Анализ данных и знаний / 6 лаба / 6

#1

x=seq(-21, 21, by = 0.21)

y=21*x+21*sin(4*3.14*x/21)+rnorm(length(x), 0, 0.42)

x

[1] -21.00 -20.79 -20.58 -20.37 -20.16 -19.95 -19.74 -19.53

[9] -19.32 -19.11 -18.90 -18.69 -18.48 -18.27 -18.06 -17.85

[17] -17.64 -17.43 -17.22 -17.01 -16.80 -16.59 -16.38 -16.17

[25] -15.96 -15.75 -15.54 -15.33 -15.12 -14.91 -14.70 -14.49

[33] -14.28 -14.07 -13.86 -13.65 -13.44 -13.23 -13.02 -12.81

[41] -12.60 -12.39 -12.18 -11.97 -11.76 -11.55 -11.34 -11.13

[49] -10.92 -10.71 -10.50 -10.29 -10.08 -9.87 -9.66 -9.45

[57] -9.24 -9.03 -8.82 -8.61 -8.40 -8.19 -7.98 -7.77

[65] -7.56 -7.35 -7.14 -6.93 -6.72 -6.51 -6.30 -6.09

[73] -5.88 -5.67 -5.46 -5.25 -5.04 -4.83 -4.62 -4.41

[81] -4.20 -3.99 -3.78 -3.57 -3.36 -3.15 -2.94 -2.73

[89] -2.52 -2.31 -2.10 -1.89 -1.68 -1.47 -1.26 -1.05

[97] -0.84 -0.63 -0.42 -0.21 0.00 0.21 0.42 0.63

[105] 0.84 1.05 1.26 1.47 1.68 1.89 2.10 2.31

[113] 2.52 2.73 2.94 3.15 3.36 3.57 3.78 3.99

[121] 4.20 4.41 4.62 4.83 5.04 5.25 5.46 5.67

[129] 5.88 6.09 6.30 6.51 6.72 6.93 7.14 7.35

[137] 7.56 7.77 7.98 8.19 8.40 8.61 8.82 9.03

[145] 9.24 9.45 9.66 9.87 10.08 10.29 10.50 10.71

[153] 10.92 11.13 11.34 11.55 11.76 11.97 12.18 12.39

[161] 12.60 12.81 13.02 13.23 13.44 13.65 13.86 14.07

[169] 14.28 14.49 14.70 14.91 15.12 15.33 15.54 15.75

[177] 15.96 16.17 16.38 16.59 16.80 17.01 17.22 17.43

[185] 17.64 17.85 18.06 18.27 18.48 18.69 18.90 19.11

[193] 19.32 19.53 19.74 19.95 20.16 20.37 20.58 20.79

[201] 21.00

y

[1] -440.3792649 -433.9138632 -426.7088879 -420.2396442

[5] -412.9561870 -406.6698131 -400.2063425 -394.8430900

[9] -388.3930377 -381.7554965 -376.8460938 -371.6531052

[13] -367.1847881 -362.1664695 -358.6181988 -353.8822876

[17] -351.6689104 -348.7795709 -346.3417657 -342.8712920

[21] -340.8099833 -337.4303270 -335.6822407 -334.5835032

[25] -332.3878510 -331.2466115 -329.2259441 -327.3574479

[29] -325.9702284 -322.9419935 -321.7801349 -318.4988396

[33] -315.4026979 -312.8169964 -310.1615348 -306.6620142

[37] -302.9092707 -298.7179711 -294.1082095 -290.2440040

[41] -283.5832884 -279.5339680 -273.3424097 -268.4380471

[45] -261.9837342 -254.8958366 -248.4884068 -240.8074303

[49] -235.3287044 -227.8008400 -220.3885780 -213.9321455

[53] -206.3775438 -199.2447923 -191.6574345 -186.2369943

[57] -179.6560782 -174.0576434 -168.4579097 -162.5591014

[61] -155.8150114 -150.8732076 -146.3167938 -142.1090635

[65] -138.5122242 -135.4998018 -130.8615641 -128.4146115

[69] -125.4085316 -122.8682069 -119.9681820 -117.6034405

[73] -116.0594542 -113.8850637 -111.6395981 -110.4171013

[77] -108.8361534 -106.8038832 -105.2538373 -102.6667084

[81] -100.5529454 -97.0964328 -95.6765832 -93.1061999

[85] -89.8155908 -86.4629631 -83.5836087 -77.8811195

[89] -74.3700806 -68.7575812 -64.8774006 -58.6044403

[93] -53.4949314 -47.0497395 -41.1503250 -33.5946323

[97] -28.3322000 -20.3571363 -14.3047658 -7.7923403

[101] 0.5012988 6.9580014 14.0424340 20.9971333

[105] 27.7277500 35.2652300 40.8481077 47.1889020

[109] 53.8143180 58.3161815 64.8952202 68.1425288

[113] 73.9014926 77.9962618 82.2316247 85.8057113

[117] 90.3072928 92.4881201 95.3842484 98.1449705

[121] 101.1565917 102.7679447 105.2097108 106.1844308

[125] 108.4693936 109.7578594 111.6379214 113.1715534

[129] 115.9417043 117.8264147 119.6171516 122.5109356

[133] 125.3437904 127.0498166 130.6818863 134.3955444

[137] 138.1672720 141.7317916 146.4709031 150.8071467

[141] 156.7570430 161.2943866 167.6544197 173.2819776

[145] 179.8427724 186.4241193 193.1001601 199.7386685

[149] 205.9826887 213.9938465 220.6805845 227.7391197

[153] 233.8278812 241.7817574 248.2196833 255.6844412

[157] 261.3666012 267.9979522 273.2099894 278.9162242

[161] 285.0156952 289.6342963 294.8004535 299.0628632

[165] 303.5249154 306.1399304 310.4683435 313.7350763

[169] 316.5194687 319.8714907 321.2851651 323.5068262

[173] 325.6691454 327.4481167 329.0985751 331.0689393

[177] 332.6747148 334.4238950 336.7384609 338.3349600

[181] 340.2820408 342.8984522 344.9454794 348.7662609

[185] 350.9490068 354.7051015 358.3159330 362.0354760

[189] 367.9647899 371.3640784 376.3579117 382.2850068

[193] 388.2131516 394.6027992 399.8146263 406.7443288

[197] 413.0690895 420.4056779 426.6744261 433.9854094

[201] 440.1198356

#2

plot(x,y, pch = 3, col =6)

#3

cor(x,y)

cor.test(x,y)

cor.test(x,y, method="spearman")

cor.test(x,y, method="kendall")

cor(x,y)

[1] 0.9983382

cor.test(x,y)

Pearson's product-moment correlation

data: x and y

t = 244.3916, df = 199, p-value < 2.2e-16

alternative hypothesis: true correlation is not equal to 0

95 percent confidence interval:

0.997805 0.998742

sample estimates:

cor

0.9983382

cor.test(x,y, method="spearman")

Spearman's rank correlation rho

data: x and y

S = 0, p-value < 2.2e-16

alternative hypothesis: true rho is not equal to 0

sample estimates:

rho

1

cor.test(x,y, method="kendall")

Kendall's rank correlation tau

data: x and y

z = 21.0825, p-value < 2.2e-16

alternative hypothesis: true tau is not equal to 0

sample estimates:

tau

1

Отсюда видно, что результаты коэффициентов корреляции сбегаются. Можно утверждать, что между выборками существует сильная линейная связь. Критерии Спирмана и Кендала говорят о том, что функция монотонна.

#4

DetCoef=function(x,y,p) {

mz=vector('numeric',length(y)-2*p)

dz=vector('numeric',length(y)-2*p)

d2z=vector('numeric',length(y)-2*p)

for(i in (p+1):(length(y)-p))

{mz[i-p]=0

for (j in (i-p):(i+p))

{mz[i-p]=mz[i-p]+y[j]

}

mz[i-p]=mz[i-p]/(2*p+1)

dz[i-p]=y[i]-mz[i-p]

}

d2z=dz*dz

se=sum(d2z)

y1=y[p:(length(y)-p)]

sy=sum((y1-mean(y1))^2)

Kd=1-se/sy

Kd;

}

p=3

DetCoef(x,y,p)

[1] 0.9999933

Из результата данной функции мы можем наблюдать сильную детерминированную связь между выборками. Эта связь наблюдается из-за того, что все выборки линейны.