Provided by: gmt_4.5.11-1build1_amd64 bug


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


       trend2d  -Fxyzmrw  -Nn_model[r]  [  xyz[w]file  ] [ -Ccondition_number ] [ -H[i][nrec] ] [
       -I[confidence_level] ] [ -V ] [ -W ] [ -:[i|o] ] [ -b[i|o][s|S|d|D[ncol]|c[var1/...]] ]  [
       -f[i|o]colinfo ]


       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).  If used, the default number of header records
              is N_HEADER_RECS.  Use -Hi if only input data should have header  records  [Default
              will  write  out header records if the input data have them]. Blank lines and lines
              starting with # are always skipped.

       -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 and/or output.
              [Default is (longitude,latitude)].  Append i to select input only or  o  to  select
              output only.  [Default affects both].

       -bi    Selects  binary  input.   Append  s  for  single precision [Default is d (double)].
              Uppercase S or D will force byte-swapping.  Optionally, append ncol, the number  of
              columns  in your binary input file if it exceeds the columns needed by the program.
              Or append c if the input  file  is  netCDF.  Optionally,  append  var1/var2/...  to
              specify  the  variables  to  be  read.   [Default  is  3  (or 4 if -W is set) input

       -bo    Selects binary output.  Append s for single  precision  [Default  is  d  (double)].
              Uppercase  S or D will force byte-swapping.  Optionally, append ncol, the number of
              desired columns in your binary output file.  [Default is 1-6 columns as set by -F].

       -f     Special formatting of input and/or output  columns  (time  or  geographical  data).
              Specify  i  or  o  to  make  this apply only to input or output [Default applies to
              both].  Give one or more columns (or column ranges) separated by commas.  Append  T
              (absolute calendar time), t (relative time in chosen TIME_UNIT since TIME_EPOCH), x
              (longitude), y (latitude), or f (floating point) to each  column  or  column  range
              item.  Shorthand -f[i|o]g means -f[i|o]0x,1y (geographic coordinates).


       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.  For evaluating
       Chebyshev polynomials, see grdmath.

       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.


       The  ASCII  output  formats  of  numerical  data  are  controlled  by  parameters  in your
       .gmtdefaults4   file.    Longitude   and   latitude    are    formatted    according    to
       OUTPUT_DEGREE_FORMAT,  whereas other values are formatted according to D_FORMAT.  Be aware
       that the format in effect can lead to loss of precision in the output, which can  lead  to
       various problems downstream.  If you find the output is not written with enough precision,
       consider switching to binary output (-bo if available) or specify more decimals using  the
       D_FORMAT setting.


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

       trend2d -F xyr -N 2 >

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

       trend2d data.xzy -F xyr -N 2r >

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

       trend2d -N 10r -I -V


       GMT(1), grdmath(1), grdtrend(1), trend1d(1)


       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.