trend2d(1) GMT trend2d(1)
NAME
trend2d - Fit a [weighted] [robust] polynomial model for z = f(x,y) to
xyz[w] data
SYNOPSIS
trend2d [ table ] -Fxyzmrw -Nn_model[+r] [ xyz[w]file ] [ -Ccondi-
tion_number ] [ -I[confidence_level] ] [ -V[level] ] [ -W ] [ [
-bbinary ] [ -dnodata ] [ -eregexp ] [ -fflags ] [ -hheaders ] [
-iflags ] [ -:[i|o] ]
Note: No space is allowed between the option flag and the associated
arguments.
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 pro-
gram will iteratively reweight the data based on a robust scale esti-
mate, in order to converge to a solution insensitive to outliers. This
may be handy when separating a aregionala field from a aresiduala which
should have non-zero mean, such as a local mountain on a regional sur-
face.
REQUIRED ARGUMENTS
-Fxyzmrw
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.
-Nn_model[+r]
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 -N4+r.
OPTIONAL ARGUMENTS
table One or more ASCII [or binary, see -bi] files containing x,y,z
[w] values in the first 3 [4] columns. If no files are speci-
fied, trend2d will read from standard input.
-Ccondition_number
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. ].
-I[confidence_level]
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[level] (more a|)
Select verbosity level [c].
-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.]
-bi[ncols][t] (more a|)
Select native binary input. [Default is 3 (or 4 if -W is set)
input columns].
-bo[ncols][type] (more a|)
Select native binary output. [Default is 1-6 columns as set by
-F].
-d[i|o]nodata (more a|)
Replace input columns that equal nodata with NaN and do the
reverse on output.
-e[~]^<i>apattern^<i>a | -e[~]/regexp/[i] (more a|)
Only accept data records that match the given pattern.
-f[i|o]colinfo (more a|)
Specify data types of input and/or output columns.
-h[i|o][n][+c][+d][+rremark][+rtitle] (more a|)
Skip or produce header record(s).
-icols[+l][+sscale][+ooffset][,^<i>a|] (more a|)
Select input columns and transformations (0 is first column).
-:[i|o] (more a|)
Swap 1st and 2nd column on input and/or output.
-^ or just -
Print a short message about the syntax of the command, then
exits (NOTE: on Windows just use -).
-+ or just +
Print an extensive usage (help) message, including the explana-
tion of any module-specific option (but not the GMT common
options), then exits.
-? or no arguments
Print a complete usage (help) message, including the explanation
of all options, then exits.
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. For evaluat-
ing Chebyshev polynomials, see grdmath.
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.
ASCII FORMAT PRECISION
The ASCII output formats of numerical data are controlled by parameters
in your gmt.conf file. Longitude and latitude are formatted according
to FORMAT_GEO_OUT, absolute time is under the control of FOR-
MAT_DATE_OUT and FORMAT_CLOCK_OUT, whereas general floating point val-
ues are formatted according to FORMAT_FLOAT_OUT. Be aware that the for-
mat in effect can lead to loss of precision in ASCII output, which can
lead to various problems downstream. If you find the output is not
written with enough precision, consider switching to binary output (-bo
if available) or specify more decimals using the FORMAT_FLOAT_OUT set-
ting.
EXAMPLES
To remove a planar trend from data.xyz by ordinary least squares, use:
gmt trend2d data.xyz -Fxyr -N2 > detrended_data.xyz
To make the above planar trend robust with respect to outliers, use:
gmt trend2d data.xzy -Fxyr -N2+r > detrended_data.xyz
To find out how many terms (up to 10 in a robust interpolant are sig-
nificant in fitting data.xyz, use:
gmt trend2d data.xyz -N10+r -I -V
SEE ALSO
gmt(1), grdmath(1), grdtrend(1), trend1d(1)
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.
COPYRIGHT
2017, P. Wessel, W. H. F. Smith, R. Scharroo, J. Luis, and F. Wobbe
5.4.2 Jun 24, 2017 trend2d(1)
gmt5 5.4.2 - Generated Thu Jun 29 16:44:59 CDT 2017