Provided by: gmt-common_5.4.5+dfsg-2_all 
NAME
trend1d - Fit a [weighted] [robust] polynomial [and/or Fourier] model for y = f(x) to
xy[w] data
SYNOPSIS
trend1d [ table ] -Fxymrw|p|P|c -Nparams [ xy[w]file ] [ -Ccondition_number ] [
-I[confidence_level] ] [ -V[level] ] [ -W ] [ -bbinary ] [ -dnodata ] [ -eregexp ] [
-fflags ] [ -hheaders ] [ -iflags ] [ -:[i|o] ]
Note: No space is allowed between the option flag and the associated arguments.
DESCRIPTION
trend1d reads x,y [and w] values from the first two [three] columns on standard input [or
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 or a mix of the two, 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
-Fxymrw|p|P|c
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. Alternatively, choose just the single selection p to output
a record with the polynomial model coefficients, P for the normalized polynomial
model coefficients, or c for the normalized Chebyshev model coefficients.
-N[p|P|f|F|c|C|s|S|x]n[,...][+llength][+oorigin][+r]
Specify the components of the (possibly mixed) model. Append one or more
comma-separated model components. Each component is of the form Tn, where T
indicates the basis function and n indicates the polynomial degree or how many
terms in the Fourier series we want to include. Choose T from p (polynomial with
intercept and powers of x up to degree n), P (just the single term x^n), f (Fourier
series with n terms), c (Cosine series with n terms), s (sine series with n terms),
F (single Fourier component of order n), C (single cosine component of order n),
and S (single sine component of order n). By default the x-origin and fundamental
period is set to the mid-point and data range, respectively. Change this using the
+oorigin and +llength modifiers. We normalize x before evaluating the basis
functions. Basically, the trigonometric bases all use the normalized x' =
(2*pi*(x-origin)/length) while the polynomials use x' = 2*(x-x_mid)/(xmax - xmin)
for stability. Finally, append +r for a robust solution [Default gives a least
squares fit]. Use -V to see a plain-text representation of the y(x) model
specified in -N.
OPTIONAL ARGUMENTS
table One or more ASCII [or binary, see -bi] files containing x,y [w] values in the first
2 [3] columns. If no files are specified, trend1d will read from standard input.
-Ccondition_number
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. ].
-I[confidence_level]
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. Note that the model terms are
added in the order they were given in -N so you should place the most important
terms first.
-V[level] (more ...)
Select verbosity level [c].
-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.]
-bi[ncols][t] (more ...)
Select native binary input. [Default is 2 (or 3 if -W is set) columns].
-bo[ncols][type] (more ...)
Select native binary output. [Default is 1-5 columns as given by -F].
-d[i|o]nodata (more ...)
Replace input columns that equal nodata with NaN and do the reverse on output.
-e[~]"pattern" | -e[~]/regexp/[i] (more ...)
Only accept data records that match the given pattern.
-f[i|o]colinfo (more ...)
Specify data types of input and/or output columns.
-h[i|o][n][+c][+d][+rremark][+rtitle] (more ...)
Skip or produce header record(s).
-icols[+l][+sscale][+ooffset][,...] (more ...)
Select input columns and transformations (0 is first column).
-:[i|o] (more ...)
Swap 1st and 2nd column on input and/or output.
-^ or just -
Print a short message about the syntax of the command, then exits (NOTE: on Windows
just use -).
-+ or just +
Print an extensive usage (help) message, including the explanation of any
module-specific option (but not the GMT common options), then exits.
-? or no arguments
Print a complete usage (help) message, including the explanation of all options,
then exits.
ASCII FORMAT PRECISION
The ASCII output formats of numerical data are controlled by parameters in your gmt.conf
file. Longitude and latitude are formatted according to FORMAT_GEO_OUT, absolute time is
under the control of FORMAT_DATE_OUT and FORMAT_CLOCK_OUT, whereas general floating point
values are formatted according to FORMAT_FLOAT_OUT. Be aware that the format in effect can
lead to loss of precision in ASCII 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 FORMAT_FLOAT_OUT setting.
REMARKS
If a polynomial model is included, then the domain of x will be shifted and scaled to [-1,
1] and the basis functions will be Chebyshev polynomials provided the polygon is of full
order (otherwise we stay with powers of x). The Chebyshev polynomials 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 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. For evaluating Chebyshev polynomials, see
gmtmath.
The -N...+r (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, use:
gmt trend1d data.xy -Fxr -Np1 > detrended_data.xy
To make the above linear trend robust with respect to outliers, use:
gmt trend1d data.xy -Fxr -Np1+r > detrended_data.xy
To fit the model y(x) = a + bx^2 + c * cos(2*pi*3*(x/l) + d * sin(2*pi*3*(x/l), with l the
fundamental period (here l = 15), try:
gmt trend1d data.xy -Fxm -NP0,P2,F3+l15 > model.xy
To find out how many terms (up to 20, say in a robust Fourier interpolant are significant
in fitting data.xy, use:
gmt trend1d data.xy -Nf20+r -I -V
SEE ALSO
gmt, gmtmath, gmtregress, grdtrend, trend2d
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.
COPYRIGHT
2019, P. Wessel, W. H. F. Smith, R. Scharroo, J. Luis, and F. Wobbe