NAME
trend1d - Fit a [weighted] [robust] polynomial [or Fourier] model for y
= f(x) to xy[w] data.
SYNOPSIS
trend1d -F<xymrw> -N[f]n_model[r] [ xy[w]file ] [ -Ccondition_# ] [
-H[nrec] ] [ -I[confidence_level] ] [ -V ] [ -W ] [ -: ] [ -bi[s][n] ]
[ -bo[s][n] ] [ -f[i|o]colinfo ]
DESCRIPTION
trend1d reads 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 variance in y.
REQUIRED ARGUMENTS
-F 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.
-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 [or binary, see -b] file containing x,y [w] values in the
first 2 [3] columns. If no file is specified, trend1d will read
from standard input.
-C Set the maximum allowed condition number for the matrix solu-
tion. trend1d 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 condition_#.
[Default: condition_# = 1.0e06. ].
-H Input file(s) has Header record(s). Number of header records can
be changed by editing your .gmtdefaults4 file. If used, GMT
default is 1 header record. Use -Hi if only input data should
have header records [Default will write out header records if
the input data have them].
-I Iteratively increase the number of model parameters, starting at
one, until n_model is reached or the reduction 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 itera-
tive robust fit]. [Default reads only the first 2 columns.]
-: Toggles between (longitude,latitude) and (latitude,longitude)
input and/or output. [Default is (longitude,latitude)]. Append
i to select input only or o to select output only. [Default
affects both].
-bi Selects binary input. Append s for single precision [Default is
double]. Append n for the number of columns in the binary
file(s).
[Default is 2 (or 3 if -W is set) columns].
-bo Selects binary output. Append s for single precision [Default is
double]. Append n for the number of columns in the binary
file(s).
-f Special formatting of input and output columns (time or geo-
graphical data) Specify i(nput) or o(utput) [Default is both
input and output]. Give one or more columns (or column ranges)
separated by commas. Append T (Absolute calendar time), t (time
relative to chosen TIME_EPOCH), x (longitude), y (latitude), g
(geographic coordinate), or f (floating point) to each column or
column range item.
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 func-
tions will be Chebyshev polynomials. These have a numerical advantage
in the form of the matrix which must be inverted and allow more accu-
rate 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 com-
pared. NOTE: The stable model coefficients are Chebyshev coefficients.
The corresponding polynomial coefficients in a + bx + cxx + ... are
also given in Verbose mode but users must realize that they are NOT
stable beyond degree 7 or 8. See Numerical Recipes for more discussion.
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 reduc-
tion in variance 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% confidence 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 Deviation and a formula from Huber [1964], and is 95% effi-
cient when the model residuals have an outlier-free normal distribu-
tion. 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, use:
trend1d data.xy -Fxr -N2 > detrended_data.xy
To make the above linear trend robust with respect to outliers, use:
trend1d data.xy -Fxr -N2r > detrended_data.xy
To find out how many terms (up to 20, say) in a robust Fourier inter-
polant are significant in fitting data.xy, use:
trend1d data.xy -Nf20r -I -V
SEE ALSO
gmt(l), grdtrend(l), trend2d(l)
REFERENCES
Huber, P. J., 1964, Robust estimation of a location parameter, Ann.
Math. Stat., 35, 73-101.
Menke, W., 1989, Geophysical Data Analysis: Discrete Inverse Theory,
Revised Edition, Academic Press, San Diego.
GMT4.0 1 Oct 2004 TREND1D(l)
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