## 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] ]

## 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
          (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.

## 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)].

          -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, grdtrend, trend1d

## 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.