1.
#annual data
Annual <- scan("http://www.calpoly.edu/~gbdurham/econ522/precip/annual_precip.csv")
Annual <- ts(Annual)
#monthly data
monthly <- scan("http://www.calpoly.edu/~gbdurham/econ522/precip/monthly_precip.csv")
monthly <- ts(monthly, frequency = 12, start = c(0,6))
2.
plot.ts(Annual)
3.
plot.ts(Annual)
s = ksmooth(1:145,Annual,bandwidth = 5)
lines(smooth(s$y),col = 3)
4.
res = Annual - s$y
par(mfrow=c(2,1))
plot(res)
hist(res)
par(mfrow=c(1,1))
acf(res)
#The residuals seem normal
4.
lamda <- 0.5
z = ((Annual^lamda)-1)/lamda
zSmooth<- ksmooth(1:145,z,bandwidth = 5)
Z_Res <- z - zSmooth$y
plot(Z_Res)
hist(Z_Res)
#Residuals at Lamda = 0.5 seem to be normal
5.
Monthly <- ts(monthly, frequency = 12, start = c(0,6))
monthplot(Monthly)
MonthSmooth<- ksmooth(1:1740,Monthly,bandwidth = 5)
Monthly_Res <- Monthly - MonthSmooth$y
plot(Monthly_Res)
hist(Monthly_Res)
#does not seem to appear normal before transformation
lamda <- 0.5
z = ((Monthly^lamda)-1)/lamda
zSmooth<- ksmooth(1:1740,Monthly,bandwidth = 5)
Z_Res <- z - zSmooth$y
plot(Z_Res)
hist(Z_Res)
#Seems to be normal after Box Cox Transformation
6.
Monthlyfactor <- factor(cycle(z))
Dummy1<- model.matrix(z~ Monthlyfactor)
Dummy1<- Dummy1[,-1]
auto.arima(z, xreg = Dummy1)
## Series: z
## ARIMA(1,0,0)(2,0,2)[12] with non-zero mean
##
## Coefficients:
## Warning in sqrt(diag(x$var.coef)): NaNs produced
## ar1 sar1 sar2 sma1 sma2 intercept Monthlyfactor2
## 0.0879 -0.2526 0.4896 0.2630 -0.4521 1.7626 -0.3676
## s.e. 0.0141 NaN NaN 0.0087 0.0154 0.1144 0.1546
## Monthlyfactor3 Monthlyfactor4 Monthlyfactor5 Monthlyfactor6
## -1.6098 -2.8217 -3.4596 -3.6535
## s.e. 0.1612 0.1618 0.1618 0.1618
## Monthlyfactor7 Monthlyfactor8 Monthlyfactor9 Monthlyfactor10
## -3.6335 -3.2443 -2.3462 -1.3704
## s.e. 0.1618 0.1618 0.1618 0.1618
## Monthlyfactor11 Monthlyfactor12
## -0.1761 0.1197
## s.e. 0.1612 0.1546
##
## sigma^2 estimated as 1.671: log likelihood=-2915.77
## AIC=5867.54 AICc=5867.93 BIC=5965.84
ARTrend<- arima(z, order = c(1,0,0), include.mean = FALSE, xreg = Dummy1)
ARTrend2<- arima(z, order = c(2,0,1), include.mean = FALSE, xreg = Dummy1)
## Warning in arima(z, order = c(2, 0, 1), include.mean = FALSE, xreg =
## Dummy1): possible convergence problem: optim gave code = 1
#AIC on ARMA (2,1) had the Lowest AIC of 5881.75
require(graphics)
tsdiag(ARTrend)
tsdiag(ARTrend2)
SARTrend<- sarima(z,1,0,0,2,0,2,12, tol = .00001)
## initial value 0.659149
## iter 2 value 0.456323
## iter 3 value 0.405773
## iter 4 value 0.393314
## iter 5 value 0.372410
## iter 6 value 0.365299
## iter 7 value 0.340690
## iter 8 value 0.324793
## iter 9 value 0.316972
## iter 10 value 0.308767
## iter 11 value 0.300280
## iter 12 value 0.294186
## iter 13 value 0.293110
## iter 14 value 0.291942
## iter 15 value 0.291885
## iter 16 value 0.291854
## iter 17 value 0.291799
## iter 18 value 0.291720
## iter 19 value 0.291544
## iter 20 value 0.291443
## iter 21 value 0.291362
## iter 22 value 0.291352
## iter 23 value 0.291348
## iter 23 value 0.291348
## iter 23 value 0.291348
## final value 0.291348
## converged
## initial value 0.289216
## iter 2 value 0.288303
## iter 3 value 0.288117
## iter 4 value 0.287693
## iter 5 value 0.287301
## iter 6 value 0.287027
## iter 7 value 0.286204
## iter 8 value 0.285661
## iter 9 value 0.285037
## iter 10 value 0.284847
## iter 11 value 0.284759
## iter 12 value 0.284710
## iter 13 value 0.284625
## iter 13 value 0.284640
## iter 14 value 0.284567
## iter 15 value 0.284507
## iter 16 value 0.284441
## iter 17 value 0.284052
## iter 18 value 0.283660
## iter 19 value 0.283076
## iter 20 value 0.282198
## iter 21 value 0.280524
## iter 22 value 0.280410
## iter 23 value 0.280284
## iter 24 value 0.280225
## iter 25 value 0.279961
## iter 26 value 0.279924
## iter 26 value 0.280343
## final value 0.279924
## converged
SARTrend2<- sarima(z,2,0,1,2,0,0,12, tol = .00001)
## initial value 0.659165
## iter 2 value 0.468578
## iter 3 value 0.410652
## iter 4 value 0.390392
## iter 5 value 0.389414
## iter 6 value 0.389269
## iter 6 value 0.389267
## iter 6 value 0.389267
## final value 0.389267
## converged
## initial value 0.390395
## iter 2 value 0.390391
## iter 3 value 0.390363
## iter 3 value 0.390362
## iter 3 value 0.390362
## final value 0.390362
## converged
#AIC and AICc of Seasonal arima with (1,0,0)(2,0,2) is smaller than the other test seasonal Arimas.