trend2d(1) GMT trend2d(1)

## NAME

trend2d - Fit a [weighted] [robust] polynomial model for z = f(x,y) to xyz[w] data

## SYNOPSIS

trend2d[table]-Fxyzmrw-Nn_model[+r] [xyz[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

trend2dreads x,y,z [and w] values from the first three [four] columns on standard input [orxyz[w]file] and fits a regression model z = f(x,y) + e by [weighted] least squares. 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,y) which significantly reduce the variance in z. n_model may be in [1,10] to fit a model of the following form (similar to grdtrend): m1 + m2*x + m3*y + m4*x*y + m5*x*x + m6*y*y + m7*x*x*x + m8*x*x*y + m9*x*y*y + m10*y*y*y. The user must specify-Nn_model, the number of model parameters to use; thus,-N4fits a bilinear trend,-N6a quadratic surface, and so on. Optionally, append+rto perform a robust fit. In this case, the pro- gram will iteratively reweight the data based on a robust scale esti- mate, in order to converge to a solution insensitive to outliers. This may be handy when separating aaregionalafield from aaresidualawhich should have non-zero mean, such as a local mountain on a regional sur- face.

## REQUIRED ARGUMENTS

-FxyzmrwSpecify up to six letters from the set {xyzmrw} in any order to create columns of ASCII [or binary] output.x= x,y= y,z= z,m= model f(x,y),r= residual z -m,w= weight used in fitting.-Nn_model[+r] Specify the number of terms in the model,n_model, and append+rto do a robust fit. E.g., a robust bilinear model is-N4+r.

## OPTIONAL ARGUMENTS

tableOne or more ASCII [or binary, see-bi] files containing x,y,z [w] values in the first 3 [4] columns. If no files are speci- fied,trend2dwill read from standard input.-Ccondition_numberSet the maximum allowed condition number for the matrix solu- tion.trend2dfits 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. ].-I[confidence_level] Iteratively 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.-V[level] (morea|) Select verbosity level [c].-WWeights are supplied in input column 4. Do a weighted least squares fit [or start with these weights when doing the itera- tive robust fit]. [Default reads only the first 3 columns.]-bi[ncols][t] (morea|) Select native binary input. [Default is 3 (or 4 if-Wis set) input columns].-bo[ncols][type] (morea|) Select native binary output. [Default is 1-6 columns as set by-F].-d[i|o]nodata(morea|) Replace input columns that equalnodatawith NaN and do the reverse on output.-e[~]^<i>apattern^<i>a|-e[~]/regexp/[i] (morea|) Only accept data records that match the given pattern.-f[i|o]colinfo(morea|) Specify data types of input and/or output columns.-h[i|o][n][+c][+d][+rremark][+rtitle] (morea|) Skip or produce header record(s).-icols[+l][+sscale][+ooffset][,^<i>a|] (morea|) Select input columns and transformations (0 is first column).-:[i|o] (morea|) 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.

## REMARKS

The domain of x and y will be shifted and scaled to [-1, 1] and the basis functions are built from Chebyshev polynomials. These have a numerical advantage in the form of the matrix which must be inverted and allow more accurate solutions. In many applications oftrend2dthe user has data located approximately along a line in the x,y plane which makes an angle with the x axis (such as data collected along a road or ship track). In this case the accuracy could be improved by a rotation of the x,y axes.trend2ddoes not search for such a rotation; instead, it may find that the matrix problem has deficient rank. However, the solution is computed using the generalized inverse and should still work out OK. The user should check the results graphically iftrend2dshows deficient rank. NOTE: The model parameters listed with-Vare Chebyshev coefficients; they are not numerically equivalent to the m#s in the equation described above. The description above is to allow the user to match-Nwith the order of the polynomial surface. For evaluat- ing Chebyshev polynomials, see grdmath. The-Nn_modelr(robust) and-I(iterative) options evaluate the signif- icance 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.

## 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 (-boif available) or specify more decimals using the FORMAT_FLOAT_OUT set- ting.

## EXAMPLES

To remove a planar trend from data.xyz by ordinary least squares, use: gmt trend2d data.xyz -Fxyr -N2 > detrended_data.xyz To make the above planar trend robust with respect to outliers, use: gmt trend2d data.xzy -Fxyr -N2+r > detrended_data.xyz To find out how many terms (up to 10 in a robust interpolant are sig- nificant in fitting data.xyz, use: gmt trend2d data.xyz -N10+r -I -V

## SEE ALSO

gmt(1),grdmath(1),grdtrend(1),trend1d(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 trend2d(1)

gmt5 5.4.2 - Generated Thu Jun 29 16:44:59 CDT 2017