## NAME

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

## SYNOPSIS

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

## DESCRIPTION

          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.

## REQUIRED ARGUMENTS

          -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.

## OPTIONS

          xy[w]file
               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
               double].

## REMARKS

          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.

## EXAMPLES

          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

## SEE ALSO

          gmt, grdtrend, trend2d

## 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.