NAME
trend2d - Fit a [weighted] [robust] polynomial model for z = f(x,y) to
xyz[w] data.
SYNOPSIS
trend2d -F<xyzmrw> -Nn_model[r] [ xyz[w]file ] [ -Ccondition_# ] [
-H[nrec] ][ -I[confidence_level] ] [ -V ] [ -W ] [ -: ] [ -bi[s][n] ] [
-bo[s][n] ] [ -f[i|o]colinfo ]
DESCRIPTION
trend2d reads x,y,z [and w] values from the first three [four] columns
on standard input [or xyz[w]file] and fits a regression model z =
f(x,y) + e by [weighted] least squares. 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,y) which significantly reduce the variance in z.
n_model may be in [1,10] to fit a model of the following form (similar
to grdtrend):
m1 + m2*x + m3*y + m4*x*y + m5*x*x + m6*y*y + m7*x*x*x + m8*x*x*y +
m9*x*y*y + m10*y*y*y.
The user must specify -Nn_model, the number of model parameters to use;
thus, -N4 fits a bilinear trend, -N6 a quadratic surface, and so on.
Optionally, append r to perform a robust fit. In this case, the program
will iteratively reweight the data based on a robust scale estimate, in
order to converge to a solution insensitive to outliers. This may be
handy when separating a "regional" field from a "residual" which should
have non-zero mean, such as a local mountain on a regional surface.
-F Specify up to six letters from the set {x y z m r w} in any
order to create columns of ASCII [or binary] output. x = x, y =
y, z = z, m = model f(x,y), r = residual z - m, w = weight used
in fitting.
-N Specify the number of terms in the model, n_model, and append r
to do a robust fit. E.g., a robust bilinear model is -N4r.
OPTIONS
xyz[w]file
ASCII [or binary, see -b] file containing x,y,z [w] values in
the first 3 [4] columns. If no file is specified, trend2d will
read from standard input.
-C Set the maximum allowed condition number for the matrix solu-
tion. trend2d 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 4. Do a weighted least
squares fit [or start with these weights when doing the itera-
tive robust fit]. [Default reads only the first 3 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 3 (or 4 if -W is set) input 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
The domain of x and y will be shifted and scaled to [-1, 1] and the
basis functions are built from Chebyshev polynomials. These have a
numerical advantage in the form of the matrix which must be inverted
and allow more accurate solutions. In many applications of trend2d the
user has data located approximately along a line in the x,y plane which
makes an angle with the x axis (such as data collected along a road or
ship track). In this case the accuracy could be improved by a rotation
of the x,y axes. trend2d does not search for such a rotation; instead,
it may find that the matrix problem has deficient rank. However, the
solution is computed using the generalized inverse and should still
work out OK. The user should check the results graphically if trend2d
shows deficient rank. NOTE: The model parameters listed with -V are
Chebyshev coefficients; they are not numerically equivalent to the m#s
in the equation described above. The description above is to allow the
user to match -N with the order of the polynomial surface.
The -Nn_modelr (robust) and -I (iterative) options evaluate the signif-
icance 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 planar trend from data.xyz by ordinary least squares, use:
trend2d data.xyz -Fxyr -N2 > detrended_data.xyz
To make the above planar trend robust with respect to outliers, use:
trend2d data.xzy -Fxyr -N2r > detrended_data.xyz
To find out how many terms (up to 10) in a robust interpolant are sig-
nificant in fitting data.xyz, use:
trend2d data.xyz -N10r -I -V
SEE ALSO
gmt(l), grdtrend(l), trend1d(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 TREND2D(l)
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