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       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] ]


       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.


              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

       -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 geographic 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].


       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.


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

       trend2d -Fxyr -N2 >

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

       trend2d data.xzy -Fxyr -N2r >

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

       trend2d -N10r -I -V


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


       Huber,  P.  J.,  1964,  Robust  estimation  of a location parameter, Ann. Math. Stat., 35,

       Menke, W., 1989, Geophysical Data Analysis:  Discrete  Inverse  Theory,  Revised  Edition,
       Academic Press, San Diego.

                                            1 Jan 2004                                 TREND2D(l)