NAME¶
trend1d - Fit a [weighted] [robust] polynomial [or Fourier] model for y = f(x)
to xy[w] data.
SYNOPSIS¶
trend1d -F<xymrw>
-N[
f]
n_model[
r] [
xy[w]file ] [
-Ccondition_# ] [
-H[
nrec] ] [
-I[
confidence_level] ] [
-V ] [
-W ] [
-: ]
[
-bi[
s][
n] ] [
-bo[
s][
n] ]
DESCRIPTION¶
trend1d reads x,y [and w] values from the first two [three] columns on
standard input [or
xy[w]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, 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¶
- -F
- 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.
- -N
- Specify the number of terms in the model, n_model, whether to fit a
Fourier ( -Nf) or polynomial [Default] model, and append r
to do a robust fit. E.g., a robust quadratic model is
-N3r.
OPTIONS¶
- xy[w]file
- ASCII [or binary, see -b] file containing x,y [w] values in the
first 2 [3] columns. If no file is specified, trend1d will read
from standard input.
- -C
- 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. ].
- -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 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.]
- -:
- 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 2 (or 3 if -W is set) columns].
- -bo
- Selects binary output. Append s for single precision [Default is
double].
If a Fourier model is selected, the domain of x will be shifted and scaled to
[-pi, pi] and the basis functions used will be 1, cos(x), sin(x), cos(2x),
sin(2x), ... If a polynomial model is selected, the domain of x will be
shifted and scaled to [-1, 1] and the basis functions will be Chebyshev
polynomials. These 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 model coefficients are Chebeshev coefficients,
NOT coefficients in a + bx + cxx + ...
The
-Nr (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, try:
trend1d data.xy
-Fxr
-N2 > detrended_data.xy
To make the above linear trend robust with respect to outliers, try:
trend1d data.xy
-Fxr
-N2
r > detrended_data.xy
To find out how many terms (up to 20, say) in a robust Fourier interpolant are
significant in fitting data.xy, try:
trend1d data.xy
-Nf20
r -I -V
SEE ALSO¶
gmt(1gmt),
grdtrend(1gmt),
trend2d(1gmt)
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.