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25 | <div class="section"> | |
26 | <div class="titlepage"><div><div><h3 class="title"> | |
27 | <a name="math_toolkit.pol_ref.discrete_quant_ref"></a><a class="link" href="discrete_quant_ref.html" title="Discrete Quantile Policies">Discrete Quantile | |
28 | Policies</a> | |
29 | </h3></div></div></div> | |
30 | <p> | |
31 | If a statistical distribution is <span class="emphasis"><em>discrete</em></span> then the random | |
32 | variable can only have integer values - this leaves us with a problem when | |
33 | calculating quantiles - we can either ignore the discreteness of the distribution | |
34 | and return a real value, or we can round to an integer. As it happens, computing | |
35 | integer values can be substantially faster than calculating a real value, | |
36 | so there are definite advantages to returning an integer, but we do then | |
37 | need to decide how best to round the result. The <code class="computeroutput"><span class="identifier">discrete_quantile</span></code> | |
38 | policy defines how discrete quantiles work, and how integer results are rounded: | |
39 | </p> | |
40 | <pre class="programlisting"><span class="keyword">enum</span> <span class="identifier">discrete_quantile_policy_type</span> | |
41 | <span class="special">{</span> | |
42 | <span class="identifier">real</span><span class="special">,</span> | |
43 | <span class="identifier">integer_round_outwards</span><span class="special">,</span> <span class="comment">// default</span> | |
44 | <span class="identifier">integer_round_inwards</span><span class="special">,</span> | |
45 | <span class="identifier">integer_round_down</span><span class="special">,</span> | |
46 | <span class="identifier">integer_round_up</span><span class="special">,</span> | |
47 | <span class="identifier">integer_round_nearest</span> | |
48 | <span class="special">};</span> | |
49 | ||
50 | <span class="keyword">template</span> <span class="special"><</span><span class="identifier">discrete_quantile_policy_type</span><span class="special">></span> | |
51 | <span class="keyword">struct</span> <span class="identifier">discrete_quantile</span><span class="special">;</span> | |
52 | </pre> | |
53 | <p> | |
54 | The values that <code class="computeroutput"><span class="identifier">discrete_quantile</span></code> | |
55 | can take have the following meanings: | |
56 | </p> | |
57 | <h6> | |
58 | <a name="math_toolkit.pol_ref.discrete_quant_ref.h0"></a> | |
59 | <span class="phrase"><a name="math_toolkit.pol_ref.discrete_quant_ref.real"></a></span><a class="link" href="discrete_quant_ref.html#math_toolkit.pol_ref.discrete_quant_ref.real">real</a> | |
60 | </h6> | |
61 | <p> | |
62 | Ignores the discreteness of the distribution, and returns a real-valued result. | |
63 | For example: | |
64 | </p> | |
65 | <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">negative_binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
66 | <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">negative_binomial_distribution</span><span class="special">;</span> | |
67 | ||
68 | <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">;</span> | |
69 | ||
70 | <span class="keyword">typedef</span> <span class="identifier">negative_binomial_distribution</span><span class="special"><</span> | |
71 | <span class="keyword">double</span><span class="special">,</span> | |
72 | <span class="identifier">policy</span><span class="special"><</span><span class="identifier">discrete_quantile</span><span class="special"><</span><span class="identifier">real</span><span class="special">></span> <span class="special">></span> | |
73 | <span class="special">></span> <span class="identifier">dist_type</span><span class="special">;</span> | |
74 | ||
75 | <span class="comment">// Lower 5% quantile:</span> | |
76 | <span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist_type</span><span class="special">(</span><span class="number">20</span><span class="special">,</span> <span class="number">0.3</span><span class="special">),</span> <span class="number">0.05</span><span class="special">);</span> | |
77 | <span class="comment">// Upper 95% quantile:</span> | |
78 | <span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist_type</span><span class="special">(</span><span class="number">20</span><span class="special">,</span> <span class="number">0.3</span><span class="special">),</span> <span class="number">0.05</span><span class="special">));</span> | |
79 | </pre> | |
80 | <p> | |
81 | Results in <code class="computeroutput"><span class="identifier">x</span> <span class="special">=</span> | |
82 | <span class="number">27.3898</span></code> and <code class="computeroutput"><span class="identifier">y</span> | |
83 | <span class="special">=</span> <span class="number">68.1584</span></code>. | |
84 | </p> | |
85 | <h6> | |
86 | <a name="math_toolkit.pol_ref.discrete_quant_ref.h1"></a> | |
87 | <span class="phrase"><a name="math_toolkit.pol_ref.discrete_quant_ref.integer_round_outwards"></a></span><a class="link" href="discrete_quant_ref.html#math_toolkit.pol_ref.discrete_quant_ref.integer_round_outwards">integer_round_outwards</a> | |
88 | </h6> | |
89 | <p> | |
90 | This is the default policy: an integer value is returned so that: | |
91 | </p> | |
92 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
93 | <li class="listitem"> | |
94 | Lower quantiles (where the probability is less than 0.5) are rounded | |
95 | down. | |
96 | </li> | |
97 | <li class="listitem"> | |
98 | Upper quantiles (where the probability is greater than 0.5) are rounded | |
99 | up. | |
100 | </li> | |
101 | </ul></div> | |
102 | <p> | |
103 | This is normally the safest rounding policy, since it ensures that both one | |
104 | and two sided intervals are guaranteed to have <span class="emphasis"><em>at least</em></span> | |
105 | the requested coverage. For example: | |
106 | </p> | |
107 | <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">negative_binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
108 | <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">negative_binomial</span><span class="special">;</span> | |
109 | ||
110 | <span class="comment">// Use the default rounding policy integer_round_outwards.</span> | |
111 | <span class="comment">// Lower quantile rounded down:</span> | |
112 | <span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="number">20</span><span class="special">,</span> <span class="number">0.3</span><span class="special">),</span> <span class="number">0.05</span><span class="special">);</span> <span class="comment">// rounded up 27 from 27.3898</span> | |
113 | <span class="comment">// Upper quantile rounded up:</span> | |
114 | <span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="number">20</span><span class="special">,</span> <span class="number">0.3</span><span class="special">),</span> <span class="number">0.05</span><span class="special">));</span> <span class="comment">// rounded down to 69 from 68.1584</span> | |
115 | </pre> | |
116 | <p> | |
117 | Results in <code class="computeroutput"><span class="identifier">x</span> <span class="special">=</span> | |
118 | <span class="number">27</span></code> (rounded down from 27.3898) and | |
119 | <code class="computeroutput"><span class="identifier">y</span> <span class="special">=</span> | |
120 | <span class="number">69</span></code> (rounded up from 68.1584). | |
121 | </p> | |
122 | <p> | |
123 | The variables x and y are now defined so that: | |
124 | </p> | |
125 | <pre class="programlisting"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="number">20</span><span class="special">),</span> <span class="identifier">x</span><span class="special">)</span> <span class="special"><=</span> <span class="number">0.05</span> | |
126 | <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="number">20</span><span class="special">),</span> <span class="identifier">y</span><span class="special">)</span> <span class="special">>=</span> <span class="number">0.95</span> | |
127 | </pre> | |
128 | <p> | |
129 | In other words we guarantee <span class="emphasis"><em>at least 90% coverage in the central | |
130 | region overall</em></span>, and also <span class="emphasis"><em>no more than 5% coverage in | |
131 | each tail</em></span>. | |
132 | </p> | |
133 | <h6> | |
134 | <a name="math_toolkit.pol_ref.discrete_quant_ref.h2"></a> | |
135 | <span class="phrase"><a name="math_toolkit.pol_ref.discrete_quant_ref.integer_round_inwards"></a></span><a class="link" href="discrete_quant_ref.html#math_toolkit.pol_ref.discrete_quant_ref.integer_round_inwards">integer_round_inwards</a> | |
136 | </h6> | |
137 | <p> | |
138 | This is the opposite of <span class="emphasis"><em>integer_round_outwards</em></span>: an integer | |
139 | value is returned so that: | |
140 | </p> | |
141 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
142 | <li class="listitem"> | |
143 | Lower quantiles (where the probability is less than 0.5) are rounded | |
144 | <span class="emphasis"><em>up</em></span>. | |
145 | </li> | |
146 | <li class="listitem"> | |
147 | Upper quantiles (where the probability is greater than 0.5) are rounded | |
148 | <span class="emphasis"><em>down</em></span>. | |
149 | </li> | |
150 | </ul></div> | |
151 | <p> | |
152 | For example: | |
153 | </p> | |
154 | <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">negative_binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
155 | <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">negative_binomial_distribution</span><span class="special">;</span> | |
156 | ||
157 | <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">;</span> | |
158 | ||
159 | <span class="keyword">typedef</span> <span class="identifier">negative_binomial_distribution</span><span class="special"><</span> | |
160 | <span class="keyword">double</span><span class="special">,</span> | |
161 | <span class="identifier">policy</span><span class="special"><</span><span class="identifier">discrete_quantile</span><span class="special"><</span><span class="identifier">integer_round_inwards</span><span class="special">></span> <span class="special">></span> | |
162 | <span class="special">></span> <span class="identifier">dist_type</span><span class="special">;</span> | |
163 | ||
164 | <span class="comment">// Lower quantile rounded up:</span> | |
165 | <span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist_type</span><span class="special">(</span><span class="number">20</span><span class="special">,</span> <span class="number">0.3</span><span class="special">),</span> <span class="number">0.05</span><span class="special">);</span> <span class="comment">// 28 rounded up from 27.3898</span> | |
166 | <span class="comment">// Upper quantile rounded down:</span> | |
167 | <span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist_type</span><span class="special">(</span><span class="number">20</span><span class="special">,</span> <span class="number">0.3</span><span class="special">),</span> <span class="number">0.05</span><span class="special">));</span> <span class="comment">// 68 rounded down from 68.1584</span> | |
168 | </pre> | |
169 | <p> | |
170 | Results in <code class="computeroutput"><span class="identifier">x</span> <span class="special">=</span> | |
171 | <span class="number">28</span></code> (rounded up from 27.3898) and <code class="computeroutput"><span class="identifier">y</span> <span class="special">=</span> <span class="number">68</span></code> | |
172 | (rounded down from 68.1584). | |
173 | </p> | |
174 | <p> | |
175 | The variables x and y are now defined so that: | |
176 | </p> | |
177 | <pre class="programlisting"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="number">20</span><span class="special">),</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">>=</span> <span class="number">0.05</span> | |
178 | <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="number">20</span><span class="special">),</span> <span class="identifier">y</span><span class="special">)</span> <span class="special"><=</span> <span class="number">0.95</span> | |
179 | </pre> | |
180 | <p> | |
181 | In other words we guarantee <span class="emphasis"><em>at no more than 90% coverage in the | |
182 | central region overall</em></span>, and also <span class="emphasis"><em>at least 5% coverage | |
183 | in each tail</em></span>. | |
184 | </p> | |
185 | <h6> | |
186 | <a name="math_toolkit.pol_ref.discrete_quant_ref.h3"></a> | |
187 | <span class="phrase"><a name="math_toolkit.pol_ref.discrete_quant_ref.integer_round_down"></a></span><a class="link" href="discrete_quant_ref.html#math_toolkit.pol_ref.discrete_quant_ref.integer_round_down">integer_round_down</a> | |
188 | </h6> | |
189 | <p> | |
190 | Always rounds down to an integer value, no matter whether it's an upper or | |
191 | a lower quantile. | |
192 | </p> | |
193 | <h6> | |
194 | <a name="math_toolkit.pol_ref.discrete_quant_ref.h4"></a> | |
195 | <span class="phrase"><a name="math_toolkit.pol_ref.discrete_quant_ref.integer_round_up"></a></span><a class="link" href="discrete_quant_ref.html#math_toolkit.pol_ref.discrete_quant_ref.integer_round_up">integer_round_up</a> | |
196 | </h6> | |
197 | <p> | |
198 | Always rounds up to an integer value, no matter whether it's an upper or | |
199 | a lower quantile. | |
200 | </p> | |
201 | <h6> | |
202 | <a name="math_toolkit.pol_ref.discrete_quant_ref.h5"></a> | |
203 | <span class="phrase"><a name="math_toolkit.pol_ref.discrete_quant_ref.integer_round_nearest"></a></span><a class="link" href="discrete_quant_ref.html#math_toolkit.pol_ref.discrete_quant_ref.integer_round_nearest">integer_round_nearest</a> | |
204 | </h6> | |
205 | <p> | |
206 | Always rounds to the nearest integer value, no matter whether it's an upper | |
207 | or a lower quantile. This will produce the requested coverage <span class="emphasis"><em>in | |
208 | the average case</em></span>, but for any specific example may results in | |
209 | either significantly more or less coverage than the requested amount. For | |
210 | example: | |
211 | </p> | |
212 | <p> | |
213 | For example: | |
214 | </p> | |
215 | <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">negative_binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
216 | <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">negative_binomial_distribution</span><span class="special">;</span> | |
217 | ||
218 | <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">;</span> | |
219 | ||
220 | <span class="keyword">typedef</span> <span class="identifier">negative_binomial_distribution</span><span class="special"><</span> | |
221 | <span class="keyword">double</span><span class="special">,</span> | |
222 | <span class="identifier">policy</span><span class="special"><</span><span class="identifier">discrete_quantile</span><span class="special"><</span><span class="identifier">integer_round_nearest</span><span class="special">></span> <span class="special">></span> | |
223 | <span class="special">></span> <span class="identifier">dist_type</span><span class="special">;</span> | |
224 | ||
225 | <span class="comment">// Lower quantile rounded (down) to nearest:</span> | |
226 | <span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist_type</span><span class="special">(</span><span class="number">20</span><span class="special">,</span> <span class="number">0.3</span><span class="special">),</span> <span class="number">0.05</span><span class="special">);</span> <span class="comment">// 27</span> | |
227 | <span class="comment">// Upper quantile rounded (down) to nearest:</span> | |
228 | <span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist_type</span><span class="special">(</span><span class="number">20</span><span class="special">,</span> <span class="number">0.3</span><span class="special">),</span> <span class="number">0.05</span><span class="special">));</span> <span class="comment">// 68</span> | |
229 | </pre> | |
230 | <p> | |
231 | Results in <code class="computeroutput"><span class="identifier">x</span> <span class="special">=</span> | |
232 | <span class="number">27</span></code> (rounded from 27.3898) and <code class="computeroutput"><span class="identifier">y</span> <span class="special">=</span> <span class="number">68</span></code> | |
233 | (rounded from 68.1584). | |
234 | </p> | |
235 | </div> | |
236 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
237 | <td align="left"></td> | |
238 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
239 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
240 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
241 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
242 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
243 | 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>) | |
244 | </p> | |
245 | </div></td> | |
246 | </tr></table> | |
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