Ubuntu Manpage: trend2d - Fit a [weighted] [robust] polynomial model for z = f(x,y) to xyz[w] data.

NAME

       trend2d - Fit a [weighted] [robust] polynomial model for z = f(x,y) to xyz[w] data.

SYNOPSIS

       trend2d  -F<xyzmrw>  -Nn_model[r] [ xyz[w]file ] [ -Ccondition_# ] [ -H[nrec] ][ -I[confidence_level] ] [
       -V ] [ -W ] [ -: ] [ -bi[s][n] ] [ -bo[s][n] ]

DESCRIPTION

       trend2d reads x,y,z [and w] values from the first three [four] columns on standard input [or  xyz[w]file]
       and  fits  a  regression  model z = f(x,y) + e by [weighted] least squares. 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,y)  which
       significantly  reduce  the  variance  in z. n_model may be in [1,10] to fit a model of the following form
       (similar to grdtrend):

       m1 + m2*x + m3*y + m4*x*y + m5*x*x + m6*y*y + m7*x*x*x + m8*x*x*y + m9*x*y*y + m10*y*y*y.

       The user must specify -Nn_model, the number of model parameters to use; thus, -N4 fits a bilinear  trend,
       -N6  a  quadratic  surface,  and  so  on. Optionally, append r to perform a robust fit. In this case, the
       program will iteratively reweight the data based on a robust scale estimate, in order to  converge  to  a
       solution  insensitive to outliers. This may be handy when separating a "regional" field from a "residual"
       which should have non-zero mean, such as a local mountain on a regional surface.

       -F     Specify up to six letters from the set {x y z m r w} in any order to create columns of  ASCII  [or
              binary]  output.  x  =  x,  y = y, z = z, m = model f(x,y), r = residual z - m, w = weight used in
              fitting.

       -N     Specify the number of terms in the model, n_model, and append r to do a robust fit. E.g., a robust
              bilinear model is -N4r.

OPTIONS

       xyz[w]file
              ASCII [or binary, see -b] file containing x,y,z [w] values in the first 3 [4] columns. If no  file
              is specified, trend2d will read from standard input.

       -C     Set  the  maximum  allowed  condition  number for the matrix solution. trend2d 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  4. Do a weighted least squares fit [or start with these
              weights when doing the iterative robust fit]. [Default reads only the first 3 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 3 (or 4 if -W is set) input columns].

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

REMARKS

The domain of x and y will be shifted and scaled to [-1, 1] and the basis functions are built from
Chebyshev polynomials. These have a numerical advantage in the form of the matrix which must be inverted
and allow more accurate solutions. In many applications of trend2d the user has data located
approximately along a line in the x,y plane which makes an angle with the x axis (such as data collected
along a road or ship track). In this case the accuracy could be improved by a rotation of the x,y axes.
trend2d does not search for such a rotation; instead, it may find that the matrix problem has deficient
rank. However, the solution is computed using the generalized inverse and should still work out OK. The
user should check the results graphically if trend2d shows deficient rank. NOTE: The model parameters
listed with -V are Chebyshev coefficients; they are not numerically equivalent to the m#s in the equation
described above. The description above is to allow the user to match -N with the order of the polynomial
surface.

The -Nn_modelr (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 planar trend from data.xyz by ordinary least squares, try:

       trend2d data.xyz -Fxyr -N2 > detrended_data.xyz

       To make the above planar trend robust with respect to outliers, try:

       trend2d data.xzy -Fxyr -N2r > detrended_data.xyz

       To find out how many terms (up to 10) in a robust interpolant are significant in fitting data.xyz, try:

       trend2d data.xyz -N10r -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                                         TREND2D(l)