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| 24 | </div> | |
| 25 | <div class="section"> | |
| 26 | <div class="titlepage"><div><div><h4 class="title"> | |
| 27 | <a name="math_toolkit.stat_tut.weg.geometric_eg"></a><a class="link" href="geometric_eg.html" title="Geometric Distribution Examples">Geometric Distribution | |
| 28 | Examples</a> | |
| 29 | </h4></div></div></div> | |
| 30 | <p> | |
| 31 | For this example, we will opt to #define two macros to control the error | |
| 32 | and discrete handling policies. For this simple example, we want to avoid | |
| 33 | throwing an exception (the default policy) and just return infinity. We | |
| 34 | want to treat the distribution as if it was continuous, so we choose a | |
| 35 | discrete_quantile policy of real, rather than the default policy integer_round_outwards. | |
| 36 | </p> | |
| 37 | <pre class="programlisting"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_OVERFLOW_ERROR_POLICY</span> <span class="identifier">ignore_error</span> | |
| 38 | <span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_DISCRETE_QUANTILE_POLICY</span> <span class="identifier">real</span> | |
| 39 | </pre> | |
| 40 | <div class="caution"><table border="0" summary="Caution"> | |
| 41 | <tr> | |
| 42 | <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../doc/src/images/caution.png"></td> | |
| 43 | <th align="left">Caution</th> | |
| 44 | </tr> | |
| 45 | <tr><td align="left" valign="top"><p> | |
| 46 | It is vital to #include distributions etc <span class="bold"><strong>after</strong></span> | |
| 47 | the above #defines | |
| 48 | </p></td></tr> | |
| 49 | </table></div> | |
| 50 | <p> | |
| 51 | After that we need some includes to provide easy access to the negative | |
| 52 | binomial distribution, and we need some std library iostream, of course. | |
| 53 | </p> | |
| 54 | <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">geometric</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
| 55 | <span class="comment">// for geometric_distribution</span> | |
| 56 | <span class="keyword">using</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">geometric_distribution</span><span class="special">;</span> <span class="comment">//</span> | |
| 57 | <span class="keyword">using</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">geometric</span><span class="special">;</span> <span class="comment">// typedef provides default type is double.</span> | |
| 58 | <span class="keyword">using</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">pdf</span><span class="special">;</span> <span class="comment">// Probability mass function.</span> | |
| 59 | <span class="keyword">using</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">cdf</span><span class="special">;</span> <span class="comment">// Cumulative density function.</span> | |
| 60 | <span class="keyword">using</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">quantile</span><span class="special">;</span> | |
| 61 | ||
| 62 | <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">negative_binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
| 63 | <span class="comment">// for negative_binomial_distribution</span> | |
| 64 | <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> <span class="comment">// typedef provides default type is double.</span> | |
| 65 | ||
| 66 | <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">normal</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
| 67 | <span class="comment">// for negative_binomial_distribution</span> | |
| 68 | <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">normal</span><span class="special">;</span> <span class="comment">// typedef provides default type is double.</span> | |
| 69 | ||
| 70 | <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> | |
| 71 | <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> | |
| 72 | <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">noshowpoint</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">left</span><span class="special">;</span> | |
| 73 | <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iomanip</span><span class="special">></span> | |
| 74 | <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">;</span> | |
| 75 | ||
| 76 | <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">limits</span><span class="special">></span> | |
| 77 | <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">;</span> | |
| 78 | </pre> | |
| 79 | <p> | |
| 80 | It is always sensible to use try and catch blocks because defaults policies | |
| 81 | are to throw an exception if anything goes wrong. | |
| 82 | </p> | |
| 83 | <p> | |
| 84 | Simple try'n'catch blocks (see below) will ensure that you get a helpful | |
| 85 | error message instead of an abrupt (and silent) program abort. | |
| 86 | </p> | |
| 87 | <h5> | |
| 88 | <a name="math_toolkit.stat_tut.weg.geometric_eg.h0"></a> | |
| 89 | <span class="phrase"><a name="math_toolkit.stat_tut.weg.geometric_eg.throwing_a_dice"></a></span><a class="link" href="geometric_eg.html#math_toolkit.stat_tut.weg.geometric_eg.throwing_a_dice">Throwing | |
| 90 | a dice</a> | |
| 91 | </h5> | |
| 92 | <p> | |
| 93 | The Geometric distribution describes the probability (<span class="emphasis"><em>p</em></span>) | |
| 94 | of a number of failures to get the first success in <span class="emphasis"><em>k</em></span> | |
| 95 | Bernoulli trials. (A <a href="http://en.wikipedia.org/wiki/Bernoulli_distribution" target="_top">Bernoulli | |
| 96 | trial</a> is one with only two possible outcomes, success of failure, | |
| 97 | and <span class="emphasis"><em>p</em></span> is the probability of success). | |
| 98 | </p> | |
| 99 | <p> | |
| 100 | Suppose an 'fair' 6-face dice is thrown repeatedly: | |
| 101 | </p> | |
| 102 | <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">success_fraction</span> <span class="special">=</span> <span class="number">1.</span><span class="special">/</span><span class="number">6</span><span class="special">;</span> <span class="comment">// success_fraction (p) = 0.1666</span> | |
| 103 | <span class="comment">// (so failure_fraction is 1 - success_fraction = 5./6 = 1- 0.1666 = 0.8333)</span> | |
| 104 | </pre> | |
| 105 | <p> | |
| 106 | If the dice is thrown repeatedly until the <span class="bold"><strong>first</strong></span> | |
| 107 | time a <span class="emphasis"><em>three</em></span> appears. The probablility distribution | |
| 108 | of the number of times it is thrown <span class="bold"><strong>not</strong></span> | |
| 109 | getting a <span class="emphasis"><em>three</em></span> (<span class="emphasis"><em>not-a-threes</em></span> | |
| 110 | number of failures to get a <span class="emphasis"><em>three</em></span>) is a geometric | |
| 111 | distribution with the success_fraction = 1/6 = 0.1666 ̇. | |
| 112 | </p> | |
| 113 | <p> | |
| 114 | We therefore start by constructing a geometric distribution with the one | |
| 115 | parameter success_fraction, the probability of success. | |
| 116 | </p> | |
| 117 | <pre class="programlisting"><span class="identifier">geometric</span> <span class="identifier">g6</span><span class="special">(</span><span class="identifier">success_fraction</span><span class="special">);</span> <span class="comment">// type double by default.</span> | |
| 118 | </pre> | |
| 119 | <p> | |
| 120 | To confirm, we can echo the success_fraction parameter of the distribution. | |
| 121 | </p> | |
| 122 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"success fraction of a six-sided dice is "</span> <span class="special"><<</span> <span class="identifier">g6</span><span class="special">.</span><span class="identifier">success_fraction</span><span class="special">()</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 123 | </pre> | |
| 124 | <p> | |
| 125 | So the probability of getting a three at the first throw (zero failures) | |
| 126 | is | |
| 127 | </p> | |
| 128 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.1667</span> | |
| 129 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.1667</span> | |
| 130 | </pre> | |
| 131 | <p> | |
| 132 | Note that the cdf and pdf are identical because the is only one throw. | |
| 133 | If we want the probability of getting the first <span class="emphasis"><em>three</em></span> | |
| 134 | on the 2nd throw: | |
| 135 | </p> | |
| 136 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.1389</span> | |
| 137 | </pre> | |
| 138 | <p> | |
| 139 | If we want the probability of getting the first <span class="emphasis"><em>three</em></span> | |
| 140 | on the 1st or 2nd throw (allowing one failure): | |
| 141 | </p> | |
| 142 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"pdf(g6, 0) + pdf(g6, 1) = "</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 143 | </pre> | |
| 144 | <p> | |
| 145 | Or more conveniently, and more generally, we can use the Cumulative Distribution | |
| 146 | Function CDF. | |
| 147 | </p> | |
| 148 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"cdf(g6, 1) = "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.3056</span> | |
| 149 | </pre> | |
| 150 | <p> | |
| 151 | If we allow many more (12) throws, the probability of getting our <span class="emphasis"><em>three</em></span> | |
| 152 | gets very high: | |
| 153 | </p> | |
| 154 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"cdf(g6, 12) = "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">12</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.9065 or 90% probability.</span> | |
| 155 | </pre> | |
| 156 | <p> | |
| 157 | If we want to be much more confident, say 99%, we can estimate the number | |
| 158 | of throws to be this sure using the inverse or quantile. | |
| 159 | </p> | |
| 160 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"quantile(g6, 0.99) = "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">g6</span><span class="special">,</span> <span class="number">0.99</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 24.26</span> | |
| 161 | </pre> | |
| 162 | <p> | |
| 163 | Note that the value returned is not an integer: if you want an integer | |
| 164 | result you should use either floor, round or ceil functions, or use the | |
| 165 | policies mechanism. | |
| 166 | </p> | |
| 167 | <p> | |
| 168 | See <a class="link" href="../../pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">understanding | |
| 169 | discrete quantiles</a>. | |
| 170 | </p> | |
| 171 | <p> | |
| 172 | The geometric distribution is related to the negative binomial    <code class="computeroutput"><span class="identifier">negative_binomial_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">RealType</span> | |
| 173 | <span class="identifier">p</span><span class="special">);</span></code> | |
| 174 | with parameter <span class="emphasis"><em>r</em></span> = 1. So we could get the same result | |
| 175 | using the negative binomial, but using the geometric the results will be | |
| 176 | faster, and may be more accurate. | |
| 177 | </p> | |
| 178 | <pre class="programlisting"><span class="identifier">negative_binomial</span> <span class="identifier">nb</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="identifier">success_fraction</span><span class="special">);</span> | |
| 179 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.1389</span> | |
| 180 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">nb</span><span class="special">,</span> <span class="number">1</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.3056</span> | |
| 181 | </pre> | |
| 182 | <p> | |
| 183 | We could also the complement to express the required probability as 1 - | |
| 184 | 0.99 = 0.01 (and get the same result): | |
| 185 | </p> | |
| 186 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"quantile(complement(g6, 1 - p)) "</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">g6</span><span class="special">,</span> <span class="number">0.01</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 24.26</span> | |
| 187 | </pre> | |
| 188 | <p> | |
| 189 | Note too that Boost.Math geometric distribution is implemented as a continuous | |
| 190 | function. Unlike other implementations (for example R) it <span class="bold"><strong>uses</strong></span> | |
| 191 | the number of failures as a <span class="bold"><strong>real</strong></span> parameter, | |
| 192 | not as an integer. If you want this integer behaviour, you may need to | |
| 193 | enforce this by rounding the parameter you pass, probably rounding down, | |
| 194 | to the nearest integer. For example, R returns the success fraction probability | |
| 195 | for all values of failures from 0 to 0.999999 thus: | |
| 196 | </p> | |
| 197 | <pre class="programlisting">   R> formatC(pgeom(0.0001,0.5, FALSE), digits=17) " 0.5" | |
| 198 | </pre> | |
| 199 | <p> | |
| 200 | So in Boost.Math the equivalent is | |
| 201 | </p> | |
| 202 | <pre class="programlisting"> <span class="identifier">geometric</span> <span class="identifier">g05</span><span class="special">(</span><span class="number">0.5</span><span class="special">);</span> <span class="comment">// Probability of success = 0.5 or 50%</span> | |
| 203 | <span class="comment">// Output all potentially significant digits for the type, here double.</span> | |
| 204 | ||
| 205 | <span class="preprocessor">#ifdef</span> <span class="identifier">BOOST_NO_CXX11_NUMERIC_LIMITS</span> | |
| 206 | <span class="keyword">int</span> <span class="identifier">max_digits10</span> <span class="special">=</span> <span class="number">2</span> <span class="special">+</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">policies</span><span class="special">::</span><span class="identifier">digits</span><span class="special"><</span><span class="keyword">double</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">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special"><></span> <span class="special">>()</span> <span class="special">*</span> <span class="number">30103UL</span><span class="special">)</span> <span class="special">/</span> <span class="number">100000UL</span><span class="special">;</span> | |
| 207 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"BOOST_NO_CXX11_NUMERIC_LIMITS is defined"</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 208 | <span class="preprocessor">#else</span> | |
| 209 | <span class="keyword">int</span> <span class="identifier">max_digits10</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">;</span> | |
| 210 | <span class="preprocessor">#endif</span> | |
| 211 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Show all potentially significant decimal digits std::numeric_limits<double>::max_digits10 = "</span> | |
| 212 | <span class="special"><<</span> <span class="identifier">max_digits10</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 213 | <span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">max_digits10</span><span class="special">);</span> <span class="comment">//</span> | |
| 214 | ||
| 215 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">g05</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// returns 0.5000346561579232, not exact 0.5.</span> | |
| 216 | </pre> | |
| 217 | <p> | |
| 218 | To get the R discrete behaviour, you simply need to round with, for example, | |
| 219 | the <code class="computeroutput"><span class="identifier">floor</span></code> function. | |
| 220 | </p> | |
| 221 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">g05</span><span class="special">,</span> <span class="identifier">floor</span><span class="special">(</span><span class="number">0.0001</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// returns exactly 0.5</span> | |
| 222 | </pre> | |
| 223 | <pre class="programlisting"><code class="computeroutput"><span class="special">></span> <span class="identifier">formatC</span><span class="special">(</span><span class="identifier">pgeom</span><span class="special">(</span><span class="number">0.9999999</span><span class="special">,</span><span class="number">0.5</span><span class="special">,</span> <span class="identifier">FALSE</span><span class="special">),</span> <span class="identifier">digits</span><span class="special">=</span><span class="number">17</span><span class="special">)</span> <span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="string">" 0.25"</span></code> | |
| 224 | <code class="computeroutput"><span class="special">></span> <span class="identifier">formatC</span><span class="special">(</span><span class="identifier">pgeom</span><span class="special">(</span><span class="number">1.999999</span><span class="special">,</span><span class="number">0.5</span><span class="special">,</span> <span class="identifier">FALSE</span><span class="special">),</span> <span class="identifier">digits</span><span class="special">=</span><span class="number">17</span><span class="special">)[</span><span class="number">1</span><span class="special">]</span> <span class="string">" 0.25"</span> <span class="identifier">k</span> <span class="special">=</span> <span class="number">1</span></code> | |
| 225 | <code class="computeroutput"><span class="special">></span> <span class="identifier">formatC</span><span class="special">(</span><span class="identifier">pgeom</span><span class="special">(</span><span class="number">1.9999999</span><span class="special">,</span><span class="number">0.5</span><span class="special">,</span> <span class="identifier">FALSE</span><span class="special">),</span> <span class="identifier">digits</span><span class="special">=</span><span class="number">17</span><span class="special">)[</span><span class="number">1</span><span class="special">]</span> <span class="string">"0.12500000000000003"</span> <span class="identifier">k</span> <span class="special">=</span> <span class="number">2</span></code> | |
| 226 | </pre> | |
| 227 | <p> | |
| 228 | shows that R makes an arbitrary round-up decision at about 1e7 from the | |
| 229 | next integer above. This may be convenient in practice, and could be replicated | |
| 230 | in C++ if desired. | |
| 231 | </p> | |
| 232 | <h5> | |
| 233 | <a name="math_toolkit.stat_tut.weg.geometric_eg.h1"></a> | |
| 234 | <span class="phrase"><a name="math_toolkit.stat_tut.weg.geometric_eg.surveying_customers_to_find_one_"></a></span><a class="link" href="geometric_eg.html#math_toolkit.stat_tut.weg.geometric_eg.surveying_customers_to_find_one_">Surveying | |
| 235 | customers to find one with a faulty product</a> | |
| 236 | </h5> | |
| 237 | <p> | |
| 238 | A company knows from warranty claims that 2% of their products will be | |
| 239 | faulty, so the 'success_fraction' of finding a fault is 0.02. It wants | |
| 240 | to interview a purchaser of faulty products to assess their 'user experience'. | |
| 241 | </p> | |
| 242 | <p> | |
| 243 | To estimate how many customers they will probably need to contact in order | |
| 244 | to find one who has suffered from the fault, we first construct a geometric | |
| 245 | distribution with probability 0.02, and then chose a confidence, say 80%, | |
| 246 | 95%, or 99% to finding a customer with a fault. Finally, we probably want | |
| 247 | to round up the result to the integer above using the <code class="computeroutput"><span class="identifier">ceil</span></code> | |
| 248 | function. (We could also use a policy, but that is hardly worthwhile for | |
| 249 | this simple application.) | |
| 250 | </p> | |
| 251 | <p> | |
| 252 | (This also assumes that each customer only buys one product: if customers | |
| 253 | bought more than one item, the probability of finding a customer with a | |
| 254 | fault obviously improves.) | |
| 255 | </p> | |
| 256 | <pre class="programlisting"><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">5</span><span class="special">);</span> | |
| 257 | <span class="identifier">geometric</span> <span class="identifier">g</span><span class="special">(</span><span class="number">0.02</span><span class="special">);</span> <span class="comment">// On average, 2 in 100 products are faulty.</span> | |
| 258 | <span class="keyword">double</span> <span class="identifier">c</span> <span class="special">=</span> <span class="number">0.95</span><span class="special">;</span> <span class="comment">// 95% confidence.</span> | |
| 259 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" quantile(g, "</span> <span class="special"><<</span> <span class="identifier">c</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="identifier">c</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 260 | ||
| 261 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"To be "</span> <span class="special"><<</span> <span class="identifier">c</span> <span class="special">*</span> <span class="number">100</span> | |
| 262 | <span class="special"><<</span> <span class="string">"% confident of finding we customer with a fault, need to survey "</span> | |
| 263 | <span class="special"><<</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="identifier">c</span><span class="special">))</span> <span class="special"><<</span> <span class="string">" customers."</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 148</span> | |
| 264 | <span class="identifier">c</span> <span class="special">=</span> <span class="number">0.99</span><span class="special">;</span> <span class="comment">// Very confident.</span> | |
| 265 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"To be "</span> <span class="special"><<</span> <span class="identifier">c</span> <span class="special">*</span> <span class="number">100</span> | |
| 266 | <span class="special"><<</span> <span class="string">"% confident of finding we customer with a fault, need to survey "</span> | |
| 267 | <span class="special"><<</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="identifier">c</span><span class="special">))</span> <span class="special"><<</span> <span class="string">" customers."</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 227</span> | |
| 268 | <span class="identifier">c</span> <span class="special">=</span> <span class="number">0.80</span><span class="special">;</span> <span class="comment">// Only reasonably confident.</span> | |
| 269 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"To be "</span> <span class="special"><<</span> <span class="identifier">c</span> <span class="special">*</span> <span class="number">100</span> | |
| 270 | <span class="special"><<</span> <span class="string">"% confident of finding we customer with a fault, need to survey "</span> | |
| 271 | <span class="special"><<</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="identifier">c</span><span class="special">))</span> <span class="special"><<</span> <span class="string">" customers."</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 79</span> | |
| 272 | </pre> | |
| 273 | <h5> | |
| 274 | <a name="math_toolkit.stat_tut.weg.geometric_eg.h2"></a> | |
| 275 | <span class="phrase"><a name="math_toolkit.stat_tut.weg.geometric_eg.basket_ball_shooters"></a></span><a class="link" href="geometric_eg.html#math_toolkit.stat_tut.weg.geometric_eg.basket_ball_shooters">Basket | |
| 276 | Ball Shooters</a> | |
| 277 | </h5> | |
| 278 | <p> | |
| 279 | According to Wikipedia, average pro basket ball players get <a href="http://en.wikipedia.org/wiki/Free_throw" target="_top">free | |
| 280 | throws</a> in the baskets 70 to 80 % of the time, but some get as high | |
| 281 | as 95%, and others as low as 50%. Suppose we want to compare the probabilities | |
| 282 | of failing to get a score only on the first or on the fifth shot? To start | |
| 283 | we will consider the average shooter, say 75%. So we construct a geometric | |
| 284 | distribution with success_fraction parameter 75/100 = 0.75. | |
| 285 | </p> | |
| 286 | <pre class="programlisting"><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="number">2</span><span class="special">);</span> | |
| 287 | <span class="identifier">geometric</span> <span class="identifier">gav</span><span class="special">(</span><span class="number">0.75</span><span class="special">);</span> <span class="comment">// Shooter averages 7.5 out of 10 in the basket.</span> | |
| 288 | </pre> | |
| 289 | <p> | |
| 290 | What is probability of getting 1st try in the basket, that is with no failures? | |
| 291 | </p> | |
| 292 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of score on 1st try = "</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">gav</span><span class="special">,</span> <span class="number">0</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.75</span> | |
| 293 | </pre> | |
| 294 | <p> | |
| 295 | This is, of course, the success_fraction probability 75%. What is the probability | |
| 296 | that the shooter only scores on the fifth shot? So there are 5-1 = 4 failures | |
| 297 | before the first success. | |
| 298 | </p> | |
| 299 | <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of score on 5th try = "</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">gav</span><span class="special">,</span> <span class="number">4</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.0029</span> | |
| 300 | </pre> | |
| 301 | <p> | |
| 302 | Now compare this with the poor and the best players success fraction. We | |
| 303 | need to constructing new distributions with the different success fractions, | |
| 304 | and then get the corresponding probability density functions values: | |
| 305 | </p> | |
| 306 | <pre class="programlisting"><span class="identifier">geometric</span> <span class="identifier">gbest</span><span class="special">(</span><span class="number">0.95</span><span class="special">);</span> | |
| 307 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of score on 5th try = "</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">gbest</span><span class="special">,</span> <span class="number">4</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 5.9e-6</span> | |
| 308 | <span class="identifier">geometric</span> <span class="identifier">gmediocre</span><span class="special">(</span><span class="number">0.50</span><span class="special">);</span> | |
| 309 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability of score on 5th try = "</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">gmediocre</span><span class="special">,</span> <span class="number">4</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.031</span> | |
| 310 | </pre> | |
| 311 | <p> | |
| 312 | So we can see the very much smaller chance (0.000006) of 4 failures by | |
| 313 | the best shooters, compared to the 0.03 of the mediocre. | |
| 314 | </p> | |
| 315 | <h5> | |
| 316 | <a name="math_toolkit.stat_tut.weg.geometric_eg.h3"></a> | |
| 317 | <span class="phrase"><a name="math_toolkit.stat_tut.weg.geometric_eg.estimating_failures"></a></span><a class="link" href="geometric_eg.html#math_toolkit.stat_tut.weg.geometric_eg.estimating_failures">Estimating | |
| 318 | failures</a> | |
| 319 | </h5> | |
| 320 | <p> | |
| 321 | Of course one man's failure is an other man's success. So a fault can be | |
| 322 | defined as a 'success'. | |
| 323 | </p> | |
| 324 | <p> | |
| 325 | If a fault occurs once after 100 flights, then one might naively say that | |
| 326 | the risk of fault is obviously 1 in 100 = 1/100, a probability of 0.01. | |
| 327 | </p> | |
| 328 | <p> | |
| 329 | This is the best estimate we can make, but while it is the truth, it is | |
| 330 | not the whole truth, for it hides the big uncertainty when estimating from | |
| 331 | a single event. "One swallow doesn't make a summer." To show | |
| 332 | the magnitude of the uncertainty, the geometric (or the negative binomial) | |
| 333 | distribution can be used. | |
| 334 | </p> | |
| 335 | <p> | |
| 336 | If we chose the popular 95% confidence in the limits, corresponding to | |
| 337 | an alpha of 0.05, because we are calculating a two-sided interval, we must | |
| 338 | divide alpha by two. | |
| 339 | </p> | |
| 340 | <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span> <span class="special">=</span> <span class="number">0.05</span><span class="special">;</span> | |
| 341 | <span class="keyword">double</span> <span class="identifier">k</span> <span class="special">=</span> <span class="number">100</span><span class="special">;</span> <span class="comment">// So frequency of occurrence is 1/100.</span> | |
| 342 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Probability is failure is "</span> <span class="special"><<</span> <span class="number">1</span><span class="special">/</span><span class="identifier">k</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 343 | <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">geometric</span><span class="special">::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span><span class="identifier">k</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span><span class="special">);</span> | |
| 344 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"geometric::find_lower_bound_on_p("</span> <span class="special"><<</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">k</span><span class="special">)</span> <span class="special"><<</span> <span class="string">", "</span> <span class="special"><<</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span> <span class="special"><<</span> <span class="string">") = "</span> | |
| 345 | <span class="special"><<</span> <span class="identifier">t</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.00025</span> | |
| 346 | <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">geometric</span><span class="special">::</span><span class="identifier">find_upper_bound_on_p</span><span class="special">(</span><span class="identifier">k</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span><span class="special">);</span> | |
| 347 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"geometric::find_upper_bound_on_p("</span> <span class="special"><<</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">k</span><span class="special">)</span> <span class="special"><<</span> <span class="string">", "</span> <span class="special"><<</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span> <span class="special"><<</span> <span class="string">") = "</span> | |
| 348 | <span class="special"><<</span> <span class="identifier">t</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.037</span> | |
| 349 | </pre> | |
| 350 | <p> | |
| 351 | So while we estimate the probability is 0.01, it might lie between 0.0003 | |
| 352 | and 0.04. Even if we relax our confidence to alpha = 90%, the bounds only | |
| 353 | contract to 0.0005 and 0.03. And if we require a high confidence, they | |
| 354 | widen to 0.00005 to 0.05. | |
| 355 | </p> | |
| 356 | <pre class="programlisting"><span class="identifier">alpha</span> <span class="special">=</span> <span class="number">0.1</span><span class="special">;</span> <span class="comment">// 90% confidence.</span> | |
| 357 | <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">geometric</span><span class="special">::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span><span class="identifier">k</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span><span class="special">);</span> | |
| 358 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"geometric::find_lower_bound_on_p("</span> <span class="special"><<</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">k</span><span class="special">)</span> <span class="special"><<</span> <span class="string">", "</span> <span class="special"><<</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span> <span class="special"><<</span> <span class="string">") = "</span> | |
| 359 | <span class="special"><<</span> <span class="identifier">t</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.0005</span> | |
| 360 | <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">geometric</span><span class="special">::</span><span class="identifier">find_upper_bound_on_p</span><span class="special">(</span><span class="identifier">k</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span><span class="special">);</span> | |
| 361 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"geometric::find_upper_bound_on_p("</span> <span class="special"><<</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">k</span><span class="special">)</span> <span class="special"><<</span> <span class="string">", "</span> <span class="special"><<</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span> <span class="special"><<</span> <span class="string">") = "</span> | |
| 362 | <span class="special"><<</span> <span class="identifier">t</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.03</span> | |
| 363 | ||
| 364 | <span class="identifier">alpha</span> <span class="special">=</span> <span class="number">0.01</span><span class="special">;</span> <span class="comment">// 99% confidence.</span> | |
| 365 | <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">geometric</span><span class="special">::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span><span class="identifier">k</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span><span class="special">);</span> | |
| 366 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"geometric::find_lower_bound_on_p("</span> <span class="special"><<</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">k</span><span class="special">)</span> <span class="special"><<</span> <span class="string">", "</span> <span class="special"><<</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span> <span class="special"><<</span> <span class="string">") = "</span> | |
| 367 | <span class="special"><<</span> <span class="identifier">t</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 5e-005</span> | |
| 368 | <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">geometric</span><span class="special">::</span><span class="identifier">find_upper_bound_on_p</span><span class="special">(</span><span class="identifier">k</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span><span class="special">);</span> | |
| 369 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"geometric::find_upper_bound_on_p("</span> <span class="special"><<</span> <span class="keyword">int</span><span class="special">(</span><span class="identifier">k</span><span class="special">)</span> <span class="special"><<</span> <span class="string">", "</span> <span class="special"><<</span> <span class="identifier">alpha</span><span class="special">/</span><span class="number">2</span> <span class="special"><<</span> <span class="string">") = "</span> | |
| 370 | <span class="special"><<</span> <span class="identifier">t</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.052</span> | |
| 371 | </pre> | |
| 372 | <p> | |
| 373 | In real life, there will usually be more than one event (fault or success), | |
| 374 | when the negative binomial, which has the neccessary extra parameter, will | |
| 375 | be needed. | |
| 376 | </p> | |
| 377 | <p> | |
| 378 | As noted above, using a catch block is always a good idea, even if you | |
| 379 | hope not to use it! | |
| 380 | </p> | |
| 381 | <pre class="programlisting"><span class="special">}</span> | |
| 382 | <span class="keyword">catch</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span><span class="special">&</span> <span class="identifier">e</span><span class="special">)</span> | |
| 383 | <span class="special">{</span> <span class="comment">// Since we have set an overflow policy of ignore_error,</span> | |
| 384 | <span class="comment">// an overflow exception should never be thrown.</span> | |
| 385 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\nMessage from thrown exception was:\n "</span> <span class="special"><<</span> <span class="identifier">e</span><span class="special">.</span><span class="identifier">what</span><span class="special">()</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
| 386 | </pre> | |
| 387 | <p> | |
| 388 | For example, without a ignore domain error policy, if we asked for | |
| 389 | </p> | |
| 390 | <pre class="programlisting"><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">g</span><span class="special">,</span> <span class="special">-</span><span class="number">1</span><span class="special">)</span></pre> | |
| 391 | <p> | |
| 392 | for example, we would get an unhelpful abort, but with a catch: | |
| 393 | </p> | |
| 394 | <pre class="programlisting">Message from thrown exception was: | |
| 395 | Error in function boost::math::pdf(const exponential_distribution<double>&, double): | |
| 396 | Number of failures argument is -1, but must be >= 0 ! | |
| 397 | </pre> | |
| 398 | <p> | |
| 399 | See full source C++ of this example at <a href="../../../../../example/geometric_examples.cpp" target="_top">geometric_examples.cpp</a> | |
| 400 | </p> | |
| 401 | <p> | |
| 402 | <a class="link" href="neg_binom_eg/neg_binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution">See | |
| 403 | negative_binomial confidence interval example.</a> | |
| 404 | </p> | |
| 405 | </div> | |
| 406 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
| 407 | <td align="left"></td> | |
| 408 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
| 409 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
| 410 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
| 411 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
| 412 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
| 413 | 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>) | |
| 414 | </p> | |
| 415 | </div></td> | |
| 416 | </tr></table> | |
| 417 | <hr> | |
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