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

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 (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)].  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].

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

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)