/
shmtvaropt.html
183 lines (129 loc) · 7.11 KB
/
shmtvaropt.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN"
"http://www.w3.org/TR/html4/strict.dtd">
<html>
<head>
<title>SHTOOLS - Localized spectral analysis</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<link rel="stylesheet" type="text/css" href="../../css/sh.css">
<link rel="icon" type="image/vnd.microsoft.icon" href="../../images/favicon.ico">
</head>
<body>
<div class="main">
<p class="centeredimage"><img src="../../images/logo.jpg" width=894 height=135 alt="SHTOOLS --- Tools for working with spherical harmonics"></p>
<table class="menu">
<tbody>
<tr>
<td><a href="http://shtools.ipgp.fr/">HOME</a></td>
<td><a href="https://github.com/SHTOOLS/SHTOOLS/releases">DOWNLOAD</a></td>
<td class="selected"><a href="../../documentation.html">DOCUMENTATION</a></td>
<td><a href="../../faq.html">FAQ</a> </td>
</tr>
</tbody>
</table>
<p class="dir">
> <a href="../../../index.html" class="dir">Home</a> > <a href="../../documentation.html" class="dir">Documentation</a> > <a href="../../localized.html" class="dir">Localized Spectral Analysis</a></p>
<PRE>
<!-- Manpage converted by man2html 3.0.1 -->
<B>SHMTVAROPT(1)</B> SHTOOLS 3.0 <B>SHMTVAROPT(1)</B>
</PRE>
<H2 class="man">SHMTVarOpt</H2 class="man"><PRE>
SHMTVarOpt - Calculate the minimum variance and corresponding optimal
weights of a localized multitaper spectral estimate.
</PRE>
<H2 class="man">SYNOPSIS</H2 class="man"><PRE>
SUBROUTINE SHMTVarOpt ( L, TAPERS, TAPER_ORDER, LWIN, KMAX, SFF,
VAR_OPT, VAR_UNIT, WEIGHT_OPT,
UNWEIGHTED_COVAR, NOCROSS )
REAL*8 TAPERS(LWIN+1, KMAX), SFF(L+WIN+1),
VAR_OPT(KMAX), VAR_UNIT(KMAX)
INTEGER L, TAPER_ORDER(KMAX), LWIN, KMAX
REAL*8, OPTIONAL WEIGHT_OPT(KMAX, KMAX),
UNWEIGHTED_COVAR(KMAX, KMAX)
INTEGER, OPTIONAL NOCROSS
</PRE>
<H2 class="man">DESCRIPTION</H2 class="man"><PRE>
<B>SHMTVarOpt</B> will determine the minimum variance that can be achieved by
a weighted multitaper spectral analysis, as is described by Wieczorek
and Simons (2006, submitted manuscript). The minimum variance is output
as a function of the number of tapers utilized, from 1 to a maximum of
KMAX, and the corresponding variance using equal weights is output for
comparison. The windowing functions are assumed to be solutions to the
spherical-cap concentration problem, as determined by a call to
<B>SHReturnTapers</B> or <B>SHReturnTapersM</B>. The minimum variance and weights are
dependent upon the form of the global unwindowed power spectrum, SFF.
If the optional argument WEIGHT_OPT is specified, then the optimal
weights will be returned as a function of the number of tapers
employed, from 1 KMAX. If UNWEIGHTED_COVAR is specified, then the
unweighted covariance matrix of the KMAX tapers (i.e., Fij) will be
output. If the optional argument NOCROSS is set to 1, then only the
diagnonal terms of Fij will be computed.
</PRE>
<H2 class="man">ARGUMENTS</H2 class="man"><PRE>
L (input) INTEGER
The angular degree to determine the minimum variance
and optimal weights.
TAPERS (input) REAL*8, DIMENSION (LWIN+1, KMAX)
A matrix of localization functions obtained from
<B>SHReturnTapers</B> or <B>SHReturnTapersM</B>.
TAPER_ORDER (input) INTEGER, DIMENSION (K)
The angular order of the windowing coefficients in
TAPERS. If this matrix was created using
<B>SHReturnTapersM</B>, then this array must be composed of
zeros.
LWIN (input) INTEGER
The spherical harmonic bandwidth of the localizing
windows.
KMAX (input) INTEGER
The maximum number of tapers to be used when
calculating the minimum variance and optimal weights.
All values from 1 to KMAX will be returned.
SFF (input) REAL*8, DIMENSION (L+LWIN+1)
The global unwindowed power spectrum of the function
to be localized.
VAR_OPT (output) REAL*8, DIMENSION (KMAX)
The minimum variance of the multitaper spectral
estimate for degree L using 1 through KMAX tapers.
VAR_UNIT (output) REAL*8, DIMENSION (KMAX)
The variance of the multitaper spectral estimate
using equal weights for degree L using 1 through KMAX
tapers.
WEIGHT_OPT (output) REAL*8, OPTIONAL, DIMENSION (KMAX, KMAX)
The optimal weights (in columns) that minimize the
multitaper spectral estimate's variance using 1
through KMAX tapers.
UNWEIGHTED_COVAR (output) REAL*8, OPTIONAL, DIMENSION (KMAX, KMAX)
The unweighted covariance matrix of the KMAX tapers
(i.e., Fij in Wieczorek and Simons 2007).
NOCROSS (input) INTEGER, OPTIONAL
If 1, only the diagonal terms of the covariance
matrix Fij will be computed. If 0, all terms will be
computed.
</PRE>
<H2 class="man">SEE ALSO</H2 class="man"><PRE>
<B>shreturntapers(1)</B>, <B>shreturntapersm(1)</B>, <B>shmtvaropt0(1)</B>
<http://shtools.ipgp.fr/>
</PRE>
<H2 class="man">REFERENCES</H2 class="man"><PRE>
Wieczorek, M. A. and F. J. Simons, Minimum variance multitaper spectral
estimation on the sphere, <B>J.</B> <B>Fourier</B> <B>Anal.</B> <B>Appl.</B>, submitted 2006.
</PRE>
<H2 class="man">COPYRIGHT AND LICENSE</H2 class="man"><PRE>
Copyright 2012 by Mark Wieczorek <wieczor@ipgp.fr>.
This is free software; you can distribute and modify it under the terms
of the revised BSD license.
SHTOOLS 3.0 2014-09-12 <B>SHMTVAROPT(1)</B>
</PRE>
<p class="dir">
> <a href="../../../index.html" class="dir">Home</a> > <a href="../../documentation.html" class="dir">Documentation</a> > <a href="../../localized.html" class="dir">Localized Spectral Analysis</a></p>
<table class="footer2" summary = "SHTOOLS; Fortran and Python spherical harmonic transform software package">
<tbody>
<tr>
<td class="c1"><a href="http://www.ipgp.fr/">Institut de Physique du Globe de Paris</a></td>
<td class="c2"><a href="http://www.sorbonne-paris-cite.fr/index.php/en">University of Sorbonne Paris Cité</a></td>
<td class="c3">© 2015 <a href="http://www.ipgp.fr/~wieczor">Mark Wieczorek</a></td>
</tr>
</tbody>
</table>
</div>
</body>
</html>