Provided by: gmt-common_5.4.3+dfsg-1_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
2018, P. Wessel, W. H. F. Smith, R. Scharroo, J. Luis, and F. Wobbe