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61<h1>Miscellaneous Statistics Functions<br>
62<small>
63[<a class="el" href="group__stat.html">Statistics Module</a>]</small>
64</h1><table border="0" cellpadding="0" cellspacing="0">
65<tr><td></td></tr>
66<tr><td colspan="2"><br><h2>Classes</h2></td></tr>
67<tr><td class="memItemLeft" nowrap align="right" valign="top">class &nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="classitpp_1_1Stat.html">itpp::Stat</a></td></tr>
68
69<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">A class for sampling a signal and calculating statistics.  <a href="classitpp_1_1Stat.html#_details">More...</a><br></td></tr>
70<tr><td colspan="2"><br><h2>Functions</h2></td></tr>
71<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="g0d54de7e2151c01c03d2a67728d9348f"></a><!-- doxytag: member="statistics::mean" ref="g0d54de7e2151c01c03d2a67728d9348f" args="(const vec &amp;v)" -->
72double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#g0d54de7e2151c01c03d2a67728d9348f">itpp::mean</a> (const <a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a> &amp;v)</td></tr>
73
74<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">The mean value. <br></td></tr>
75<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="gaaf3e3c616250ca3ca5978f48cbb5dd1"></a><!-- doxytag: member="statistics::mean" ref="gaaf3e3c616250ca3ca5978f48cbb5dd1" args="(const cvec &amp;v)" -->
76std::complex&lt; double &gt;&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#gaaf3e3c616250ca3ca5978f48cbb5dd1">itpp::mean</a> (const <a class="el" href="classitpp_1_1Vec.html#e83c1408740e41a7e29c383b71d4d544">cvec</a> &amp;v)</td></tr>
77
78<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">The mean value. <br></td></tr>
79<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="g7afe5ec5250289ad54c4781a7dd33106"></a><!-- doxytag: member="statistics::mean" ref="g7afe5ec5250289ad54c4781a7dd33106" args="(const svec &amp;v)" -->
80double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#g7afe5ec5250289ad54c4781a7dd33106">itpp::mean</a> (const <a class="el" href="classitpp_1_1Vec.html#7d53e7d4b228c818e8ade5c59c4ed9d9">svec</a> &amp;v)</td></tr>
81
82<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">The mean value. <br></td></tr>
83<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="g4b54d91471c62a9e0762fa1f11c54747"></a><!-- doxytag: member="statistics::mean" ref="g4b54d91471c62a9e0762fa1f11c54747" args="(const ivec &amp;v)" -->
84double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#g4b54d91471c62a9e0762fa1f11c54747">itpp::mean</a> (const <a class="el" href="classitpp_1_1Vec.html#b03757d874926a9be91535e71af1656e">ivec</a> &amp;v)</td></tr>
85
86<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">The mean value. <br></td></tr>
87<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="gf6b99073aa2aa917bed339de7c346f9b"></a><!-- doxytag: member="statistics::mean" ref="gf6b99073aa2aa917bed339de7c346f9b" args="(const mat &amp;m)" -->
88double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#gf6b99073aa2aa917bed339de7c346f9b">itpp::mean</a> (const <a class="el" href="classitpp_1_1Mat.html#6bba394f181c76fda12759568986c613">mat</a> &amp;m)</td></tr>
89
90<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">The mean value. <br></td></tr>
91<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="g29d3c842d5588b90771a0165d70049d4"></a><!-- doxytag: member="statistics::mean" ref="g29d3c842d5588b90771a0165d70049d4" args="(const cmat &amp;m)" -->
92std::complex&lt; double &gt;&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#g29d3c842d5588b90771a0165d70049d4">itpp::mean</a> (const <a class="el" href="classitpp_1_1Mat.html#6fbac4b7184807da188e5b85d42f038b">cmat</a> &amp;m)</td></tr>
93
94<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">The mean value. <br></td></tr>
95<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="ga25d48abcbd380b02e9cc754f3ffcca5"></a><!-- doxytag: member="statistics::mean" ref="ga25d48abcbd380b02e9cc754f3ffcca5" args="(const smat &amp;m)" -->
96double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#ga25d48abcbd380b02e9cc754f3ffcca5">itpp::mean</a> (const <a class="el" href="classitpp_1_1Mat.html#debd320317a18c022d3ca856af6d3687">smat</a> &amp;m)</td></tr>
97
98<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">The mean value. <br></td></tr>
99<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="gc370b4cfdc6baa037b7442ae6e2c4b5c"></a><!-- doxytag: member="statistics::mean" ref="gc370b4cfdc6baa037b7442ae6e2c4b5c" args="(const imat &amp;m)" -->
100double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#gc370b4cfdc6baa037b7442ae6e2c4b5c">itpp::mean</a> (const <a class="el" href="classitpp_1_1Mat.html#9e0dd2f635e4bd0b28c09685d9c4f3bb">imat</a> &amp;m)</td></tr>
101
102<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">The mean value. <br></td></tr>
103<tr><td class="memTemplParams" nowrap colspan="2"><a class="anchor" name="gb694fa4cd6dcdb3b4b2dfbb7dbf14eb5"></a><!-- doxytag: member="statistics::geometric_mean" ref="gb694fa4cd6dcdb3b4b2dfbb7dbf14eb5" args="(const Vec&lt; T &gt; &amp;v)" -->
104template&lt;class T &gt; </td></tr>
105<tr><td class="memTemplItemLeft" nowrap align="right" valign="top">double&nbsp;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="group__statistics.html#gb694fa4cd6dcdb3b4b2dfbb7dbf14eb5">itpp::geometric_mean</a> (const Vec&lt; T &gt; &amp;v)</td></tr>
106
107<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">The geometric mean of a vector. <br></td></tr>
108<tr><td class="memTemplParams" nowrap colspan="2"><a class="anchor" name="g520b83173c482245fb13e718f94a6c82"></a><!-- doxytag: member="statistics::geometric_mean" ref="g520b83173c482245fb13e718f94a6c82" args="(const Mat&lt; T &gt; &amp;m)" -->
109template&lt;class T &gt; </td></tr>
110<tr><td class="memTemplItemLeft" nowrap align="right" valign="top">double&nbsp;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="group__statistics.html#g520b83173c482245fb13e718f94a6c82">itpp::geometric_mean</a> (const Mat&lt; T &gt; &amp;m)</td></tr>
111
112<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">The geometric mean of a matrix. <br></td></tr>
113<tr><td class="memTemplParams" nowrap colspan="2"><a class="anchor" name="gfa60b547d34225cdd35adccf91775d9f"></a><!-- doxytag: member="statistics::median" ref="gfa60b547d34225cdd35adccf91775d9f" args="(const Vec&lt; T &gt; &amp;v)" -->
114template&lt;class T &gt; </td></tr>
115<tr><td class="memTemplItemLeft" nowrap align="right" valign="top">double&nbsp;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="group__statistics.html#gfa60b547d34225cdd35adccf91775d9f">itpp::median</a> (const Vec&lt; T &gt; &amp;v)</td></tr>
116
117<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">The median. <br></td></tr>
118<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="g8017c2e680dbd2f56c8bbf1572337b29"></a><!-- doxytag: member="statistics::norm" ref="g8017c2e680dbd2f56c8bbf1572337b29" args="(const cvec &amp;v)" -->
119double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#g8017c2e680dbd2f56c8bbf1572337b29">itpp::norm</a> (const <a class="el" href="classitpp_1_1Vec.html#e83c1408740e41a7e29c383b71d4d544">cvec</a> &amp;v)</td></tr>
120
121<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Calculate the 2-norm: norm(v)=sqrt(sum(abs(v).^2)). <br></td></tr>
122<tr><td class="memTemplParams" nowrap colspan="2"><a class="anchor" name="g2cd659a539e6a85709a6de1a143e4618"></a><!-- doxytag: member="statistics::norm" ref="g2cd659a539e6a85709a6de1a143e4618" args="(const Vec&lt; T &gt; &amp;v)" -->
123template&lt;class T &gt; </td></tr>
124<tr><td class="memTemplItemLeft" nowrap align="right" valign="top">double&nbsp;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="group__statistics.html#g2cd659a539e6a85709a6de1a143e4618">itpp::norm</a> (const Vec&lt; T &gt; &amp;v)</td></tr>
125
126<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Calculate the 2-norm: norm(v)=sqrt(sum(abs(v).^2)). <br></td></tr>
127<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="g516b973d7e8a792c9a22e0b42d3bb00b"></a><!-- doxytag: member="statistics::norm" ref="g516b973d7e8a792c9a22e0b42d3bb00b" args="(const cvec &amp;v, int p)" -->
128double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#g516b973d7e8a792c9a22e0b42d3bb00b">itpp::norm</a> (const <a class="el" href="classitpp_1_1Vec.html#e83c1408740e41a7e29c383b71d4d544">cvec</a> &amp;v, int p)</td></tr>
129
130<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Calculate the p-norm: norm(v,p)=sum(abs(v).^2)^(1/p). <br></td></tr>
131<tr><td class="memTemplParams" nowrap colspan="2"><a class="anchor" name="g1ca85f4f5c9bf6260a6b41e10a878e31"></a><!-- doxytag: member="statistics::norm" ref="g1ca85f4f5c9bf6260a6b41e10a878e31" args="(const Vec&lt; T &gt; &amp;v, int p)" -->
132template&lt;class T &gt; </td></tr>
133<tr><td class="memTemplItemLeft" nowrap align="right" valign="top">double&nbsp;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="group__statistics.html#g1ca85f4f5c9bf6260a6b41e10a878e31">itpp::norm</a> (const Vec&lt; T &gt; &amp;v, int p)</td></tr>
134
135<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Calculate the p-norm: norm(v,p)=sum(abs(v).^2)^(1/p). <br></td></tr>
136<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="g01d46dbf6c159d444b0dac31e80b21fc"></a><!-- doxytag: member="statistics::norm" ref="g01d46dbf6c159d444b0dac31e80b21fc" args="(const cvec &amp;v, const std::string &amp;s)" -->
137double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#g01d46dbf6c159d444b0dac31e80b21fc">itpp::norm</a> (const <a class="el" href="classitpp_1_1Vec.html#e83c1408740e41a7e29c383b71d4d544">cvec</a> &amp;v, const std::string &amp;s)</td></tr>
138
139<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Calculate the Frobenius norm for s = "fro" (equal to 2-norm). <br></td></tr>
140<tr><td class="memTemplParams" nowrap colspan="2"><a class="anchor" name="gc2a8340331beb29e8f97baa00b2e31b8"></a><!-- doxytag: member="statistics::norm" ref="gc2a8340331beb29e8f97baa00b2e31b8" args="(const Vec&lt; T &gt; &amp;v, const std::string &amp;s)" -->
141template&lt;class T &gt; </td></tr>
142<tr><td class="memTemplItemLeft" nowrap align="right" valign="top">double&nbsp;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="group__statistics.html#gc2a8340331beb29e8f97baa00b2e31b8">itpp::norm</a> (const Vec&lt; T &gt; &amp;v, const std::string &amp;s)</td></tr>
143
144<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Calculate the Frobenius norm for s = "fro" (equal to 2-norm). <br></td></tr>
145<tr><td class="memItemLeft" nowrap align="right" valign="top">double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#g33961993e1531da45cc4a3c02a6dd124">itpp::norm</a> (const <a class="el" href="classitpp_1_1Mat.html#6bba394f181c76fda12759568986c613">mat</a> &amp;m, int p)</td></tr>
146
147<tr><td class="memItemLeft" nowrap align="right" valign="top">double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#g831b04981a856a72f8f669bff13fd1e3">itpp::norm</a> (const <a class="el" href="classitpp_1_1Mat.html#6fbac4b7184807da188e5b85d42f038b">cmat</a> &amp;m, int p)</td></tr>
148
149<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="g23a0b74b4713026fbeebc7cc0d7e19bd"></a><!-- doxytag: member="statistics::norm" ref="g23a0b74b4713026fbeebc7cc0d7e19bd" args="(const mat &amp;m, const std::string &amp;s)" -->
150double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#g23a0b74b4713026fbeebc7cc0d7e19bd">itpp::norm</a> (const <a class="el" href="classitpp_1_1Mat.html#6bba394f181c76fda12759568986c613">mat</a> &amp;m, const std::string &amp;s)</td></tr>
151
152<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Calculate the Frobenius norm of a matrix for s = "fro". <br></td></tr>
153<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="ged268405b2969894507b862c8e2052f3"></a><!-- doxytag: member="statistics::norm" ref="ged268405b2969894507b862c8e2052f3" args="(const cmat &amp;m, const std::string &amp;s)" -->
154double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#ged268405b2969894507b862c8e2052f3">itpp::norm</a> (const <a class="el" href="classitpp_1_1Mat.html#6fbac4b7184807da188e5b85d42f038b">cmat</a> &amp;m, const std::string &amp;s)</td></tr>
155
156<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Calculate the Frobenius norm of a matrix for s = "fro". <br></td></tr>
157<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="g9245a3ae0b0ea33e57991d449abd1c9e"></a><!-- doxytag: member="statistics::variance" ref="g9245a3ae0b0ea33e57991d449abd1c9e" args="(const cvec &amp;v)" -->
158double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#g9245a3ae0b0ea33e57991d449abd1c9e">itpp::variance</a> (const <a class="el" href="classitpp_1_1Vec.html#e83c1408740e41a7e29c383b71d4d544">cvec</a> &amp;v)</td></tr>
159
160<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">The variance of the elements in the vector. Normalized with N-1 to be unbiased. <br></td></tr>
161<tr><td class="memTemplParams" nowrap colspan="2"><a class="anchor" name="g7a4799948ff9779eaeecabde0ce5fcbb"></a><!-- doxytag: member="statistics::variance" ref="g7a4799948ff9779eaeecabde0ce5fcbb" args="(const Vec&lt; T &gt; &amp;v)" -->
162template&lt;class T &gt; </td></tr>
163<tr><td class="memTemplItemLeft" nowrap align="right" valign="top">double&nbsp;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="group__statistics.html#g7a4799948ff9779eaeecabde0ce5fcbb">itpp::variance</a> (const Vec&lt; T &gt; &amp;v)</td></tr>
164
165<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">The variance of the elements in the vector. Normalized with N-1 to be unbiased. <br></td></tr>
166<tr><td class="memTemplParams" nowrap colspan="2"><a class="anchor" name="ga45f7193097f878a35b65b591690d0d0"></a><!-- doxytag: member="statistics::energy" ref="ga45f7193097f878a35b65b591690d0d0" args="(const Vec&lt; T &gt; &amp;v)" -->
167template&lt;class T &gt; </td></tr>
168<tr><td class="memTemplItemLeft" nowrap align="right" valign="top">double&nbsp;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="group__statistics.html#ga45f7193097f878a35b65b591690d0d0">itpp::energy</a> (const Vec&lt; T &gt; &amp;v)</td></tr>
169
170<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Calculate the energy: squared 2-norm. energy(v)=sum(abs(v).^2). <br></td></tr>
171<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="g38b936ea401045e0667c193fb0e0da8c"></a><!-- doxytag: member="statistics::within_tolerance" ref="g38b936ea401045e0667c193fb0e0da8c" args="(double x, double xref, double tol=1e&#45;14)" -->
172bool&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#g38b936ea401045e0667c193fb0e0da8c">itpp::within_tolerance</a> (double x, double xref, double tol=1e-14)</td></tr>
173
174<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Return true if the input value <code>x</code> is within the tolerance <code>tol</code> of the reference value <code>xref</code>. <br></td></tr>
175<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="gd45b87b809750f8e464f2d518c7b9d89"></a><!-- doxytag: member="statistics::within_tolerance" ref="gd45b87b809750f8e464f2d518c7b9d89" args="(std::complex&lt; double &gt; x, std::complex&lt; double &gt; xref, double tol=1e&#45;14)" -->
176bool&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#gd45b87b809750f8e464f2d518c7b9d89">itpp::within_tolerance</a> (std::complex&lt; double &gt; x, std::complex&lt; double &gt; xref, double tol=1e-14)</td></tr>
177
178<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Return true if the input value <code>x</code> is within the tolerance <code>tol</code> of the reference value <code>xref</code>. <br></td></tr>
179<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="gd87b1fc7b3e0a18bd2d450dc2883a169"></a><!-- doxytag: member="statistics::within_tolerance" ref="gd87b1fc7b3e0a18bd2d450dc2883a169" args="(const vec &amp;x, const vec &amp;xref, double tol=1e&#45;14)" -->
180bool&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#gd87b1fc7b3e0a18bd2d450dc2883a169">itpp::within_tolerance</a> (const <a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a> &amp;x, const <a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a> &amp;xref, double tol=1e-14)</td></tr>
181
182<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Return true if the input vector <code>x</code> is elementwise within the tolerance <code>tol</code> of the reference vector <code>xref</code>. <br></td></tr>
183<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="g5acad6199520ccfc17fc956791e116f9"></a><!-- doxytag: member="statistics::within_tolerance" ref="g5acad6199520ccfc17fc956791e116f9" args="(const cvec &amp;x, const cvec &amp;xref, double tol=1e&#45;14)" -->
184bool&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#g5acad6199520ccfc17fc956791e116f9">itpp::within_tolerance</a> (const <a class="el" href="classitpp_1_1Vec.html#e83c1408740e41a7e29c383b71d4d544">cvec</a> &amp;x, const <a class="el" href="classitpp_1_1Vec.html#e83c1408740e41a7e29c383b71d4d544">cvec</a> &amp;xref, double tol=1e-14)</td></tr>
185
186<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Return true if the input vector <code>x</code> is elementwise within the tolerance <code>tol</code> of the reference vector <code>xref</code>. <br></td></tr>
187<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="gb35370c1a742e895ec4584e6fd6100ab"></a><!-- doxytag: member="statistics::within_tolerance" ref="gb35370c1a742e895ec4584e6fd6100ab" args="(const mat &amp;X, const mat &amp;Xref, double tol=1e&#45;14)" -->
188bool&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#gb35370c1a742e895ec4584e6fd6100ab">itpp::within_tolerance</a> (const <a class="el" href="classitpp_1_1Mat.html#6bba394f181c76fda12759568986c613">mat</a> &amp;X, const <a class="el" href="classitpp_1_1Mat.html#6bba394f181c76fda12759568986c613">mat</a> &amp;Xref, double tol=1e-14)</td></tr>
189
190<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Return true if the input matrix <code>X</code> is elementwise within the tolerance <code>tol</code> of the reference matrix <code>Xref</code>. <br></td></tr>
191<tr><td class="memItemLeft" nowrap align="right" valign="top"><a class="anchor" name="g5a534abd4196839b8669b5679d5ab7c8"></a><!-- doxytag: member="statistics::within_tolerance" ref="g5a534abd4196839b8669b5679d5ab7c8" args="(const cmat &amp;X, const cmat &amp;Xref, double tol=1e&#45;14)" -->
192bool&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#g5a534abd4196839b8669b5679d5ab7c8">itpp::within_tolerance</a> (const <a class="el" href="classitpp_1_1Mat.html#6fbac4b7184807da188e5b85d42f038b">cmat</a> &amp;X, const <a class="el" href="classitpp_1_1Mat.html#6fbac4b7184807da188e5b85d42f038b">cmat</a> &amp;Xref, double tol=1e-14)</td></tr>
193
194<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Return true if the input matrix <code>X</code> is elementwise within the tolerance <code>tol</code> of the reference matrix <code>Xref</code>. <br></td></tr>
195<tr><td class="memItemLeft" nowrap align="right" valign="top">double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#gde571eebaa7dbd8289b6d7184fe625ca">itpp::moment</a> (const <a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a> &amp;x, const int r)</td></tr>
196
197<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Calculate the central moment of vector x.  <a href="#gde571eebaa7dbd8289b6d7184fe625ca"></a><br></td></tr>
198<tr><td class="memItemLeft" nowrap align="right" valign="top">double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#g43f3c117ed93400ceed5bdacf37f49ea">itpp::skewness</a> (const <a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a> &amp;x)</td></tr>
199
200<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Calculate the skewness excess of the input vector x.  <a href="#g43f3c117ed93400ceed5bdacf37f49ea"></a><br></td></tr>
201<tr><td class="memItemLeft" nowrap align="right" valign="top">double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#g00f842b0e850195093781692a6aaaac3">itpp::kurtosisexcess</a> (const <a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a> &amp;x)</td></tr>
202
203<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Calculate the kurtosis excess of the input vector x.  <a href="#g00f842b0e850195093781692a6aaaac3"></a><br></td></tr>
204<tr><td class="memItemLeft" nowrap align="right" valign="top">double&nbsp;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__statistics.html#g49dd4e3746500b1587bfd96fb54439f0">itpp::kurtosis</a> (const <a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a> &amp;x)</td></tr>
205
206<tr><td class="mdescLeft">&nbsp;</td><td class="mdescRight">Calculate the kurtosis of the input vector x.  <a href="#g49dd4e3746500b1587bfd96fb54439f0"></a><br></td></tr>
207</table>
208<hr><h2>Function Documentation</h2>
209<a class="anchor" name="g49dd4e3746500b1587bfd96fb54439f0"></a><!-- doxytag: member="itpp::kurtosis" ref="g49dd4e3746500b1587bfd96fb54439f0" args="(const vec &amp;x)" -->
210<div class="memitem">
211<div class="memproto">
212      <table class="memname">
213        <tr>
214          <td class="memname">double itpp::kurtosis           </td>
215          <td>(</td>
216          <td class="paramtype">const <a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a> &amp;&nbsp;</td>
217          <td class="paramname"> <em>x</em>          </td>
218          <td>&nbsp;)&nbsp;</td>
219          <td><code> [inline]</code></td>
220        </tr>
221      </table>
222</div>
223<div class="memdoc">
224
225<p>
226Calculate the kurtosis of the input vector x.
227<p>
228The kurtosis is a measure of peakedness of a distribution. The kurtosis is defined as <p class="formulaDsp">
229<img class="formulaDsp" alt="\[ \gamma_2 = \frac{\mathrm{E}[x-\mu]^4}{\sigma^4} \]" src="form_387.png">
230<p>
231 where <img class="formulaInl" alt="$\mu$" src="form_68.png"> is the mean and <img class="formulaInl" alt="$\sigma$" src="form_377.png"> the standard deviation. For a Gaussian variable, the kurtusis is 3.<p>
232See also the definition of kurtosisexcess.
233<p>References <a class="el" href="misc__stat_8cpp-source.html#l00220">itpp::kurtosisexcess()</a>.</p>
234
235</div>
236</div><p>
237<a class="anchor" name="g00f842b0e850195093781692a6aaaac3"></a><!-- doxytag: member="itpp::kurtosisexcess" ref="g00f842b0e850195093781692a6aaaac3" args="(const vec &amp;x)" -->
238<div class="memitem">
239<div class="memproto">
240      <table class="memname">
241        <tr>
242          <td class="memname">double itpp::kurtosisexcess           </td>
243          <td>(</td>
244          <td class="paramtype">const <a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a> &amp;&nbsp;</td>
245          <td class="paramname"> <em>x</em>          </td>
246          <td>&nbsp;)&nbsp;</td>
247          <td></td>
248        </tr>
249      </table>
250</div>
251<div class="memdoc">
252
253<p>
254Calculate the kurtosis excess of the input vector x.
255<p>
256The kurtosis excess is a measure of peakedness of a distribution. The kurtosis excess is defined as <p class="formulaDsp">
257<img class="formulaDsp" alt="\[ \gamma_2 = \frac{\mathrm{E}[x-\mu]^4}{\sigma^4} - 3 \]" src="form_383.png">
258<p>
259 where <img class="formulaInl" alt="$\mu$" src="form_68.png"> is the mean and <img class="formulaInl" alt="$\sigma$" src="form_377.png"> the standard deviation.<p>
260The kurtosis excess is estimated as <p class="formulaDsp">
261<img class="formulaDsp" alt="\[ \gamma_2 = \frac{k_4}{{k_2}^2} \]" src="form_384.png">
262<p>
263 where <p class="formulaDsp">
264<img class="formulaDsp" alt="\[ k_2 = \frac{n}{n-1} m_2 \]" src="form_379.png">
265<p>
266 and <p class="formulaDsp">
267<img class="formulaDsp" alt="\[ k_4 = \frac{n^2 [(n+1)m_4 - 3(n-1){m_2}^2]}{(n-1)(n-2)(n-3)} \]" src="form_385.png">
268<p>
269 Here <img class="formulaInl" alt="$m_2$" src="form_381.png"> is the sample variance and <img class="formulaInl" alt="$m_4$" src="form_386.png"> is the 4th sample central moment.
270<p>References <a class="el" href="misc__stat_8cpp-source.html#l00174">itpp::moment()</a>, and <a class="el" href="misc__stat_8h-source.html#l00236">itpp::variance()</a>.</p>
271
272<p>Referenced by <a class="el" href="misc__stat_8h-source.html#l00378">itpp::kurtosis()</a>.</p>
273
274</div>
275</div><p>
276<a class="anchor" name="gde571eebaa7dbd8289b6d7184fe625ca"></a><!-- doxytag: member="itpp::moment" ref="gde571eebaa7dbd8289b6d7184fe625ca" args="(const vec &amp;x, const int r)" -->
277<div class="memitem">
278<div class="memproto">
279      <table class="memname">
280        <tr>
281          <td class="memname">double itpp::moment           </td>
282          <td>(</td>
283          <td class="paramtype">const <a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a> &amp;&nbsp;</td>
284          <td class="paramname"> <em>x</em>, </td>
285        </tr>
286        <tr>
287          <td class="paramkey"></td>
288          <td></td>
289          <td class="paramtype">const int&nbsp;</td>
290          <td class="paramname"> <em>r</em></td><td>&nbsp;</td>
291        </tr>
292        <tr>
293          <td></td>
294          <td>)</td>
295          <td></td><td></td><td></td>
296        </tr>
297      </table>
298</div>
299<div class="memdoc">
300
301<p>
302Calculate the central moment of vector x.
303<p>
304The <img class="formulaInl" alt="$r$" src="form_281.png">th sample central moment of the samples in the vector <img class="formulaInl" alt="$ \mathbf{x} $" src="form_374.png"> is defined as<p>
305<p class="formulaDsp">
306<img class="formulaDsp" alt="\[ m_r = \mathrm{E}[x-\mu]^r = \frac{1}{n} \sum_{i=0}^{n-1} (x_i - \mu)^r \]" src="form_375.png">
307<p>
308 where <img class="formulaInl" alt="$\mu$" src="form_68.png"> is the sample mean.
309<p>References <a class="el" href="misc__stat_8cpp-source.html#l00071">itpp::mean()</a>, and <a class="el" href="log__exp_8h-source.html#l00193">itpp::pow()</a>.</p>
310
311<p>Referenced by <a class="el" href="misc__stat_8cpp-source.html#l00220">itpp::kurtosisexcess()</a>, and <a class="el" href="misc__stat_8cpp-source.html#l00210">itpp::skewness()</a>.</p>
312
313</div>
314</div><p>
315<a class="anchor" name="g831b04981a856a72f8f669bff13fd1e3"></a><!-- doxytag: member="itpp::norm" ref="g831b04981a856a72f8f669bff13fd1e3" args="(const cmat &amp;m, int p)" -->
316<div class="memitem">
317<div class="memproto">
318      <table class="memname">
319        <tr>
320          <td class="memname">double itpp::norm           </td>
321          <td>(</td>
322          <td class="paramtype">const <a class="el" href="classitpp_1_1Mat.html#6fbac4b7184807da188e5b85d42f038b">cmat</a> &amp;&nbsp;</td>
323          <td class="paramname"> <em>m</em>, </td>
324        </tr>
325        <tr>
326          <td class="paramkey"></td>
327          <td></td>
328          <td class="paramtype">int&nbsp;</td>
329          <td class="paramname"> <em>p</em> = <code>2</code></td><td>&nbsp;</td>
330        </tr>
331        <tr>
332          <td></td>
333          <td>)</td>
334          <td></td><td></td><td></td>
335        </tr>
336      </table>
337</div>
338<div class="memdoc">
339
340<p>
341Calculate the p-norm of a complex matrix<p>
342p = 1: max(svd(m)) p = 2: max(sum(abs(X)))<p>
343Default if no p is given is the 2-norm
344<p>References <a class="el" href="fix__functions_8cpp-source.html#l00076">itpp::abs()</a>, <a class="el" href="itassert_8h-source.html#l00094">it_assert</a>, <a class="el" href="tcp_8h-source.html#l00117">itpp::max()</a>, <a class="el" href="matfunc_8h-source.html#l00077">itpp::sum()</a>, and <a class="el" href="svd_8cpp-source.html#l00218">itpp::svd()</a>.</p>
345
346</div>
347</div><p>
348<a class="anchor" name="g33961993e1531da45cc4a3c02a6dd124"></a><!-- doxytag: member="itpp::norm" ref="g33961993e1531da45cc4a3c02a6dd124" args="(const mat &amp;m, int p)" -->
349<div class="memitem">
350<div class="memproto">
351      <table class="memname">
352        <tr>
353          <td class="memname">double itpp::norm           </td>
354          <td>(</td>
355          <td class="paramtype">const <a class="el" href="classitpp_1_1Mat.html#6bba394f181c76fda12759568986c613">mat</a> &amp;&nbsp;</td>
356          <td class="paramname"> <em>m</em>, </td>
357        </tr>
358        <tr>
359          <td class="paramkey"></td>
360          <td></td>
361          <td class="paramtype">int&nbsp;</td>
362          <td class="paramname"> <em>p</em> = <code>2</code></td><td>&nbsp;</td>
363        </tr>
364        <tr>
365          <td></td>
366          <td>)</td>
367          <td></td><td></td><td></td>
368        </tr>
369      </table>
370</div>
371<div class="memdoc">
372
373<p>
374Calculate the p-norm of a real matrix<p>
375p = 1: max(svd(m)) p = 2: max(sum(abs(X)))<p>
376Default if no p is given is the 2-norm
377<p>References <a class="el" href="fix__functions_8cpp-source.html#l00076">itpp::abs()</a>, <a class="el" href="itassert_8h-source.html#l00094">it_assert</a>, <a class="el" href="tcp_8h-source.html#l00117">itpp::max()</a>, <a class="el" href="matfunc_8h-source.html#l00077">itpp::sum()</a>, and <a class="el" href="svd_8cpp-source.html#l00218">itpp::svd()</a>.</p>
378
379</div>
380</div><p>
381<a class="anchor" name="g43f3c117ed93400ceed5bdacf37f49ea"></a><!-- doxytag: member="itpp::skewness" ref="g43f3c117ed93400ceed5bdacf37f49ea" args="(const vec &amp;x)" -->
382<div class="memitem">
383<div class="memproto">
384      <table class="memname">
385        <tr>
386          <td class="memname">double itpp::skewness           </td>
387          <td>(</td>
388          <td class="paramtype">const <a class="el" href="classitpp_1_1Vec.html#02e1bb55f60f3c2eb7a020eb1c2cfcf4">vec</a> &amp;&nbsp;</td>
389          <td class="paramname"> <em>x</em>          </td>
390          <td>&nbsp;)&nbsp;</td>
391          <td></td>
392        </tr>
393      </table>
394</div>
395<div class="memdoc">
396
397<p>
398Calculate the skewness excess of the input vector x.
399<p>
400The skewness is a measure of the degree of asymmetry of distribution. Negative skewness means that the distribution is spread more to the left of the mean than to the right, and vice versa if the skewness is positive.<p>
401The skewness of the samples in the vector <img class="formulaInl" alt="$ \mathbf{x} $" src="form_374.png"> is <p class="formulaDsp">
402<img class="formulaDsp" alt="\[ \gamma_1 = \frac{\mathrm{E}[x-\mu]^3}{\sigma^3} \]" src="form_376.png">
403<p>
404 where <img class="formulaInl" alt="$\mu$" src="form_68.png"> is the mean and <img class="formulaInl" alt="$\sigma$" src="form_377.png"> the standard deviation.<p>
405The skewness is estimated as <p class="formulaDsp">
406<img class="formulaDsp" alt="\[ \gamma_1 = \frac{k_3}{{k_2}^{3/2}} \]" src="form_378.png">
407<p>
408 where <p class="formulaDsp">
409<img class="formulaDsp" alt="\[ k_2 = \frac{n}{n-1} m_2 \]" src="form_379.png">
410<p>
411 and <p class="formulaDsp">
412<img class="formulaDsp" alt="\[ k_3 = \frac{n^2}{(n-1)(n-2)} m_3 \]" src="form_380.png">
413<p>
414 Here <img class="formulaInl" alt="$m_2$" src="form_381.png"> is the sample variance and <img class="formulaInl" alt="$m_3$" src="form_382.png"> is the 3rd sample central moment.
415<p>References <a class="el" href="misc__stat_8cpp-source.html#l00174">itpp::moment()</a>, <a class="el" href="log__exp_8h-source.html#l00193">itpp::pow()</a>, and <a class="el" href="misc__stat_8h-source.html#l00236">itpp::variance()</a>.</p>
416
417</div>
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420<hr size="1"><address style="text-align: right;"><small>Generated on Tue Jun 2 10:02:14 2009 for mixpp by&nbsp;
421<a href="http://www.doxygen.org/index.html">
422<img src="doxygen.png" alt="doxygen" align="middle" border="0"></a> 1.5.8 </small></address>
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