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 solu-
              tion.  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 itera-
              tive 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].

       -O     Selects Overlay plot mode [Default initializes a new  plot  sys-
              tem].


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  func-
       tions  will be Chebyshev polynomials.  These have a numerical advantage
       in the form of the matrix which must be inverted and allow  more  accu-
       rate  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 com-
       pared.  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 confi-
       dence_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  distri-
       bution.   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 inter-
       polant are significant in fitting data.xy, try:

       trend1d data.xy -Nf20r -I -V


SEE ALSO

       gmt(l), grdtrend(l), trend2d(l)


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.



VERSION                              DATE                           TREND1D(l)

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