165 lines
8.8 KiB
C#
165 lines
8.8 KiB
C#
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using BotSharp.Algorithm.HiddenMarkovModel.MathUtils.Distribution;
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using System;
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using System.Collections.Generic;
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using System.Linq;
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using System.Text;
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namespace BotSharp.Algorithm.HiddenMarkovModel.MathUtils.Statistics
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{
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/// <summary>
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/// Bootstrapping is an alternative approach (to CLT) for constructing confidence intervals
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///
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/// The term bootstrapping comes from the phrase "pulling oneself up by one's bootstraps", which is a metaphor for accomplishing an impossible task without any outside help.
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///
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/// Bootstrapping can be used for constructing confidence intervals for statistic such as median, for which the standard error based on CLT cannot be directly obtained.
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///
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/// Bootstrapping scheme:
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/// 1. take a boostrap sample - a random sample taken with replacement from the original sample, of the same size as the original bootstrapSample.
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/// 2. calculate the boostrap statistic - a statistic such as mean, median, proportion, etc. computed on the bootstrap samples
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/// 3. repeat steps 1 and 2 many times to create a bootstrap distribution - a distribution of bootstrap statistics
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/// </summary>
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public class Bootstrapping
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{
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/// <summary>
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/// Return a set of simulated bootstrap statistics that form the bootstrap distribution for means via simulation given the original sample
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/// </summary>
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/// <param name="originalSampleMean">point estimate of sample mean from the original sample</param>
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/// <param name="originalSampleStdDev">standard deviation of the original sample</param>
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/// <param name="originalSampleSize">size of the original sample</param>
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/// <param name="bootstrapSampleCount">The number of bootstrap samples collected to form the bootstrap distribution</param>
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/// <returns></returns>
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public static double[] SimulateSampleMeans(double originalSampleMean, double originalSampleStdDev, int originalSampleSize, int bootstrapSampleCount)
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{
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Gaussian distribution = new Gaussian(originalSampleMean, originalSampleStdDev);
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double[] bootstrapMeans = new double[bootstrapSampleCount];
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double[] bootstrapSample = new double[originalSampleSize];
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for (int i = 0; i < bootstrapSampleCount; ++i)
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{
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for (int j = 0; j < originalSampleSize; ++j)
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{
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bootstrapSample[j] = distribution.Next();
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}
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bootstrapMeans[i] = Mean.GetMean(bootstrapSample);
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}
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return bootstrapMeans;
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}
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/// <summary>
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/// Return a set of simulated bootstrap statistics that form the bootstrap distribution for means via simulation given the original sample
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/// </summary>
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/// <param name="originalSample">The original bootstrap sample</param>
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/// <param name="bootstrapSampleCount">The number of bootstrap samples collected to form the bootstrap distribution</param>
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/// <returns></returns>
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public static double[] SimulateSampleMeans(double[] originalSample, int bootstrapSampleCount)
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{
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double originalSampleMean = Mean.GetMean(originalSample);
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double originalSampleStdDev = StdDev.GetStdDev(originalSample, originalSampleMean);
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return SimulateSampleMeans(originalSampleMean, originalSampleStdDev, originalSample.Length, bootstrapSampleCount);
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}
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/// <summary>
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/// Return a set of simulated bootstrap statistics that form the bootstrap distribution for medians via simulation given the original sample
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/// </summary>
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/// <param name="originalSampleMean">point estimate of sample mean from the original sample</param>
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/// <param name="originalSampleStdDev">standard deviation of the original sample</param>
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/// <param name="originalSampleSize">size of the original sample</param>
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/// <param name="bootstrapSampleCount">The number of bootstrap samples collected to form the bootstrap distribution</param>
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/// <returns></returns>
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public static double[] SimulateSampleMedians(double originalSampleMean, double originalSampleStdDev, int originalSampleSize, int bootstrapSampleCount)
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{
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Gaussian distribution = new Gaussian(originalSampleMean, originalSampleStdDev);
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double[] bootstrapMedians = new double[bootstrapSampleCount];
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double[] bootstrapSample = new double[originalSampleSize];
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for (int i = 0; i < bootstrapSampleCount; ++i)
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{
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for (int j = 0; j < originalSampleSize; ++j)
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{
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bootstrapSample[j] = distribution.Next();
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}
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bootstrapMedians[i] = Median.GetMedian(bootstrapSample);
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}
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return bootstrapMedians;
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}
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/// <summary>
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/// Return a set of simulated bootstrap statistics that form the bootstrap distribution for medians via simulation given the original sample
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/// </summary>
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/// <param name="originalSample">The original sample</param>
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/// <param name="bootstrapSampleCount">The number of bootstrap samples collected to form the bootstrap distribution</param>
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/// <returns></returns>
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public static double[] SimulateSampleMedians(double[] originalSample, int bootstrapSampleCount)
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{
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double originalSampleMedian = Median.GetMedian(originalSample);
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double originalSampleStdDev = StdDev.GetStdDev(originalSample, originalSampleMedian);
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return SimulateSampleMedians(originalSampleMedian, originalSampleStdDev, originalSample.Length, bootstrapSampleCount);
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}
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/// <summary>
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/// Return the confidence interval for median given the original sample
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/// </summary>
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/// <param name="originalSample"></param>
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/// <param name="bootstrapSampleCount"></param>
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/// <param name="confidence_level"></param>
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/// <returns></returns>
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public static double[] GetConfidenceIntervalForMedian(double[] originalSample, int bootstrapSampleCount, double confidence_level)
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{
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double[] bootstrapMedians = SimulateSampleMedians(originalSample, bootstrapSampleCount);
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double bootstrap_mean = Mean.GetMean(bootstrapMedians);
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double bootstrap_SE = StdDev.GetStdDev(bootstrapMedians, bootstrap_mean); //standard deviation of sample median in the bootstrap distribution
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double p1 = (1 - confidence_level) / 2;
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double p2 = 1 - p1;
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double z1 = Gaussian.GetQuantile(p1);
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double z2 = Gaussian.GetQuantile(p2);
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return new double[] { bootstrap_mean + z1 * bootstrap_SE, bootstrap_mean + z2 * bootstrap_SE };
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}
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/// <summary>
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/// Two-sided or one-sided test for a single median
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///
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/// Given that:
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/// H_0 : median = null_value
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/// H_A : median != null_value
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///
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/// By Central Limit Theorem:
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/// sample_median ~ N(mu, SE)
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///
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/// p-value = (sample_median is at least ||null_value - point_estimate|| away from the null_value) | median = null_value)
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/// if(p-value < significance_level) reject H_0
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/// </summary>
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/// <param name="originalSample">The original sample</param>
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/// <param name="bootstrapSampleCount"></param>
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/// <param name="null_value"></param>
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/// <param name="significance_level"></param>
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/// <param name="one_sided"></param>
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/// <returns></returns>
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public static bool RejectH0_ForMedian(double[] originalSample, int bootstrapSampleCount, double null_value, out double pValue, double significance_level = 0.05, bool one_sided = false)
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{
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double[] bootstrapMedians = SimulateSampleMedians(originalSample, bootstrapSampleCount);
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double bootstrap_mean = Mean.GetMean(bootstrapMedians);
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double bootstrap_SE = StdDev.GetStdDev(bootstrapMedians, bootstrap_mean);
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return HypothesisTesting.RejectH0(bootstrap_mean, null_value, bootstrap_SE, originalSample.Length, out pValue, significance_level, one_sided);
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}
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public static double[] GetConfidenceIntervalForMean(double[] originalSample, int bootstrapSampleCount, double confidence_level)
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{
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double[] bootstrapMeans = SimulateSampleMeans(originalSample, bootstrapSampleCount);
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double bootstrap_mean = Mean.GetMean(bootstrapMeans);
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double bootstrap_SE = StdDev.GetStdDev(bootstrapMeans, bootstrap_mean);
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double p1 = (1 - confidence_level) / 2;
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double p2 = 1 - p1;
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double z1 = Gaussian.GetQuantile(p1);
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double z2 = Gaussian.GetQuantile(p2);
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return new double[] { bootstrap_mean + z1 * bootstrap_SE, bootstrap_mean + z2 * bootstrap_SE };
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}
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}
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}
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