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**PROGRAM:**

**NAME**

trend1d - Fit a [weighted] [robust] polynomial [and/or Fourier] model for y = f(x) to

xy[w] data

**SYNOPSIS**

**trend1d**[

__table__]

**xymrw|p**[

**p**|

**P**|

**f**|

**F**|

**c**|

**C**|

**s**|

**S**|

**x**]

__n__[,...][

**+l**

__length__][

**+o**

__origin__][

**+r**] [

__xy[w]file__]

[

__condition_number__] [ [

__confidence_level__] ] [ [

__level__] ] [ ] [

**-b**<binary> ] [

**-d**<nodata> ]

[

**-f**<flags> ] [

**-h**<headers> ] [

**-i**<flags> ] [

**-:**[

**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**

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

**-Fp**(i.e., no other of the 5 letters)

to output only the model coefficients.

**-N[p|P|f|F|c|C|s|S|x]**

__n__

**[,...][+l**

__length__

**][+o**

__origin__

**][+r]**

Specify the components of the (possibly mixed) model. Append one or more

comma-separated model components. Each component is of the form

**T**

__n__, where

**T**

indicates the basis function and

__n__indicates 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

__terms__),

**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__-origin and

fundamental period is set to the mid-point and data range, respectively. Change

this using the

**+o**

__origin__and

**+l**

__length__modifiers. We normalize

__x__before evaluating

the basis functions. Basically, the trigonometric bases all use the normalized x'

= (2*pi*(x-

__origin__)/

__length__) while the polynomials use x' = 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

specified 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.

**-C**

__condition_number__

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. ].

**-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**

**...)**

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 iterative robust fit]. [Default reads only the

first 2 columns.]

**-bi[**

__ncols__

**][t]**

**(more**

**...)**

Select native binary input. [Default is 2 (or 3 if

**-W**is set) columns].

**-bo[**

__ncols__

**][**

__type__

**]**

**(more**

**...)**

Select native binary output. [Default is 1-5 columns as given by

**-F**].

**-d[i|o]**

__nodata__

**(more**

**...)**

Replace input columns that equal

__nodata__with NaN and do the reverse on output.

**-f[i|o]**

__colinfo__

**(more**

**...)**

Specify data types of input and/or output columns.

**-h[i|o][**

__n__

**][+c][+d][+r**

__remark__

**][+r**

__title__

**]**

**(more**

**...)**

Skip or produce header record(s).

**-i**

__cols__

**[l][s**

__scale__

**][o**

__offset__

**][,**

__...__

**]**

**(more**

**...)**

Select input columns (0 is first column).

**-:[i|o]**

**(more**

**...)**

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

use just

**-**).

**-+**

**or**

**just**

**+**

Print an extensive usage (help) message, including the explanation 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 options, then

exits.

**--version**

Print GMT version and exit.

**--show-datadir**

Print full path to GMT share directory and exit.

**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, whereas other

values are formatted according to FORMAT_FLOAT_OUT. Be aware that the format in effect can

lead to loss of precision in the 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.

**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 polynomials 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 magnitude 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 + ... 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

**-N**...

**+r**(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, 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 interpolant are significant

in fitting data.xy, use:

gmt trend1d data.xy -Nf20+r -I -V

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