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<HTML>
<HEAD>
<TITLE>trend1d</TITLE>
</HEAD>
<BODY>
<H1>trend1d</H1>
<HR>
<PRE>
<!-- Manpage converted by man2html 3.0.1 -->
trend1d - Fit a [weighted] [robust] polynomial [or
Fourier] model for y = f(x) to xy[w] data.
</PRE>
<H2>SYNOPSIS</H2><PRE>
<B>trend1d</B> <B>-F</B><I><xymrw></I> <B>-N</B>[<B>f</B>]<I>n</I><B>_</B><I>model</I>[<B>r</B>] [ <I>xy[w]file</I> ] [ <B>-C</B><I>condi</I>
<I>tion</I><B>_</B><I>#</I> ] [ <B>-H</B>[<I>nrec</I>] ] [ <B>-I</B>[<I>confidence</I><B>_</B><I>level</I>] ] [ <B>-V</B> ] [ <B>-W</B>
] [ <B>-:</B> ] [ <B>-bi</B>[<B>s</B>][<I>n</I>] ] [ <B>-bo</B>[<B>s</B>] ]
</PRE>
<H2>DESCRIPTION</H2><PRE>
<B>trend1d</B> reads x,y [and w] values from the first two
[three] columns on standard input [or <I>xy[w]file</I>] 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 variance in y.
</PRE>
<H2>REQUIRED ARGUMENTS</H2><PRE>
<B>-F</B> Specify up to five letters from the set {x y m r w}
in any order to create columns of ASCII [or binary]
output. x = x, y = y, m = model f(x), r = residual
y - m, w = weight used in fitting.
<B>-N</B> Specify the number of terms in the model, <I>n</I><B>_</B><I>model</I>,
whether to fit a Fourier (<B>-Nf</B>) or polynomial
[Default] model, and append <B>r</B> to do a robust fit.
E.g., a robust quadratic model is <B>-N</B><I>3</I><B>r</B>.
</PRE>
<H2>OPTIONS</H2><PRE>
<I>xy[w]file</I>
ASCII [or binary, see <B>-b</B>] file containing x,y [w]
values in the first 2 [3] columns. If no file is
specified, <B>trend1d</B> will read from standard input.
<B>-C</B> Set the maximum allowed condition number for the
matrix solution. <B>trend1d</B> fits a damped least
squares model, retaining only that part of the
eigenvalue spectrum such that the ratio of the
largest eigenvalue to the smallest eigenvalue is
<I>condition</I><B>_</B><I>#</I>. [Default: <I>condition</I><B>_</B><I>#</I> = 1.0e06. ].
<B>-H</B> Input file(s) has Header record(s). Number of
header records can be changed by editing your .gmt
defaults file. If used, <B><A HREF="GMT.html">GMT</A></B> default is 1 header
record.
<B>-I</B> Iteratively increase the number of model parame
ters, starting at one, until <I>n</I><B>_</B><I>model</I> is reached or
the reduction in variance of the model is not sig
nificant at the <I>confidence</I><B>_</B><I>level</I> level. You may set
<B>-I</B> only, without an attached number; in this case
and 1. See remarks section.
<B>-V</B> Selects verbose mode, which will send progress
reports to stderr [Default runs "silently"].
<B>-W</B> 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.]
<B>-:</B> Toggles between (longitude,latitude) and (lati
tude,longitude) input/output. [Default is (longi
tude,latitude)]. Applies to geographic coordinates
only.
<B>-bi</B> Selects binary input. Append <B>s</B> for single precision
[Default is double]. Append <I>n</I> for the number of
columns in the binary file(s). [Default is 2 (or 3
if <B>-W</B> is set) columns].
<B>-bo</B> Selects binary output. Append <B>s</B> for single preci
sion [Default is double].
</PRE>
<H2>REMARKS</H2><PRE>
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), <B>cos(2x)</B>, <B>sin(2x)</B>, ... 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 coefficients in a + bx + cxx + ...
The <B>-Nr</B> (robust) and <B>-I</B> (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 <B>-I</B> option. The user
may be surprised to find that in most cases the reduction
in variance achieved by increasing the number of terms in
a model is not significant at a very high degree of confi
dence. For example, with 120 degrees of freedom, Chi-
Squared must decrease by 26% or more to be significant at
the 95% confidence level. If you want to keep iterating as
long as Chi-Squared is decreasing, set <I>confidence</I><B>_</B><I>level</I> to
zero.
A low confidence limit (such as the default value of 0.51)
outliers. The weight is based on the Median Absolute Devi
ation and a formula from Huber [1964], and is 95% effi
cient when the model residuals have an outlier-free normal
distribution. 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.
</PRE>
<H2>EXAMPLES</H2><PRE>
To remove a linear trend from data.xy by ordinary least
squares, try:
<B>trend1d</B> data.xy <B>-F</B>xr <B>-N</B>2 > detrended_data.xy
To make the above linear trend robust with respect to out
liers, try:
<B>trend1d</B> data.xy <B>-F</B>xr <B>-N</B>2<B>r</B> > 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:
<B>trend1d</B> data.xy <B>-Nf</B>20<B>r</B> <B>-I</B> <B>-V</B>
</PRE>
<H2>SEE ALSO</H2><PRE>
<I>gmt</I>(l), <I><A HREF="grdtrend.html">grdtrend</A></I>(l), <I><A HREF="trend2d.html">trend2d</A></I>(l)
</PRE>
<H2>REFERENCES</H2><PRE>
Huber, P. J., 1964, Robust estimation of a location param
eter, <I>Ann.</I> <I>Math.</I> <I>Stat.,</I> <I>35,</I> 73-101.
Menke, W., 1989, Geophysical Data Analysis: Discrete
Inverse Theory, Revised Edition, Academic Press, San
Diego.
</PRE>
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