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<title>TREND2D</title>
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<h1 align="center">TREND2D</h1>
<a href="#NAME">NAME</a><br>
<a href="#SYNOPSIS">SYNOPSIS</a><br>
<a href="#DESCRIPTION">DESCRIPTION</a><br>
<a href="#OPTIONS">OPTIONS</a><br>
<a href="#REMARKS">REMARKS</a><br>
<a href="#ASCII FORMAT PRECISION">ASCII FORMAT PRECISION</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">trend2d −
Fit a [weighted] [robust] polynomial model for z = f(x,y) to
xyz[w] data.</p>
<h2>SYNOPSIS
<a name="SYNOPSIS"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em"><b>trend2d
−Fxyzmrw −N</b><i>n_model</i>[<b>r</b>] [
<i>xyz[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>trend2d</b>
reads x,y,z [and w] values from the first three [four]
columns on standard input [or <i>xyz[w]file</i>] 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):</p>
<p style="margin-left:11%; margin-top: 1em">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.</p>
<p style="margin-left:11%; margin-top: 1em">The user must
specify <b>−N</b><i>n_model</i>, the number of model
parameters to use; thus, <b>−N</b><i>4</i> fits a
bilinear trend, <b>−N</b><i>6</i> a quadratic surface,
and so on. Optionally, append <b>r</b> to perform a robust
fit. In this case, the program will iteratively reweight the
data based on a robust scale estimate, in order to converge
to a solution insensitive to outliers. This may be handy
when separating a "regional" field from a
"residual" which should have non-zero mean, such
as a local mountain on a regional surface.</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="3%">
<p><b>−F</b></p></td>
<td width="8%"></td>
<td width="78%">
<p>Specify up to six letters from the set {x y z m r w} 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.</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>, and append <b>r</b> to do a robust fit.
E.g., a robust bilinear model is
<b>−N</b><i>4</i><b>r</b>.</p> </td></tr>
</table>
<h2>OPTIONS
<a name="OPTIONS"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em"><i>xyz[w]file</i></p>
<p style="margin-left:22%;">ASCII [or binary, see
<b>−b</b>] file containing x,y,z [w] values in the
first 3 [4] columns. If no file is specified, <b>trend2d</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>trend2d</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 4. Do a weighted
least squares fit [or start with these weights when doing
the iterative robust fit]. [Default reads only the first 3
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 3 (or 4 if
<b>−W</b> is set) input 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 1-6 columns as set 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>REMARKS
<a name="REMARKS"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em">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 of <b>trend2d</b> the 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. <b>trend2d</b> does 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 if
<b>trend2d</b> shows deficient rank. NOTE: The model
parameters listed with <b>−V</b> are 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 <b>−N</b> with the order of
the polynomial surface. For evaluating Chebyshev
polynomials, see <b><A HREF="grdmath.html">grdmath</A></b>.</p>
<p style="margin-left:11%; margin-top: 1em">The
<b>−N</b><i>n_model</i><b>r</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>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>EXAMPLES
<a name="EXAMPLES"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em">To remove a
planar trend from data.xyz by ordinary least squares,
use:</p>
<p style="margin-left:11%; margin-top: 1em"><b>trend2d</b>
data.xyz <b>−F</b> xyr <b>−N</b> 2 >
detrended_data.xyz</p>
<p style="margin-left:11%; margin-top: 1em">To make the
above planar trend robust with respect to outliers, use:</p>
<p style="margin-left:11%; margin-top: 1em"><b>trend2d</b>
data.xzy <b>−F</b> xyr <b>−N</b> 2<b>r</b> >
detrended_data.xyz</p>
<p style="margin-left:11%; margin-top: 1em">To find out how
many terms (up to 10) in a robust interpolant are
significant in fitting data.xyz, use:</p>
<p style="margin-left:11%; margin-top: 1em"><b>trend2d</b>
data.xyz <b>−N</b> 10<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="grdmath.html">grdmath</A></i>(1), <i><A HREF="grdtrend.html">grdtrend</A></i>(1), <i><A HREF="trend1d.html">trend1d</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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