trend1d - Fit a [weighted] [robust] polynomial [or Fourier] model for y = f(x) to xy[w] data.

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] ] [-f[i|o]colinfo]

trend1dreads x,y [and w] values from the first two [three] columns on standard input [orxy[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.

-FSpecify 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.-NSpecify the number of terms in the model,n_model, whether to fit a Fourier (-Nf) or polynomial [Default] model, and appendrto do a robust fit. E.g., a robust quadratic model is-N3r.

xy[w]fileASCII [or binary, see-b] file containing x,y [w] values in the first 2 [3] columns. If no file is specified,trend1dwill read from standard input.-CSet the maximum allowed condition number for the matrix solu- tion.trend1dfits 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 iscondition_#. [Default:condition_#= 1.0e06. ].-HInput file(s) has Header record(s). Number of header records can be changed by editing your .gmtdefaults4 file. If used,GMTdefault is 1 header record. Use-Hiif only input data should have header records [Default will write out header records if the input data have them].-IIteratively increase the number of model parameters, starting at one, untiln_modelis reached or the reduction in variance of the model is not significant at theconfidence_levellevel. You may set-Ionly, 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.-VSelects verbose mode, which will send progress reports to stderr [Default runs "silently"].-WWeights 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.]-:Toggles between (longitude,latitude) and (latitude,longitude) input and/or output. [Default is (longitude,latitude)]. Appendito select input only oroto select output only. [Default affects both].-biSelects binary input. Appendsfor single precision [Default is double]. Appendnfor the number of columns in the binary file(s). [Default is 2 (or 3 if-Wis set) columns].-boSelects binary output. Appendsfor single precision [Default is double]. Appendnfor the number of columns in the binary file(s).-fSpecial formatting of input and output columns (time or geo- graphical data) Specifyi(nput) oro(utput) [Default is both input and output]. Give one or more columns (or column ranges) separated by commas. AppendT(Absolute calendar time),t(time relative to chosen TIME_EPOCH),x(longitude),y(latitude),g(geographic coordinate), orf(floating point) to each column or column range item.

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 func- tions will be Chebyshev polynomials. These have a numerical advantage in the form of the matrix which must be inverted and allow more accu- rate 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 com- pared. 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. 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-Ioption. 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, setconfidence_levelto 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.

To remove a linear trend from data.xy by ordinary least squares, use:trend1ddata.xy-Fxr-N2 > detrended_data.xy To make the above linear trend robust with respect to outliers, use:trend1ddata.xy-Fxr-N2r> detrended_data.xy To find out how many terms (up to 20, say) in a robust Fourier inter- polant are significant in fitting data.xy, use:trend1ddata.xy-Nf20r-I-V

gmt(l),grdtrend(l),trend2d(l)

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. GMT4.0 1 Oct 2004 TREND1D(l)

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