1math::statistics(n)            Tcl Math Library            math::statistics(n)
2
3
4
5______________________________________________________________________________
6

NAME

8       math::statistics - Basic statistical functions and procedures
9

SYNOPSIS

11       package require Tcl  8
12
13       package require math::statistics  0.2
14
15       ::math::statistics::mean data
16
17       ::math::statistics::min data
18
19       ::math::statistics::max data
20
21       ::math::statistics::number data
22
23       ::math::statistics::stdev data
24
25       ::math::statistics::var data
26
27       ::math::statistics::median data
28
29       ::math::statistics::basic-stats data
30
31       ::math::statistics::histogram limits values
32
33       ::math::statistics::corr data1 data2
34
35       ::math::statistics::interval-mean-stdev data confidence
36
37       ::math::statistics::t-test-mean data est_mean est_stdev confidence
38
39       ::math::statistics::test-normal data confidence
40
41       ::math::statistics::lillieforsFit data
42
43       ::math::statistics::quantiles data confidence
44
45       ::math::statistics::quantiles limits counts confidence
46
47       ::math::statistics::autocorr data
48
49       ::math::statistics::crosscorr data1 data2
50
51       ::math::statistics::mean-histogram-limits mean stdev number
52
53       ::math::statistics::minmax-histogram-limits min max number
54
55       ::math::statistics::linear-model xdata ydata intercept
56
57       ::math::statistics::linear-residuals xdata ydata intercept
58
59       ::math::statistics::test-2x2 n11 n21 n12 n22
60
61       ::math::statistics::print-2x2 n11 n21 n12 n22
62
63       ::math::statistics::control-xbar data ?nsamples?
64
65       ::math::statistics::control-Rchart data ?nsamples?
66
67       ::math::statistics::test-xbar control data
68
69       ::math::statistics::test-Rchart control data
70
71       ::math::statistics::pdf-normal mean stdev value
72
73       ::math::statistics::pdf-exponential mean value
74
75       ::math::statistics::pdf-uniform xmin xmax value
76
77       ::math::statistics::cdf-normal mean stdev value
78
79       ::math::statistics::cdf-exponential mean value
80
81       ::math::statistics::cdf-uniform xmin xmax value
82
83       ::math::statistics::cdf-students-t degrees value
84
85       ::math::statistics::random-normal mean stdev number
86
87       ::math::statistics::random-exponential mean number
88
89       ::math::statistics::random-uniform xmin xmax value
90
91       ::math::statistics::histogram-uniform xmin xmax limits number
92
93       ::math::statistics::filter varname data expression
94
95       ::math::statistics::map varname data expression
96
97       ::math::statistics::samplescount varname list expression
98
99       ::math::statistics::subdivide
100
101       ::math::statistics::plot-scale canvas xmin xmax ymin ymax
102
103       ::math::statistics::plot-xydata canvas xdata ydata tag
104
105       ::math::statistics::plot-xyline canvas xdata ydata tag
106
107       ::math::statistics::plot-tdata canvas tdata tag
108
109       ::math::statistics::plot-tline canvas tdata tag
110
111       ::math::statistics::plot-histogram canvas counts limits tag
112
113_________________________________________________________________
114

DESCRIPTION

116       The  math::statistics  package  contains  functions  and procedures for
117       basic statistical data analysis, such as:
118
119       ·      Descriptive  statistical  parameters  (mean,  minimum,  maximum,
120              standard deviation)
121
122       ·      Estimates  of  the  distribution  in  the form of histograms and
123              quantiles
124
125       ·      Basic testing of hypotheses
126
127       ·      Probability and cumulative density functions It is meant to help
128              in  developing  data  analysis applications or doing ad hoc data
129              analysis, it is not in itself a  full  application,  nor  is  it
130              intended  to  rival with full (non-)commercial statistical pack‐
131              ages.
132
133       The purpose of this document is to describe the implemented  procedures
134       and  provide some examples of their usage. As there is ample literature
135       on the algorithms involved, we refer to relevant text  books  for  more
136       explanations.   The  package  contains  a fairly large number of public
137       procedures. They can be distinguished in  three  sets:  general  proce‐
138       dures,  procedures  that  deal with specific statistical distributions,
139       list procedures to select or transform data and simple plotting  proce‐
140       dures  (these require Tk).  Note: The data that need to be analyzed are
141       always contained in a simple list. Missing values  are  represented  as
142       empty list elements.
143

GENERAL PROCEDURES

145       The general statistical procedures are:
146
147       ::math::statistics::mean data
148              Determine the mean value of the given list of data.
149
150              data - List of data
151
152
153       ::math::statistics::min data
154              Determine the minimum value of the given list of data.
155
156              data - List of data
157
158
159       ::math::statistics::max data
160              Determine the maximum value of the given list of data.
161
162              data - List of data
163
164
165       ::math::statistics::number data
166              Determine the number of non-missing data in the given list
167
168              data - List of data
169
170
171       ::math::statistics::stdev data
172              Determine the standard deviation of the data in the given list
173
174              data - List of data
175
176
177       ::math::statistics::var data
178              Determine the variance of the data in the given list
179
180              data - List of data
181
182
183       ::math::statistics::median data
184              Determine  the  median  of the data in the given list (Note that
185              this requires sorting the data, which may be a costly operation)
186
187              data - List of data
188
189
190       ::math::statistics::basic-stats data
191              Determine a list of all the descriptive parameters: mean,  mini‐
192              mum, maximum, number of data, standard deviation and variance.
193
194              (This routine is called whenever either or all of the basic sta‐
195              tistical parameters are required.  Hence  all  calculations  are
196              done and the relevant values are returned.)
197
198              data - List of data
199
200
201       ::math::statistics::histogram limits values
202              Determine  histogram  information  for  the  given list of data.
203              Returns a list consisting of the number of values that fall into
204              each interval.  (The first interval consists of all values lower
205              than the first limit, the last interval consists of  all  values
206              greater  than  the  last limit.  There is one more interval than
207              there are limits.)
208
209              limits - List of upper  limits  (in  ascending  order)  for  the
210              intervals of the histogram.
211
212              values - List of data
213
214
215       ::math::statistics::corr data1 data2
216              Determine the correlation coefficient between two sets of data.
217
218              data1 - First list of data
219
220              data2 - Second list of data
221
222
223       ::math::statistics::interval-mean-stdev data confidence
224              Return the interval containing the mean value and one containing
225              the standard  deviation  with  a  certain  level  of  confidence
226              (assuming a normal distribution)
227
228              data - List of raw data values (small sample)
229
230              confidence - Confidence level (0.95 or 0.99 for instance)
231
232
233       ::math::statistics::t-test-mean data est_mean est_stdev confidence
234              Test  whether  the  mean value of a sample is in accordance with
235              the estimated normal distribution with a certain level of confi‐
236              dence.   Returns  1  if  the  test  succeeds or 0 if the mean is
237              unlikely to fit the given distribution.
238
239              data - List of raw data values (small sample)
240
241              est_mean - Estimated mean of the distribution
242
243              est_stdev - Estimated stdev of the distribution
244
245              confidence - Confidence level (0.95 or 0.99 for instance)
246
247
248       ::math::statistics::test-normal data confidence
249              Test whether the given data follow a normal distribution with  a
250              certain level of confidence.  Returns 1 if the data are normally
251              distributed within the level of confidence, returns  0  if  not.
252              The underlying test is the Lilliefors test.
253
254              data - List of raw data values
255
256              confidence - Confidence level (one of 0.80, 0.90, 0.95 or 0.99)
257
258
259       ::math::statistics::lillieforsFit data
260              Returns  the  goodness of fit to a normal distribution according
261              to Lilliefors. The higher the number, the more likely  the  data
262              are indeed normally distributed. The test requires at least five
263              data points.
264
265              data - List of raw data values
266
267
268       ::math::statistics::quantiles data confidence
269              Return the quantiles for a given set of data
270
271              data - List of raw data values
272
273              confidence - Confidence level (0.95 or 0.99 for instance)
274
275
276       ::math::statistics::quantiles limits counts confidence
277              Return the quantiles based on histogram information (alternative
278              to the call with two arguments)
279
280              limits - List of upper limits from histogram
281
282              counts - List of counts for for each interval in histogram
283
284              confidence -  Confidence level (0.95 or 0.99 for instance)
285
286
287       ::math::statistics::autocorr data
288              Return  the autocorrelation function as a list of values (assum‐
289              ing equidistance between samples, about 1/2 of the number of raw
290              data)
291
292              The correlation is determined in such a way that the first value
293              is always 1 and all others are equal to or smaller than  1.  The
294              number of values involved will diminish as the "time" (the index
295              in the list of returned values) increases
296
297              data - Raw data for which the autocorrelation must be determined
298
299
300       ::math::statistics::crosscorr data1 data2
301              Return the  cross-correlation  function  as  a  list  of  values
302              (assuming  equidistance between samples, about 1/2 of the number
303              of raw data)
304
305              The correlation is determined in such a way that the values  can
306              never  exceed 1 in magnitude. The number of values involved will
307              diminish as the "time" (the index in the list of  returned  val‐
308              ues) increases.
309
310              data1 - First list of data
311
312              data2 - Second list of data
313
314
315       ::math::statistics::mean-histogram-limits mean stdev number
316              Determine reasonable limits based on mean and standard deviation
317              for a histogram
318
319              Convenience function - the result is suitable for the  histogram
320              function.
321
322              mean - Mean of the data
323
324              stdev - Standard deviation
325
326              number - Number of limits to generate (defaults to 8)
327
328
329       ::math::statistics::minmax-histogram-limits min max number
330              Determine reasonable limits based on a minimum and maximum for a
331              histogram
332
333              Convenience function - the result is suitable for the  histogram
334              function.
335
336              min - Expected minimum
337
338              max - Expected maximum
339
340              number - Number of limits to generate (defaults to 8)
341
342
343       ::math::statistics::linear-model xdata ydata intercept
344              Determine  the  coefficients for a linear regression between two
345              series of data (the model: Y = A  +  B*X).  Returns  a  list  of
346              parameters describing the fit
347
348              xdata - List of independent data
349
350              ydata - List of dependent data to be fitted
351
352              intercept - (Optional) compute the intercept (1, default) or fit
353              to a line through the origin (0)
354
355              The result consists of the following list:
356
357              ·      (Estimate of) Intercept A
358
359              ·      (Estimate of) Slope B
360
361              ·      Standard deviation of Y relative to fit
362
363              ·      Correlation coefficient R2
364
365              ·      Number of degrees of freedom df
366
367              ·      Standard error of the intercept A
368
369              ·      Significance level of A
370
371              ·      Standard error of the slope B
372
373              ·      Significance level of B
374
375       ::math::statistics::linear-residuals xdata ydata intercept
376              Determine the difference between actual data and predicted  from
377              the linear model.
378
379              Returns  a  list  of the differences between the actual data and
380              the predicted values.
381
382              xdata - List of independent data
383
384              ydata - List of dependent data to be fitted
385
386              intercept - (Optional) compute the intercept (1, default) or fit
387              to a line through the origin (0)
388
389       ::math::statistics::test-2x2 n11 n21 n12 n22
390              Determine  if two set of samples, each from a binomial distribu‐
391              tion, differ significantly or not (implying a different  parame‐
392              ter).
393
394              Returns  the "chi-square" value, which can be used to the deter‐
395              mine the significance.
396
397              n11 - Number of outcomes with the first  value  from  the  first
398              sample.
399
400              n21  -  Number  of outcomes with the first value from the second
401              sample.
402
403              n12 - Number of outcomes with the second value  from  the  first
404              sample.
405
406              n22  -  Number of outcomes with the second value from the second
407              sample.
408
409
410       ::math::statistics::print-2x2 n11 n21 n12 n22
411              Determine if two set of samples, each from a binomial  distribu‐
412              tion,  differ significantly or not (implying a different parame‐
413              ter).
414
415              Returns a short report, useful in an interactive session.
416
417              n11 - Number of outcomes with the first  value  from  the  first
418              sample.
419
420              n21  -  Number  of outcomes with the first value from the second
421              sample.
422
423              n12 - Number of outcomes with the second value  from  the  first
424              sample.
425
426              n22  -  Number of outcomes with the second value from the second
427              sample.
428
429
430       ::math::statistics::control-xbar data ?nsamples?
431              Determine the control limits for an xbar chart.  The  number  of
432              data in each subsample defaults to 4. At least 20 subsamples are
433              required.
434
435              Returns the mean, the lower limit, the upper limit and the  num‐
436              ber of data per subsample.
437
438              data - List of observed data
439
440              nsamples - Number of data per subsample
441
442
443       ::math::statistics::control-Rchart data ?nsamples?
444              Determine  the control limits for an R chart. The number of data
445              in each subsample defaults to 4.  At  least  20  subsamples  are
446              required.
447
448              Returns the mean range, the lower limit, the upper limit and the
449              number of data per subsample.
450
451              data - List of observed data
452
453              nsamples - Number of data per subsample
454
455
456       ::math::statistics::test-xbar control data
457              Determine if the data exceed the control  limits  for  the  xbar
458              chart.
459
460              Returns a list of subsamples (their indices) that indeed violate
461              the limits.
462
463              control - Control limits as returned by the "control-xbar"  pro‐
464              cedure
465
466              data - List of observed data
467
468
469       ::math::statistics::test-Rchart control data
470              Determine if the data exceed the control limits for the R chart.
471
472              Returns a list of subsamples (their indices) that indeed violate
473              the limits.
474
475              control - Control limits as  returned  by  the  "control-Rchart"
476              procedure
477
478              data - List of observed data
479
480

STATISTICAL DISTRIBUTIONS

482       In  the  literature  a large number of probability distributions can be
483       found. The statistics package supports:
484
485       ·      The normal or Gaussian distribution
486
487       ·      The uniform distribution - equal probability for all data within
488              a given interval
489
490       ·      The  exponential  distribution  -  useful as a model for certain
491              extreme-value distributions.
492
493       ·      PM - binomial, Poisson, chi-squared, student's T, F.  In princi‐
494              ple for each distribution one has procedures for:
495
496       ·      The probability density (pdf-*)
497
498       ·      The cumulative density (cdf-*)
499
500       ·      Quantiles for the given distribution (quantiles-*)
501
502       ·      Histograms for the given distribution (histogram-*)
503
504       ·      List of random values with the given distribution (random-*) The
505              following procedures have been implemented:
506
507       ::math::statistics::pdf-normal mean stdev value
508              Return the probability of a given value for a  normal  distribu‐
509              tion with given mean and standard deviation.
510
511              mean - Mean value of the distribution
512
513              stdev - Standard deviation of the distribution
514
515              value - Value for which the probability is required
516
517
518       ::math::statistics::pdf-exponential mean value
519              Return  the probability of a given value for an exponential dis‐
520              tribution with given mean.
521
522              mean - Mean value of the distribution
523
524              value - Value for which the probability is required
525
526
527       ::math::statistics::pdf-uniform xmin xmax value
528              Return the probability of a given value for a uniform  distribu‐
529              tion with given extremes.
530
531              xmin - Minimum value of the distribution
532
533              xmin - Maximum value of the distribution
534
535              value - Value for which the probability is required
536
537
538       ::math::statistics::cdf-normal mean stdev value
539              Return  the cumulative probability of a given value for a normal
540              distribution with given mean and standard deviation, that is the
541              probability for values up to the given one.
542
543              mean - Mean value of the distribution
544
545              stdev - Standard deviation of the distribution
546
547              value - Value for which the probability is required
548
549
550       ::math::statistics::cdf-exponential mean value
551              Return  the cumulative probability of a given value for an expo‐
552              nential distribution with given mean.
553
554              mean - Mean value of the distribution
555
556              value - Value for which the probability is required
557
558
559       ::math::statistics::cdf-uniform xmin xmax value
560              Return the cumulative probability of a given value for a uniform
561              distribution with given extremes.
562
563              xmin - Minimum value of the distribution
564
565              xmin - Maximum value of the distribution
566
567              value - Value for which the probability is required
568
569
570       ::math::statistics::cdf-students-t degrees value
571              Return  the  cumulative  probability of a given value for a Stu‐
572              dent's t distribution with given number of degrees.
573
574              degrees - Number of degrees of freedom
575
576              value - Value for which the probability is required
577
578
579       ::math::statistics::random-normal mean stdev number
580              Return a list of "number" random values satisfying a normal dis‐
581              tribution with given mean and standard deviation.
582
583              mean - Mean value of the distribution
584
585              stdev - Standard deviation of the distribution
586
587              number - Number of values to be returned
588
589
590       ::math::statistics::random-exponential mean number
591              Return  a  list of "number" random values satisfying an exponen‐
592              tial distribution with given mean.
593
594              mean - Mean value of the distribution
595
596              number - Number of values to be returned
597
598
599       ::math::statistics::random-uniform xmin xmax value
600              Return a list of "number" random  values  satisfying  a  uniform
601              distribution with given extremes.
602
603              xmin - Minimum value of the distribution
604
605              xmin - Maximum value of the distribution
606
607              number - Number of values to be returned
608
609
610       ::math::statistics::histogram-uniform xmin xmax limits number
611              Return the expected histogram for a uniform distribution.
612
613              xmin - Minimum value of the distribution
614
615              xmax - Maximum value of the distribution
616
617              limits - Upper limits for the buckets in the histogram
618
619              number - Total number of "observations" in the histogram
620
621              TO DO: more function descriptions to be added
622

DATA MANIPULATION

624       The data manipulation procedures act on lists or lists of lists:
625
626       ::math::statistics::filter varname data expression
627              Return  a  list  consisting  of  the  data for which the logical
628              expression is true (this command works analogously to  the  com‐
629              mand foreach).
630
631              varname - Name of the variable used in the expression
632
633              data - List of data
634
635              expression - Logical expression using the variable name
636
637
638       ::math::statistics::map varname data expression
639              Return  a  list  consisting of the data that are transformed via
640              the expression.
641
642              varname - Name of the variable used in the expression
643
644              data - List of data
645
646              expression - Expression to be used to transform (map) the data
647
648
649       ::math::statistics::samplescount varname list expression
650              Return a list consisting of the counts of all data in  the  sub‐
651              lists of the "list" argument for which the expression is true.
652
653              varname - Name of the variable used in the expression
654
655              data - List of sublists, each containing the data
656
657              expression  -  Logical  expression to test the data (defaults to
658              "true").
659
660
661       ::math::statistics::subdivide
662              Routine PM - not implemented yet
663

PLOT PROCEDURES

665       The following simple plotting procedures are available:
666
667       ::math::statistics::plot-scale canvas xmin xmax ymin ymax
668              Set the scale for a plot in the given canvas. All plot  routines
669              expect  this  function to be called first. There is no automatic
670              scaling provided.
671
672              canvas - Canvas widget to use
673
674              xmin - Minimum x value
675
676              xmax - Maximum x value
677
678              ymin - Minimum y value
679
680              ymax - Maximum y value
681
682
683       ::math::statistics::plot-xydata canvas xdata ydata tag
684              Create a simple XY plot in the given canvas - the data are shown
685              as  a  collection of dots. The tag can be used to manipulate the
686              appearance.
687
688              canvas - Canvas widget to use
689
690              xdata - Series of independent data
691
692              ydata - Series of dependent data
693
694              tag - Tag to give to the plotted data (defaults to xyplot)
695
696
697       ::math::statistics::plot-xyline canvas xdata ydata tag
698              Create a simple XY plot in the given canvas - the data are shown
699              as a line through the data points. The tag can be used to manip‐
700              ulate the appearance.
701
702              canvas - Canvas widget to use
703
704              xdata - Series of independent data
705
706              ydata - Series of dependent data
707
708              tag - Tag to give to the plotted data (defaults to xyplot)
709
710
711       ::math::statistics::plot-tdata canvas tdata tag
712              Create a simple XY plot in the given canvas - the data are shown
713              as  a  collection of dots. The horizontal coordinate is equal to
714              the index. The tag can be used  to  manipulate  the  appearance.
715              This  type of presentation is suitable for autocorrelation func‐
716              tions for instance or for inspecting the  time-dependent  behav‐
717              iour.
718
719              canvas - Canvas widget to use
720
721              tdata - Series of dependent data
722
723              tag - Tag to give to the plotted data (defaults to xyplot)
724
725
726       ::math::statistics::plot-tline canvas tdata tag
727              Create a simple XY plot in the given canvas - the data are shown
728              as a line. See plot-tdata for an explanation.
729
730              canvas - Canvas widget to use
731
732              tdata - Series of dependent data
733
734              tag - Tag to give to the plotted data (defaults to xyplot)
735
736
737       ::math::statistics::plot-histogram canvas counts limits tag
738              Create a simple histogram in the given canvas
739
740              canvas - Canvas widget to use
741
742              counts - Series of bucket counts
743
744              limits - Series of upper limits for the buckets
745
746              tag - Tag to give to the plotted data (defaults to xyplot)
747
748

THINGS TO DO

750       The following procedures are yet to be implemented:
751
752       ·      F-test-stdev
753
754       ·      interval-mean-stdev
755
756       ·      histogram-normal
757
758       ·      histogram-exponential
759
760       ·      test-histogram
761
762       ·      test-corr
763
764       ·      quantiles-*
765
766       ·      fourier-coeffs
767
768       ·      fourier-residuals
769
770       ·      onepar-function-fit
771
772       ·      onepar-function-residuals
773
774       ·      plot-linear-model
775
776       ·      subdivide
777

EXAMPLES

779       The code below is a small example of how you can examine a set of data:
780
781       # Simple example:
782       # - Generate data (as a cheap way of getting some)
783       # - Perform statistical analysis to describe the data
784       #
785       package require math::statistics
786
787       #
788       # Two auxiliary procs
789       #
790       proc pause {time} {
791          set wait 0
792          after [expr {$time*1000}] {set ::wait 1}
793          vwait wait
794       }
795
796       proc print-histogram {counts limits} {
797          foreach count $counts limit $limits {
798             if { $limit != {} } {
799                puts [format "<%12.4g\t%d" $limit $count]
800                set prev_limit $limit
801             } else {
802                puts [format ">%12.4g\t%d" $prev_limit $count]
803             }
804          }
805       }
806
807       #
808       # Our source of arbitrary data
809       #
810       proc generateData { data1 data2 } {
811          upvar 1 $data1 _data1
812          upvar 1 $data2 _data2
813
814          set d1 0.0
815          set d2 0.0
816          for { set i 0 } { $i < 100 } { incr i } {
817             set d1 [expr {10.0-2.0*cos(2.0*3.1415926*$i/24.0)+3.5*rand()}]
818             set d2 [expr {0.7*$d2+0.3*$d1+0.7*rand()}]
819             lappend _data1 $d1
820             lappend _data2 $d2
821          }
822          return {}
823       }
824
825       #
826       # The analysis session
827       #
828       package require Tk
829       console show
830       canvas .plot1
831       canvas .plot2
832       pack   .plot1 .plot2 -fill both -side top
833
834       generateData data1 data2
835
836       puts "Basic statistics:"
837       set b1 [::math::statistics::basic-stats $data1]
838       set b2 [::math::statistics::basic-stats $data2]
839       foreach label {mean min max number stdev var} v1 $b1 v2 $b2 {
840          puts "$label\t$v1\t$v2"
841       }
842       puts "Plot the data as function of \"time\" and against each other"
843       ::math::statistics::plot-scale .plot1  0 100  0 20
844       ::math::statistics::plot-scale .plot2  0 20   0 20
845       ::math::statistics::plot-tline .plot1 $data1
846       ::math::statistics::plot-tline .plot1 $data2
847       ::math::statistics::plot-xydata .plot2 $data1 $data2
848
849       puts "Correlation coefficient:"
850       puts [::math::statistics::corr $data1 $data2]
851
852       pause 2
853       puts "Plot histograms"
854       ::math::statistics::plot-scale .plot2  0 20 0 100
855       set limits         [::math::statistics::minmax-histogram-limits 7 16]
856       set histogram_data [::math::statistics::histogram $limits $data1]
857       ::math::statistics::plot-histogram .plot2 $histogram_data $limits
858
859       puts "First series:"
860       print-histogram $histogram_data $limits
861
862       pause 2
863       set limits         [::math::statistics::minmax-histogram-limits 0 15 10]
864       set histogram_data [::math::statistics::histogram $limits $data2]
865       ::math::statistics::plot-histogram .plot2 $histogram_data $limits d2
866
867       puts "Second series:"
868       print-histogram $histogram_data $limits
869
870       puts "Autocorrelation function:"
871       set  autoc [::math::statistics::autocorr $data1]
872       puts [::math::statistics::map $autoc {[format "%.2f" $x]}]
873       puts "Cross-correlation function:"
874       set  crossc [::math::statistics::crosscorr $data1 $data2]
875       puts [::math::statistics::map $crossc {[format "%.2f" $x]}]
876
877       ::math::statistics::plot-scale .plot1  0 100 -1  4
878       ::math::statistics::plot-tline .plot1  $autoc "autoc"
879       ::math::statistics::plot-tline .plot1  $crossc "crossc"
880
881       puts "Quantiles: 0.1, 0.2, 0.5, 0.8, 0.9"
882       puts "First:  [::math::statistics::quantiles $data1 {0.1 0.2 0.5 0.8 0.9}]"
883       puts "Second: [::math::statistics::quantiles $data2 {0.1 0.2 0.5 0.8 0.9}]"
884
885
886       If you run this example, then the following should be clear:
887
888       ·      There is a strong correlation between two time series,  as  dis‐
889              played  by  the raw data and especially by the correlation func‐
890              tions.
891
892       ·      Both time series show a significant periodic component
893
894       ·      The histograms are not very useful in identifying the nature  of
895              the time series - they do not show the periodic nature.
896

KEYWORDS

898       data analysis, mathematics, statistics
899
900
901
902math                                  0.2                  math::statistics(n)
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