TREND2D(1) Generic Mapping Tools TREND2D(1)NAMEtrend2d - Fit a [weighted] [robust] polynomial model for z = f(x,y) to
xyz[w] data.
SYNOPSIStrend2d-Fxyzmrw -Nn_model[r] [ xyz[w]file ] [ -Ccondition_number ] [
-H[i][nrec] ] [ -I[confidence_level] ] [ -V ] [ -W ] [ -:[i|o] ] [
-b[i|o][s|S|d|D[ncol]|c[var1/...]] ] [ -f[i|o]colinfo ]
DESCRIPTIONtrend2d 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 "regional" field from a "residual" which
should have non-zero mean, such as a local mountain on a regional sur‐
face.
-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). If used, the default number
of header records is N_HEADER_RECS. Use -Hi if only input data
should have header records [Default will write out header
records if the input data have them]. Blank lines and lines
starting with # are always skipped.
-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
d (double)]. Uppercase S or D will force byte-swapping.
Optionally, append ncol, the number of columns in your binary
input file if it exceeds the columns needed by the program. Or
append c if the input file is netCDF. Optionally, append
var1/var2/... to specify the variables to be read. [Default is
3 (or 4 if -W is set) input columns].
-bo Selects binary output. Append s for single precision [Default
is d (double)]. Uppercase S or D will force byte-swapping.
Optionally, append ncol, the number of desired columns in your
binary output file. [Default is 1-6 columns as set by -F].
-f Special formatting of input and/or output columns (time or geo‐
graphical data). Specify i or o to make this apply only to
input or output [Default applies to both]. Give one or more
columns (or column ranges) separated by commas. Append T (abso‐
lute calendar time), t (relative time in chosen TIME_UNIT since
TIME_EPOCH), x (longitude), y (latitude), or f (floating point)
to each column or column range item. Shorthand -f[i|o]g means
-f[i|o]0x,1y (geographic coordinates).
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 evalu‐
ating 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
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 confi‐
dence_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 distri‐
bution. 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 .gmtdefaults4 file. Longitude and latitude are formatted
according to OUTPUT_DEGREE_FORMAT, whereas other values are formatted
according to D_FORMAT. Be aware that the format in effect can lead to
loss of precision in the output, which can lead to various problems
downstream. If you find the output is not written with enough preci‐
sion, consider switching to binary output (-bo if available) or specify
more decimals using the D_FORMAT setting.
EXAMPLES
To remove a planar trend from data.xyz by ordinary least squares, use:
trend2d data.xyz -F xyr -N 2 > detrended_data.xyz
To make the above planar trend robust with respect to outliers, use:
trend2d data.xzy -F xyr -N 2r > 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 -N 10r -I -V
SEE ALSOGMT(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.
GMT 4.5.14 1 Nov 2015 TREND2D(1)
