1TREND2D(1) Generic Mapping Tools TREND2D(1)
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6 trend2d - Fit a [weighted] [robust] polynomial model for z = f(x,y) to
7 xyz[w] data.
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10 trend2d -Fxyzmrw -Nn_model[r] [ xyz[w]file ] [ -Ccondition_number ] [
11 -H[i][nrec] ] [ -I[confidence_level] ] [ -V ] [ -W ] [ -:[i|o] ] [
12 -b[i|o][s|S|d|D[ncol]|c[var1/...]] ] [ -f[i|o]colinfo ]
13
15 trend2d reads x,y,z [and w] values from the first three [four] columns
16 on standard input [or xyz[w]file] and fits a regression model z =
17 f(x,y) + e by [weighted] least squares. The fit may be made robust by
18 iterative reweighting of the data. The user may also search for the
19 number of terms in f(x,y) which significantly reduce the variance in z.
20 n_model may be in [1,10] to fit a model of the following form (similar
21 to grdtrend):
22 m1 + m2*x + m3*y + m4*x*y + m5*x*x + m6*y*y + m7*x*x*x + m8*x*x*y +
24 m9*x*y*y + m10*y*y*y.
25 The user must specify -Nn_model, the number of model parameters to use;
27 thus, -N4 fits a bilinear trend, -N6 a quadratic surface, and so on.
28 Optionally, append r to perform a robust fit. In this case, the pro‐
29 gram will iteratively reweight the data based on a robust scale esti‐
30 mate, in order to converge to a solution insensitive to outliers. This
31 may be handy when separating a "regional" field from a "residual" which
32 should have non-zero mean, such as a local mountain on a regional sur‐
33 face.
34 -F Specify up to six letters from the set {x y z m r w} in any
36 order to create columns of ASCII [or binary] output. x = x, y =
37 y, z = z, m = model f(x,y), r = residual z - m, w = weight used
38 in fitting.
39 -N Specify the number of terms in the model, n_model, and append r
41 to do a robust fit. E.g., a robust bilinear model is -N4r.
42
44 xyz[w]file
45 ASCII [or binary, see -b] file containing x,y,z [w] values in
46 the first 3 [4] columns. If no file is specified, trend2d will
47 read from standard input.
48 -C Set the maximum allowed condition number for the matrix solu‐
50 tion. trend2d fits a damped least squares model, retaining only
51 that part of the eigenvalue spectrum such that the ratio of the
52 largest eigenvalue to the smallest eigenvalue is condition_#.
53 [Default: condition_# = 1.0e06. ].
54 -H Input file(s) has header record(s). If used, the default number
56 of header records is N_HEADER_RECS. Use -Hi if only input data
57 should have header records [Default will write out header
58 records if the input data have them]. Blank lines and lines
59 starting with # are always skipped.
60 -I Iteratively increase the number of model parameters, starting at
62 one, until n_model is reached or the reduction in variance of
63 the model is not significant at the confidence_level level. You
64 may set -I only, without an attached number; in this case the
65 fit will be iterative with a default confidence level of 0.51.
66 Or choose your own level between 0 and 1. See remarks section.
67 -V Selects verbose mode, which will send progress reports to stderr
69 [Default runs "silently"].
70 -W Weights are supplied in input column 4. Do a weighted least
72 squares fit [or start with these weights when doing the itera‐
73 tive robust fit]. [Default reads only the first 3 columns.]
74 -: Toggles between (longitude,latitude) and (latitude,longitude)
76 input and/or output. [Default is (longitude,latitude)]. Append
77 i to select input only or o to select output only. [Default
78 affects both].
79 -bi Selects binary input. Append s for single precision [Default is
81 d (double)]. Uppercase S or D will force byte-swapping.
82 Optionally, append ncol, the number of columns in your binary
83 input file if it exceeds the columns needed by the program. Or
84 append c if the input file is netCDF. Optionally, append
85 var1/var2/... to specify the variables to be read. [Default is
86 3 (or 4 if -W is set) input columns].
87 -bo Selects binary output. Append s for single precision [Default
89 is d (double)]. Uppercase S or D will force byte-swapping.
90 Optionally, append ncol, the number of desired columns in your
91 binary output file. [Default is 1-6 columns as set by -F].
92 -f Special formatting of input and/or output columns (time or geo‐
94 graphical data). Specify i or o to make this apply only to
95 input or output [Default applies to both]. Give one or more
96 columns (or column ranges) separated by commas. Append T (abso‐
97 lute calendar time), t (relative time in chosen TIME_UNIT since
98 TIME_EPOCH), x (longitude), y (latitude), or f (floating point)
99 to each column or column range item. Shorthand -f[i|o]g means
100 -f[i|o]0x,1y (geographic coordinates).
101
103 The domain of x and y will be shifted and scaled to [-1, 1] and the
104 basis functions are built from Chebyshev polynomials. These have a
105 numerical advantage in the form of the matrix which must be inverted
106 and allow more accurate solutions. In many applications of trend2d the
107 user has data located approximately along a line in the x,y plane which
108 makes an angle with the x axis (such as data collected along a road or
109 ship track). In this case the accuracy could be improved by a rotation
110 of the x,y axes. trend2d does not search for such a rotation; instead,
111 it may find that the matrix problem has deficient rank. However, the
112 solution is computed using the generalized inverse and should still
113 work out OK. The user should check the results graphically if trend2d
114 shows deficient rank. NOTE: The model parameters listed with -V are
115 Chebyshev coefficients; they are not numerically equivalent to the m#s
116 in the equation described above. The description above is to allow the
117 user to match -N with the order of the polynomial surface. For evalu‐
118 ating Chebyshev polynomials, see grdmath.
119 The -Nn_modelr (robust) and -I (iterative) options evaluate the signif‐
121 icance of the improvement in model misfit Chi-Squared by an F test.
122 The default confidence limit is set at 0.51; it can be changed with the
123 -I option. The user may be surprised to find that in most cases the
124 reduction in variance achieved by increasing the number of terms in a
125 model is not significant at a very high degree of confidence. For
126 example, with 120 degrees of freedom, Chi-Squared must decrease by 26%
127 or more to be significant at the 95% confidence level. If you want to
128 keep iterating as long as Chi-Squared is decreasing, set confi‐
129 dence_level to zero.
130 A low confidence limit (such as the default value of 0.51) is needed to
132 make the robust method work. This method iteratively reweights the
133 data to reduce the influence of outliers. The weight is based on the
134 Median Absolute Deviation and a formula from Huber [1964], and is 95%
135 efficient when the model residuals have an outlier-free normal distri‐
136 bution. This means that the influence of outliers is reduced only
137 slightly at each iteration; consequently the reduction in Chi-Squared
138 is not very significant. If the procedure needs a few iterations to
139 successfully attenuate their effect, the significance level of the F
140 test must be kept low.
141
143 The ASCII output formats of numerical data are controlled by parameters
144 in your .gmtdefaults4 file. Longitude and latitude are formatted
145 according to OUTPUT_DEGREE_FORMAT, whereas other values are formatted
146 according to D_FORMAT. Be aware that the format in effect can lead to
147 loss of precision in the output, which can lead to various problems
148 downstream. If you find the output is not written with enough preci‐
149 sion, consider switching to binary output (-bo if available) or specify
150 more decimals using the D_FORMAT setting.
151
153 To remove a planar trend from data.xyz by ordinary least squares, use:
154 trend2d data.xyz -Fxyr -N2 > detrended_data.xyz
156 To make the above planar trend robust with respect to outliers, use:
158 trend2d data.xzy -Fxyr -N2r > detrended_data.xyz
160 To find out how many terms (up to 10) in a robust interpolant are sig‐
162 nificant in fitting data.xyz, use:
163 trend2d data.xyz -N10r -I -V
165
167 GMT(1), grdmath(1), grdtrend(1), trend1d(1)
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170 Huber, P. J., 1964, Robust estimation of a location parameter, Ann.
171 Math. Stat., 35, 73-101.
172 Menke, W., 1989, Geophysical Data Analysis: Discrete Inverse Theory,
174 Revised Edition, Academic Press, San Diego.
175GMT 4.5.6 10 Mar 2011 TREND2D(1)