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] ]
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
solution. 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 .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 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 iterative robust
fit]. [Default reads only the first 3 columns.]
- -:
- Toggles between (longitude,latitude) and
(latitude,longitude) input/output. [Default is (longitude,latitude)].
Applies to geographic coordinates only.
- -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].
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 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 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% efficient 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.
EXAMPLES¶
To remove a planar trend from data.xyz by ordinary least squares, try:
trend2d data.xyz
-Fxyr
-N2 > detrended_data.xyz
To make the above planar trend robust with respect to outliers, try:
trend2d data.xzy
-Fxyr
-N2
r > detrended_data.xyz
To find out how many terms (up to 10) in a robust interpolant are significant in
fitting data.xyz, try:
trend2d data.xyz
-N10
r -I -V
SEE ALSO¶
gmt(1gmt),
grdtrend(1gmt),
trend1d(1gmt)
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.