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| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.policy.pol_tutorial.understand_dis_quant"></a><a class="link" href="understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions"> |
| Understanding Quantiles of Discrete Distributions</a> |
| </h4></div></div></div> |
| <p> |
| Discrete distributions present us with a problem when calculating the quantile: |
| we are starting from a continuous real-valued variable - the probability |
| - but the result (the value of the random variable) should really be discrete. |
| </p> |
| <p> |
| Consider for example a Binomial distribution, with a sample size of 50, |
| and a success fraction of 0.5. There are a variety of ways we can plot |
| a discrete distribution, but if we plot the PDF as a step-function then |
| it looks something like this: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/binomial_pdf.png" alt="binomial_pdf"></span> |
| </p> |
| <p> |
| Now lets suppose that the user asks for a the quantile that corresponds |
| to a probability of 0.05, if we zoom in on the CDF for that region here's |
| what we see: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/binomial_quantile_1.png" alt="binomial_quantile_1"></span> |
| </p> |
| <p> |
| As can be seen there is no random variable that corresponds to a probability |
| of exactly 0.05, so we're left with two choices as shown in the figure: |
| </p> |
| <div class="itemizedlist"><ul type="disc"> |
| <li> |
| We could round the result down to 18. |
| </li> |
| <li> |
| We could round the result up to 19. |
| </li> |
| </ul></div> |
| <p> |
| In fact there's actually a third choice as well: we could "pretend" |
| that the distribution was continuous and return a real valued result: in |
| this case we would calculate a result of approximately 18.701 (this accurately |
| reflects the fact that the result is nearer to 19 than 18). |
| </p> |
| <p> |
| By using policies we can offer any of the above as options, but that still |
| leaves the question: <span class="emphasis"><em>What is actually the right thing to do?</em></span> |
| </p> |
| <p> |
| And in particular: <span class="emphasis"><em>What policy should we use by default?</em></span> |
| </p> |
| <p> |
| In coming to an answer we should realise that: |
| </p> |
| <div class="itemizedlist"><ul type="disc"> |
| <li> |
| Calculating an integer result is often much faster than calculating |
| a real-valued result: in fact in our tests it was up to 20 times faster. |
| </li> |
| <li> |
| Normally people calculate quantiles so that they can perform a test |
| of some kind: <span class="emphasis"><em>"If the random variable is less than N |
| then we can reject our null-hypothesis with 90% confidence."</em></span> |
| </li> |
| </ul></div> |
| <p> |
| So there is a genuine benefit to calculating an integer result as well |
| as it being "the right thing to do" from a philosophical point |
| of view. What's more if someone asks for a quantile at 0.05, then we can |
| normally assume that they are asking for <span class="emphasis"><em><span class="bold"><strong>at |
| least</strong></span> 95% of the probability to the right of the value chosen, |
| and <span class="bold"><strong>no more than</strong></span> 5% of the probability |
| to the left of the value chosen.</em></span> |
| </p> |
| <p> |
| In the above binomial example we would therefore round the result down |
| to 18. |
| </p> |
| <p> |
| The converse applies to upper-quantiles: If the probability is greater |
| than 0.5 we would want to round the quantile up, <span class="emphasis"><em>so that <span class="bold"><strong>at least</strong></span> the requested probability is to the left |
| of the value returned, and <span class="bold"><strong>no more than</strong></span> |
| 1 - the requested probability is to the right of the value returned.</em></span> |
| </p> |
| <p> |
| Likewise for two-sided intervals, we would round lower quantiles down, |
| and upper quantiles up. This ensures that we have <span class="emphasis"><em>at least the |
| requested probability in the central region</em></span> and <span class="emphasis"><em>no |
| more than 1 minus the requested probability in the tail areas.</em></span> |
| </p> |
| <p> |
| For example, taking our 50 sample binomial distribution with a success |
| fraction of 0.5, if we wanted a two sided 90% confidence interval, then |
| we would ask for the 0.05 and 0.95 quantiles with the results <span class="emphasis"><em>rounded |
| outwards</em></span> so that <span class="emphasis"><em>at least 90% of the probability</em></span> |
| is in the central area: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/binomial_pdf_3.png" alt="binomial_pdf_3"></span> |
| </p> |
| <p> |
| So far so good, but there is in fact a trap waiting for the unwary here: |
| </p> |
| <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> |
| </pre> |
| <p> |
| returns 18 as the result, which is what we would expect from the graph |
| above, and indeed there is no x greater than 18 for which: |
| </p> |
| <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> |
| </pre> |
| <p> |
| However: |
| </p> |
| <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> |
| </pre> |
| <p> |
| returns 31, and indeed while there is no x less than 31 for which: |
| </p> |
| <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> |
| </pre> |
| <p> |
| We might naively expect that for this symmetrical distribution the result |
| would be 32 (since 32 = 50 - 18), but we need to remember that the cdf |
| of the binomial is <span class="emphasis"><em>inclusive</em></span> of the random variable. |
| So while the left tail area <span class="emphasis"><em>includes</em></span> the quantile |
| returned, the right tail area always excludes an upper quantile value: |
| since that "belongs" to the central area. |
| </p> |
| <p> |
| Look at the graph above to see what's going on here: the lower quantile |
| of 18 belongs to the left tail, so any value <= 18 is in the left tail. |
| The upper quantile of 31 on the other hand belongs to the central area, |
| so the tail area actually starts at 32, so any value > 31 is in the |
| right tail. |
| </p> |
| <p> |
| Therefore if U and L are the upper and lower quantiles respectively, then |
| a random variable X is in the tail area - where we would reject the null |
| hypothesis if: |
| </p> |
| <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> |
| </pre> |
| <p> |
| And the a variable X is inside the central region if: |
| </p> |
| <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> |
| </pre> |
| <p> |
| The moral here is to <span class="emphasis"><em>always be very careful with your comparisons |
| when dealing with a discrete distribution</em></span>, and if in doubt, |
| <span class="emphasis"><em>base your comparisons on CDF's instead</em></span>. |
| </p> |
| <a name="math_toolkit.policy.pol_tutorial.understand_dis_quant.other_rounding_policies_are_available"></a><h6> |
| <a name="id1254402"></a> |
| <a class="link" href="understand_dis_quant.html#math_toolkit.policy.pol_tutorial.understand_dis_quant.other_rounding_policies_are_available">Other |
| Rounding Policies are Available</a> |
| </h6> |
| <p> |
| As you would expect from a section on policies, you won't be surprised |
| to know that other rounding options are available: |
| </p> |
| <div class="variablelist"> |
| <p class="title"><b></b></p> |
| <dl> |
| <dt><span class="term">integer_round_outwards</span></dt> |
| <dd> |
| <p> |
| This is the default policy as described above: lower quantiles are |
| rounded down (probability < 0.5), and upper quantiles (probability |
| > 0.5) are rounded up. |
| </p> |
| <p> |
| This gives <span class="emphasis"><em>no more than</em></span> the requested probability |
| in the tails, and <span class="emphasis"><em>at least</em></span> the requested probability |
| in the central area. |
| </p> |
| </dd> |
| <dt><span class="term">integer_round_inwards</span></dt> |
| <dd> |
| <p> |
| This is the exact opposite of the default policy: lower quantiles |
| are rounded up (probability < 0.5), and upper quantiles (probability |
| > 0.5) are rounded down. |
| </p> |
| <p> |
| This gives <span class="emphasis"><em>at least</em></span> the requested probability |
| in the tails, and <span class="emphasis"><em>no more than</em></span> the requested |
| probability in the central area. |
| </p> |
| </dd> |
| <dt><span class="term">integer_round_down</span></dt> |
| <dd><p> |
| This policy will always round the result down no matter whether it |
| is an upper or lower quantile |
| </p></dd> |
| <dt><span class="term">integer_round_up</span></dt> |
| <dd><p> |
| This policy will always round the result up no matter whether it |
| is an upper or lower quantile |
| </p></dd> |
| <dt><span class="term">integer_round_nearest</span></dt> |
| <dd><p> |
| This policy will always round the result to the nearest integer no |
| matter whether it is an upper or lower quantile |
| </p></dd> |
| <dt><span class="term">real</span></dt> |
| <dd><p> |
| This policy will return a real valued result for the quantile of |
| a discrete distribution: this is generally much slower than finding |
| an integer result but does allow for more sophisticated rounding |
| policies. |
| </p></dd> |
| </dl> |
| </div> |
| <p> |
| </p> |
| <p> |
| To understand how the rounding policies for the discrete distributions |
| can be used, we'll use the 50-sample binomial distribution with a success |
| fraction of 0.5 once again, and calculate all the possible quantiles |
| at 0.05 and 0.95. |
| </p> |
| <p> |
| </p> |
| <p> |
| Begin by including the needed headers (and some using statements for |
| conciseness): |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> |
| <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> |
| <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> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iomanip</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> |
| <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="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></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Next we'll bring the needed declarations into scope, and define distribution |
| types for all the available rounding policies: |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="comment">// Avoid |
| </span><span class="comment">// using namespace std; // and |
| </span><span class="comment">// using namespace boost::math; |
| </span><span class="comment">// to avoid potential ambiguity of names, like binomial. |
| </span><span class="comment">// using namespace boost::math::policies; is small risk, but |
| </span><span class="comment">// the necessary items are brought into scope thus: |
| </span> |
| <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="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> |
| <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> |
| |
| <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> |
| <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> |
| <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> |
| <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> |
| <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> |
| <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> |
| |
| <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> |
| <span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><</span> |
| <span class="keyword">double</span><span class="special">,</span> |
| <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> |
| <span class="identifier">binom_round_outwards</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><</span> |
| <span class="keyword">double</span><span class="special">,</span> |
| <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> |
| <span class="identifier">binom_round_inwards</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><</span> |
| <span class="keyword">double</span><span class="special">,</span> |
| <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> |
| <span class="identifier">binom_round_down</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><</span> |
| <span class="keyword">double</span><span class="special">,</span> |
| <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> |
| <span class="identifier">binom_round_up</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><</span> |
| <span class="keyword">double</span><span class="special">,</span> |
| <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> |
| <span class="identifier">binom_round_nearest</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">binomial_distribution</span><span class="special"><</span> |
| <span class="keyword">double</span><span class="special">,</span> |
| <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> |
| <span class="identifier">binom_real_quantile</span><span class="special">;</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Now let's set to work calling those quantiles: |
| </p> |
| <p> |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> |
| <span class="special">{</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> |
| <span class="string">"Testing rounding policies for a 50 sample binomial distribution,\n"</span> |
| <span class="string">"with a success fraction of 0.5.\n\n"</span> |
| <span class="string">"Lower quantiles are calculated at p = 0.05\n\n"</span> |
| <span class="string">"Upper quantiles at p = 0.95.\n\n"</span><span class="special">;</span> |
| |
| <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> |
| <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> |
| <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> |
| <span class="special"><<</span> <span class="string">"Upper Quantile"</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| |
| <span class="comment">// Test integer_round_outwards: |
| </span> <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> |
| <span class="special"><<</span> <span class="string">"integer_round_outwards"</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> |
| <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> |
| <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> |
| <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> |
| <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| |
| <span class="comment">// Test integer_round_inwards: |
| </span> <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> |
| <span class="special"><<</span> <span class="string">"integer_round_inwards"</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> |
| <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> |
| <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> |
| <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> |
| <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| |
| <span class="comment">// Test integer_round_down: |
| </span> <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> |
| <span class="special"><<</span> <span class="string">"integer_round_down"</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> |
| <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> |
| <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> |
| <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> |
| <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| |
| <span class="comment">// Test integer_round_up: |
| </span> <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> |
| <span class="special"><<</span> <span class="string">"integer_round_up"</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> |
| <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> |
| <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> |
| <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> |
| <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| |
| <span class="comment">// Test integer_round_nearest: |
| </span> <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> |
| <span class="special"><<</span> <span class="string">"integer_round_nearest"</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> |
| <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> |
| <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> |
| <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> |
| <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| |
| <span class="comment">// Test real: |
| </span> <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> |
| <span class="special"><<</span> <span class="string">"real"</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> |
| <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> |
| <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> |
| <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> |
| <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="special">}</span> <span class="comment">// int main()</span></pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <p> |
| Which produces the program output: |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting">policy_eg_10.vcxproj -> J:\Cpp\MathToolkit\test\Math_test\Release\policy_eg_10.exe |
| Testing rounding policies for a 50 sample binomial distribution, |
| with a success fraction of 0.5. |
| |
| Lower quantiles are calculated at p = 0.05 |
| |
| Upper quantiles at p = 0.95. |
| |
| Policy Lower Quantile Upper Quantile |
| integer_round_outwards 18 31 |
| integer_round_inwards 19 30 |
| integer_round_down 18 30 |
| integer_round_up 19 31 |
| integer_round_nearest 19 30 |
| real 18.701 30.299 |
| </pre> |
| <p> |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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>) |
| </p> |
| </div></td> |
| </tr></table> |
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