TREND2D(l)																TREND2D(l)

NAME

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

SYNOPSIS

       trend2d	-F<xyzmrw> -Nn_model[r] [ xyz[w]file ] [ -Ccondition_# ] [ -H[nrec] ][ -I[confidence_level] ] [ -V ] [ -W ] [ -: ] [ -bi[s][n] ] [
       -bo[s][n] ]

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 (sim-
       ilar 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 "resid-
       ual" 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.

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 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 .gmtdefaults file. If used, GMT  default
	      is 1 header record.

       -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/output. [Default  is	(longitude,latitude)].	 Applies  to  geo-
	      graphic coordinates only.

       -bi    Selects  binary input. Append s for single precision [Default is double].  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].

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.

       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.

EXAMPLES

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

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

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

       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, try:

       trend2d data.xyz -N10r -I -V

SEE ALSO

       gmt(1gmt), grdtrend(1gmt), trend1d(1gmt)

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.

								    1 Jan 2004								TREND2D(l)