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2 1 (4/13) : ruby 2 / 49

3 2 ( ) : gnuplot 3 / 49

4 IIJ / 4 / 49

5 1 ( ) / 5 / 49

6 ( ) 6 / 49

7 (summary statistics) : (mean) (median) (mode) : (range) (variance) (standard deviation) 7 / 49

8 (mean): x = 1 n (median): { xr+1 m, m = 2r + 1 x median = (x r + x r+1 )/2 m, m = 2r n i=1 (mode): x i f(x) mode median mean median mode mean x 8 / 49

9 (percentiles) pth-percentile: p% median = 50th-percentile total observations (%) sorted variable x 9 / 49

10 (range): (variance): σ 2 = 1 n (x i x) 2 n i=1 (standatd deviation): σ 68% (mean ± stddev) 95% (mean ± 2stddev) f(x) 1 mean median exp(-x**2/2) σ % x 95% 10 / 49

11 (variance): σ 2 = 1 n (x i x) 2 n i=1 σ 2 = 1 n (x i x) 2 n i=1 = 1 n (x 2 i n 2x i x + x 2 ) i=1 = 1 n n ( x 2 i 2 x n x i + n x 2 ) i=1 i=1 = 1 n x 2 i n 2 x2 + x 2 i=1 = 1 n x 2 i n x2 i=1 11 / 49

12 : 12 / 49

13 : 1/N ( ) 1/N : 1 13 / 49

14 : ( ) ( ) (population): (sample) : ( ) : ( ) population samples estimate estimate 14 / 49

15 ( ) N(µ, σ/ n) n 15 / 49

16 (normal distribution) N(µ, σ) 2 : µ σ f(x) 1 mean median exp(-x**2/2) σ % x 95% 16 / 49

17 (sample mean): x x = 1 n n i=1 x i (sample variance): s 2 s 2 = 1 n 1 n (x i x) 2 i=1 (sample standard deviation): s : n (n 1) (degree of freedom): x 1 17 / 49

18 (standard error) : (SE) SE = σ/ n n 1/ n ( ) N(µ, σ) µ SE = σ/ n 18 / 49

19 (sample variance): s 2 s 2 = 1 n 1 n (x i x) 2 i=1 (n 1) x µ S 2 σ 2 x µ N(µ, σ/ n) (n 1)/n E(S 2 ) = n 1 n σ2 σ 2 = n n 1 S2 = 1 n 1 n (x i x) 2 i=1 19 / 49

20 normalized traffic volume cdf time (sec) normalized traffic volume / 49

21 : sample data from a book: P. K. Janert Gnuplot in Action # Minutes Count :2,355 :171.3 :14.1 : / 49

22 : (2) count finish time (minutes) 22 / 49

23 : (3) rank finish time (minutes) 23 / 49

24 XY XY : 0 ( ) XY 3D ( : ) 24 / 49

25 25 / 49

26 X Y 4 normalized traffic volume time (sec) 26 / 49

27 (1/2) X : Y : frequency normalized traffic volume 27 / 49

28 (2/2) ( ) ( ) 28 / 49

29 (probability density function; pdf) 1 : X x f(x) = P [X = x] pdf normalized traffic volume 29 / 49

30 (cumulative distribution function; cdf) : x f(x) = P [X = x] : x F (x) = P [X <= x] cdf normalized traffic volume 30 / 49

31 CDF CDF CDF 1800 ping rtt 18 ping rtt histogram histogram response time (msec) response time (msec) CDF samples 100 samples response time (msec) ( ) ( )100 ( )CDF 31 / 49

32 (interquartile range) interquartile range (IQR): ( - ) ( 50%) ( ): ( ) : 25/50/75-percentiles : min/max inner fance (Q 1 1.5IQR, Q IQR) max upper quartile mean median lower quartile min 32 / 49

33 (original vs 100 samples) : min max CDF original 100 samples samples 100 samples response time (msec) 33 / 49

34 (scatter plots) 2 X : X Y : Y X Y : ( ) 0.7 ( ) 0.0 ( ) / 49

35 gnuplot grace GUI gnuplot Mac: gnuplot Homebrew/MacPorts (XQuatrz ) Windows: windows 35 / 49

36 : filename = ARGV[0] count = 0 file = open(filename) while text = file.gets count += 1 end file.close puts count count.rb $ ruby count.rb foo.txt Ruby #!/usr/bin/env ruby count = 0 ARGF.each_line do line count += 1 end puts count 36 / 49

37 : : P. K. Janert Gnuplot in Action 37 / 49

38 : ( ) # regular expression to read minutes and count re = /^(\d+)\s+(\d+)/ sum = 0 # sum of data n = 0 # the number of data ARGF.each_line do line if re.match(line) min = $1.to_i cnt = $2.to_i sum += min * cnt n += cnt end end mean = Float(sum) / n printf "n:%d mean:%.1f\n", n, mean % ruby mean.rb marathon.txt n:2355 mean: / 49

39 : : σ 2 = 1 n n i=1 (x i x) 2 # regular expression to read minutes and count re = /^(\d+)\s+(\d+)/ data = Array.new sum = 0 # sum of data n = 0 # the number of data ARGF.each_line do line if re.match(line) min = $1.to_i cnt = $2.to_i sum += min * cnt n += cnt for i in 1.. cnt data.push min end end end mean = Float(sum) / n sqsum = 0.0 data.each do i sqsum += (i - mean)**2 end var = sqsum / n stddev = Math.sqrt(var) printf "n:%d mean:%.1f variance:%.1f stddev:%.1f\n", n, mean, var, stddev % ruby stddev.rb marathon.txt n:2355 mean:171.3 variance:199.9 stddev: / 49

40 : : σ 2 = 1 n n i=1 x2 i x2 # regular expression to read minutes and count re = /^(\d+)\s+(\d+)/ sum = 0 # sum of data n = 0 # the number of data sqsum = 0 # sum of squares ARGF.each_line do line if re.match(line) min = $1.to_i cnt = $2.to_i sum += min * cnt n += cnt sqsum += min**2 * cnt end end mean = Float(sum) / n var = Float(sqsum) / n - mean**2 stddev = Math.sqrt(var) printf "n:%d mean:%.1f variance:%.1f stddev:%.1f\n", n, mean, var, stddev % ruby stddev2.rb marathon.txt n:2355 mean:171.3 variance:199.9 stddev: / 49

41 : # regular expression to read minutes and count re = /^(\d+)\s+(\d+)/ data = Array.new ARGF.each_line do line if re.match(line) min = $1.to_i cnt = $2.to_i for i in 1.. cnt data.push min end end end data.sort! # just in case data is not sorted n = data.length # number of array elements r = n / 2 # when n is odd, n/2 is rounded down if n % 2!= 0 median = data[r] else median = (data[r - 1] + data[r])/2 end printf "r:%d median:%d\n", r, median % ruby median.rb marathon.txt r:1177 median: / 49

42 : gnuplot gnuplot 42 / 49

43 plot "marathon.txt" using 1:2 with boxes ( ) set boxwidth 1 set xlabel "finish time (minutes)" set ylabel "count" set yrange [0:180] set grid y plot "marathon.txt" using 1:2 with boxes notitle "marathon.txt" using 1:2 count finish time (minutes) 43 / 49

44 : CDF : # Minutes Count : # Minutes Count CumulativeCount / 49

45 : CDF (2) ruby code: re = /^(\d+)\s+(\d+)/ cum = 0 ARGF.each_line do line begin if re.match(line) # matched time, cnt = $~.captures cum += cnt.to_i puts "#{time}\t#{cnt}\t#{cum}" end end end gnuplot command: set xlabel "finish time (minutes)" set ylabel "CDF" set grid y plot "marathon-cdf.txt" using 1:($3 / 2355) with lines notitle 45 / 49

46 CDF CDF finish time (minutes) 46 / 49

47 : gnuplot> set terminal png gnuplot> set output "plotfile.png" gnuplot> replot gnuplot> load "scriptfile" gnuplot> quit 47 / 49

48 2 ( ) : gnuplot 48 / 49

49 3 (4/27) : 49 / 49

¥¤¥ó¥¿¡¼¥Í¥Ã¥È·×Â¬¤È¥Ç¡¼¥¿²òÀÏ Âè2²ó

¥¤¥ó¥¿¡¼¥Í¥Ã¥È·×Â¬¤È¥Ç¡¼¥¿²òÀÏ Âè2²ó 2 212 4 13 1 (4/6) : ruby 2 / 35 ( ) : gnuplot 3 / 35 ( ) 4 / 35 (summary statistics) : (mean) (median) (mode) : (range) (variance) (standard deviation) 5 / 35 (mean): x = 1 n (median): { xr+1 m, m = 2r