<!-- Creator : groff version 1.19.2 -->
<!-- CreationDate: Wed Jan 2 08:40:10 2013 -->
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
"http://www.w3.org/TR/html4/loose.dtd">
<html>
<head>
<meta name="generator" content="groff -Thtml, see www.gnu.org">
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta name="Content-Style" content="text/css">
<style type="text/css">
p { margin-top: 0; margin-bottom: 0; }
pre { margin-top: 0; margin-bottom: 0; }
table { margin-top: 0; margin-bottom: 0; }
</style>
<title>TREND2D</title>
</head>
<body bgcolor="#ffffff">
<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>
<a name="NAME"></a>
<h2>NAME</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>
<a name="SYNOPSIS"></a>
<h2>SYNOPSIS</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>
<a name="DESCRIPTION"></a>
<h2>DESCRIPTION</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 style="margin-top: 1em" valign="top"><b>−F</b></p> </td>
<td width="8%"></td>
<td width="78%">
<p style="margin-top: 1em" valign="top">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 valign="top" align="left">
<td width="11%"></td>
<td width="3%">
<p style="margin-top: 1em" valign="top"><b>−N</b></p> </td>
<td width="8%"></td>
<td width="78%">
<p style="margin-top: 1em" valign="top">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>
</table>
<a name="OPTIONS"></a>
<h2>OPTIONS</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 style="margin-top: 1em" valign="top"><b>−C</b></p> </td>
<td width="7%"></td>
<td width="78%">
<p style="margin-top: 1em" valign="top">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 valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p style="margin-top: 1em" valign="top"><b>−H</b></p> </td>
<td width="7%"></td>
<td width="78%">
<p style="margin-top: 1em" valign="top">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 valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p style="margin-top: 1em" valign="top"><b>−I</b></p> </td>
<td width="7%"></td>
<td width="78%">
<p style="margin-top: 1em" valign="top">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 valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p style="margin-top: 1em" valign="top"><b>−V</b></p> </td>
<td width="7%"></td>
<td width="78%">
<p style="margin-top: 1em" valign="top">Selects verbose
mode, which will send progress reports to stderr [Default
runs "silently"].</p></td>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p style="margin-top: 1em" valign="top"><b>−W</b></p> </td>
<td width="7%"></td>
<td width="78%">
<p style="margin-top: 1em" valign="top">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 valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p style="margin-top: 1em" valign="top"><b>−:</b></p> </td>
<td width="7%"></td>
<td width="78%">
<p style="margin-top: 1em" valign="top">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 valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p style="margin-top: 1em" valign="top"><b>−bi</b></p> </td>
<td width="7%"></td>
<td width="78%">
<p style="margin-top: 1em" valign="top">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 valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p style="margin-top: 1em" valign="top"><b>−bo</b></p> </td>
<td width="7%"></td>
<td width="78%">
<p style="margin-top: 1em" valign="top">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 valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p style="margin-top: 1em" valign="top"><b>−f</b></p> </td>
<td width="7%"></td>
<td width="78%">
<p style="margin-top: 1em" valign="top">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>
</table>
<a name="REMARKS"></a>
<h2>REMARKS</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>
<a name="ASCII FORMAT PRECISION"></a>
<h2>ASCII FORMAT PRECISION</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>
<a name="EXAMPLES"></a>
<h2>EXAMPLES</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>
<a name="SEE ALSO"></a>
<h2>SEE ALSO</h2>
<p style="margin-left:11%; margin-top: 1em"><i><A HREF="GMT.html">GMT</A></i>(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>
<a name="REFERENCES"></a>
<h2>REFERENCES</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>
<hr>
</body>
</html>
distrib
>
Fedora
>
18
>
x86_64
>
media
>
updates
>
by-pkgid
>
fa8759c1f2240ce030f5ac6d5041a2a1
>
files
>
472
