NAME
trend1d - Fit a [weighted] [robust] polynomial [or Fourier]
model for y = f(x) to ascii xy[w] data.
SYNOPSIS
trend1d -N[f]n_model[r] [ xy[w]file ] [ -F<xymrw> ] [ -Ccon-
dition_# ] [ -H ] [ -I[confidence_level] ] [ -V ] [ -W ] [ -
: ]
DESCRIPTION
trend1d reads ascii x,y [and w] values from the first two
[three] columns on standard input [or xy[w]file] and fits a
regression model y = f(x) + e by [weighted] least squares.
The functional form of f(x) may be chosen as polynomial or
Fourier, and the fit may be made robust by iterative
reweighting of the data. The user may also search for the
number of terms in f(x) which significantly reduce the vari-
ance in y.
REQUIRED ARGUMENTS
-N Specify the number of terms in the model, n_model,
whether to fit a Fourier (-Nf) or polynomial [Default]
model, and append r to do a robust fit. E.g., a robust
quadratic model is -N3r.
OPTIONS
xy[w]file
ASCII file containing x,y [w] values in the first 2 [3]
columns. If no file is specified, trend1d will read
from standard input.
-F Specify up to five letters from the set {x y m r w} in
any order to create columns of ASCII output. x = x, y
= y, m = model f(x), r = residual y - m, w = weight
used in fitting.
-C Set the maximum allowed condition number for the
matrix solution. trend1d fits a damped least squares
model, retaining only that part of the eigenvalue spec-
trum such that the ratio of the largest eigenvalue to
the smallest eigenvalue is condition_#. [Default: con-
dition_# = 1.0e06. ].
-H Input file(s) has Header record(s). Number of header
records can be changed by editing your .gmtdefaults
file. If used, GMT default is 1 header record.
-I Iteratively increase the number of model parameters,
starting at one, until n_model is reached or the reduc-
tion in variance of the model is not significant at the
confidence_level level. You may set -I only, without
an attached number; in this case the fit will be
iterative with a default confidence level of 0.51. Or
choose your own level between 0 and 1. See remarks
section.
-V Selects verbose mode, which will send progress reports
to stderr [Default runs "silently"].
-W Weights are supplied in input column 3. Do a weighted
least squares fit [or start with these weights when
doing the iterative robust fit]. [Default reads only
the first 2 columns.]
- : Toggles between (longitude,latitude) and
(latitude,longitude) input/output. [Default is
(longitude,latitude)]
REMARKS
If a Fourier model is selected, the domain of x will be
shifted and scaled to [-pi, pi] and the basis functions used
will be 1, cos(x), sin(x), cos(2x), sin(2x), ... If a
polynomial model is selected, the domain of x will be
shifted and scaled to [-1, 1] and the basis functions will
be Chebyshev polynomials. These have a numerical advantage
in the form of the matrix which must be inverted and allow
more accurate solutions. The Chebyshev polynomial of degree
n has n+1 extrema in [-1, 1], at all of which its value is
either -1 or +1. Therefore the magnitude of the polynomial
model coefficients can be directly compared. NOTE: The
model coefficients are Chebeshev coefficients, NOT coeffi-
cients in a + bx + cxx + ...
The -Nr (robust) and -I (iterative) options evaluate the
significance of the improvement in model misfit Chi-Squared
by an F test. The default confidence limit is set at 0.51;
it can be changed with the -I option. The user may be
surprised to find that in most cases the reduction in vari-
ance achieved by increasing the number of terms in a model
is not significant at a very high degree of confidence. For
example, with 120 degrees of freedom, Chi-Squared must
decrease by 26% or more to be significant at the 95% confi-
dence level. If you want to keep iterating as long as Chi-
Squared is decreasing, set confidence_level to zero.
A low confidence limit (such as the default value of 0.51)
is needed to make the robust method work. This method
iteratively reweights the data to reduce the influence of
outliers. The weight is based on the Median Absolute Devia-
tion and a formula from Huber [1964], and is 95% efficient
when the model residuals have an outlier-free normal distri-
bution. This means that the influence of outliers is
reduced only slightly at each iteration; consequently the
reduction in Chi-Squared is not very significant. If the
procedure needs a few iterations to successfully attenuate
their effect, the significance level of the F test must be
kept low.
EXAMPLES
To remove a linear trend from data.xy by ordinary least
squares, try:
trend1d data.xy -Fxr -N2 > detrended_data.xy
To make the above linear trend robust with respect to
outliers, try:
trend1d data.xy -Fxr -N2r > detrended_data.xy
To find out how many terms (up to 20, say) in a robust
Fourier interpolant are significant in fitting data.xy, try:
trend1d data.xy -Nf20r -I -V
SEE ALSO
gmt, grdtrend, trend2d
REFERENCES
Wessel, P., and W. H. F. Smith, 1995, The Generic Mapping
Tools (GMT) version 3.0 Technical Reference & Cookbook,
SOEST/NOAA.
Wessel, P., and W. H. F. Smith, 1995, New Version of the
Generic Mapping Tools Released, EOS Trans. AGU, 76, p. 329.
Wessel, P., and W. H. F. Smith, 1995, New Version of the
Generic Mapping Tools Released,
http://www.agu.org/eos_elec/95154e.html, Copyright 1995 by
the American Geophysical Union.
Wessel, P., and W. H. F. Smith, 1991, Free Software Helps
Map and Display Data, EOS Trans. AGU, 72, p. 441.
Huber, P. J., 1964, Robust estimation of a location parame-
ter, Ann. Math. Stat., 35, 73-101.
Menke, W., 1989, Geophysical Data Analysis: Discrete
Inverse Theory, Revised Edition, Academic Press, San Diego.