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


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


      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 "silently"].

      -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)].

      -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) columns].

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


      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 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 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, grdtrend, trend2d


      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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