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       trend1d  -  Fit  a [weighted] [robust] polynomial [or Fourier] model for y = f(x) to xy[w]


       trend1d -F<xymrw> -N[f]n_model[r]  [  xy[w]file  ]  [  -Ccondition_#  ]  [  -H[nrec]  ]  [
       -I[confidence_level] ] [ -V ] [ -W ] [ -: ] [ -bi[s][n] ] [ -bo[s][n] ]


       trend1d  reads x,y [and w] values from the first two [three] columns on standard input [or
       xy[w]file] and fits a regression model y = f(x) +  e  by  [weighted]  least  squares.  The
       functional  form  of  f(x) may be chosen as polynomial or Fourier, and 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) which significantly reduce the variance in y.


       -F     Specify  up to five letters from the set {x y m r w} in any order to create columns
              of ASCII [or binary] output. x = x, y = y, m = model f(x), r = residual y - m, w  =
              weight used in fitting.

       -N     Specify  the  number of terms in the model, n_model, whether to fit a Fourier (-Nf)
              or polynomial [Default] model, and append r to do a  robust  fit.  E.g.,  a  robust
              quadratic model is -N3r.


              ASCII  [or  binary,  see  -b]  file  containing  x,y  [w] values in the first 2 [3]
              columns. If no file is specified, trend1d will read from standard input.

       -C     Set the maximum allowed condition number for the matrix solution.  trend1d  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 3. Do a weighted least squares fit [or start
              with these weights when doing the iterative robust fit]. [Default  reads  only  the
              first 2 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 2 (or 3 if -W is set)

       -bo    Selects binary output. Append s for single precision [Default is double].


       If a Fourier model is selected, the domain of x will be shifted and scaled  to  [-pi,  pi]
       and  the  basis  functions  used  will  be  1,  cos(x), sin(x), cos(2x), sin(2x), ... If a
       polynomial model is selected, the domain of x will be shifted and scaled to  [-1,  1]  and
       the basis functions will be Chebyshev polynomials. These have a numerical advantage in the
       form of the matrix which  must  be  inverted  and  allow  more  accurate  solutions.   The
       Chebyshev  polynomial of degree n has n+1 extrema in [-1, 1], at all of which its value is
       either -1 or +1. Therefore the magnitude of  the  polynomial  model  coefficients  can  be
       directly   compared.   NOTE:  The  model  coefficients  are  Chebeshev  coefficients,  NOT
       coefficients in a + bx + cxx + ...

       The -Nr (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

       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 linear trend from data.xy by ordinary least squares, try:

       trend1d data.xy -Fxr -N2 > detrended_data.xy

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

       trend1d data.xy -Fxr -N2r > detrended_data.xy

       To find out how many terms (up to 20, say) in a robust Fourier interpolant are significant
       in fitting data.xy, try:

       trend1d data.xy -Nf20r -I -V


       gmt(1gmt), grdtrend(1gmt), trend2d(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                                 TREND1D(l)