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][n] ] [ -f[i|o]colinfo ]


       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 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  .gmtdefaults4  file.  If  used,  GMT
              default  is  1  header record. Use -Hi if only input data should
              have header records [Default will write out  header  records  if
              the input data have them].

       -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 and/or output. [Default is (longitude,latitude)].   Append
              i  to  select  input  only  or o to select output only. [Default
              affects both].

       -bi    Selects binary input. Append s for single precision [Default  is
              double].   Append  n  for  the  number  of columns in the binary
              [Default is 2 (or 3 if -W is set) columns].

       -bo    Selects binary output. Append s for single precision [Default is
              double].   Append  n  for  the  number  of columns in the binary

       -f     Special formatting of input and output  columns  (time  or  geo-
              graphical  data)  Specify  i(nput)  or o(utput) [Default is both
              input and output].  Give one or more columns (or column  ranges)
              separated by commas.  Append T (Absolute calendar time), t (time
              relative to chosen TIME_EPOCH), x (longitude), y  (latitude),  g
              (geographic coordinate), or f (floating point) to each column or
              column range item.


       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 stable model coefficients are Chebyshev coefficients.
       The corresponding polynomial coefficients in a + bx +  cxx  +  ...  are
       also  given  in  Verbose  mode but users must realize that they are NOT
       stable beyond degree 7 or 8. See Numerical Recipes for more discussion.

       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 reduc-
       tion  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% effi-
       cient when the model residuals have an  outlier-free  normal  distribu-
       tion.  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, use:

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

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

       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, use:

       trend1d data.xy -Nf20r -I -V


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


       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.

GMT4.0                            1 Oct 2004                        TREND1D(l)

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