1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
|
<!DOCTYPE html>
<!--[if lt IE 7]> <html class="no-js lt-ie9 lt-ie8 lt-ie7"> <![endif]-->
<!--[if IE 7]> <html class="no-js lt-ie9 lt-ie8"> <![endif]-->
<!--[if IE 8]> <html class="no-js lt-ie9"> <![endif]-->
<!--[if gt IE 8]><!--> <html class="no-js"> <!--<![endif]-->
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1">
<title>Basic Statistics - MLlib - Spark 1.3.0 Documentation</title>
<link rel="stylesheet" href="css/bootstrap.min.css">
<style>
body {
padding-top: 60px;
padding-bottom: 40px;
}
</style>
<meta name="viewport" content="width=device-width">
<link rel="stylesheet" href="css/bootstrap-responsive.min.css">
<link rel="stylesheet" href="css/main.css">
<script src="js/vendor/modernizr-2.6.1-respond-1.1.0.min.js"></script>
<link rel="stylesheet" href="css/pygments-default.css">
</head>
<body>
<!--[if lt IE 7]>
<p class="chromeframe">You are using an outdated browser. <a href="http://browsehappy.com/">Upgrade your browser today</a> or <a href="http://www.google.com/chromeframe/?redirect=true">install Google Chrome Frame</a> to better experience this site.</p>
<![endif]-->
<!-- This code is taken from http://twitter.github.com/bootstrap/examples/hero.html -->
<div class="navbar navbar-fixed-top" id="topbar">
<div class="navbar-inner">
<div class="container">
<div class="brand"><a href="index.html">
<img src="img/spark-logo-hd.png" style="height:50px;"/></a><span class="version">1.3.0</span>
</div>
<ul class="nav">
<!--TODO(andyk): Add class="active" attribute to li some how.-->
<li><a href="index.html">Overview</a></li>
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Programming Guides<b class="caret"></b></a>
<ul class="dropdown-menu">
<li><a href="quick-start.html">Quick Start</a></li>
<li><a href="programming-guide.html">Spark Programming Guide</a></li>
<li class="divider"></li>
<li><a href="streaming-programming-guide.html">Spark Streaming</a></li>
<li><a href="sql-programming-guide.html">DataFrames and SQL</a></li>
<li><a href="mllib-guide.html">MLlib (Machine Learning)</a></li>
<li><a href="graphx-programming-guide.html">GraphX (Graph Processing)</a></li>
<li><a href="bagel-programming-guide.html">Bagel (Pregel on Spark)</a></li>
</ul>
</li>
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown">API Docs<b class="caret"></b></a>
<ul class="dropdown-menu">
<li><a href="api/scala/index.html#org.apache.spark.package">Scala</a></li>
<li><a href="api/java/index.html">Java</a></li>
<li><a href="api/python/index.html">Python</a></li>
</ul>
</li>
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Deploying<b class="caret"></b></a>
<ul class="dropdown-menu">
<li><a href="cluster-overview.html">Overview</a></li>
<li><a href="submitting-applications.html">Submitting Applications</a></li>
<li class="divider"></li>
<li><a href="spark-standalone.html">Spark Standalone</a></li>
<li><a href="running-on-mesos.html">Mesos</a></li>
<li><a href="running-on-yarn.html">YARN</a></li>
<li class="divider"></li>
<li><a href="ec2-scripts.html">Amazon EC2</a></li>
</ul>
</li>
<li class="dropdown">
<a href="api.html" class="dropdown-toggle" data-toggle="dropdown">More<b class="caret"></b></a>
<ul class="dropdown-menu">
<li><a href="configuration.html">Configuration</a></li>
<li><a href="monitoring.html">Monitoring</a></li>
<li><a href="tuning.html">Tuning Guide</a></li>
<li><a href="job-scheduling.html">Job Scheduling</a></li>
<li><a href="security.html">Security</a></li>
<li><a href="hardware-provisioning.html">Hardware Provisioning</a></li>
<li><a href="hadoop-third-party-distributions.html">3<sup>rd</sup>-Party Hadoop Distros</a></li>
<li class="divider"></li>
<li><a href="building-spark.html">Building Spark</a></li>
<li><a href="https://cwiki.apache.org/confluence/display/SPARK/Contributing+to+Spark">Contributing to Spark</a></li>
<li><a href="https://cwiki.apache.org/confluence/display/SPARK/Supplemental+Spark+Projects">Supplemental Projects</a></li>
</ul>
</li>
</ul>
<!--<p class="navbar-text pull-right"><span class="version-text">v1.3.0</span></p>-->
</div>
</div>
</div>
<div class="container" id="content">
<h1 class="title"><a href="mllib-guide.html">MLlib</a> - Basic Statistics</h1>
<ul id="markdown-toc">
<li><a href="#summary-statistics">Summary statistics</a></li>
<li><a href="#correlations">Correlations</a></li>
<li><a href="#stratified-sampling">Stratified sampling</a></li>
<li><a href="#hypothesis-testing">Hypothesis testing</a></li>
<li><a href="#random-data-generation">Random data generation</a></li>
</ul>
<p><code>\[
\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\x}{\mathbf{x}}
\newcommand{\y}{\mathbf{y}}
\newcommand{\wv}{\mathbf{w}}
\newcommand{\av}{\mathbf{\alpha}}
\newcommand{\bv}{\mathbf{b}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\id}{\mathbf{I}}
\newcommand{\ind}{\mathbf{1}}
\newcommand{\0}{\mathbf{0}}
\newcommand{\unit}{\mathbf{e}}
\newcommand{\one}{\mathbf{1}}
\newcommand{\zero}{\mathbf{0}}
\]</code></p>
<h2 id="summary-statistics">Summary statistics</h2>
<p>We provide column summary statistics for <code>RDD[Vector]</code> through the function <code>colStats</code>
available in <code>Statistics</code>.</p>
<div class="codetabs">
<div data-lang="scala">
<p><a href="api/scala/index.html#org.apache.spark.mllib.stat.Statistics$"><code>colStats()</code></a> returns an instance of
<a href="api/scala/index.html#org.apache.spark.mllib.stat.MultivariateStatisticalSummary"><code>MultivariateStatisticalSummary</code></a>,
which contains the column-wise max, min, mean, variance, and number of nonzeros, as well as the
total count.</p>
<div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vector</span>
<span class="k">import</span> <span class="nn">org.apache.spark.mllib.stat.</span><span class="o">{</span><span class="nc">MultivariateStatisticalSummary</span><span class="o">,</span> <span class="nc">Statistics</span><span class="o">}</span>
<span class="k">val</span> <span class="n">observations</span><span class="k">:</span> <span class="kt">RDD</span><span class="o">[</span><span class="kt">Vector</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span> <span class="c1">// an RDD of Vectors</span>
<span class="c1">// Compute column summary statistics.</span>
<span class="k">val</span> <span class="n">summary</span><span class="k">:</span> <span class="kt">MultivariateStatisticalSummary</span> <span class="o">=</span> <span class="nc">Statistics</span><span class="o">.</span><span class="n">colStats</span><span class="o">(</span><span class="n">observations</span><span class="o">)</span>
<span class="n">println</span><span class="o">(</span><span class="n">summary</span><span class="o">.</span><span class="n">mean</span><span class="o">)</span> <span class="c1">// a dense vector containing the mean value for each column</span>
<span class="n">println</span><span class="o">(</span><span class="n">summary</span><span class="o">.</span><span class="n">variance</span><span class="o">)</span> <span class="c1">// column-wise variance</span>
<span class="n">println</span><span class="o">(</span><span class="n">summary</span><span class="o">.</span><span class="n">numNonzeros</span><span class="o">)</span> <span class="c1">// number of nonzeros in each column</span></code></pre></div>
</div>
<div data-lang="java">
<p><a href="api/java/org/apache/spark/mllib/stat/Statistics.html"><code>colStats()</code></a> returns an instance of
<a href="api/java/org/apache/spark/mllib/stat/MultivariateStatisticalSummary.html"><code>MultivariateStatisticalSummary</code></a>,
which contains the column-wise max, min, mean, variance, and number of nonzeros, as well as the
total count.</p>
<div class="highlight"><pre><code class="language-java" data-lang="java"><span class="kn">import</span> <span class="nn">org.apache.spark.api.java.JavaRDD</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.api.java.JavaSparkContext</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vector</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.stat.MultivariateStatisticalSummary</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.stat.Statistics</span><span class="o">;</span>
<span class="n">JavaSparkContext</span> <span class="n">jsc</span> <span class="o">=</span> <span class="o">...</span>
<span class="n">JavaRDD</span><span class="o"><</span><span class="n">Vector</span><span class="o">></span> <span class="n">mat</span> <span class="o">=</span> <span class="o">...</span> <span class="c1">// an RDD of Vectors</span>
<span class="c1">// Compute column summary statistics.</span>
<span class="n">MultivariateStatisticalSummary</span> <span class="n">summary</span> <span class="o">=</span> <span class="n">Statistics</span><span class="o">.</span><span class="na">colStats</span><span class="o">(</span><span class="n">mat</span><span class="o">.</span><span class="na">rdd</span><span class="o">());</span>
<span class="n">System</span><span class="o">.</span><span class="na">out</span><span class="o">.</span><span class="na">println</span><span class="o">(</span><span class="n">summary</span><span class="o">.</span><span class="na">mean</span><span class="o">());</span> <span class="c1">// a dense vector containing the mean value for each column</span>
<span class="n">System</span><span class="o">.</span><span class="na">out</span><span class="o">.</span><span class="na">println</span><span class="o">(</span><span class="n">summary</span><span class="o">.</span><span class="na">variance</span><span class="o">());</span> <span class="c1">// column-wise variance</span>
<span class="n">System</span><span class="o">.</span><span class="na">out</span><span class="o">.</span><span class="na">println</span><span class="o">(</span><span class="n">summary</span><span class="o">.</span><span class="na">numNonzeros</span><span class="o">());</span> <span class="c1">// number of nonzeros in each column</span></code></pre></div>
</div>
<div data-lang="python">
<p><a href="api/python/pyspark.mllib.stat.Statistics-class.html#colStats"><code>colStats()</code></a> returns an instance of
<a href="api/python/pyspark.mllib.stat.MultivariateStatisticalSummary-class.html"><code>MultivariateStatisticalSummary</code></a>,
which contains the column-wise max, min, mean, variance, and number of nonzeros, as well as the
total count.</p>
<div class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">from</span> <span class="nn">pyspark.mllib.stat</span> <span class="kn">import</span> <span class="n">Statistics</span>
<span class="n">sc</span> <span class="o">=</span> <span class="o">...</span> <span class="c"># SparkContext</span>
<span class="n">mat</span> <span class="o">=</span> <span class="o">...</span> <span class="c"># an RDD of Vectors</span>
<span class="c"># Compute column summary statistics.</span>
<span class="n">summary</span> <span class="o">=</span> <span class="n">Statistics</span><span class="o">.</span><span class="n">colStats</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span>
<span class="k">print</span> <span class="n">summary</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
<span class="k">print</span> <span class="n">summary</span><span class="o">.</span><span class="n">variance</span><span class="p">()</span>
<span class="k">print</span> <span class="n">summary</span><span class="o">.</span><span class="n">numNonzeros</span><span class="p">()</span></code></pre></div>
</div>
</div>
<h2 id="correlations">Correlations</h2>
<p>Calculating the correlation between two series of data is a common operation in Statistics. In MLlib
we provide the flexibility to calculate pairwise correlations among many series. The supported
correlation methods are currently Pearson’s and Spearman’s correlation.</p>
<div class="codetabs">
<div data-lang="scala">
<p><a href="api/scala/index.html#org.apache.spark.mllib.stat.Statistics$"><code>Statistics</code></a> provides methods to
calculate correlations between series. Depending on the type of input, two <code>RDD[Double]</code>s or
an <code>RDD[Vector]</code>, the output will be a <code>Double</code> or the correlation <code>Matrix</code> respectively.</p>
<div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="k">import</span> <span class="nn">org.apache.spark.SparkContext</span>
<span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg._</span>
<span class="k">import</span> <span class="nn">org.apache.spark.mllib.stat.Statistics</span>
<span class="k">val</span> <span class="n">sc</span><span class="k">:</span> <span class="kt">SparkContext</span> <span class="o">=</span> <span class="o">...</span>
<span class="k">val</span> <span class="n">seriesX</span><span class="k">:</span> <span class="kt">RDD</span><span class="o">[</span><span class="kt">Double</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span> <span class="c1">// a series</span>
<span class="k">val</span> <span class="n">seriesY</span><span class="k">:</span> <span class="kt">RDD</span><span class="o">[</span><span class="kt">Double</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span> <span class="c1">// must have the same number of partitions and cardinality as seriesX</span>
<span class="c1">// compute the correlation using Pearson's method. Enter "spearman" for Spearman's method. If a </span>
<span class="c1">// method is not specified, Pearson's method will be used by default. </span>
<span class="k">val</span> <span class="n">correlation</span><span class="k">:</span> <span class="kt">Double</span> <span class="o">=</span> <span class="nc">Statistics</span><span class="o">.</span><span class="n">corr</span><span class="o">(</span><span class="n">seriesX</span><span class="o">,</span> <span class="n">seriesY</span><span class="o">,</span> <span class="s">"pearson"</span><span class="o">)</span>
<span class="k">val</span> <span class="n">data</span><span class="k">:</span> <span class="kt">RDD</span><span class="o">[</span><span class="kt">Vector</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span> <span class="c1">// note that each Vector is a row and not a column</span>
<span class="c1">// calculate the correlation matrix using Pearson's method. Use "spearman" for Spearman's method.</span>
<span class="c1">// If a method is not specified, Pearson's method will be used by default. </span>
<span class="k">val</span> <span class="n">correlMatrix</span><span class="k">:</span> <span class="kt">Matrix</span> <span class="o">=</span> <span class="nc">Statistics</span><span class="o">.</span><span class="n">corr</span><span class="o">(</span><span class="n">data</span><span class="o">,</span> <span class="s">"pearson"</span><span class="o">)</span></code></pre></div>
</div>
<div data-lang="java">
<p><a href="api/java/org/apache/spark/mllib/stat/Statistics.html"><code>Statistics</code></a> provides methods to
calculate correlations between series. Depending on the type of input, two <code>JavaDoubleRDD</code>s or
a <code>JavaRDD<Vector></code>, the output will be a <code>Double</code> or the correlation <code>Matrix</code> respectively.</p>
<div class="highlight"><pre><code class="language-java" data-lang="java"><span class="kn">import</span> <span class="nn">org.apache.spark.api.java.JavaDoubleRDD</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.api.java.JavaSparkContext</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.*</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.stat.Statistics</span><span class="o">;</span>
<span class="n">JavaSparkContext</span> <span class="n">jsc</span> <span class="o">=</span> <span class="o">...</span>
<span class="n">JavaDoubleRDD</span> <span class="n">seriesX</span> <span class="o">=</span> <span class="o">...</span> <span class="c1">// a series</span>
<span class="n">JavaDoubleRDD</span> <span class="n">seriesY</span> <span class="o">=</span> <span class="o">...</span> <span class="c1">// must have the same number of partitions and cardinality as seriesX</span>
<span class="c1">// compute the correlation using Pearson's method. Enter "spearman" for Spearman's method. If a </span>
<span class="c1">// method is not specified, Pearson's method will be used by default. </span>
<span class="n">Double</span> <span class="n">correlation</span> <span class="o">=</span> <span class="n">Statistics</span><span class="o">.</span><span class="na">corr</span><span class="o">(</span><span class="n">seriesX</span><span class="o">.</span><span class="na">srdd</span><span class="o">(),</span> <span class="n">seriesY</span><span class="o">.</span><span class="na">srdd</span><span class="o">(),</span> <span class="s">"pearson"</span><span class="o">);</span>
<span class="n">JavaRDD</span><span class="o"><</span><span class="n">Vector</span><span class="o">></span> <span class="n">data</span> <span class="o">=</span> <span class="o">...</span> <span class="c1">// note that each Vector is a row and not a column</span>
<span class="c1">// calculate the correlation matrix using Pearson's method. Use "spearman" for Spearman's method.</span>
<span class="c1">// If a method is not specified, Pearson's method will be used by default. </span>
<span class="n">Matrix</span> <span class="n">correlMatrix</span> <span class="o">=</span> <span class="n">Statistics</span><span class="o">.</span><span class="na">corr</span><span class="o">(</span><span class="n">data</span><span class="o">.</span><span class="na">rdd</span><span class="o">(),</span> <span class="s">"pearson"</span><span class="o">);</span></code></pre></div>
</div>
<div data-lang="python">
<p><a href="api/python/pyspark.mllib.stat.Statistics-class.html"><code>Statistics</code></a> provides methods to
calculate correlations between series. Depending on the type of input, two <code>RDD[Double]</code>s or
an <code>RDD[Vector]</code>, the output will be a <code>Double</code> or the correlation <code>Matrix</code> respectively.</p>
<div class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">from</span> <span class="nn">pyspark.mllib.stat</span> <span class="kn">import</span> <span class="n">Statistics</span>
<span class="n">sc</span> <span class="o">=</span> <span class="o">...</span> <span class="c"># SparkContext</span>
<span class="n">seriesX</span> <span class="o">=</span> <span class="o">...</span> <span class="c"># a series</span>
<span class="n">seriesY</span> <span class="o">=</span> <span class="o">...</span> <span class="c"># must have the same number of partitions and cardinality as seriesX</span>
<span class="c"># Compute the correlation using Pearson's method. Enter "spearman" for Spearman's method. If a </span>
<span class="c"># method is not specified, Pearson's method will be used by default. </span>
<span class="k">print</span> <span class="n">Statistics</span><span class="o">.</span><span class="n">corr</span><span class="p">(</span><span class="n">seriesX</span><span class="p">,</span> <span class="n">seriesY</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">"pearson"</span><span class="p">)</span>
<span class="n">data</span> <span class="o">=</span> <span class="o">...</span> <span class="c"># an RDD of Vectors</span>
<span class="c"># calculate the correlation matrix using Pearson's method. Use "spearman" for Spearman's method.</span>
<span class="c"># If a method is not specified, Pearson's method will be used by default. </span>
<span class="k">print</span> <span class="n">Statistics</span><span class="o">.</span><span class="n">corr</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">"pearson"</span><span class="p">)</span></code></pre></div>
</div>
</div>
<h2 id="stratified-sampling">Stratified sampling</h2>
<p>Unlike the other statistics functions, which reside in MLlib, stratified sampling methods,
<code>sampleByKey</code> and <code>sampleByKeyExact</code>, can be performed on RDD’s of key-value pairs. For stratified
sampling, the keys can be thought of as a label and the value as a specific attribute. For example
the key can be man or woman, or document ids, and the respective values can be the list of ages
of the people in the population or the list of words in the documents. The <code>sampleByKey</code> method
will flip a coin to decide whether an observation will be sampled or not, therefore requires one
pass over the data, and provides an <em>expected</em> sample size. <code>sampleByKeyExact</code> requires significant
more resources than the per-stratum simple random sampling used in <code>sampleByKey</code>, but will provide
the exact sampling size with 99.99% confidence. <code>sampleByKeyExact</code> is currently not supported in
python.</p>
<div class="codetabs">
<div data-lang="scala">
<p><a href="api/scala/index.html#org.apache.spark.rdd.PairRDDFunctions"><code>sampleByKeyExact()</code></a> allows users to
sample exactly $\lceil f_k \cdot n_k \rceil \, \forall k \in K$ items, where $f_k$ is the desired
fraction for key $k$, $n_k$ is the number of key-value pairs for key $k$, and $K$ is the set of
keys. Sampling without replacement requires one additional pass over the RDD to guarantee sample
size, whereas sampling with replacement requires two additional passes.</p>
<div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="k">import</span> <span class="nn">org.apache.spark.SparkContext</span>
<span class="k">import</span> <span class="nn">org.apache.spark.SparkContext._</span>
<span class="k">import</span> <span class="nn">org.apache.spark.rdd.PairRDDFunctions</span>
<span class="k">val</span> <span class="n">sc</span><span class="k">:</span> <span class="kt">SparkContext</span> <span class="o">=</span> <span class="o">...</span>
<span class="k">val</span> <span class="n">data</span> <span class="k">=</span> <span class="o">...</span> <span class="c1">// an RDD[(K, V)] of any key value pairs</span>
<span class="k">val</span> <span class="n">fractions</span><span class="k">:</span> <span class="kt">Map</span><span class="o">[</span><span class="kt">K</span>, <span class="kt">Double</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span> <span class="c1">// specify the exact fraction desired from each key</span>
<span class="c1">// Get an exact sample from each stratum</span>
<span class="k">val</span> <span class="n">approxSample</span> <span class="k">=</span> <span class="n">data</span><span class="o">.</span><span class="n">sampleByKey</span><span class="o">(</span><span class="n">withReplacement</span> <span class="k">=</span> <span class="kc">false</span><span class="o">,</span> <span class="n">fractions</span><span class="o">)</span>
<span class="k">val</span> <span class="n">exactSample</span> <span class="k">=</span> <span class="n">data</span><span class="o">.</span><span class="n">sampleByKeyExact</span><span class="o">(</span><span class="n">withReplacement</span> <span class="k">=</span> <span class="kc">false</span><span class="o">,</span> <span class="n">fractions</span><span class="o">)</span></code></pre></div>
</div>
<div data-lang="java">
<p><a href="api/java/org/apache/spark/api/java/JavaPairRDD.html"><code>sampleByKeyExact()</code></a> allows users to
sample exactly $\lceil f_k \cdot n_k \rceil \, \forall k \in K$ items, where $f_k$ is the desired
fraction for key $k$, $n_k$ is the number of key-value pairs for key $k$, and $K$ is the set of
keys. Sampling without replacement requires one additional pass over the RDD to guarantee sample
size, whereas sampling with replacement requires two additional passes.</p>
<div class="highlight"><pre><code class="language-java" data-lang="java"><span class="kn">import</span> <span class="nn">java.util.Map</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.api.java.JavaPairRDD</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.api.java.JavaSparkContext</span><span class="o">;</span>
<span class="n">JavaSparkContext</span> <span class="n">jsc</span> <span class="o">=</span> <span class="o">...</span>
<span class="n">JavaPairRDD</span><span class="o"><</span><span class="n">K</span><span class="o">,</span> <span class="n">V</span><span class="o">></span> <span class="n">data</span> <span class="o">=</span> <span class="o">...</span> <span class="c1">// an RDD of any key value pairs</span>
<span class="n">Map</span><span class="o"><</span><span class="n">K</span><span class="o">,</span> <span class="n">Object</span><span class="o">></span> <span class="n">fractions</span> <span class="o">=</span> <span class="o">...</span> <span class="c1">// specify the exact fraction desired from each key</span>
<span class="c1">// Get an exact sample from each stratum</span>
<span class="n">JavaPairRDD</span><span class="o"><</span><span class="n">K</span><span class="o">,</span> <span class="n">V</span><span class="o">></span> <span class="n">approxSample</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="na">sampleByKey</span><span class="o">(</span><span class="kc">false</span><span class="o">,</span> <span class="n">fractions</span><span class="o">);</span>
<span class="n">JavaPairRDD</span><span class="o"><</span><span class="n">K</span><span class="o">,</span> <span class="n">V</span><span class="o">></span> <span class="n">exactSample</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="na">sampleByKeyExact</span><span class="o">(</span><span class="kc">false</span><span class="o">,</span> <span class="n">fractions</span><span class="o">);</span></code></pre></div>
</div>
<div data-lang="python">
<p><a href="api/python/pyspark.rdd.RDD-class.html#sampleByKey"><code>sampleByKey()</code></a> allows users to
sample approximately $\lceil f_k \cdot n_k \rceil \, \forall k \in K$ items, where $f_k$ is the
desired fraction for key $k$, $n_k$ is the number of key-value pairs for key $k$, and $K$ is the
set of keys.</p>
<p><em>Note:</em> <code>sampleByKeyExact()</code> is currently not supported in Python.</p>
<div class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">sc</span> <span class="o">=</span> <span class="o">...</span> <span class="c"># SparkContext</span>
<span class="n">data</span> <span class="o">=</span> <span class="o">...</span> <span class="c"># an RDD of any key value pairs</span>
<span class="n">fractions</span> <span class="o">=</span> <span class="o">...</span> <span class="c"># specify the exact fraction desired from each key as a dictionary</span>
<span class="n">approxSample</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">sampleByKey</span><span class="p">(</span><span class="bp">False</span><span class="p">,</span> <span class="n">fractions</span><span class="p">);</span></code></pre></div>
</div>
</div>
<h2 id="hypothesis-testing">Hypothesis testing</h2>
<p>Hypothesis testing is a powerful tool in statistics to determine whether a result is statistically
significant, whether this result occurred by chance or not. MLlib currently supports Pearson’s
chi-squared ( $\chi^2$) tests for goodness of fit and independence. The input data types determine
whether the goodness of fit or the independence test is conducted. The goodness of fit test requires
an input type of <code>Vector</code>, whereas the independence test requires a <code>Matrix</code> as input.</p>
<p>MLlib also supports the input type <code>RDD[LabeledPoint]</code> to enable feature selection via chi-squared
independence tests.</p>
<div class="codetabs">
<div data-lang="scala">
<p><a href="api/scala/index.html#org.apache.spark.mllib.stat.Statistics$"><code>Statistics</code></a> provides methods to
run Pearson’s chi-squared tests. The following example demonstrates how to run and interpret
hypothesis tests.</p>
<div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="k">import</span> <span class="nn">org.apache.spark.SparkContext</span>
<span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg._</span>
<span class="k">import</span> <span class="nn">org.apache.spark.mllib.regression.LabeledPoint</span>
<span class="k">import</span> <span class="nn">org.apache.spark.mllib.stat.Statistics._</span>
<span class="k">val</span> <span class="n">sc</span><span class="k">:</span> <span class="kt">SparkContext</span> <span class="o">=</span> <span class="o">...</span>
<span class="k">val</span> <span class="n">vec</span><span class="k">:</span> <span class="kt">Vector</span> <span class="o">=</span> <span class="o">...</span> <span class="c1">// a vector composed of the frequencies of events</span>
<span class="c1">// compute the goodness of fit. If a second vector to test against is not supplied as a parameter, </span>
<span class="c1">// the test runs against a uniform distribution. </span>
<span class="k">val</span> <span class="n">goodnessOfFitTestResult</span> <span class="k">=</span> <span class="nc">Statistics</span><span class="o">.</span><span class="n">chiSqTest</span><span class="o">(</span><span class="n">vec</span><span class="o">)</span>
<span class="n">println</span><span class="o">(</span><span class="n">goodnessOfFitTestResult</span><span class="o">)</span> <span class="c1">// summary of the test including the p-value, degrees of freedom, </span>
<span class="c1">// test statistic, the method used, and the null hypothesis.</span>
<span class="k">val</span> <span class="n">mat</span><span class="k">:</span> <span class="kt">Matrix</span> <span class="o">=</span> <span class="o">...</span> <span class="c1">// a contingency matrix</span>
<span class="c1">// conduct Pearson's independence test on the input contingency matrix</span>
<span class="k">val</span> <span class="n">independenceTestResult</span> <span class="k">=</span> <span class="nc">Statistics</span><span class="o">.</span><span class="n">chiSqTest</span><span class="o">(</span><span class="n">mat</span><span class="o">)</span>
<span class="n">println</span><span class="o">(</span><span class="n">independenceTestResult</span><span class="o">)</span> <span class="c1">// summary of the test including the p-value, degrees of freedom...</span>
<span class="k">val</span> <span class="n">obs</span><span class="k">:</span> <span class="kt">RDD</span><span class="o">[</span><span class="kt">LabeledPoint</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span> <span class="c1">// (feature, label) pairs.</span>
<span class="c1">// The contingency table is constructed from the raw (feature, label) pairs and used to conduct</span>
<span class="c1">// the independence test. Returns an array containing the ChiSquaredTestResult for every feature </span>
<span class="c1">// against the label.</span>
<span class="k">val</span> <span class="n">featureTestResults</span><span class="k">:</span> <span class="kt">Array</span><span class="o">[</span><span class="kt">ChiSqTestResult</span><span class="o">]</span> <span class="k">=</span> <span class="nc">Statistics</span><span class="o">.</span><span class="n">chiSqTest</span><span class="o">(</span><span class="n">obs</span><span class="o">)</span>
<span class="k">var</span> <span class="n">i</span> <span class="k">=</span> <span class="mi">1</span>
<span class="n">featureTestResults</span><span class="o">.</span><span class="n">foreach</span> <span class="o">{</span> <span class="n">result</span> <span class="k">=></span>
<span class="n">println</span><span class="o">(</span><span class="n">s</span><span class="s">"Column $i:\n$result"</span><span class="o">)</span>
<span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="o">}</span> <span class="c1">// summary of the test</span></code></pre></div>
</div>
<div data-lang="java">
<p><a href="api/java/org/apache/spark/mllib/stat/Statistics.html"><code>Statistics</code></a> provides methods to
run Pearson’s chi-squared tests. The following example demonstrates how to run and interpret
hypothesis tests.</p>
<div class="highlight"><pre><code class="language-java" data-lang="java"><span class="kn">import</span> <span class="nn">org.apache.spark.api.java.JavaRDD</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.api.java.JavaSparkContext</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.*</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.regression.LabeledPoint</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.stat.Statistics</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.stat.test.ChiSqTestResult</span><span class="o">;</span>
<span class="n">JavaSparkContext</span> <span class="n">jsc</span> <span class="o">=</span> <span class="o">...</span>
<span class="n">Vector</span> <span class="n">vec</span> <span class="o">=</span> <span class="o">...</span> <span class="c1">// a vector composed of the frequencies of events</span>
<span class="c1">// compute the goodness of fit. If a second vector to test against is not supplied as a parameter, </span>
<span class="c1">// the test runs against a uniform distribution. </span>
<span class="n">ChiSqTestResult</span> <span class="n">goodnessOfFitTestResult</span> <span class="o">=</span> <span class="n">Statistics</span><span class="o">.</span><span class="na">chiSqTest</span><span class="o">(</span><span class="n">vec</span><span class="o">);</span>
<span class="c1">// summary of the test including the p-value, degrees of freedom, test statistic, the method used, </span>
<span class="c1">// and the null hypothesis.</span>
<span class="n">System</span><span class="o">.</span><span class="na">out</span><span class="o">.</span><span class="na">println</span><span class="o">(</span><span class="n">goodnessOfFitTestResult</span><span class="o">);</span>
<span class="n">Matrix</span> <span class="n">mat</span> <span class="o">=</span> <span class="o">...</span> <span class="c1">// a contingency matrix</span>
<span class="c1">// conduct Pearson's independence test on the input contingency matrix</span>
<span class="n">ChiSqTestResult</span> <span class="n">independenceTestResult</span> <span class="o">=</span> <span class="n">Statistics</span><span class="o">.</span><span class="na">chiSqTest</span><span class="o">(</span><span class="n">mat</span><span class="o">);</span>
<span class="c1">// summary of the test including the p-value, degrees of freedom...</span>
<span class="n">System</span><span class="o">.</span><span class="na">out</span><span class="o">.</span><span class="na">println</span><span class="o">(</span><span class="n">independenceTestResult</span><span class="o">);</span>
<span class="n">JavaRDD</span><span class="o"><</span><span class="n">LabeledPoint</span><span class="o">></span> <span class="n">obs</span> <span class="o">=</span> <span class="o">...</span> <span class="c1">// an RDD of labeled points</span>
<span class="c1">// The contingency table is constructed from the raw (feature, label) pairs and used to conduct</span>
<span class="c1">// the independence test. Returns an array containing the ChiSquaredTestResult for every feature </span>
<span class="c1">// against the label.</span>
<span class="n">ChiSqTestResult</span><span class="o">[]</span> <span class="n">featureTestResults</span> <span class="o">=</span> <span class="n">Statistics</span><span class="o">.</span><span class="na">chiSqTest</span><span class="o">(</span><span class="n">obs</span><span class="o">.</span><span class="na">rdd</span><span class="o">());</span>
<span class="kt">int</span> <span class="n">i</span> <span class="o">=</span> <span class="mi">1</span><span class="o">;</span>
<span class="k">for</span> <span class="o">(</span><span class="n">ChiSqTestResult</span> <span class="n">result</span> <span class="o">:</span> <span class="n">featureTestResults</span><span class="o">)</span> <span class="o">{</span>
<span class="n">System</span><span class="o">.</span><span class="na">out</span><span class="o">.</span><span class="na">println</span><span class="o">(</span><span class="s">"Column "</span> <span class="o">+</span> <span class="n">i</span> <span class="o">+</span> <span class="s">":"</span><span class="o">);</span>
<span class="n">System</span><span class="o">.</span><span class="na">out</span><span class="o">.</span><span class="na">println</span><span class="o">(</span><span class="n">result</span><span class="o">);</span> <span class="c1">// summary of the test</span>
<span class="n">i</span><span class="o">++;</span>
<span class="o">}</span></code></pre></div>
</div>
<div data-lang="python">
<p><a href="api/python/index.html#pyspark.mllib.stat.Statistics$"><code>Statistics</code></a> provides methods to
run Pearson’s chi-squared tests. The following example demonstrates how to run and interpret
hypothesis tests.</p>
<div class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">from</span> <span class="nn">pyspark</span> <span class="kn">import</span> <span class="n">SparkContext</span>
<span class="kn">from</span> <span class="nn">pyspark.mllib.linalg</span> <span class="kn">import</span> <span class="n">Vectors</span><span class="p">,</span> <span class="n">Matrices</span>
<span class="kn">from</span> <span class="nn">pyspark.mllib.regresssion</span> <span class="kn">import</span> <span class="n">LabeledPoint</span>
<span class="kn">from</span> <span class="nn">pyspark.mllib.stat</span> <span class="kn">import</span> <span class="n">Statistics</span>
<span class="n">sc</span> <span class="o">=</span> <span class="n">SparkContext</span><span class="p">()</span>
<span class="n">vec</span> <span class="o">=</span> <span class="n">Vectors</span><span class="o">.</span><span class="n">dense</span><span class="p">(</span><span class="o">...</span><span class="p">)</span> <span class="c"># a vector composed of the frequencies of events</span>
<span class="c"># compute the goodness of fit. If a second vector to test against is not supplied as a parameter,</span>
<span class="c"># the test runs against a uniform distribution.</span>
<span class="n">goodnessOfFitTestResult</span> <span class="o">=</span> <span class="n">Statistics</span><span class="o">.</span><span class="n">chiSqTest</span><span class="p">(</span><span class="n">vec</span><span class="p">)</span>
<span class="k">print</span> <span class="n">goodnessOfFitTestResult</span> <span class="c"># summary of the test including the p-value, degrees of freedom,</span>
<span class="c"># test statistic, the method used, and the null hypothesis.</span>
<span class="n">mat</span> <span class="o">=</span> <span class="n">Matrices</span><span class="o">.</span><span class="n">dense</span><span class="p">(</span><span class="o">...</span><span class="p">)</span> <span class="c"># a contingency matrix</span>
<span class="c"># conduct Pearson's independence test on the input contingency matrix</span>
<span class="n">independenceTestResult</span> <span class="o">=</span> <span class="n">Statistics</span><span class="o">.</span><span class="n">chiSqTest</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span>
<span class="k">print</span> <span class="n">independenceTestResult</span> <span class="c"># summary of the test including the p-value, degrees of freedom...</span>
<span class="n">obs</span> <span class="o">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">parallelize</span><span class="p">(</span><span class="o">...</span><span class="p">)</span> <span class="c"># LabeledPoint(feature, label) .</span>
<span class="c"># The contingency table is constructed from an RDD of LabeledPoint and used to conduct</span>
<span class="c"># the independence test. Returns an array containing the ChiSquaredTestResult for every feature</span>
<span class="c"># against the label.</span>
<span class="n">featureTestResults</span> <span class="o">=</span> <span class="n">Statistics</span><span class="o">.</span><span class="n">chiSqTest</span><span class="p">(</span><span class="n">obs</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">result</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">featureTestResults</span><span class="p">):</span>
<span class="k">print</span> <span class="s">"Column $d:"</span> <span class="o">%</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<span class="k">print</span> <span class="n">result</span></code></pre></div>
</div>
</div>
<h2 id="random-data-generation">Random data generation</h2>
<p>Random data generation is useful for randomized algorithms, prototyping, and performance testing.
MLlib supports generating random RDDs with i.i.d. values drawn from a given distribution:
uniform, standard normal, or Poisson.</p>
<div class="codetabs">
<div data-lang="scala">
<p><a href="api/scala/index.html#org.apache.spark.mllib.random.RandomRDDs"><code>RandomRDDs</code></a> provides factory
methods to generate random double RDDs or vector RDDs.
The following example generates a random double RDD, whose values follows the standard normal
distribution <code>N(0, 1)</code>, and then map it to <code>N(1, 4)</code>.</p>
<div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="k">import</span> <span class="nn">org.apache.spark.SparkContext</span>
<span class="k">import</span> <span class="nn">org.apache.spark.mllib.random.RandomRDDs._</span>
<span class="k">val</span> <span class="n">sc</span><span class="k">:</span> <span class="kt">SparkContext</span> <span class="o">=</span> <span class="o">...</span>
<span class="c1">// Generate a random double RDD that contains 1 million i.i.d. values drawn from the</span>
<span class="c1">// standard normal distribution `N(0, 1)`, evenly distributed in 10 partitions.</span>
<span class="k">val</span> <span class="n">u</span> <span class="k">=</span> <span class="n">normalRDD</span><span class="o">(</span><span class="n">sc</span><span class="o">,</span> <span class="mi">1000000L</span><span class="o">,</span> <span class="mi">10</span><span class="o">)</span>
<span class="c1">// Apply a transform to get a random double RDD following `N(1, 4)`.</span>
<span class="k">val</span> <span class="n">v</span> <span class="k">=</span> <span class="n">u</span><span class="o">.</span><span class="n">map</span><span class="o">(</span><span class="n">x</span> <span class="k">=></span> <span class="mf">1.0</span> <span class="o">+</span> <span class="mf">2.0</span> <span class="o">*</span> <span class="n">x</span><span class="o">)</span></code></pre></div>
</div>
<div data-lang="java">
<p><a href="api/java/index.html#org.apache.spark.mllib.random.RandomRDDs"><code>RandomRDDs</code></a> provides factory
methods to generate random double RDDs or vector RDDs.
The following example generates a random double RDD, whose values follows the standard normal
distribution <code>N(0, 1)</code>, and then map it to <code>N(1, 4)</code>.</p>
<div class="highlight"><pre><code class="language-java" data-lang="java"><span class="kn">import</span> <span class="nn">org.apache.spark.SparkContext</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">org.apache.spark.api.JavaDoubleRDD</span><span class="o">;</span>
<span class="kn">import</span> <span class="nn">static</span> <span class="n">org</span><span class="o">.</span><span class="na">apache</span><span class="o">.</span><span class="na">spark</span><span class="o">.</span><span class="na">mllib</span><span class="o">.</span><span class="na">random</span><span class="o">.</span><span class="na">RandomRDDs</span><span class="o">.*;</span>
<span class="n">JavaSparkContext</span> <span class="n">jsc</span> <span class="o">=</span> <span class="o">...</span>
<span class="c1">// Generate a random double RDD that contains 1 million i.i.d. values drawn from the</span>
<span class="c1">// standard normal distribution `N(0, 1)`, evenly distributed in 10 partitions.</span>
<span class="n">JavaDoubleRDD</span> <span class="n">u</span> <span class="o">=</span> <span class="n">normalJavaRDD</span><span class="o">(</span><span class="n">jsc</span><span class="o">,</span> <span class="mi">1000000L</span><span class="o">,</span> <span class="mi">10</span><span class="o">);</span>
<span class="c1">// Apply a transform to get a random double RDD following `N(1, 4)`.</span>
<span class="n">JavaDoubleRDD</span> <span class="n">v</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="na">map</span><span class="o">(</span>
<span class="k">new</span> <span class="n">Function</span><span class="o"><</span><span class="n">Double</span><span class="o">,</span> <span class="n">Double</span><span class="o">>()</span> <span class="o">{</span>
<span class="kd">public</span> <span class="n">Double</span> <span class="nf">call</span><span class="o">(</span><span class="n">Double</span> <span class="n">x</span><span class="o">)</span> <span class="o">{</span>
<span class="k">return</span> <span class="mf">1.0</span> <span class="o">+</span> <span class="mf">2.0</span> <span class="o">*</span> <span class="n">x</span><span class="o">;</span>
<span class="o">}</span>
<span class="o">});</span></code></pre></div>
</div>
<div data-lang="python">
<p><a href="api/python/pyspark.mllib.random.RandomRDDs-class.html"><code>RandomRDDs</code></a> provides factory
methods to generate random double RDDs or vector RDDs.
The following example generates a random double RDD, whose values follows the standard normal
distribution <code>N(0, 1)</code>, and then map it to <code>N(1, 4)</code>.</p>
<div class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">from</span> <span class="nn">pyspark.mllib.random</span> <span class="kn">import</span> <span class="n">RandomRDDs</span>
<span class="n">sc</span> <span class="o">=</span> <span class="o">...</span> <span class="c"># SparkContext</span>
<span class="c"># Generate a random double RDD that contains 1 million i.i.d. values drawn from the</span>
<span class="c"># standard normal distribution `N(0, 1)`, evenly distributed in 10 partitions.</span>
<span class="n">u</span> <span class="o">=</span> <span class="n">RandomRDDs</span><span class="o">.</span><span class="n">uniformRDD</span><span class="p">(</span><span class="n">sc</span><span class="p">,</span> <span class="il">1000000L</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="c"># Apply a transform to get a random double RDD following `N(1, 4)`.</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="n">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mf">1.0</span> <span class="o">+</span> <span class="mf">2.0</span> <span class="o">*</span> <span class="n">x</span><span class="p">)</span></code></pre></div>
</div>
</div>
</div> <!-- /container -->
<script src="js/vendor/jquery-1.8.0.min.js"></script>
<script src="js/vendor/bootstrap.min.js"></script>
<script src="js/main.js"></script>
<!-- MathJax Section -->
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
TeX: { equationNumbers: { autoNumber: "AMS" } }
});
</script>
<script>
// Note that we load MathJax this way to work with local file (file://), HTTP and HTTPS.
// We could use "//cdn.mathjax...", but that won't support "file://".
(function(d, script) {
script = d.createElement('script');
script.type = 'text/javascript';
script.async = true;
script.onload = function(){
MathJax.Hub.Config({
tex2jax: {
inlineMath: [ ["$", "$"], ["\\\\(","\\\\)"] ],
displayMath: [ ["$$","$$"], ["\\[", "\\]"] ],
processEscapes: true,
skipTags: ['script', 'noscript', 'style', 'textarea', 'pre']
}
});
};
script.src = ('https:' == document.location.protocol ? 'https://' : 'http://') +
'cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML';
d.getElementsByTagName('head')[0].appendChild(script);
}(document));
</script>
</body>
</html>
|
