library(TSA)

Exhibit 1.

Stimulating 48 chi-square distributed values with 2 degree of freedoms.

set.seed(1)
chi2Data = rchisq(n=48, df = 2)
chi2Data

##  [1] 0.3102827 3.7648032 3.6090250 1.6723553 2.4450873 2.3167105 1.9800399
##  [8] 0.6147466 0.1892382 0.3144031 0.6216108 0.9374638 0.1363947 2.4985843
## [15] 2.0162627 2.7218900 2.4119764 1.1082308 2.4140052 3.1249573 0.2615523
## [22] 0.4381074 0.6944394 0.8599189 0.2924151 1.6834339 0.4848175 3.1606987
## [29] 2.3705121 0.7808625 0.4621734 1.9421776 0.2631446 7.5287682 1.6422509
## [36] 0.3217521 0.1915028 4.0533800 5.7617003 0.5480962 0.8056038 0.9452761
## [43] 0.8429060 0.2250805 4.9721928 6.4326576 1.7855267 0.2479961

plot(chi2Data, ylab='Values',type='o', main = "Time Series Plot of 48 Random Chi-square Values - Simulation 1")

Repeating the above process severally with new simulation each time.

set.seed(2)
chi2Data = rchisq(n=48, df = 2)
chi2Data

##  [1]  0.13379914  0.10229738  0.04028197  1.18554413  0.24805534
##  [6]  0.68974337  0.21882289  1.67786343  2.87032735  0.52174972
## [11]  0.54903888  6.85098698  0.07212130  0.02295962  2.50348957
## [16]  0.52678012  2.74026729  0.20322753  0.42268784  0.13088216
## [21]  3.20164927  1.27588766  1.72915482  0.67711670  5.50376913
## [26]  2.07408870  5.79746877  0.61464093  0.15737342  0.61349495
## [31]  1.37283441  0.84120790  2.40900097  4.33272195  2.69401721
## [36]  0.67235638  1.52192461  0.80969526  0.42601487  0.12857682
## [41]  0.65600344  3.04240104  4.63624859  4.54391989  4.79065313
## [46] 11.41917591  0.16711767  0.33052796

plot(chi2Data, ylab='Values', type='o', main = "Time Series Plot of 48 Random Chi-square Values - Simulation 2")

set.seed(3)
chi2Data = rchisq(n=48, df = 2)
chi2Data

##  [1] 0.10227867 0.46854638 0.03434340 1.52890879 1.01788908 1.21401347
##  [7] 2.80188482 1.89979167 0.34492938 5.75437358 3.79743381 2.21923152
## [13] 0.59518147 0.29335560 0.16792412 0.34922218 1.26801407 0.31537134
## [19] 0.43222539 0.19105662 2.94341899 0.90066330 0.34480834 2.56734177
## [25] 2.29392936 2.31278903 0.56401115 2.25389603 5.25846359 1.18358795
## [31] 0.09613006 2.42158522 0.60913251 2.88720420 0.98134730 1.64167728
## [37] 0.41193307 2.29205283 0.18287499 0.50328148 1.07734628 0.97734245
## [43] 2.80685371 3.32690236 0.44590874 2.97292468 1.41384222 1.35472329

plot(chi2Data, ylab='Values', type='o', main = "Time Series Plot of 48 Random Chi-square Values - Simulation 3")

set.seed(4)
chi2Data = rchisq(n=48, df = 2)
chi2Data

##  [1] 1.33002900 0.37994874 0.29813591 4.65073628 2.21233300 5.51028797
##  [7] 8.49525527 0.96466374 0.93718516 1.10844768 0.24817828 0.93846310
## [13] 1.48798484 5.20049448 0.63988577 2.61910291 0.73858513 3.52290482
## [19] 1.22880675 3.04093840 0.21780155 1.05336165 5.51778440 1.53069037
## [25] 0.51739022 0.44965321 0.04585698 1.27320735 3.66318255 3.86939091
## [31] 0.32679856 2.28577369 3.36228859 1.28287070 0.18917697 0.33411625
## [37] 0.97531463 0.79197355 3.12899907 0.13994107 2.65339163 1.88254232
## [43] 0.77245032 1.72248156 4.60774702 0.79921990 2.28113660 0.54567093

plot(chi2Data, ylab='Values', type='o', main = "Time Series Plot of 48 Random Chi-square Values - Simulation 4")

set.seed(5)
chi2Data = rchisq(n=48, df = 2)
chi2Data

##  [1] 0.16436974 0.35649916 1.10165686 4.88485782 0.32910837 1.22161881
##  [7] 0.16777783 1.20485147 3.48966961 0.18753777 0.93003492 0.05863721
## [13] 0.59159115 2.67897709 2.77556345 4.15346771 2.24926674 2.49364135
## [19] 0.62801979 5.74408293 0.61803285 0.15759515 0.11269173 6.58726716
## [25] 3.46191511 4.18598406 2.79847500 3.48458534 0.43179826 0.18714373
## [31] 7.58500595 1.95315146 0.13897258 2.67793562 1.50324819 0.84882794
## [37] 1.28310534 2.96762396 0.33815339 0.43866358 2.34906792 0.84708713
## [43] 8.90546305 0.73219066 3.62401319 2.01703371 1.59360942 1.86652557

plot(chi2Data, ylab='Values', type='o', main = "Time Series Plot of 48 Random Chi-square Values - Simulation 5")

The plots have no patterns.
The plots are not symmetric.
They have lots of outliers.
99.7% of the observations are not within 3 standard deviation.

All these simulations of random chi-square distributed processes seem to be random and nonnormal.

Exhibit 2.

Stimulating 48 t-distributed values with 5 degree of freedoms.

set.seed(6)
tData = rt(n=48, df = 5)
tData

##  [1]  0.387806040  1.798674460 -1.141354053 -0.782801148 -0.727638412
##  [6] -1.070491557 -0.522914430 -1.024070742  3.699266473  1.152383425
## [11]  1.130557500 -1.325652916  1.500540680 -0.639393825 -2.107441881
## [16] -1.494389717 -0.457895026 -2.616364452  0.883014320  0.373231933
## [21] -0.168635842  0.214340473 -0.900774639  0.905792656 -1.072382985
## [26] -0.122733746  0.200213477 -2.749963852  0.894578403 -1.423755828
## [31] -0.539347135 -0.029989506  1.731023625  0.966398665  0.727625273
## [36] -0.031925360 -0.009499812 -0.242327376 -0.626094461  2.271923198
## [41] -1.394862711 -1.001773487 -1.072616943 -0.507959288 -0.087536547
## [46]  3.449846614 -1.327486145  0.058806823

plot(tData, ylab='Values', type='o', main = "Time Series Plot of 48 Random t-Values - Simulation 1")

Repeating the above process severally with new simulation each time.

set.seed(7)
tData = rt(n=48, df = 5)
tData

##  [1]  4.432866409  0.542618738 -0.155049385 -0.124063866  2.174072270
##  [6]  1.681277264  0.216878860  0.764455417  0.890157093 -0.135391272
## [11]  2.867074733  1.697816456  0.405644114 -0.445955014  0.494376232
## [16] -0.054687571  1.830453115 -0.114156702  1.062250947 -1.260895051
## [21] -0.190845581 -0.007588192  1.519782541  1.206742163  0.286979786
## [26]  1.105062807  0.486593443  0.256158394  0.610833682  1.129661117
## [31]  0.717844983  0.430178290  1.408389691 -0.717981846  0.631871360
## [36]  0.992051484  0.052766856 -0.169803139  0.575209637 -0.002353689
## [41] -0.397570689  1.425670573 -0.459663937 -2.037361254 -0.232731403
## [46] -0.117065466 -2.499862639  0.813868086

plot(tData, ylab='Values', type='o', main = "Time Series Plot of 48 Random t-Values - Simulation 2")

set.seed(8)
tData = rt(n=48, df = 5)
tData

##  [1] -0.07290740 -0.64351347  0.78982497 -0.64472337  0.91115940
##  [6] -1.17960185 -0.18255067 -0.01235775 -0.93022761  1.79827255
## [11] -1.47395084  0.63490018 -0.51791017 -0.33784793 -0.07997034
## [16]  0.89178608  0.10582708  2.05664212 -3.27593550  1.88360156
## [21]  0.86352296  0.08371531  2.13329671 -0.54442812  0.37419810
## [26] -0.17657303 -0.35965982 -0.78606766 -0.45515305 -0.38454066
## [31]  0.67040333  0.26569450  1.08910529  0.19252348 -1.48892469
## [36]  0.54399455  4.53258863 -1.67132008 -0.62874169 -2.68439191
## [41]  1.39991582  1.87455953  0.23316722 -0.34085663  1.06887537
## [46]  2.44587585  0.54044643 -0.43395684

plot(tData, ylab='Values', type='o', main = "Time Series Plot of 48 Random t-Values - Simulation 3")

set.seed(9)
tData = rt(n=48, df = 5)
tData

##  [1] -1.20519861 -1.40361236 -0.93354078 -0.02229274 -0.01894893
##  [6]  2.93377663  1.01753468  1.41231887  0.97225340  1.37211791
## [11]  0.07040706  1.21887424 -1.42618851  1.10708273  0.05468504
## [16]  1.85508720  0.14617349  0.22586406  2.08485786  1.61367320
## [21]  0.20243793 -0.87933594 -0.23129085  0.23704518  1.75123617
## [26]  0.54702137  0.24443470 -1.04556118  1.01721220 -0.50325709
## [31] -0.21337458  1.47326049 -0.59403205  1.05865203  0.92286741
## [36]  1.21355225  2.35884003 -1.19964399 -0.07151976 -0.58556532
## [41] -1.43435879  1.15884317  0.27532533 -0.84223179 -2.71065238
## [46] -1.53202438  0.20681262  0.66956163

plot(tData, ylab='Values', type='o', main = "Time Series Plot of 48 Random t-Values - Simulation 4")

set.seed(10)
tData = rt(n=48, df = 5)
tData

##  [1]  0.02241932 -1.07273566  0.76072274 -0.25812356  0.97207866
##  [6] -0.19742819  0.80351704 -1.14678324  1.89059639 -0.68289489
## [11] -1.12021546 -0.55193480  0.24193730 -2.13127882  0.83255230
## [16] -1.42280977 -2.22163755 -0.93925475 -0.71540590 -1.09300306
## [21]  0.03578411  0.08911051  0.87105941  0.44250151 -0.84878595
## [26] -0.65476931 -1.64974853 -0.99801779  1.07901248 -0.44872844
## [31] -1.22803988 -2.54126677  0.21869923 -1.47788575 -0.76432156
## [36]  0.71943804 -0.41390917  0.20307005  0.63908102 -0.09695765
## [41] -0.06716334  0.76639649 -1.36529900  0.11553529 -0.74174363
## [46] -0.92897985 -0.59039814  1.78594140

plot(tData, ylab='Values', type='o', main = "Time Series Plot of 48 Random t-Values - Simulation 5")

The plots have no patterns.
They have lots of outliers.
99.7% of the observations are not within 3 standard deviation.

All these simulations of random t distributed processes seem to be random and nonnormal.

Exhibit 3.

data(tempdub)
sss = data.frame(tempdub)
plot(tempdub, ylab='Temperature', type='l', main = "Time Series Plot for the Dubuque temperature Series")
points(y=tempdub, x=time(tempdub), pch=as.vector(season(tempdub)))