Time Series-1

时间序列第一次作业

第二题

1. 根据时序图可以发现该时间序列有明显的趋势和周期性，初步判断是非平稳的。
2. 如图所示
3. 该时间序列的自相关系数随着阶数的增加并没有收敛到0，反而出现周期性。因此可以看出该时间序列是非平稳的。

data <- read.csv('zye22.csv')
tsdata <- ts(data$CO2,start = 1975,frequency = 12)
print(tsdata)

        Jan    Feb    Mar    Apr    May    Jun    Jul    Aug    Sep    Oct
1975 330.45 330.97 331.64 332.87 333.61 333.55 331.90 330.05 328.58 328.31
1976 331.63 332.46 333.36 334.45 334.82 334.32 333.05 330.87 329.24 328.87
1977 332.81 333.23 334.55 335.82 336.44 335.99 334.65 332.41 331.32 330.73
1978 334.66 335.07 336.33 337.39 337.65 337.57 336.25 334.39 332.44 332.25
1979 335.89 336.44 337.63 338.54 339.06 338.95 337.41 335.71 333.68 333.69
1980 337.81 338.16 339.88 340.57 341.19 340.87 339.25 337.19 335.49 336.63
        Nov    Dec
1975 329.41 330.63
1976 330.18 331.50
1977 332.05 333.53
1978 333.59 334.76
1979 335.05 336.53
1980 337.74 338.36

par(mfrow = c(1,2))
plot(tsdata)
# 样本自相关系数如下
c = acf(tsdata)

print(c$acf)

, , 1

             [,1]
 [1,]  1.00000000
 [2,]  0.90750778
 [3,]  0.72171377
 [4,]  0.51251814
 [5,]  0.34982244
 [6,]  0.24689637
 [7,]  0.20309427
 [8,]  0.21020799
 [9,]  0.26428810
[10,]  0.36433219
[11,]  0.48471672
[12,]  0.58456166
[13,]  0.60197891
[14,]  0.51841257
[15,]  0.36856286
[16,]  0.20671211
[17,]  0.08138070
[18,]  0.00135460
[19,] -0.03247916

第七题

1. 观察时序图可以发现该时间序列数据并没有明显的趋势和周期性，初步判断是平稳的
2. 随着阶数越来越大，自相关系数逐渐减少，并在2个标准差的范围之内。因此可以认为自相关系数很小。该时间序列是平稳的。
3. 对该时间序列做纯随机检验，H0：所有的自相关系数都为0.H1：所有的自相关系数不全为零。显著性水平α = 0.05，P值为6.344e-08，远远小于0.05.因此可以拒绝原假设，接受备择假设，认为该时间序列并非纯随机序列，仍有有用的信息未被提取出来。还可以继续进行分析

data1 <- read.csv('zye27.csv')
tsdata1 <- ts(data1$死亡率,start =1915, frequency = 1)
print(tsdata1)

Time Series:
Start = 1915 
End = 2004 
Frequency = 1 
 [1] 0.5215052 0.4248284 0.4250311 0.4771938 0.8280212 0.6156186 0.3666270
 [8] 0.4308883 0.2810287 0.4646245 0.2693951 0.5779049 0.5661151 0.5077584
[15] 0.7507175 0.6808395 0.7661091 0.4561473 0.4977496 0.4193273 0.6095514
[22] 0.4573370 0.5705478 0.3478996 0.3874993 0.5824285 0.2391033 0.2367445
[29] 0.2626158 0.4240934 0.3652750 0.3750758 0.4090056 0.3891676 0.2402610
[36] 0.1589496 0.4393373 0.5094681 0.3743465 0.4339828 0.4130557 0.3288928
[43] 0.5186648 0.5486504 0.5469111 0.4963494 0.5308929 0.5957761 0.5570584
[50] 0.5731325 0.5005416 0.5431269 0.5593657 0.6911693 0.4403485 0.5676662
[57] 0.5969114 0.4735537 0.5923935 0.5975556 0.6334127 0.6057115 0.7046107
[64] 0.4805263 0.7026860 0.7009017 0.6030854 0.6980919 0.5976560 0.8023421
[71] 0.6017109 0.5993127 0.6025625 0.7016625 0.4995714 0.4980918 0.4975690
[78] 0.6001830 0.3339542 0.2744370 0.3209428 0.5406671 0.4050209 0.2885961
[85] 0.3275942 0.3132606 0.2575562 0.2138386 0.1861856 0.1592713

par(mfrow = c(1,2))
plot(tsdata1)
# 样本自相关系数如下
b = acf(tsdata1,lag.max = 50)

print(b$acf)

, , 1

             [,1]
 [1,]  1.00000000
 [2,]  0.56076761
 [3,]  0.46196842
 [4,]  0.37628563
 [5,]  0.39972314
 [6,]  0.32391693
 [7,]  0.21044131
 [8,]  0.25533506
 [9,]  0.18721880
[10,]  0.21006106
[11,]  0.05404696
[12,]  0.08762089
[13,] -0.06481815
[14,] -0.09531789
[15,] -0.15266937
[16,] -0.16984920
[17,] -0.11922124
[18,] -0.13976241
[19,] -0.18842019
[20,] -0.28441719
[21,] -0.22236954
[22,] -0.20626166
[23,] -0.20272664
[24,] -0.25407346
[25,] -0.22858437
[26,] -0.18260723
[27,] -0.19885069
[28,] -0.22658010
[29,] -0.14321498
[30,] -0.19883059
[31,] -0.24102495
[32,] -0.20328289
[33,] -0.18534691
[34,] -0.13835885
[35,] -0.23415755
[36,] -0.20783024
[37,] -0.23855855
[38,] -0.16941025
[39,] -0.16804499
[40,] -0.12554065
[41,] -0.09679402
[42,] -0.14916868
[43,] -0.08185914
[44,] -0.07889550
[45,]  0.07922571
[46,]  0.03410389
[47,]  0.03423551
[48,]  0.08881282
[49,]  0.17885920
[50,]  0.17050772
[51,]  0.12019900

# 纯随机检验
a = Box.test(tsdata1,type = 'Ljung-Box')
print(a$p.value)

[1] 6.34389e-08