trend1d(1) GMT trend1d(1)
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 ] [ -Ccondi-
tion_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 normal-
ized polynomial model coefficients, or c for the normalized
Chebyshev model coefficients.
-N[p|P|f|F|c|C|s|S|x]n[,a|][+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 indi-
cates 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-ori-
gin 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 nor-
malized xa = (2*pi*(x-origin)/length) while the polynomials use
xa = 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 speci-
fied 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 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. ].
-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 a|)
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 itera-
tive robust fit]. [Default reads only the first 2 columns.]
-bi[ncols][t] (more a|)
Select native binary input. [Default is 2 (or 3 if -W is set)
columns].
-bo[ncols][type] (more a|)
Select native binary output. [Default is 1-5 columns as given by
-F].
-d[i|o]nodata (more a|)
Replace input columns that equal nodata with NaN and do the
reverse on output.
-e[~]^<i>apattern^<i>a | -e[~]/regexp/[i] (more a|)
Only accept data records that match the given pattern.
-f[i|o]colinfo (more a|)
Specify data types of input and/or output columns.
-h[i|o][n][+c][+d][+rremark][+rtitle] (more a|)
Skip or produce header record(s).
-icols[+l][+sscale][+ooffset][,^<i>a|] (more a|)
Select input columns and transformations (0 is first column).
-:[i|o] (more a|)
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 explana-
tion 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 FOR-
MAT_DATE_OUT and FORMAT_CLOCK_OUT, whereas general floating point val-
ues are formatted according to FORMAT_FLOAT_OUT. Be aware that the for-
mat 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 set-
ting.
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 polyno-
mials 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 magni-
tude 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 + a| 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 -Na|+r (robust) and -I (iterative) options evaluate the signifi-
cance 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
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% 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.
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 inter-
polant are significant in fitting data.xy, use:
gmt trend1d data.xy -Nf20+r -I -V
SEE ALSO
gmt(1), gmtmath(1), gmtregress(1), grdtrend(1), trend2d(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.
COPYRIGHT
2017, P. Wessel, W. H. F. Smith, R. Scharroo, J. Luis, and F. Wobbe
5.4.2 Jun 24, 2017 trend1d(1)
gmt5 5.4.2 - Generated Thu Jun 29 16:44:05 CDT 2017