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25 | <div class="section"> | |
26 | <div class="titlepage"><div><div><h3 class="title"> | |
27 | <a name="math_toolkit.pol_tutorial.understand_dis_quant"></a><a class="link" href="understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">Understanding | |
28 | Quantiles of Discrete Distributions</a> | |
29 | </h3></div></div></div> | |
30 | <p> | |
31 | Discrete distributions present us with a problem when calculating the quantile: | |
32 | we are starting from a continuous real-valued variable - the probability | |
33 | - but the result (the value of the random variable) should really be discrete. | |
34 | </p> | |
35 | <p> | |
36 | Consider for example a Binomial distribution, with a sample size of 50, and | |
37 | a success fraction of 0.5. There are a variety of ways we can plot a discrete | |
38 | distribution, but if we plot the PDF as a step-function then it looks something | |
39 | like this: | |
40 | </p> | |
41 | <p> | |
42 | <span class="inlinemediaobject"><img src="../../../graphs/binomial_pdf.png"></span> | |
43 | </p> | |
44 | <p> | |
45 | Now lets suppose that the user asks for a the quantile that corresponds to | |
46 | a probability of 0.05, if we zoom in on the CDF for that region here's what | |
47 | we see: | |
48 | </p> | |
49 | <p> | |
50 | <span class="inlinemediaobject"><img src="../../../graphs/binomial_quantile_1.png"></span> | |
51 | </p> | |
52 | <p> | |
53 | As can be seen there is no random variable that corresponds to a probability | |
54 | of exactly 0.05, so we're left with two choices as shown in the figure: | |
55 | </p> | |
56 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
57 | <li class="listitem"> | |
58 | We could round the result down to 18. | |
59 | </li> | |
60 | <li class="listitem"> | |
61 | We could round the result up to 19. | |
62 | </li> | |
63 | </ul></div> | |
64 | <p> | |
65 | In fact there's actually a third choice as well: we could "pretend" | |
66 | that the distribution was continuous and return a real valued result: in | |
67 | this case we would calculate a result of approximately 18.701 (this accurately | |
68 | reflects the fact that the result is nearer to 19 than 18). | |
69 | </p> | |
70 | <p> | |
71 | By using policies we can offer any of the above as options, but that still | |
72 | leaves the question: <span class="emphasis"><em>What is actually the right thing to do?</em></span> | |
73 | </p> | |
74 | <p> | |
75 | And in particular: <span class="emphasis"><em>What policy should we use by default?</em></span> | |
76 | </p> | |
77 | <p> | |
78 | In coming to an answer we should realise that: | |
79 | </p> | |
80 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
81 | <li class="listitem"> | |
82 | Calculating an integer result is often much faster than calculating a | |
83 | real-valued result: in fact in our tests it was up to 20 times faster. | |
84 | </li> | |
85 | <li class="listitem"> | |
86 | Normally people calculate quantiles so that they can perform a test of | |
87 | some kind: <span class="emphasis"><em>"If the random variable is less than N then | |
88 | we can reject our null-hypothesis with 90% confidence."</em></span> | |
89 | </li> | |
90 | </ul></div> | |
91 | <p> | |
92 | So there is a genuine benefit to calculating an integer result as well as | |
93 | it being "the right thing to do" from a philosophical point of | |
94 | view. What's more if someone asks for a quantile at 0.05, then we can normally | |
95 | assume that they are asking for <span class="emphasis"><em><span class="bold"><strong>at least</strong></span> | |
96 | 95% of the probability to the right of the value chosen, and <span class="bold"><strong>no | |
97 | more than</strong></span> 5% of the probability to the left of the value chosen.</em></span> | |
98 | </p> | |
99 | <p> | |
100 | In the above binomial example we would therefore round the result down to | |
101 | 18. | |
102 | </p> | |
103 | <p> | |
104 | The converse applies to upper-quantiles: If the probability is greater than | |
105 | 0.5 we would want to round the quantile up, <span class="emphasis"><em>so that <span class="bold"><strong>at | |
106 | least</strong></span> the requested probability is to the left of the value returned, | |
107 | and <span class="bold"><strong>no more than</strong></span> 1 - the requested probability | |
108 | is to the right of the value returned.</em></span> | |
109 | </p> | |
110 | <p> | |
111 | Likewise for two-sided intervals, we would round lower quantiles down, and | |
112 | upper quantiles up. This ensures that we have <span class="emphasis"><em>at least the requested | |
113 | probability in the central region</em></span> and <span class="emphasis"><em>no more than 1 | |
114 | minus the requested probability in the tail areas.</em></span> | |
115 | </p> | |
116 | <p> | |
117 | For example, taking our 50 sample binomial distribution with a success fraction | |
118 | of 0.5, if we wanted a two sided 90% confidence interval, then we would ask | |
119 | for the 0.05 and 0.95 quantiles with the results <span class="emphasis"><em>rounded outwards</em></span> | |
120 | so that <span class="emphasis"><em>at least 90% of the probability</em></span> is in the central | |
121 | area: | |
122 | </p> | |
123 | <p> | |
124 | <span class="inlinemediaobject"><img src="../../../graphs/binomial_pdf_3.png"></span> | |
125 | </p> | |
126 | <p> | |
127 | So far so good, but there is in fact a trap waiting for the unwary here: | |
128 | </p> | |
129 | <pre class="programlisting"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binomial</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.05</span><span class="special">);</span> | |
130 | </pre> | |
131 | <p> | |
132 | returns 18 as the result, which is what we would expect from the graph above, | |
133 | and indeed there is no x greater than 18 for which: | |
134 | </p> | |
135 | <pre class="programlisting"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">binomial</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="identifier">x</span><span class="special">)</span> <span class="special"><=</span> <span class="number">0.05</span><span class="special">;</span> | |
136 | </pre> | |
137 | <p> | |
138 | However: | |
139 | </p> | |
140 | <pre class="programlisting"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binomial</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.95</span><span class="special">);</span> | |
141 | </pre> | |
142 | <p> | |
143 | returns 31, and indeed while there is no x less than 31 for which: | |
144 | </p> | |
145 | <pre class="programlisting"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">binomial</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">>=</span> <span class="number">0.95</span><span class="special">;</span> | |
146 | </pre> | |
147 | <p> | |
148 | We might naively expect that for this symmetrical distribution the result | |
149 | would be 32 (since 32 = 50 - 18), but we need to remember that the cdf of | |
150 | the binomial is <span class="emphasis"><em>inclusive</em></span> of the random variable. So | |
151 | while the left tail area <span class="emphasis"><em>includes</em></span> the quantile returned, | |
152 | the right tail area always excludes an upper quantile value: since that "belongs" | |
153 | to the central area. | |
154 | </p> | |
155 | <p> | |
156 | Look at the graph above to see what's going on here: the lower quantile of | |
157 | 18 belongs to the left tail, so any value <= 18 is in the left tail. The | |
158 | upper quantile of 31 on the other hand belongs to the central area, so the | |
159 | tail area actually starts at 32, so any value > 31 is in the right tail. | |
160 | </p> | |
161 | <p> | |
162 | Therefore if U and L are the upper and lower quantiles respectively, then | |
163 | a random variable X is in the tail area - where we would reject the null | |
164 | hypothesis if: | |
165 | </p> | |
166 | <pre class="programlisting"><span class="identifier">X</span> <span class="special"><=</span> <span class="identifier">L</span> <span class="special">||</span> <span class="identifier">X</span> <span class="special">></span> <span class="identifier">U</span> | |
167 | </pre> | |
168 | <p> | |
169 | And the a variable X is inside the central region if: | |
170 | </p> | |
171 | <pre class="programlisting"><span class="identifier">L</span> <span class="special"><</span> <span class="identifier">X</span> <span class="special"><=</span> <span class="identifier">U</span> | |
172 | </pre> | |
173 | <p> | |
174 | The moral here is to <span class="emphasis"><em>always be very careful with your comparisons | |
175 | when dealing with a discrete distribution</em></span>, and if in doubt, <span class="emphasis"><em>base | |
176 | your comparisons on CDF's instead</em></span>. | |
177 | </p> | |
178 | <h5> | |
179 | <a name="math_toolkit.pol_tutorial.understand_dis_quant.h0"></a> | |
180 | <span class="phrase"><a name="math_toolkit.pol_tutorial.understand_dis_quant.other_rounding_policies_are_avai"></a></span><a class="link" href="understand_dis_quant.html#math_toolkit.pol_tutorial.understand_dis_quant.other_rounding_policies_are_avai">Other | |
181 | Rounding Policies are Available</a> | |
182 | </h5> | |
183 | <p> | |
184 | As you would expect from a section on policies, you won't be surprised to | |
185 | know that other rounding options are available: | |
186 | </p> | |
187 | <div class="variablelist"> | |
188 | <p class="title"><b></b></p> | |
189 | <dl class="variablelist"> | |
190 | <dt><span class="term">integer_round_outwards</span></dt> | |
191 | <dd> | |
192 | <p> | |
193 | This is the default policy as described above: lower quantiles are | |
194 | rounded down (probability < 0.5), and upper quantiles (probability | |
195 | > 0.5) are rounded up. | |
196 | </p> | |
197 | <p> | |
198 | This gives <span class="emphasis"><em>no more than</em></span> the requested probability | |
199 | in the tails, and <span class="emphasis"><em>at least</em></span> the requested probability | |
200 | in the central area. | |
201 | </p> | |
202 | </dd> | |
203 | <dt><span class="term">integer_round_inwards</span></dt> | |
204 | <dd> | |
205 | <p> | |
206 | This is the exact opposite of the default policy: lower quantiles are | |
207 | rounded up (probability < 0.5), and upper quantiles (probability | |
208 | > 0.5) are rounded down. | |
209 | </p> | |
210 | <p> | |
211 | This gives <span class="emphasis"><em>at least</em></span> the requested probability | |
212 | in the tails, and <span class="emphasis"><em>no more than</em></span> the requested probability | |
213 | in the central area. | |
214 | </p> | |
215 | </dd> | |
216 | <dt><span class="term">integer_round_down</span></dt> | |
217 | <dd><p> | |
218 | This policy will always round the result down no matter whether it | |
219 | is an upper or lower quantile | |
220 | </p></dd> | |
221 | <dt><span class="term">integer_round_up</span></dt> | |
222 | <dd><p> | |
223 | This policy will always round the result up no matter whether it is | |
224 | an upper or lower quantile | |
225 | </p></dd> | |
226 | <dt><span class="term">integer_round_nearest</span></dt> | |
227 | <dd><p> | |
228 | This policy will always round the result to the nearest integer no | |
229 | matter whether it is an upper or lower quantile | |
230 | </p></dd> | |
231 | <dt><span class="term">real</span></dt> | |
232 | <dd><p> | |
233 | This policy will return a real valued result for the quantile of a | |
234 | discrete distribution: this is generally much slower than finding an | |
235 | integer result but does allow for more sophisticated rounding policies. | |
236 | </p></dd> | |
237 | </dl> | |
238 | </div> | |
239 | <p> | |
240 | To understand how the rounding policies for the discrete distributions can | |
241 | be used, we'll use the 50-sample binomial distribution with a success fraction | |
242 | of 0.5 once again, and calculate all the possible quantiles at 0.05 and 0.95. | |
243 | </p> | |
244 | <p> | |
245 | Begin by including the needed headers (and some using statements for conciseness): | |
246 | </p> | |
247 | <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> | |
248 | <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
249 | <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">left</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">fixed</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">scientific</span><span class="special">;</span> | |
250 | <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iomanip</span><span class="special">></span> | |
251 | <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">;</span> | |
252 | <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">;</span> | |
253 | ||
254 | <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">binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
255 | </pre> | |
256 | <p> | |
257 | Next we'll bring the needed declarations into scope, and define distribution | |
258 | types for all the available rounding policies: | |
259 | </p> | |
260 | <pre class="programlisting"><span class="comment">// Avoid </span> | |
261 | <span class="comment">// using namespace std; // and </span> | |
262 | <span class="comment">// using namespace boost::math;</span> | |
263 | <span class="comment">// to avoid potential ambiguity of names, like binomial.</span> | |
264 | <span class="comment">// using namespace boost::math::policies; is small risk, but</span> | |
265 | <span class="comment">// the necessary items are brought into scope thus:</span> | |
266 | ||
267 | <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">binomial_distribution</span><span class="special">;</span> | |
268 | <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">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special">;</span> | |
269 | <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">policies</span><span class="special">::</span><span class="identifier">discrete_quantile</span><span class="special">;</span> | |
270 | ||
271 | <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">policies</span><span class="special">::</span><span class="identifier">integer_round_outwards</span><span class="special">;</span> | |
272 | <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">policies</span><span class="special">::</span><span class="identifier">integer_round_down</span><span class="special">;</span> | |
273 | <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">policies</span><span class="special">::</span><span class="identifier">integer_round_up</span><span class="special">;</span> | |
274 | <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">policies</span><span class="special">::</span><span class="identifier">integer_round_nearest</span><span class="special">;</span> | |
275 | <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">policies</span><span class="special">::</span><span class="identifier">integer_round_inwards</span><span class="special">;</span> | |
276 | <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">policies</span><span class="special">::</span><span class="identifier">real</span><span class="special">;</span> | |
277 | ||
278 | <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">binomial_distribution</span><span class="special">;</span> <span class="comment">// Not std::binomial_distribution.</span> | |
279 | ||
280 | <span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><</span> | |
281 | <span class="keyword">double</span><span class="special">,</span> | |
282 | <span class="identifier">policy</span><span class="special"><</span><span class="identifier">discrete_quantile</span><span class="special"><</span><span class="identifier">integer_round_outwards</span><span class="special">></span> <span class="special">></span> <span class="special">></span> | |
283 | <span class="identifier">binom_round_outwards</span><span class="special">;</span> | |
284 | ||
285 | <span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><</span> | |
286 | <span class="keyword">double</span><span class="special">,</span> | |
287 | <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> <span class="special">></span> | |
288 | <span class="identifier">binom_round_inwards</span><span class="special">;</span> | |
289 | ||
290 | <span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><</span> | |
291 | <span class="keyword">double</span><span class="special">,</span> | |
292 | <span class="identifier">policy</span><span class="special"><</span><span class="identifier">discrete_quantile</span><span class="special"><</span><span class="identifier">integer_round_down</span><span class="special">></span> <span class="special">></span> <span class="special">></span> | |
293 | <span class="identifier">binom_round_down</span><span class="special">;</span> | |
294 | ||
295 | <span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><</span> | |
296 | <span class="keyword">double</span><span class="special">,</span> | |
297 | <span class="identifier">policy</span><span class="special"><</span><span class="identifier">discrete_quantile</span><span class="special"><</span><span class="identifier">integer_round_up</span><span class="special">></span> <span class="special">></span> <span class="special">></span> | |
298 | <span class="identifier">binom_round_up</span><span class="special">;</span> | |
299 | ||
300 | <span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><</span> | |
301 | <span class="keyword">double</span><span class="special">,</span> | |
302 | <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> <span class="special">></span> | |
303 | <span class="identifier">binom_round_nearest</span><span class="special">;</span> | |
304 | ||
305 | <span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><</span> | |
306 | <span class="keyword">double</span><span class="special">,</span> | |
307 | <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> <span class="special">></span> | |
308 | <span class="identifier">binom_real_quantile</span><span class="special">;</span> | |
309 | </pre> | |
310 | <p> | |
311 | Now let's set to work calling those quantiles: | |
312 | </p> | |
313 | <pre class="programlisting"><span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> | |
314 | <span class="special">{</span> | |
315 | <span class="identifier">cout</span> <span class="special"><<</span> | |
316 | <span class="string">"Testing rounding policies for a 50 sample binomial distribution,\n"</span> | |
317 | <span class="string">"with a success fraction of 0.5.\n\n"</span> | |
318 | <span class="string">"Lower quantiles are calculated at p = 0.05\n\n"</span> | |
319 | <span class="string">"Upper quantiles at p = 0.95.\n\n"</span><span class="special">;</span> | |
320 | ||
321 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
322 | <span class="special"><<</span> <span class="string">"Policy"</span><span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
323 | <span class="special"><<</span> <span class="string">"Lower Quantile"</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
324 | <span class="special"><<</span> <span class="string">"Upper Quantile"</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
325 | ||
326 | <span class="comment">// Test integer_round_outwards:</span> | |
327 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
328 | <span class="special"><<</span> <span class="string">"integer_round_outwards"</span> | |
329 | <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
330 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_round_outwards</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.05</span><span class="special">)</span> | |
331 | <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
332 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_round_outwards</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.95</span><span class="special">)</span> | |
333 | <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
334 | ||
335 | <span class="comment">// Test integer_round_inwards:</span> | |
336 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
337 | <span class="special"><<</span> <span class="string">"integer_round_inwards"</span> | |
338 | <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
339 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_round_inwards</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.05</span><span class="special">)</span> | |
340 | <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
341 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_round_inwards</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.95</span><span class="special">)</span> | |
342 | <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
343 | ||
344 | <span class="comment">// Test integer_round_down:</span> | |
345 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
346 | <span class="special"><<</span> <span class="string">"integer_round_down"</span> | |
347 | <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
348 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_round_down</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.05</span><span class="special">)</span> | |
349 | <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
350 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_round_down</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.95</span><span class="special">)</span> | |
351 | <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
352 | ||
353 | <span class="comment">// Test integer_round_up:</span> | |
354 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
355 | <span class="special"><<</span> <span class="string">"integer_round_up"</span> | |
356 | <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
357 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_round_up</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.05</span><span class="special">)</span> | |
358 | <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
359 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_round_up</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.95</span><span class="special">)</span> | |
360 | <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
361 | ||
362 | <span class="comment">// Test integer_round_nearest:</span> | |
363 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
364 | <span class="special"><<</span> <span class="string">"integer_round_nearest"</span> | |
365 | <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
366 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_round_nearest</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.05</span><span class="special">)</span> | |
367 | <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
368 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_round_nearest</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.95</span><span class="special">)</span> | |
369 | <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
370 | ||
371 | <span class="comment">// Test real:</span> | |
372 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">25</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
373 | <span class="special"><<</span> <span class="string">"real"</span> | |
374 | <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
375 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_real_quantile</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.05</span><span class="special">)</span> | |
376 | <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">18</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> | |
377 | <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binom_real_quantile</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">),</span> <span class="number">0.95</span><span class="special">)</span> | |
378 | <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
379 | <span class="special">}</span> <span class="comment">// int main()</span> | |
380 | </pre> | |
381 | <p> | |
382 | Which produces the program output: | |
383 | </p> | |
384 | <pre class="programlisting"> policy_eg_10.vcxproj -> J:\Cpp\MathToolkit\test\Math_test\Release\policy_eg_10.exe | |
385 | Testing rounding policies for a 50 sample binomial distribution, | |
386 | with a success fraction of 0.5. | |
387 | ||
388 | Lower quantiles are calculated at p = 0.05 | |
389 | ||
390 | Upper quantiles at p = 0.95. | |
391 | ||
392 | Policy Lower Quantile Upper Quantile | |
393 | integer_round_outwards 18 31 | |
394 | integer_round_inwards 19 30 | |
395 | integer_round_down 18 30 | |
396 | integer_round_up 19 31 | |
397 | integer_round_nearest 19 30 | |
398 | real 18.701 30.299 | |
399 | </pre> | |
400 | </div> | |
401 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
402 | <td align="left"></td> | |
403 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
404 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
405 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
406 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
407 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
408 | 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>) | |
409 | </p> | |
410 | </div></td> | |
411 | </tr></table> | |
412 | <hr> | |
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