From 31aa2ed85031b163dcb8d4a650dd08e55d1f4a02 Mon Sep 17 00:00:00 2001 From: Alexandros Giavaras Date: Sun, 19 Apr 2020 21:11:15 +0100 Subject: [PATCH 01/12] Add Markove chain Monte Carlo example --- README.md | 1 + .../java/examples/mc/example3/Example3.java | 35 ++++++++++ src/main/java/examples/mc/example3/example.md | 64 +++++++++++++++++++ 3 files changed, 100 insertions(+) create mode 100644 src/main/java/examples/mc/example3/Example3.java create mode 100644 src/main/java/examples/mc/example3/example.md diff --git a/README.md b/README.md index da5d699..f790564 100644 --- a/README.md +++ b/README.md @@ -53,6 +53,7 @@ You can start using Jstat by adding the following dependency to your project: ### Monte Carlo - Example 1: Monte Carlo Simulation for the Monty Hall Problem - Example 2: Calculate the area of a circle using Monte Carlo integration +- Example 3: Markov chain Monte Carlo for pi calculation ### Plotting diff --git a/src/main/java/examples/mc/example3/Example3.java b/src/main/java/examples/mc/example3/Example3.java new file mode 100644 index 0000000..1eee42b --- /dev/null +++ b/src/main/java/examples/mc/example3/Example3.java @@ -0,0 +1,35 @@ +package examples.mc.example3; + +import java.util.concurrent.ThreadLocalRandom; + +public class Example3 { + + static public void main(String[] args){ + + + + final int N_ITERATIONS = 10000; + final double DELTA = 0.1; + double x = 1.0; + double y = 1.0; + double area_under_curve = 0.0; + + for(int itr=0; itr < N_ITERATIONS; ++itr){ + + double del_x = ThreadLocalRandom.current().nextDouble(-DELTA, DELTA); + double del_y = ThreadLocalRandom.current().nextDouble(-DELTA, DELTA); + + if(Math.abs(x + del_x) < 1.0 && Math.abs(y + del_y) < 1.0){ + x += del_x; + y += del_y; + } + + if(x*x + y*y < 1.0){ + area_under_curve += 1; + } + } + + System.out.println("Pi is: " + 4.0* area_under_curve/(double)N_ITERATIONS); + + } +} diff --git a/src/main/java/examples/mc/example3/example.md b/src/main/java/examples/mc/example3/example.md new file mode 100644 index 0000000..f984955 --- /dev/null +++ b/src/main/java/examples/mc/example3/example.md @@ -0,0 +1,64 @@ +# Markov Chain Monte Carlo For Pi Calculation + +## Contents + * [Overview](#overview) + * [Import files](#include_files) + * [The main function](#m_func) + * [Results](#results) + * [Source Code](#source_code) + +## Overview + +## Import files + + ``` +package examples.mc.example3; + +import java.util.concurrent.ThreadLocalRandom; + ``` + +## The main function + +``` +public class Example3 { + + static public void main(String[] args){ + + + + final int N_ITERATIONS = 10000; + final double DELTA = 0.1; + double x = 1.0; + double y = 1.0; + double area_under_curve = 0.0; + + for(int itr=0; itr < N_ITERATIONS; ++itr){ + + double del_x = ThreadLocalRandom.current().nextDouble(-DELTA, DELTA); + double del_y = ThreadLocalRandom.current().nextDouble(-DELTA, DELTA); + + if(Math.abs(x + del_x) < 1.0 && Math.abs(y + del_y) < 1.0){ + x += del_x; + y += del_y; + } + + if(x*x + y*y < 1.0){ + area_under_curve += 1; + } + } + + System.out.println("Pi is: " + 4.0* area_under_curve/(double)N_ITERATIONS); + + } +} +``` + +## Results + +``` +Pi is: 3.0244 +``` + + ## Source Code + + Example3.java \ No newline at end of file From a50119213f17f4ffe4fdb737a304cf1f6c35e03e Mon Sep 17 00:00:00 2001 From: Alexandros Giavaras Date: Sun, 26 Apr 2020 20:22:30 +0100 Subject: [PATCH 02/12] #69 Update ListUtils API --- src/main/java/utils/ListUtils.java | 25 ++++++++++++++++++++++--- 1 file changed, 22 insertions(+), 3 deletions(-) diff --git a/src/main/java/utils/ListUtils.java b/src/main/java/utils/ListUtils.java index 6cfab63..4f3ed53 100644 --- a/src/main/java/utils/ListUtils.java +++ b/src/main/java/utils/ListUtils.java @@ -1,5 +1,6 @@ package utils; +import org.apache.commons.math3.distribution.AbstractIntegerDistribution; import org.apache.commons.math3.distribution.AbstractRealDistribution; import java.util.ArrayList; @@ -11,7 +12,8 @@ public class ListUtils { /** - * Create pairs of consisting of one element from list1 (Pair.first) and one element from list 2 (Pair.second) + * Create pairs of consisting of one element from list1 (Pair.first) + * and one element from list 2 (Pair.second) */ public static List> zip(List lst1, List lst2){ @@ -29,12 +31,12 @@ public static List> zip(List lst1, List lst2){ return result; } - public static double[] toDoubleArray(final List list){ + public static double[] toDoubleArray(final List list){ double[] array = new double[list.size()]; for(int i=0; i randomSample(int size, AbstractRealDistribution dist) return sample; } + /** + * Generate a random sample from the given distribution + * @param size the size of the sample + * @param dist the distribution to create the sample from + * @return sample list of values sampled from AbstractRealDistribution + */ + public static List randomSample(int size, AbstractIntegerDistribution dist){ + List sample = new ArrayList<>(size); + + for(int i=0; i int doPartition(List list, Comparator comparator, int start, int end){ From d0f3267c0ba642a8a25652e9fa8d1f2ae6a902a9 Mon Sep 17 00:00:00 2001 From: Alexandros Giavaras Date: Sun, 26 Apr 2020 20:23:19 +0100 Subject: [PATCH 03/12] #69 Add example 8 --- README.md | 1 + .../examples/stats/example8/Example8.java | 31 ++++++++++++++++ .../java/examples/stats/example8/example.md | 37 +++++++++++++++++++ 3 files changed, 69 insertions(+) create mode 100644 src/main/java/examples/stats/example8/Example8.java create mode 100644 src/main/java/examples/stats/example8/example.md diff --git a/README.md b/README.md index f790564..c0046ee 100644 --- a/README.md +++ b/README.md @@ -49,6 +49,7 @@ You can start using Jstat by adding the following dependency to your project: - Example 5: Calculate descriptive statistics metrics - Example 6: Hypothesis tests on the mean of a normal distribution with known variance - Example 7: Type II error and the sample size +- Example 8: Simulate the Central Limit Theorem ### Monte Carlo - Example 1: Monte Carlo Simulation for the Monty Hall Problem diff --git a/src/main/java/examples/stats/example8/Example8.java b/src/main/java/examples/stats/example8/Example8.java new file mode 100644 index 0000000..f1d7255 --- /dev/null +++ b/src/main/java/examples/stats/example8/Example8.java @@ -0,0 +1,31 @@ +package examples.stats.example8; + +import io.CSVFileWriter; +import org.apache.commons.math3.distribution.UniformIntegerDistribution; +import stats.Statistics; +import utils.ListUtils; + +import java.util.ArrayList; +import java.util.List; + +public class Example8 { + + public static void main(String[] args){ + + final int N_SIMS = 1000; + List avg = new ArrayList<>(N_SIMS); + + for(int i =0; i sample = ListUtils.randomSample(i+1, new UniformIntegerDistribution(1,6)); + double mean = Statistics.calculate(ListUtils.toDoubleArray(sample), Statistics.Metrics.MEAN); + avg.add(mean); + } + + + CSVFileWriter writer = new CSVFileWriter(new String("averages.csv")); + writer.writeColumnNames("Column1", "Column2"); + + + } +} diff --git a/src/main/java/examples/stats/example8/example.md b/src/main/java/examples/stats/example8/example.md new file mode 100644 index 0000000..f369df0 --- /dev/null +++ b/src/main/java/examples/stats/example8/example.md @@ -0,0 +1,37 @@ +# Example 8: Simulate The Central Limit Theorem + + ## Contents + * [Overview](#overview) + * [Central Limit Theorem](#goodness_of_fit) + * [Import files](#include_files) + * [The main function](#m_func) + * [Results](#results) + * [Source Code](#source_code) + + ## Overview + + ### Central Limit Theorem + + + ## Import files + + ``` + + ``` + + ## The main function + + ``` + + + ``` + + ## Results + + ``` + + ``` + + ## Source Code + + Example8.java \ No newline at end of file From 88e414cbe7942213906fb9959607cc68901be422 Mon Sep 17 00:00:00 2001 From: Alexandros Giavaras Date: Sun, 26 Apr 2020 20:23:47 +0100 Subject: [PATCH 04/12] #69 Add io package --- src/main/java/io/CSVFileWriter.java | 64 +++++++++++++++++++++++++++++ 1 file changed, 64 insertions(+) create mode 100644 src/main/java/io/CSVFileWriter.java diff --git a/src/main/java/io/CSVFileWriter.java b/src/main/java/io/CSVFileWriter.java new file mode 100644 index 0000000..d7f7e0c --- /dev/null +++ b/src/main/java/io/CSVFileWriter.java @@ -0,0 +1,64 @@ +package io; +import tech.tablesaw.io.csv.CsvWriteOptions; +import tech.tablesaw.io.csv.CsvWriter; + +import java.io.File; +import java.io.FileWriter; + +public class CSVFileWriter { + + public static final char DEFUALT_DELIMITER = ','; + public static final char DEFAULT_COMMENT = '#'; + public CSVFileWriter(String filename){ + + this.filename = filename; + this.delimiter = CSVFileWriter.DEFUALT_DELIMITER; + this.commentDelimiter = CSVFileWriter.DEFAULT_COMMENT; + + } + + public void writeColumnNames(String... names){ + + // first create file object for file placed at location + // specified by filepath + File file = new File(filename); + + try{ + + // create FileWriter object with file as parameter + FileWriter outputfile = new FileWriter(file); + outputfile.write(this.createColumns(names)); + outputfile.close(); + } + catch (Exception e){ + + } + } + + /** + * + * @param names List of names that correspond to the collumns + * @return + */ + private String createColumns(String... names){ + + StringBuilder builder = new StringBuilder(names.length); + + int counter = 0; + builder.append(this.commentDelimiter); + for(String name:names){ + builder.append(name); + + if(counter != names.length - 1){ + builder.append(this.delimiter); + } + } + + return builder.toString(); + } + + private char delimiter; + private char commentDelimiter; + private String filename; + +} From 401d26813cfa6908ad4360973d2d155c38b18f2c Mon Sep 17 00:00:00 2001 From: Alexandros Giavaras Date: Mon, 27 Apr 2020 09:08:41 +0100 Subject: [PATCH 05/12] Update CSV file writer --- src/main/java/io/CSVFileWriter.java | 90 ++++++++++++++++++++++++++--- 1 file changed, 82 insertions(+), 8 deletions(-) diff --git a/src/main/java/io/CSVFileWriter.java b/src/main/java/io/CSVFileWriter.java index d7f7e0c..8f4f038 100644 --- a/src/main/java/io/CSVFileWriter.java +++ b/src/main/java/io/CSVFileWriter.java @@ -4,35 +4,69 @@ import java.io.File; import java.io.FileWriter; +import java.io.IOException; +import java.util.List; public class CSVFileWriter { public static final char DEFUALT_DELIMITER = ','; public static final char DEFAULT_COMMENT = '#'; + public CSVFileWriter(String filename){ this.filename = filename; this.delimiter = CSVFileWriter.DEFUALT_DELIMITER; this.commentDelimiter = CSVFileWriter.DEFAULT_COMMENT; + File file = new File(filename); + + try { + this.outputfile = new FileWriter(file); + } + catch(IOException e){ + } } + /** + * + * @param names List of strings representing the column names + */ public void writeColumnNames(String... names){ - // first create file object for file placed at location - // specified by filepath - File file = new File(filename); + try{ + this.outputfile.write(this.createColumns(names)); + this.outputfile.close(); + } + catch (IOException e){ + + } + } + + /** + * + * @param row Write a row of doubles + */ + public void writeRow(Double...row){ try{ + this.outputfile.write(this.createRow(row)); + this.outputfile.close(); + } + catch (IOException e){ - // create FileWriter object with file as parameter - FileWriter outputfile = new FileWriter(file); - outputfile.write(this.createColumns(names)); - outputfile.close(); } - catch (Exception e){ + } + + public void writeDoubleRow(List row){ + + try{ + this.outputfile.write(this.createDoubleRow(row)); + this.outputfile.close(); + } + catch (IOException e){ } + } /** @@ -52,6 +86,45 @@ private String createColumns(String... names){ if(counter != names.length - 1){ builder.append(this.delimiter); } + counter++; + } + + return builder.toString(); + } + + private String createRow(Double... row){ + + StringBuilder builder = new StringBuilder(row.length); + + int counter = 0; + for(Double item:row){ + + builder.append(item.toString()); + + if(counter != row.length - 1){ + builder.append(this.delimiter); + } + counter++; + + } + + return builder.toString(); + } + + private String createDoubleRow(List row){ + + StringBuilder builder = new StringBuilder(row.size()); + + int counter = 0; + for(Double item:row){ + + builder.append(item.toString()); + + if(counter != row.size() - 1){ + builder.append(this.delimiter); + } + counter++; + } return builder.toString(); @@ -60,5 +133,6 @@ private String createColumns(String... names){ private char delimiter; private char commentDelimiter; private String filename; + private FileWriter outputfile; } From c9c36cf39c2c3762038547cd58d166e4b1af53a4 Mon Sep 17 00:00:00 2001 From: Alexandros Giavaras Date: Mon, 27 Apr 2020 09:09:15 +0100 Subject: [PATCH 06/12] Update Example8 --- .../examples/stats/example8/Example8.java | 22 +++++++++++++------ 1 file changed, 15 insertions(+), 7 deletions(-) diff --git a/src/main/java/examples/stats/example8/Example8.java b/src/main/java/examples/stats/example8/Example8.java index f1d7255..4115df3 100644 --- a/src/main/java/examples/stats/example8/Example8.java +++ b/src/main/java/examples/stats/example8/Example8.java @@ -5,6 +5,7 @@ import stats.Statistics; import utils.ListUtils; +import java.nio.file.Paths; import java.util.ArrayList; import java.util.List; @@ -12,20 +13,27 @@ public class Example8 { public static void main(String[] args){ + String path = Paths.get("src/main/java/examples/stats/example8") + .toAbsolutePath() + .toString(); + + System.out.println("Path is: "+path); + final int N_SIMS = 1000; List avg = new ArrayList<>(N_SIMS); for(int i =0; i sample = ListUtils.randomSample(i+1, new UniformIntegerDistribution(1,6)); - double mean = Statistics.calculate(ListUtils.toDoubleArray(sample), Statistics.Metrics.MEAN); - avg.add(mean); - } + List sample = ListUtils.randomSample(i+1, + new UniformIntegerDistribution(1,6)); + double mean = Statistics.calculate(ListUtils.toDoubleArray(sample), + Statistics.Metrics.MEAN); - CSVFileWriter writer = new CSVFileWriter(new String("averages.csv")); - writer.writeColumnNames("Column1", "Column2"); - + avg.add(mean); + CSVFileWriter writer = new CSVFileWriter(new String( path +"/" + "averages" + i + ".csv")); + writer.writeDoubleRow(avg); + } } } From 95cbbdbbc766396beb95d070b0adf31e5d344e95 Mon Sep 17 00:00:00 2001 From: Alexandros Giavaras Date: Mon, 27 Apr 2020 09:09:52 +0100 Subject: [PATCH 07/12] Update API --- src/main/java/visualizations/Histograms.java | 11 +++++++++++ 1 file changed, 11 insertions(+) diff --git a/src/main/java/visualizations/Histograms.java b/src/main/java/visualizations/Histograms.java index 5252850..dd4a944 100644 --- a/src/main/java/visualizations/Histograms.java +++ b/src/main/java/visualizations/Histograms.java @@ -6,6 +6,9 @@ import tech.tablesaw.plotly.Plot; import tech.tablesaw.plotly.api.Histogram; import tech.tablesaw.plotly.api.Histogram2D; +import utils.ListUtils; + +import java.util.List; /** * Class that plots basic Histograms. @@ -17,6 +20,7 @@ public class HistogramOptions { public String chartTitle; public String xAxisName; public String yAxisName; + public int numBins; } /** @@ -35,6 +39,13 @@ public static void plotHistogram(IVector x, HistogramOptions options) { Plot.show(Histogram.create(options.chartTitle, table, options.xAxisName)); } + public static void plot(List data, HistogramOptions options){ + + DoubleColumn xCol = DoubleColumn.create(options.xAxisName, ListUtils.toDoubleArray(data)); + Table table = Table.create(xCol); + Plot.show(Histogram.create(options.chartTitle, table, options.xAxisName)); + } + /** * Plots a histogram given the chart title, * data set in Table format, From 301998ddfc8594d5e77c80832befab810709885d Mon Sep 17 00:00:00 2001 From: Alexandros Giavaras Date: Tue, 28 Apr 2020 09:19:01 +0100 Subject: [PATCH 08/12] Add path to examples --- src/main/java/base/CommonConstants.java | 12 ++++++++++++ 1 file changed, 12 insertions(+) diff --git a/src/main/java/base/CommonConstants.java b/src/main/java/base/CommonConstants.java index 3fac0b2..c005a42 100644 --- a/src/main/java/base/CommonConstants.java +++ b/src/main/java/base/CommonConstants.java @@ -1,10 +1,22 @@ package base; +import java.nio.file.Paths; + public final class CommonConstants { public static double getTol(){return CommonConstants.tol;} public static void setTol(double tol){CommonConstants.tol = tol;} + public static String examplesPath(){ + String path = "src/main/java/examples"; + return path; + } + + public static String mcExamplesPath(){ + String path = CommonConstants.examplesPath(); + return path + "/mc"; + } + protected static double tol = 1.0e-8; private CommonConstants(){} From 5f05ac81eb3d48426b902474f7a0d837a5e90aeb Mon Sep 17 00:00:00 2001 From: Alexandros Giavaras Date: Tue, 28 Apr 2020 09:19:34 +0100 Subject: [PATCH 09/12] Fix bugs and API update --- src/main/java/io/CSVFileWriter.java | 31 +++++++++++++++++++++++++++-- 1 file changed, 29 insertions(+), 2 deletions(-) diff --git a/src/main/java/io/CSVFileWriter.java b/src/main/java/io/CSVFileWriter.java index 8f4f038..307ab8e 100644 --- a/src/main/java/io/CSVFileWriter.java +++ b/src/main/java/io/CSVFileWriter.java @@ -1,6 +1,7 @@ package io; import tech.tablesaw.io.csv.CsvWriteOptions; import tech.tablesaw.io.csv.CsvWriter; +import utils.Pair; import java.io.File; import java.io.FileWriter; @@ -35,7 +36,7 @@ public void writeColumnNames(String... names){ try{ this.outputfile.write(this.createColumns(names)); - this.outputfile.close(); + //this.outputfile.close(); } catch (IOException e){ @@ -57,6 +58,32 @@ public void writeRow(Double...row){ } } + public void writeRow(Integer idx, Double value){ + + try{ + + String rowVal = idx.toString() + this.delimiter + value.toString() + "\n"; + this.outputfile.write(rowVal); + //this.outputfile.close(); + } + catch (IOException e){ + + } + } + + public void writeRow(Pair row){ + + try{ + + String rowVal = row.first.toString() + this.delimiter + row.second.toString(); + this.outputfile.write(rowVal); + this.outputfile.close(); + } + catch (IOException e){ + + } + } + public void writeDoubleRow(List row){ try{ @@ -89,7 +116,7 @@ private String createColumns(String... names){ counter++; } - return builder.toString(); + return builder.toString() + "\n"; } private String createRow(Double... row){ From d42ba8b136b7d81d62a726750ae65499e8a70bf7 Mon Sep 17 00:00:00 2001 From: Alexandros Giavaras Date: Tue, 28 Apr 2020 09:20:09 +0100 Subject: [PATCH 10/12] Add save images --- src/main/java/examples/stats/example8/Example8.java | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/src/main/java/examples/stats/example8/Example8.java b/src/main/java/examples/stats/example8/Example8.java index 4115df3..9ff23ed 100644 --- a/src/main/java/examples/stats/example8/Example8.java +++ b/src/main/java/examples/stats/example8/Example8.java @@ -32,8 +32,10 @@ public static void main(String[] args){ avg.add(mean); - CSVFileWriter writer = new CSVFileWriter(new String( path +"/" + "averages" + i + ".csv")); - writer.writeDoubleRow(avg); + if(i==10 || i == 100 || i == 500 || i==999) { + CSVFileWriter writer = new CSVFileWriter(new String(path + "/" + "averages" + i + ".csv")); + writer.writeDoubleRow(avg); + } } } } From f2b422a006b76b3e0b78f13ca1b44da0efea4559 Mon Sep 17 00:00:00 2001 From: Alexandros Giavaras Date: Tue, 28 Apr 2020 09:20:41 +0100 Subject: [PATCH 11/12] Add mc example4 --- .../java/examples/mc/example4/Example4.java | 55 + src/main/java/examples/mc/example4/example.md | 40 + .../java/examples/mc/example4/positions.csv | 9791 +++++++++++++++++ 3 files changed, 9886 insertions(+) create mode 100644 src/main/java/examples/mc/example4/Example4.java create mode 100644 src/main/java/examples/mc/example4/example.md create mode 100644 src/main/java/examples/mc/example4/positions.csv diff --git a/src/main/java/examples/mc/example4/Example4.java b/src/main/java/examples/mc/example4/Example4.java new file mode 100644 index 0000000..fa21efc --- /dev/null +++ b/src/main/java/examples/mc/example4/Example4.java @@ -0,0 +1,55 @@ +package examples.mc.example4; + +import base.CommonConstants; +import io.CSVFileWriter; +import org.apache.commons.math3.distribution.AbstractRealDistribution; +import org.apache.commons.math3.distribution.NormalDistribution; + +import java.util.ArrayList; +import java.util.List; + +public class Example4 { + + + public static double target(double x){ + + if(x < 0.){ + return 0.0; + } + + return Math.exp(-x); + } + + public static void main(String[] args){ + + final int N_ITERATIONS = 10000; + List pos = new ArrayList<>(N_ITERATIONS); + for(int i=0; i Acknowledgements + This example was taken from fiveMinuteStats. + ## Overview + + ### Metropolis algorithm + + + ## Import files + + ``` + + ``` + + ## The main function + + ``` + + + ``` + + ## Results + + ``` + + ``` + + ## Source Code + + Example4.java \ No newline at end of file diff --git a/src/main/java/examples/mc/example4/positions.csv b/src/main/java/examples/mc/example4/positions.csv new file mode 100644 index 0000000..a4fe364 --- /dev/null +++ b/src/main/java/examples/mc/example4/positions.csv @@ -0,0 +1,9791 @@ +#ITERATION,POSITION +0,2.0 +1,1.6315655935662816 +2,2.3994463168834796 +3,2.308947971668913 +4,2.1376640295825364 +5,2.1511221223521417 +6,1.5701268982768735 +7,0.7235951153530489 +8,1.0028578867721611 +9,-0.22839456272195857 +10,-1.1695752086938431 +11,-0.4954349426186849 +12,-0.45395896365130406 +13,1.091244646341302 +14,2.3861608241228764 +15,2.52750725241468 +16,1.3717319639928587 +17,0.04461916570675073 +18,0.8805944959364791 +19,0.49763102189765634 +20,0.7278665518329079 +21,-0.6669344174776646 +22,-0.2368523525744724 +23,-1.1109367131851169 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+9777,98.74940384205212 +9778,101.79585782929327 +9779,101.30967145086424 +9780,100.7872294460126 +9781,99.19152515101815 +9782,98.83267560392902 +9783,96.84420860761256 +9784,94.86577539601463 +9785,94.51773806857263 +9786,92.82236841317008 +9787,93.5349277330323 +9788,94.13645067204742 +9789,93.2 \ No newline at end of file From 9504e6571ea1fc033579e7d14b384b41194de4e1 Mon Sep 17 00:00:00 2001 From: Alexandros Giavaras Date: Tue, 28 Apr 2020 09:20:57 +0100 Subject: [PATCH 12/12] Update example --- README.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/README.md b/README.md index c0046ee..41a578d 100644 --- a/README.md +++ b/README.md @@ -55,6 +55,7 @@ You can start using Jstat by adding the following dependency to your project: - Example 1: Monte Carlo Simulation for the Monty Hall Problem - Example 2: Calculate the area of a circle using Monte Carlo integration - Example 3: Markov chain Monte Carlo for pi calculation +- Example 4: The Metropolis algorithm ### Plotting @@ -68,5 +69,6 @@ You can start using Jstat by adding the following dependency to your project: - Example 1: Load data from an Elasticsearch index +