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

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][n] ]

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)].  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) 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

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                                         TREND1D(l)