1TREND2D(1) Generic Mapping Tools TREND2D(1)
2
6 trend2d - Fit a [weighted] [robust] polynomial model for z = f(x,y) to
7 xyz[w] data.
8
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). Number of header records
56 can be changed by editing your .gmtdefaults4 file. If used, GMT
57 default is 1 header record. Use -Hi if only input data should
58 have header records [Default will write out header records if
59 the input data have them]. Blank lines and lines starting with #
60 are always skipped.
61 -I Iteratively increase the number of model parameters, starting at
63 one, until n_model is reached or the reduction in variance of
64 the model is not significant at the confidence_level level. You
65 may set -I only, without an attached number; in this case the
66 fit will be iterative with a default confidence level of 0.51.
67 Or choose your own level between 0 and 1. See remarks section.
68 -V Selects verbose mode, which will send progress reports to stderr
70 [Default runs "silently"].
71 -W Weights are supplied in input column 4. Do a weighted least
73 squares fit [or start with these weights when doing the itera‐
74 tive robust fit]. [Default reads only the first 3 columns.]
75 -: Toggles between (longitude,latitude) and (latitude,longitude)
77 input and/or output. [Default is (longitude,latitude)]. Append
78 i to select input only or o to select output only. [Default
79 affects both].
80 -bi Selects binary input. Append s for single precision [Default is
82 d (double)]. Uppercase S or D will force byte-swapping.
83 Optionally, append ncol, the number of columns in your binary
84 input file if it exceeds the columns needed by the program. Or
85 append c if the input file is netCDF. Optionally, append
86 var1/var2/... to specify the variables to be read. [Default is
87 3 (or 4 if -W is set) input columns].
88 -bo Selects binary output. Append s for single precision [Default
90 is d (double)]. Uppercase S or D will force byte-swapping.
91 Optionally, append ncol, the number of desired columns in your
92 binary output file. [Default is 1-6 columns as set by -F].
93 -f Special formatting of input and/or output columns (time or geo‐
95 graphical data). Specify i or o to make this apply only to
96 input or output [Default applies to both]. Give one or more
97 columns (or column ranges) separated by commas. Append T (abso‐
98 lute calendar time), t (relative time in chosen TIME_UNIT since
99 TIME_EPOCH), x (longitude), y (latitude), or f (floating point)
100 to each column or column range item. Shorthand -f[i|o]g means
101 -f[i|o]0x,1y (geographic coordinates).
102
104 The domain of x and y will be shifted and scaled to [-1, 1] and the
105 basis functions are built from Chebyshev polynomials. These have a
106 numerical advantage in the form of the matrix which must be inverted
107 and allow more accurate solutions. In many applications of trend2d the
108 user has data located approximately along a line in the x,y plane which
109 makes an angle with the x axis (such as data collected along a road or
110 ship track). In this case the accuracy could be improved by a rotation
111 of the x,y axes. trend2d does not search for such a rotation; instead,
112 it may find that the matrix problem has deficient rank. However, the
113 solution is computed using the generalized inverse and should still
114 work out OK. The user should check the results graphically if trend2d
115 shows deficient rank. NOTE: The model parameters listed with -V are
116 Chebyshev coefficients; they are not numerically equivalent to the m#s
117 in the equation described above. The description above is to allow the
118 user to match -N with the order of the polynomial surface. For evalu‐
119 ating Chebyshev polynomials, see grdmath.
120 The -Nn_modelr (robust) and -I (iterative) options evaluate the signif‐
122 icance of the improvement in model misfit Chi-Squared by an F test.
123 The default confidence limit is set at 0.51; it can be changed with the
124 -I option. The user may be surprised to find that in most cases the
125 reduction in variance achieved by increasing the number of terms in a
126 model is not significant at a very high degree of confidence. For
127 example, with 120 degrees of freedom, Chi-Squared must decrease by 26%
128 or more to be significant at the 95% confidence level. If you want to
129 keep iterating as long as Chi-Squared is decreasing, set confi‐
130 dence_level to zero.
131 A low confidence limit (such as the default value of 0.51) is needed to
133 make the robust method work. This method iteratively reweights the
134 data to reduce the influence of outliers. The weight is based on the
135 Median Absolute Deviation and a formula from Huber [1964], and is 95%
136 efficient when the model residuals have an outlier-free normal distri‐
137 bution. This means that the influence of outliers is reduced only
138 slightly at each iteration; consequently the reduction in Chi-Squared
139 is not very significant. If the procedure needs a few iterations to
140 successfully attenuate their effect, the significance level of the F
141 test must be kept low.
142
144 The ASCII output formats of numerical data are controlled by parameters
145 in your .gmtdefaults4 file. Longitude and latitude are formatted
146 according to OUTPUT_DEGREE_FORMAT, whereas other values are formatted
147 according to D_FORMAT. Be aware that the format in effect can lead to
148 loss of precision in the output, which can lead to various problems
149 downstream. If you find the output is not written with enough preci‐
150 sion, consider switching to binary output (-bo if available) or specify
151 more decimals using the D_FORMAT setting.
152
154 To remove a planar trend from data.xyz by ordinary least squares, use:
155 trend2d data.xyz -Fxyr -N2 > detrended_data.xyz
157 To make the above planar trend robust with respect to outliers, use:
159 trend2d data.xzy -Fxyr -N2r > detrended_data.xyz
161 To find out how many terms (up to 10) in a robust interpolant are sig‐
163 nificant in fitting data.xyz, use:
164 trend2d data.xyz -N10r -I -V
166
168 GMT(1), grdmath(1), grdtrend(1), trend1d(1)
169
171 Huber, P. J., 1964, Robust estimation of a location parameter, Ann.
172 Math. Stat., 35, 73-101.
173 Menke, W., 1989, Geophysical Data Analysis: Discrete Inverse Theory,
175 Revised Edition, Academic Press, San Diego.
176GMT 4.3.1 15 May 2008 TREND2D(1)