Sinusoidal Least-Squares Fit

Numeric Summary

The Sine-Component Linear Fit is part of the Numeric Summary in most of the spectral procedures. When this option is selected, a least-squares time-domain sinusoidal fit is made to the incoming data.

This option is selected in the Format menu of the Numeric Summary. The Add Sine Component Linear Fit Summary and Add Sine Component Linear Fit Details menu items control the amount of fit information displayed in the report. The estimated parameters can be reported with either 90% Confidence, 95% Confidence, or 99% Confidence limits.

Suboptimal Least-Squares

A linear sinusoidal fit is a suboptimal regression because the frequencies are locked at the values determined by the spectral procedure. Only the amplitudes and phases are allowed to vary. For this analysis to be meaningful, it must be possible to model the data with a combination of narrowband components. The data must be wide-sense stationary where sinusoids serve as valid narrowband models. For the fit to be useful, the component count must be also correct and the frequencies must be accurately determined. The count of spectral components and the set of spectral frequencies automatically come from the spectral procedure.

Reconstructed Covariance Matrix

After a linear fit is made, a covariance matrix is reconstructed as if a non-linear fit had iterated to this same set of parameters. This makes it possible to derive standard errors and confidence limits for all parameters, including the frequencies. The reasoning is that the frequencies are determined by a different estimation procedure (specific to the spectral algorithm), and that composite estimation, however non-optimal, should be reflected in the statistics.

Fit Summary

The Add Sine Component Linear Fit Summary produces a table of frequencies, amplitudes, phases (sine-based), and powers (TISA). Although the frequencies originate from the spectral procedure, the amplitudes and phases derive from the least-squares fit. Also, an analytic TISA (time-integral squared amplitude) power is computed for each component. This is a time-domain integral based on each component's amplitude, frequency, and phase and the time range represented in the data.

Absolute and relative percents are also given for the component powers. These are often the quantities of interest when comparing strengths of signal components. The summed power reported in the table is merely the sum of the component powers. It is not the power of the composite signal that would result from the addition of the components. In most instances, this sum will be lower than the TISA power of the incoming data.

The fit statistics reported are the rē (r-squared) correlation coefficient, a degree-of-freedom adjusted rē, the standard error, and the F-statistic. As a fit becomes more ideal, the rē values approach 1.0 (0.0 represents a complete lack of fit), the standard error decreases toward zero, and the F-statistic goes toward infinity. Be very suspicious of low rē fits since this may indicate an incorrect component count, inaccurate spectral frequencies, or a spectrum that cannot be modeled with sinusoids. The report also lists the data, model, and error powers from the fit. In a good fit, most of the power in the data is accounted by the model rather than the error. The ratio that is reported is a signal/noise estimate (in power). This should be close to the S/N ratio (in power) of the signal.

Fit Details

The Add Sine Component Linear Fit Details adds a statistical breakdown of each sinusoid in the fit. For each, the estimate, standard error, t-value, confidence limits, and probabilities are given for the respective amplitude, frequency, and phase parameters. Note that these statistics reflect a suboptimum fit. A full non-linear optimization (where the frequencies are allowed to also vary) will almost always result in an improved goodness of fit.

The Non-Linear Optimization option is recommended for refining all sinusoidal fits.