Curve
fitting in PAST includes a range of linear and non-linear functions.Curve
fitting in PAST includes a range of linear and non-linear functions.
Linear regression can be performed with two different algorithms: standard
(least-squares) regression and the "Reduced Major Axis" method.
Least-squares regression keeps the x values
fixed, and it finds the line that minimizes the squared errors in the y
values. Reduced Major Axis minimizes both the x and the y errors
simultaneously. Both x and y values can also be log-transformed,
in effect fitting the data to the "allometric" function y=10bxa.
An allometric slope value around 1.0 indicates that an "isometric" fit
may be more applicable to the data than an allometric fit. Values for the
regression slope and intercepts, their errors, a 
2
correlation value, Pearson's r coefficient, and the probability that the
columns are not correlated are given.
In addition, the sum of up to six sinusoids (not necessarily harmonically related) with frequencies specified by the user, but with unknown amplitudes and phases, can be fitted to bivariate data. This method can be useful for modeling periodicities in time series, such as annual growth cycles or climatic cycles, usually in combination with spectral analysis (see below). The algorithm is based on a least-squares criterion and singular value decomposition (Press et al. 1992). Frequencies can also be estimated by trial and error, by adjusting the frequency so that amplitude is maximized.
Further, PAST allows fitting of data to the logistic equation y=a/(1+be-cx), using Levenberg-Marquardt nonlinear optimization (Press et al. 1992). The logistic equation can model growth with saturation, and it was used by Sepkoski (1984) to describe the proposed stabilization of marine diversity in the late Palaeozoic. Another option is fitting to the von Bertalanffy growth equation y=a(1-be-cx). This equation is used for modeling growth of multi-celled animals (Brown and Rothery 1993).
Searching for periodicities in time series (data sampled as a function of time) has been an important and controversial subject in paleontology in the last few decades, and we have therefore implemented two methods for such analysis in the program: spectral analysis and autocorrelation. Spectral (harmonic) analysis of time series can be performed using the Lomb periodogram algorithm, which is more appropriate than the standard Fast Fourier Transform for paleontological data (which are often unevenly sampled; Press et al. 1992). Evenly-spaced data are of course also accepted. In addition to the plotting of the periodogram, the highest peak in the spectrum is presented with its frequency and power value, together with a probability that the peak could occur from random data. The data set can be optionally detrended (linear component removed) prior to analysis. Applications include detection of Milankovitch cycles in isotopic data (Muller and MacDonald 2000) and searching for periodicities in diversity curves (Raup and Sepkoski 1984). Autocorrelation (Davis 1986) can be carried out on evenly sampled temporal-stratigraphical data. A predominantly zero autocorrelation signifies random data — periodicities turn up as peaks.
