TREND2D(1)                         GMT4.1.3                         TREND2D(1)



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

SYNOPSIS
       trend2d -Fxyzmrw -Nn_model[r] [ xyz[w]file ] [ -Ccondition_number ] [
       -H[i][nrec] ][ -I[confidence_level] ] [ -V ] [ -W ] [ -:[i|o] ] [
       -b[i|o][s|S|d|D][ncol] ] [ -f[i|o]colinfo ]

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 .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 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 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
              d (double)].  Uppercase S (or D) will force byte-swapping.
              Optionally, append ncol, the number of columns in your binary
              file if it exceeds the columns needed by the program.  [Default
              is 3 (or 4 if -W is set) input columns].

       -bo    Selects binary output.  Append s for single precision [Default
              is d (double)].  Uppercase S (or D) will force byte-swapping.
              Optionally, append ncol, the number of desired columns in your
              binary output file.  [Default is 1-6 columns as set by -F].

       -f     Special formatting of input and output columns (time or
              geographical 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), or f (floating point) to each column or column range
              item.  Shorthand -f[i|o]g means -f[i|o]0x,1y (geographic
              coordinates).

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.  For
       evaluating Chebyshev polynomials, see grdmath.

       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.

ASCII FORMAT PRECISION
       The ASCII output formats of numerical data are controlled by parameters
       in your .gmtdefaults4 file.  Longitude and latitude are formatted
       according to OUTPUT_DEGREE_FORMAT, whereas other values are formatted
       according to D_FORMAT.  Be aware that the format in effect can lead to
       loss of precision in the output, which can lead to various problems
       downstream.  If you find the output is not written with enough
       precision, consider switching to binary output (-bo if available) or
       specify more decimals using the D_FORMAT setting.

EXAMPLES
       To remove a planar trend from data.xyz by ordinary least squares, use:

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

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

       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, use:

       trend2d data.xyz -N10r -I -V

SEE ALSO
       GMT(1), grdmath(1), grdtrend(1), trend1d(1)

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.



GMT4.1.3                          1 Jun 2006                        TREND2D(1)
