Linear regression of 1-D data sets
gmt regress [ table ] [ -A[min/max/inc][+f[n|p]] ] [ -Clevel ] [ -Ex|y|o|r ] [ -Fflags ] [ -N1|2|r|w ] [ -S[r] ] [ -T[min/max/]inc[+n] | -Tfile|list ] [ -W[w][x][y][r] ] [ -V[level] ] [ -aflags ] [ -bbinary ] [ -dnodata ] [ -eregexp ] [ -ggaps ] [ -hheaders ] [ -iflags ] [ -oflags ] [ -qflags ] [ --PAR=value ]
Note: No space is allowed between the option flag and the associated arguments.
regress reads one or more data tables [or stdin] and determines the best linear regression model y = a + b* x for each segment using the chosen parameters. The user may specify which data and model components should be reported. By default, the model will be evaluated at the input points, but alternatively you can specify an equidistant range over which to evaluate the model, or turn off evaluation completely. Instead of determining the best fit we can perform a scan of all possible regression lines (for a range of slope angles) and examine how the chosen misfit measure varies with slope. This is particularly useful when analyzing data with many outliers. Note: If you actually need to work with log10 of x or y you can accomplish that transformation during read by using the -i option.
- One or more ASCII (or binary, see -bi[ncols][type]) data table file(s) holding a number of data columns. If no tables are given then we read from standard input. The first two columns are expected to contain the required x and y data. Depending on your -W and -E settings we may expect an additional 1-3 columns with error estimates of one of both of the data coordinates, and even their correlation.
- There are two uses for this setting: (1) Instead of determining a best-fit regression we explore the full range of regressions. Examine all possible regression lines with slope angles between min and max, using steps of inc degrees [-90/+90/1]. For each slope, the optimum intercept is determined based on your regression type (-E) and misfit norm (-N) settings. For each data segment we report the four columns angle, E, slope, intercept, for the range of specified angles. The best model parameters within this range are written into the segment header and reported in verbose information mode (-Vi). (2) Except for -N2, append +f to force the best regression to only consider the given restricted range of angles [all angles]. As shortcuts for negative or positive slopes, just use +fn or +fp, respectively.
- Set the confidence level (in %) to use for the optional calculation of confidence bands on the regression . This is only used if -F includes the output column c.
- Type of linear regression, i.e., select the type of misfit we should calculate. Choose from x (regress x on y; i.e., the misfit is measured horizontally from data point to regression line), y (regress y on x; i.e., the misfit is measured vertically [Default]), o (orthogonal regression; i.e., the misfit is measured from data point orthogonally to nearest point on the line), or r (Reduced Major Axis regression; i.e., the misfit is the product of both vertical and horizontal misfits) [y].
- Append a combination of the columns you wish returned; the output order will match the order specified. Choose from x (observed x), y (observed y), m (model prediction), r (residual = data minus model), c (symmetrical confidence interval on the regression; see -C for specifying the level), z (standardized residuals or so-called z-scores) and w (outlier weights 0 or 1; for -Nw these are the Reweighted Least Squares weights) [xymrczw]. As an alternative to evaluating the model, just give -Fp and we instead write a single record with the model parameters npoints xmean ymean angle misfit slope intercept sigma_slope sigma_intercept.
- Selects the norm to use for the misfit calculation. Choose among 1 (L-1 measure; the mean of the absolute residuals), 2 (Least-squares; the mean of the squared residuals), r (LMS; The least median of the squared residuals), or w (RLS; Reweighted Least Squares: the mean of the squared residuals after outliers identified via LMS have been removed) [Default is 2]. Traditional regression uses L-2 while L-1 and in particular LMS are more robust in how they handle outliers. As alluded to, RLS implies an initial LMS regression which is then used to identify outliers in the data, assign these a zero weight, and then redo the regression using a L-2 norm.
- Restricts which records will be output. By default all data records will be output in the format specified by -F. Use -S to exclude data points identified as outliers by the regression. Alternatively, use -Sr to reverse this and only output the outlier records.
- -T[min/max/]inc[+n] | -Tfile|list
- Evaluate the best-fit regression model at the equidistant points implied by the arguments. If only -Tinc is given instead we will reset min and max to the extreme x-values for each segment. To skip the model evaluation entirely, simply provide -T0. For details on array creation, see Generate 1D Array.
- Specifies weighted regression and which weights will be provided. Append x if giving 1-sigma uncertainties in the x-observations, y if giving 1-sigma uncertainties in y, and r if giving correlations between x and y observations, in the order these columns appear in the input (after the two required and leading x, y columns). Giving both x and y (and optionally r) implies an orthogonal regression, otherwise giving x requires -Ex and y requires -Ey. We convert uncertainties in x and y to regression weights via the relationship weight = 1/sigma. Use -Ww if the we should interpret the input columns to have precomputed weights instead. Note: Residuals with respect to the regression line will be scaled by the given weights. Most norms will then square this weighted residual (-N1 is the only exception).
- -V[level] (more …)
- Select verbosity level [w].
- -a[col=]name[,…] (more …)
- Set aspatial column associations col=name.
- -bi[ncols][t] (more …)
- Select native binary format for primary input.
- -bo[ncols][type] (more …)
- Select native binary output. [Default is same as input].
- -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.
- -g[a]x|y|d|X|Y|D|[col]zgap[+n|p] (more …)
- Determine data gaps and line breaks.
- -h[i|o][n][+c][+d][+msegheader][+rremark][+ttitle] (more …)
- Skip or produce header record(s).
- -icols[+l][+sscale][+ooffset][,…][,t[word]] (more …)
- Select input columns and transformations (0 is first column, t is trailing text, append word to read one word only).
- -ocols[,…][,t[word]] (more …)
- Select output columns (0 is first column; t is trailing text, append word to write one word only).
- -q[i|o][~]rows[+ccol][+a|f|s] (more …)
- Select input or output rows or data range(s) [all].
- -^ or just -
- Print a short message about the syntax of the command, then exit (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 exit.
- -? or no arguments
- Print a complete usage (help) message, including the explanation of all options, then exit.
- Temporarily override a GMT default setting; repeatable. See gmt.conf for parameters.
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.
Generate 1D Array¶
We will demonstrate the use of options for creating 1-D arrays via gmtmath. Make an evenly spaced coordinate array from min to max in steps of inc, e.g.,:
gmt math -o0 -T3.1/4.2/0.1 T = 3.1 3.2 3.3 3.4 3.5 3.6 3.7
Append +b if we should take log2 of min and max, get their nearest integers, build an equidistant log2-array using inc integer increments in log2, then undo the log2 conversion. E.g., -T3/20/1+b will produce this sequence:
gmt math -o0 -T3/20/1+b T = 4 8 16
Append +l if we should take log10 of min and max and build an array where inc can be 1 (every magnitude), 2, (1, 2, 5 times magnitude) or 3 (1-9 times magnitude). E.g., -T7/135/2+l will produce this sequence:
gmt math -o0 -T7/135/2+l T = 10 20 50 100
For output values less frequently than every magnitude, use a negative integer inc:
gmt math -o0 -T1e-4/1e4/-2+l T = 0.0001 0.01 1 100 10000
Append +n if inc is meant to indicate the number of equidistant coordinates instead. To have exactly 5 equidistant values from 3.44 and 7.82, run:
gmt math -o0 -T3.44/7.82/5+n T = 3.44 4.535 5.63 6.725 7.82
Alternatively, give a file with output coordinates in the first column, or provide a comma-separated list of specific coordinates, such as the first 6 Fibonacci numbers:
gmt math -o0 -T0,1,1,2,3,5 T = 0 1 1 2 3 5
If you only want a single value then you must append a comma to distinguish the list from the setting of inc.
If the module allows you to set up an absolute time series, append a valid time unit from the list year, month, week, day, hour, minute, and second to the given increment; add +t to ensure time column (or use -f). Note: The internal time unit is still controlled independently by TIME_UNIT. The first 7 days of March 2020:
gmt math -o0 -T2020-03-01T/2020-03-07T/1d T = 2020-03-01T00:00:00 2020-03-02T00:00:00 2020-03-03T00:00:00 2020-03-04T00:00:00 2020-03-05T00:00:00 2020-03-06T00:00:00 2020-03-07T00:00:00
A few modules allow for +a which will paste the coordinate array to the output table.
Likewise, if the module allows you to set up a spatial distance series (with distances computed from the first two data columns), specify a new increment as inc with a geospatial distance unit from the list degree (arc), minute (arc), second (arc), meter, foot, kilometer, Miles (statute), nautical miles, or survey foot; see -j for calculation mode. To interpolate Cartesian distances instead, you must use the special unit c.
Finally, if you are only providing an increment and will obtain min and max from the data, then it is possible (max - min)/inc is not an integer, as required. If so, then inc will be adjusted to fit the range. Alternatively, append +e to keep inc exact and adjust max instead (keeping min fixed).
The output segment header will contain all the various statistics we compute for each segment. These are in order: N (number of points), x0 (weighted mean x), y0 (weighted mean y), angle (of line), E (misfit), slope, intercept, sigma_slope, and sigma_intercept. More the standard regression (-Ey) we also report the Pearsonian correlation (r) and coefficient of determination (R).
Note: Below are some examples of valid syntax for this module.
The examples that use remote files (file names starting with
can be cut and pasted into your terminal for testing.
Other commands requiring input files are just dummy examples of the types
of uses that are common but cannot be run verbatim as written.
To return the coordinates on the best-fit orthogonal regression line through the data in the remote file hertzsprung-russell.txt, try:
gmt regress @hertzsprung-russell.txt -Eo -Fxm
To do a standard least-squares regression on the x-y data in points.txt and return x, y, and model prediction with 99% confidence intervals, try
gmt regress points.txt -Fxymc -C99 > points_regressed.txt
To just get the slope for the above regression, try
slope=`gmt regress points.txt -Fp -o5`
To do a reweighted least-squares regression on the data rough.txt and return x, y, model prediction and the RLS weights, try
gmt regress rough.txt -Fxymw > points_regressed.txt
To do an orthogonal least-squares regression on the data crazy.txt but first take the logarithm of both x and y, then return x, y, model prediction and the normalized residuals (z-scores), try
gmt regress crazy.txt -Eo -Fxymz -i0-1l > points_regressed.txt
To examine how the orthogonal LMS misfits vary with angle between 0 and 90 in steps of 0.2 degrees for the same file, try
gmt regress points.txt -A0/90/0.2 -Eo -Nr > points_analysis.txt
To force an orthogonal LMS to pick the best solution with a positive slope, try
gmt regress points.txt -A+fp -Eo -Nr > best_pos_slope.txt
Draper, N. R., and H. Smith, 1998, Applied regression analysis, 3rd ed., 736 pp., John Wiley and Sons, New York.
Rousseeuw, P. J., and A. M. Leroy, 1987, Robust regression and outlier detection, 329 pp., John Wiley and Sons, New York.
York, D., N. M. Evensen, M. L. Martinez, and J. De Basebe Delgado, 2004, Unified equations for the slope, intercept, and standard errors of the best straight line, Am. J. Phys., 72(3), 367-375.