TREND2D

trend2d − Fit a [weighted] [robust] polynomial model for z = f(x,y) to xyz[w] data.

trend2d −F<xyzmrw> −Nn_model[r] [ xyz[w]file ] [ −Ccondition_# ] [ −H[nrec] ][ −I[confidence_level] ] [ −V ] [ −W ] [ −: ] [ −bi[s][n] ] [ −bo[s][n] ] [ −f[i|o]colinfo ]

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 program will iteratively reweight the data based on a robust scale estimate, 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 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 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.

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 condition_#. [Default: condition_# = 1.0e06. ].

−H

Input file(s) has Header record(s). Number of header records can be changed by editing your .gmtdefaults4 file. If used, GMT default is 1 header record. Use −Hi if only input data should have header records [Default will write out header records if the input data have them].

−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 iterative 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 double]. Uppercase S (or D) will force byte-swapping. 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 precision [Default is double]. Uppercase S (or D) will force byte-swapping. Append n for the number of columns in the binary file(s).

−f

Special formatting of input and output columns (time or geographical data). Specify i(nput) or o(utput) [Default is both input and output]. Give one or more columns (or column ranges) separated by commas. Append T (Absolute calendar time), t (time relative to chosen 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).

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 evaluating Chebyshev polynomials, see grdmath.

The −Nn_modelr (robust) and −I (iterative) options evaluate 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 Deviation and a formula from Huber [1964], and is 95% efficient 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.

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 precision, consider switching to binary output (−bo if available) or specify more decimals using the D_FORMAT setting.

To remove a planar trend from data.xyz by ordinary least squares, use:

trend2d data.xyz −Fxyr −N2 > detrended_data.xyz

To make the above planar trend robust with respect to outliers, use:

trend2d data.xzy −Fxyr −N2r > detrended_data.xyz

To find out how many terms (up to 10) in a robust interpolant are significant in fitting data.xyz, use:

trend2d data.xyz −N10r −I −V

GMT(l), grdmath(l), grdtrend(l), trend1d(l)

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.