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<title>TREND1D</title>
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<h1 align="center">TREND1D</h1>
<a href="#NAME">NAME</a><br>
<a href="#SYNOPSIS">SYNOPSIS</a><br>
<a href="#DESCRIPTION">DESCRIPTION</a><br>
<a href="#REQUIRED ARGUMENTS">REQUIRED ARGUMENTS</a><br>
<a href="#OPTIONS">OPTIONS</a><br>
<a href="#ASCII FORMAT PRECISION">ASCII FORMAT PRECISION</a><br>
<a href="#REMARKS">REMARKS</a><br>
<a href="#EXAMPLES">EXAMPLES</a><br>
<a href="#SEE ALSO">SEE ALSO</a><br>
<a href="#REFERENCES">REFERENCES</a><br>
<hr>
<h2>NAME
<a name="NAME"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em">trend1d −
Fit a [weighted] [robust] polynomial [or Fourier] model for
y = f(x) to xy[w] data.</p>
<h2>SYNOPSIS
<a name="SYNOPSIS"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em"><b>trend1d
−Fxymrw −N</b>[<b>f</b>]<i>n_model</i>[<b>r</b>]
[ <i>xy[w]file</i> ] [
<b>−C</b><i>condition_number</i> ] [
<b>−H</b>[<b>i</b>][<i>nrec</i>] ] [
<b>−I</b>[<i>confidence_level</i>] ] [ <b>−V</b>
] [ <b>−W</b> ] [ <b>−:</b>[<b>i</b>|<b>o</b>] ]
[
<b>−b</b>[<b>i</b>|<b>o</b>][<b>s</b>|<b>S</b>|<b>d</b>|<b>D</b>[<i>ncol</i>]|<b>c</b>[<i>var1</i><b>/</b><i>...</i>]]
] [ <b>−f</b>[<b>i</b>|<b>o</b>]<i>colinfo</i> ]</p>
<h2>DESCRIPTION
<a name="DESCRIPTION"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em"><b>trend1d</b>
reads x,y [and w] values from the first two [three] columns
on standard input [or <i>xy[w]file</i>] 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.</p>
<h2>REQUIRED ARGUMENTS
<a name="REQUIRED ARGUMENTS"></a>
</h2>
<table width="100%" border="0" rules="none" frame="void"
cellspacing="0" cellpadding="0">
<tr valign="top" align="left">
<td width="11%"></td>
<td width="3%">
<p style="margin-top: 1em"><b>−F</b></p></td>
<td width="8%"></td>
<td width="78%">
<p style="margin-top: 1em">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.</p></td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="3%">
<p><b>−N</b></p></td>
<td width="8%"></td>
<td width="78%">
<p>Specify the number of terms in the model,
<i>n_model</i>, whether to fit a Fourier (<b>−Nf</b>)
or polynomial [Default] model, and append <b>r</b> to do a
robust fit. E.g., a robust quadratic model is
<b>−N</b><i>3</i><b>r</b>.</p> </td></tr>
</table>
<h2>OPTIONS
<a name="OPTIONS"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em"><i>xy[w]file</i></p>
<p style="margin-left:22%;">ASCII [or binary, see
<b>−b</b>] file containing x,y [w] values in the first
2 [3] columns. If no file is specified, <b>trend1d</b> will
read from standard input.</p>
<table width="100%" border="0" rules="none" frame="void"
cellspacing="0" cellpadding="0">
<tr valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p><b>−C</b></p></td>
<td width="7%"></td>
<td width="78%">
<p>Set the maximum allowed condition number for the matrix
solution. <b>trend1d</b> 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 <i>condition_#</i>. [Default:
<i>condition_#</i> = 1.0e06. ].</p></td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p><b>−H</b></p></td>
<td width="7%"></td>
<td width="78%">
<p>Input file(s) has header record(s). If used, the default
number of header records is <b><A HREF="gmtdefaults.html#N_HEADER_RECS">N_HEADER_RECS</A></b>. Use
<b>−Hi</b> if only input data should have header
records [Default will write out header records if the input
data have them]. Blank lines and lines starting with # are
always skipped.</p></td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p><b>−I</b></p></td>
<td width="7%"></td>
<td width="78%">
<p>Iteratively increase the number of model parameters,
starting at one, until <i>n_model</i> is reached or the
reduction in variance of the model is not significant at the
<i>confidence_level</i> level. You may set <b>−I</b>
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.</p> </td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p><b>−V</b></p></td>
<td width="7%"></td>
<td width="78%">
<p>Selects verbose mode, which will send progress reports
to stderr [Default runs "silently"].</p></td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p><b>−W</b></p></td>
<td width="7%"></td>
<td width="78%">
<p>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.]</p> </td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p><b>−:</b></p></td>
<td width="7%"></td>
<td width="78%">
<p>Toggles between (longitude,latitude) and
(latitude,longitude) input and/or output. [Default is
(longitude,latitude)]. Append <b>i</b> to select input only
or <b>o</b> to select output only. [Default affects
both].</p> </td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p><b>−bi</b></p></td>
<td width="7%"></td>
<td width="78%">
<p>Selects binary input. Append <b>s</b> for single
precision [Default is <b>d</b> (double)]. Uppercase <b>S</b>
or <b>D</b> will force byte-swapping. Optionally, append
<i>ncol</i>, the number of columns in your binary input file
if it exceeds the columns needed by the program. Or append
<b>c</b> if the input file is netCDF. Optionally, append
<i>var1</i><b>/</b><i>var2</i><b>/</b><i>...</i> to specify
the variables to be read. [Default is 2 (or 3 if
<b>−W</b> is set) columns].</p></td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p><b>−bo</b></p></td>
<td width="7%"></td>
<td width="78%">
<p>Selects binary output. Append <b>s</b> for single
precision [Default is <b>d</b> (double)]. Uppercase <b>S</b>
or <b>D</b> will force byte-swapping. Optionally, append
<i>ncol</i>, the number of desired columns in your binary
output file. [Default is 1-5 columns as given by
<b>−F</b>].</p> </td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p><b>−f</b></p></td>
<td width="7%"></td>
<td width="78%">
<p>Special formatting of input and/or output columns (time
or geographical data). Specify <b>i</b> or <b>o</b> to make
this apply only to input or output [Default applies to
both]. Give one or more columns (or column ranges) separated
by commas. Append <b>T</b> (absolute calendar time),
<b>t</b> (relative time in chosen <b><A HREF="gmtdefaults.html#TIME_UNIT">TIME_UNIT</A></b> since
<b><A HREF="gmtdefaults.html#TIME_EPOCH">TIME_EPOCH</A></b>), <b>x</b> (longitude), <b>y</b>
(latitude), or <b>f</b> (floating point) to each column or
column range item. Shorthand
<b>−f</b>[<b>i</b>|<b>o</b>]<b>g</b> means
<b>−f</b>[<b>i</b>|<b>o</b>]0<b>x</b>,1<b>y</b>
(geographic coordinates).</p></td></tr>
</table>
<h2>ASCII FORMAT PRECISION
<a name="ASCII FORMAT PRECISION"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em">The ASCII
output formats of numerical data are controlled by
parameters in your .gmtdefaults4 file. Longitude and
latitude are formatted according to
<b><A HREF="gmtdefaults.html#OUTPUT_DEGREE_FORMAT">OUTPUT_DEGREE_FORMAT</A></b>, whereas other values are
formatted according to <b><A HREF="gmtdefaults.html#D_FORMAT">D_FORMAT</A></b>. 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 (<b>−bo</b> if
available) or specify more decimals using the
<b><A HREF="gmtdefaults.html#D_FORMAT">D_FORMAT</A></b> setting.</p>
<h2>REMARKS
<a name="REMARKS"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em">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 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 <b><A HREF="gmtmath.html">gmtmath</A></b>.</p>
<p style="margin-left:11%; margin-top: 1em">The
<b>−Nr</b> (robust) and <b>−I</b> (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
<b>−I</b> 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
<i>confidence_level</i> to zero.</p>
<p style="margin-left:11%; margin-top: 1em">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.</p>
<h2>EXAMPLES
<a name="EXAMPLES"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em">To remove a
linear trend from data.xy by ordinary least squares,
use:</p>
<p style="margin-left:11%; margin-top: 1em"><b>trend1d</b>
data.xy <b>−F</b> xr <b>−N</b> 2 >
detrended_data.xy</p>
<p style="margin-left:11%; margin-top: 1em">To make the
above linear trend robust with respect to outliers, use:</p>
<p style="margin-left:11%; margin-top: 1em"><b>trend1d</b>
data.xy <b>−F</b> xr <b>−N</b> 2<b>r</b> >
detrended_data.xy</p>
<p style="margin-left:11%; margin-top: 1em">To find out how
many terms (up to 20, say) in a robust Fourier interpolant
are significant in fitting data.xy, use:</p>
<p style="margin-left:11%; margin-top: 1em"><b>trend1d</b>
data.xy <b>−Nf</b> 20<b>r −I −V</b></p>
<h2>SEE ALSO
<a name="SEE ALSO"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em"><b><i><A HREF="GMT.html">GMT</A></i></b>(1),
<i><A HREF="gmtmath.html">gmtmath</A></i>(1), <i><A HREF="grdtrend.html">grdtrend</A></i>(1), <i><A HREF="trend2d.html">trend2d</A></i>(1)</p>
<h2>REFERENCES
<a name="REFERENCES"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em">Huber, P. J.,
1964, Robust estimation of a location parameter, <i>Ann.
Math. Stat., 35,</i> 73-101.</p>
<p style="margin-left:11%; margin-top: 1em">Menke, W.,
1989, Geophysical Data Analysis: Discrete Inverse Theory,
Revised Edition, Academic Press, San Diego.</p>
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