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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">&lt;</span><span class="identifier">discrete_quantile_policy_type</span><span class="special">&gt;</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">&lt;</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">&gt;</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">&lt;</span>
71 <span class="keyword">double</span><span class="special">,</span>
72 <span class="identifier">policy</span><span class="special">&lt;</span><span class="identifier">discrete_quantile</span><span class="special">&lt;</span><span class="identifier">real</span><span class="special">&gt;</span> <span class="special">&gt;</span>
73 <span class="special">&gt;</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">&lt;</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">&gt;</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">&lt;=</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">&gt;=</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">&lt;</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">&gt;</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">&lt;</span>
160 <span class="keyword">double</span><span class="special">,</span>
161 <span class="identifier">policy</span><span class="special">&lt;</span><span class="identifier">discrete_quantile</span><span class="special">&lt;</span><span class="identifier">integer_round_inwards</span><span class="special">&gt;</span> <span class="special">&gt;</span>
162 <span class="special">&gt;</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">&gt;=</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">&lt;=</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">&lt;</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">&gt;</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">&lt;</span>
221 <span class="keyword">double</span><span class="special">,</span>
222 <span class="identifier">policy</span><span class="special">&lt;</span><span class="identifier">discrete_quantile</span><span class="special">&lt;</span><span class="identifier">integer_round_nearest</span><span class="special">&gt;</span> <span class="special">&gt;</span>
223 <span class="special">&gt;</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>
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240 Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, 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>)
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