction (MSER) process, implemented as a sparse matrix inversion. As with the other LSSA methods, FOS avoids the major shortcoming of discrete Fourier analysis, and can achieve highly accurate identifications of embedded periodicities and excels with unequally spaced data; the fast orthogonal search method has also been applied to other problems such as nonlinear system identification.

Chen and Donoho's "basis pursuit" method

Chen and Donoho have developed a procedure called basis pursuit for fitting a sparse set of sinusoids or other functions from an over-complete set. The method defines an optimal solution as the one that minimizes the L1 norm of the coefficients, so that the problem can be cast as a linear programming problem, for which efficient solution methods are available.

Palmer's Chi-squared method

Palmer has developed a method for finding the best-fit function to any chosen number of harmonics, allowing more freedom to find non-sinusoidal harmonic functions. This method is a fast technique (FFT-based) for doing weighted least-squares analysis on arbitrarily spaced data with non-uniform standard errors. Source code that implements this technique is available. Because data are often not sampled at uniformly spaced discrete times, this method "grids" the data by sparsely filling a time series array at the sample times. All intervening grid points receive zero statistical weight, equivalent to having infinite error bars at times between samples.

Applications

The most useful feature of the LSSA method is enabling incomplete records to be spectrally analyzed, without the need to manipulate the record or to invent otherwise non-existent data.

Magnitudes in the LSSA spectrum depict the contribution of a frequency or period to the variance of the time series. Generally, spectral magnitudes defined in the above manner enable the output's straightforward significance level regime. Alternatively, magnitudes in the Vaníček spectrum can also be expressed in dB. Note that magnitudes in the Vaníček spectrum follow β-distribution.

Inverse transformation of Vaníček's LSSA is possible, as is most easily seen by writing the forward transform as a matrix; the matrix inverse (when the matrix is not singular) or pseudo-inverse will then be an inverse transformation; the inverse will exactly match the original data if the chosen sinusoids are mutually independent at the sample points and their number is equal to the number of data points. No such inverse procedure is known for the periodogram method.

Implementation

The LSSA can be implemented in less than a page of MATLAB code. In essence:

"to compute the least-squares spectrum we must compute m spectral values ... which involves performing the least-squares approximation m times, each time to get [the spectral power] for a different frequency"

I.e., for each frequency in a desired set of frequencies, sine and cosine functions are evaluated at the times corresponding to the data samples, and dot products of the data vector with the sinusoid vectors are taken and appropriately normalized; following the method known as Lomb/Scargle periodogram, a time shift is calculated for each frequency to orthogonalize the sine and cosine components before the dot product, as described by Craymer; finally, a power is computed from those two amplitude components. This same process implements a discrete Fourier transform when the data are uniformly spaced in time and the frequencies chosen correspond to integer numbers of cycles over the finite data record.

This method treats each sinusoidal component independently, or out of context, even though they may not be orthogonal on the data points; it is Vaníček's original method. In contrast, as Craymer explains, it is also possible to perform a full simultaneous or in-context least-squares fit by solving a matrix equation, partitioning the total data variance between the specified sinusoid frequencies. Such a matrix least-squares solution is natively available in MATLAB as the backslash operator.

Craymer explains that the simultaneous or in-context method, as opposed to the independent or out-of-context version (as well as the periodogram version due to Lomb), cannot fit more components (sines and cosines) than there are data samples, and further that:

"...serious repercussions can also arise if the selected frequencies result in some of the Fourier components (trig functions) becoming nearly linearly dependent with each other, thereby producing an ill-conditioned or near singular N. To avoid such ill conditioning it becomes necessary to either select a different set of frequencies to be estimated (e.g., equally spaced frequencies) or simply neglect the correlations in N (i.e., the off-diagonal blocks) and estimate the inverse least squares transform separately for the individual frequencies..."

Lomb's periodogram method, on the other hand, can use an arbitrarily high number of, or density of, frequency components, as in a standard periodogram; that is, the frequency domain can be over-sampled by an arbitrary factor.

In Fourier analysis, such as the Fourier transform or the discrete Fourier transform, the sinusoids being fitted to the data are all mutually orthogonal, so there is no distinction between the simple out-of-context dot-product-based projection onto basis functions versus an in-context simultaneous least-squares fit; that is, no matrix inversion is required to least-squares partition the variance between orthogonal sinusoids of different frequencies. This method is usually preferred for its efficient fast Fourier transform implementation, when complete data records with equally spaced samples are available.

See also

• Orthogonal functions
• Sinusoidal model
• Spectral density
• Spectral density estimation, for competing alternatives

External links

• LSSA software freeware download (via ftp), FORTRAN, Vaníček's method, from the Natural Resources Canada.