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| 24 | </div> | |
| 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> | |
| 413 | <div class="spirit-nav"> | |
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