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] ]


      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

      -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 "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)].

      -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


      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, grdtrend, trend1d


      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.

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