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25 | <div class="section"> | |
26 | <div class="titlepage"><div><div><h4 class="title"> | |
27 | <a name="math_toolkit.dist_ref.dists.logistic_dist"></a><a class="link" href="logistic_dist.html" title="Logistic Distribution">Logistic | |
28 | Distribution</a> | |
29 | </h4></div></div></div> | |
30 | <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">logistic</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre> | |
31 | <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> | |
32 | ||
33 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span> | |
34 | <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span> | |
35 | <span class="keyword">class</span> <span class="identifier">logistic_distribution</span><span class="special">;</span> | |
36 | ||
37 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">></span> | |
38 | <span class="keyword">class</span> <span class="identifier">logistic_distribution</span> | |
39 | <span class="special">{</span> | |
40 | <span class="keyword">public</span><span class="special">:</span> | |
41 | <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> | |
42 | <span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span> | |
43 | <span class="comment">// Construct:</span> | |
44 | <span class="identifier">logistic_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> | |
45 | <span class="comment">// Accessors:</span> | |
46 | <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// location.</span> | |
47 | <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// scale.</span> | |
48 | ||
49 | <span class="special">};</span> | |
50 | ||
51 | <span class="keyword">typedef</span> <span class="identifier">logistic_distribution</span><span class="special"><></span> <span class="identifier">logistic</span><span class="special">;</span> | |
52 | ||
53 | <span class="special">}}</span> <span class="comment">// namespaces</span> | |
54 | </pre> | |
55 | <p> | |
56 | The logistic distribution is a continous probability distribution. It has | |
57 | two parameters - location and scale. The cumulative distribution function | |
58 | of the logistic distribution appears in logistic regression and feedforward | |
59 | neural networks. Among other applications, United State Chess Federation | |
60 | and FIDE use it to calculate chess ratings. | |
61 | </p> | |
62 | <p> | |
63 | The following graph shows how the distribution changes as the parameters | |
64 | change: | |
65 | </p> | |
66 | <p> | |
67 | <span class="inlinemediaobject"><img src="../../../../graphs/logistic_pdf.svg" align="middle"></span> | |
68 | </p> | |
69 | <h5> | |
70 | <a name="math_toolkit.dist_ref.dists.logistic_dist.h0"></a> | |
71 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.logistic_dist.member_functions"></a></span><a class="link" href="logistic_dist.html#math_toolkit.dist_ref.dists.logistic_dist.member_functions">Member | |
72 | Functions</a> | |
73 | </h5> | |
74 | <pre class="programlisting"><span class="identifier">logistic_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">u</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">s</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> | |
75 | </pre> | |
76 | <p> | |
77 | Constructs a logistic distribution with location <span class="emphasis"><em>u</em></span> | |
78 | and scale <span class="emphasis"><em>s</em></span>. | |
79 | </p> | |
80 | <p> | |
81 | Requires <code class="computeroutput"><span class="identifier">scale</span> <span class="special">></span> | |
82 | <span class="number">0</span></code>, otherwise a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> | |
83 | is raised. | |
84 | </p> | |
85 | <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> | |
86 | </pre> | |
87 | <p> | |
88 | Returns the location of this distribution. | |
89 | </p> | |
90 | <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> | |
91 | </pre> | |
92 | <p> | |
93 | Returns the scale of this distribution. | |
94 | </p> | |
95 | <h5> | |
96 | <a name="math_toolkit.dist_ref.dists.logistic_dist.h1"></a> | |
97 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.logistic_dist.non_member_accessors"></a></span><a class="link" href="logistic_dist.html#math_toolkit.dist_ref.dists.logistic_dist.non_member_accessors">Non-member | |
98 | Accessors</a> | |
99 | </h5> | |
100 | <p> | |
101 | All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor | |
102 | functions</a> that are generic to all distributions are supported: | |
103 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, | |
104 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, | |
105 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>, | |
106 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>, | |
107 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>, | |
108 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, | |
109 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>, | |
110 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>. | |
111 | </p> | |
112 | <p> | |
113 | The domain of the random variable is [-[max_value], +[min_value]]. However, | |
114 | the pdf and cdf support inputs of +∞ and -∞ | |
115 | as special cases if RealType permits. | |
116 | </p> | |
117 | <p> | |
118 | At <code class="computeroutput"><span class="identifier">p</span><span class="special">=</span><span class="number">1</span></code> and <code class="computeroutput"><span class="identifier">p</span><span class="special">=</span><span class="number">0</span></code>, the quantile | |
119 | function returns the result of +<a class="link" href="../../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a> | |
120 | and -<a class="link" href="../../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>, | |
121 | while the complement quantile function returns the result of -<a class="link" href="../../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a> | |
122 | and +<a class="link" href="../../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a> | |
123 | respectively. | |
124 | </p> | |
125 | <h5> | |
126 | <a name="math_toolkit.dist_ref.dists.logistic_dist.h2"></a> | |
127 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.logistic_dist.accuracy"></a></span><a class="link" href="logistic_dist.html#math_toolkit.dist_ref.dists.logistic_dist.accuracy">Accuracy</a> | |
128 | </h5> | |
129 | <p> | |
130 | The logistic distribution is implemented in terms of the <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">exp</span></code> and the <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">log</span></code> | |
131 | functions, so its accuracy is related to the accurate implementations of | |
132 | those functions on a given platform. When calculating the quantile with | |
133 | a non-zero <span class="emphasis"><em>position</em></span> parameter catastrophic cancellation | |
134 | errors can occur: in such cases, only a low <span class="emphasis"><em>absolute error</em></span> | |
135 | can be guaranteed. | |
136 | </p> | |
137 | <h5> | |
138 | <a name="math_toolkit.dist_ref.dists.logistic_dist.h3"></a> | |
139 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.logistic_dist.implementation"></a></span><a class="link" href="logistic_dist.html#math_toolkit.dist_ref.dists.logistic_dist.implementation">Implementation</a> | |
140 | </h5> | |
141 | <div class="informaltable"><table class="table"> | |
142 | <colgroup> | |
143 | <col> | |
144 | <col> | |
145 | </colgroup> | |
146 | <thead><tr> | |
147 | <th> | |
148 | <p> | |
149 | Function | |
150 | </p> | |
151 | </th> | |
152 | <th> | |
153 | <p> | |
154 | Implementation Notes | |
155 | </p> | |
156 | </th> | |
157 | </tr></thead> | |
158 | <tbody> | |
159 | <tr> | |
160 | <td> | |
161 | <p> | |
162 | ||
163 | </p> | |
164 | </td> | |
165 | <td> | |
166 | <p> | |
167 | Using the relation: pdf = e<sup>-(x-u)/s</sup> / (s*(1+e<sup>-(x-u)/s</sup>)<sup>2</sup>) | |
168 | </p> | |
169 | </td> | |
170 | </tr> | |
171 | <tr> | |
172 | <td> | |
173 | <p> | |
174 | cdf | |
175 | </p> | |
176 | </td> | |
177 | <td> | |
178 | <p> | |
179 | Using the relation: p = 1/(1+e<sup>-(x-u)/s</sup>) | |
180 | </p> | |
181 | </td> | |
182 | </tr> | |
183 | <tr> | |
184 | <td> | |
185 | <p> | |
186 | cdf complement | |
187 | </p> | |
188 | </td> | |
189 | <td> | |
190 | <p> | |
191 | Using the relation: q = 1/(1+e<sup>(x-u)/s</sup>) | |
192 | </p> | |
193 | </td> | |
194 | </tr> | |
195 | <tr> | |
196 | <td> | |
197 | <p> | |
198 | quantile | |
199 | </p> | |
200 | </td> | |
201 | <td> | |
202 | <p> | |
203 | Using the relation: x = u - s*log(1/p-1) | |
204 | </p> | |
205 | </td> | |
206 | </tr> | |
207 | <tr> | |
208 | <td> | |
209 | <p> | |
210 | quantile from the complement | |
211 | </p> | |
212 | </td> | |
213 | <td> | |
214 | <p> | |
215 | Using the relation: x = u + s*log(p/1-p) | |
216 | </p> | |
217 | </td> | |
218 | </tr> | |
219 | <tr> | |
220 | <td> | |
221 | <p> | |
222 | mean | |
223 | </p> | |
224 | </td> | |
225 | <td> | |
226 | <p> | |
227 | u | |
228 | </p> | |
229 | </td> | |
230 | </tr> | |
231 | <tr> | |
232 | <td> | |
233 | <p> | |
234 | mode | |
235 | </p> | |
236 | </td> | |
237 | <td> | |
238 | <p> | |
239 | The same as the mean. | |
240 | </p> | |
241 | </td> | |
242 | </tr> | |
243 | <tr> | |
244 | <td> | |
245 | <p> | |
246 | skewness | |
247 | </p> | |
248 | </td> | |
249 | <td> | |
250 | <p> | |
251 | 0 | |
252 | </p> | |
253 | </td> | |
254 | </tr> | |
255 | <tr> | |
256 | <td> | |
257 | <p> | |
258 | kurtosis excess | |
259 | </p> | |
260 | </td> | |
261 | <td> | |
262 | <p> | |
263 | 6/5 | |
264 | </p> | |
265 | </td> | |
266 | </tr> | |
267 | <tr> | |
268 | <td> | |
269 | <p> | |
270 | variance | |
271 | </p> | |
272 | </td> | |
273 | <td> | |
274 | <p> | |
275 | (π*s)<sup>2</sup> / 3 | |
276 | </p> | |
277 | </td> | |
278 | </tr> | |
279 | </tbody> | |
280 | </table></div> | |
281 | </div> | |
282 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
283 | <td align="left"></td> | |
284 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
285 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
286 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
287 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
288 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
289 | file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) | |
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