trend2d
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 ] [ -Ccondi
tion_# ] [ -H[nrec] ][ -I[confidence_level] ] [ -V ] [ -W
] [ -: ] [ -bi[s][n] ] [ -bo[s] ]
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 reweight
ing 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 fol
lowing 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 per
form a robust fit. In this case, the program will itera
tively reweight the data based on a robust scale estimate,
in order to converge to a solution insensitive to out
liers. 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 fit
ting.
-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
header records can be changed by editing your .gmt
defaults file. If used, GMT default is 1 header
record.
-I Iteratively increase the number of model parame
ters, starting at one, until n_model is reached or
the reduction in variance of the model is not sig
nificant 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 (lati
tude,longitude) input/output. [Default is (longi
tude,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 preci
sion [Default is double].
REMARKS
The domain of x and y will be shifted and scaled to [-1,
1] and the basis functions are built from Chebyshev poly
nomials. 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 col
lected along a road or ship track). In this case the accu
racy 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. How
ever, 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 Cheby
shev coefficients; they are not numerically equivalent to
the m#s in the equation described above. The description
The -Nn_modelr (robust) and -I (iterative) options evalu
ate 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 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.
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 out
liers, try:
trend2d data.xzy -Fxyr -N2r > detrended_data.xyz
To find out how many terms (up to 10) in a robust inter
polant are significant in fitting data.xyz, try:
trend2d data.xyz -N10r -I -V
SEE ALSO
gmt(l), grdtrend(l), trend1d(l)
REFERENCES
Huber, P. J., 1964, Robust estimation of a location param
eter, Ann. Math. Stat., 35, 73-101.
Menke, W., 1989, Geophysical Data Analysis: Discrete
Inverse Theory, Revised Edition, Academic Press, San
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