Question 1
set.seed(8675309)
x = rnorm(500)
ar2 = arima.sim(list( ar=c(.85,.13), ma = c(-0.8, 0.7)),499)
plot(acf(x,lag.max = 50))
plot(ar2, axes=TRUE, xlab="Time")
ACF<-ARMAacf(ar=c(.85,.13),ma = c(-0.8, 0.7), lag.max=50)
plot(ACF, type="h", xlab="lag")
Question 2
x<- read.csv("/Users/YaseenAbdulridha/Downloads/1_10.csv")
# Create time series object
xt<- ts(x)
# Plot
plot.ts(xt)
# Fit random walk models with and without drift. Prefer?
NO_DRIFT<- rwf(x, h=499, drift=FALSE, level=c(80,95), fan=FALSE, lambda=NULL)
DRIFT<- rwf(x, h=499, drift=TRUE, level=c(80,95), fan=FALSE, lambda=NULL)
plot(NO_DRIFT$fitted)
plot(DRIFT$fitted)
# PLOT / ACF
plot(DRIFT$residuals)
DDrift<- DRIFT$residuals[-1,1]
acf(DDrift, lag.max = 50)
# Create Quarterly DUmmies fit the model
Model<- data.frame(xt)
seq
## function (...)
## UseMethod("seq")
## <bytecode: 0x7fe421107628>
## <environment: namespace:base>
Quarter<- c()
while(length(Quarter)<498)
(
for (i in 1:4)
{
Quarter <- c(Quarter,i)
}
)
Model<- cbind(Model, Quarter[-1])
q1<- ifelse(Model[,2] == 1,1,0)
q2<- ifelse(Model[,2] == 2,1,0)
q3<- ifelse(Model[,2] == 3,1,0)
q4<- ifelse(Model[,2] == 4,1,0)
Regression<- lm(xt~ lag(xt,1)+q1+q2+q3)
summary(Regression)
## Warning in summary.lm(Regression): essentially perfect fit: summary may be
## unreliable
##
## Call:
## lm(formula = xt ~ lag(xt, 1) + q1 + q2 + q3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.444e-14 -1.140e-16 5.300e-17 2.620e-16 2.122e-15
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.817e-15 1.575e-16 -2.424e+01 <2e-16 ***
## lag(xt, 1) 1.000e+00 9.624e-18 1.039e+17 <2e-16 ***
## q1 -1.742e-17 2.033e-16 -8.600e-02 0.932
## q2 -6.503e-17 2.031e-16 -3.200e-01 0.749
## q3 -2.834e-16 2.028e-16 -1.397e+00 0.163
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.604e-15 on 494 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 2.707e+33 on 4 and 494 DF, p-value: < 2.2e-16
# test that q1 is zero/ pvalue/ decision
# P value is very large
# Use individual T TEst
#. test Joint hypothesis that coefficeints
#Use F test
#use f test to test for joint significance of extra variables
# Plot residuals and ACF for residuals
plot(Regression$residuals)
acf(Regression$residuals, lag.max = 50)
Question 3
Y<- read.csv("/Users/YaseenAbdulridha/Downloads/1_11.csv")
#Create Time series object
Yt<- ts(Y)
# Plot Data and Acf
plot.ts(Yt)
acf(Yt, lag.max = 50)
# Lag Plot
lag1.plot(Yt, max.lag = 6)
# Fit Arma models
fit<- arima(Yt, order=c(1,0,0))
fit_1<- arima(Yt, order=c(2,0,0))
fit_2<- arima(Yt, order=c(3,0,0))
fit_3<- arima(Yt, order=c(4,0,0))
fit_4<- arima(Yt, order=c(5,0,0))
fit_5<- arima(Yt, order=c(6,0,0))
# Which do you prefer? Why?
ACF2<- ARMAacf(ar = fit_3$coef, lag.max = 50)
plot(ACF2, type = "h")
Question 4
# Read data and construct time series
Z<- read.csv("/Users/YaseenAbdulridha/Downloads/1_13.csv")
Zt<- ts(Z)
# Plot
plot.ts(Zt)
# Kernel smoother
plot(Zt)
lines( ksmooth( time(Zt), Zt, "normal", bandwidth=12), lwd=2, col=3 )
lines( ksmooth( time(Zt), Zt, "normal", bandwidth=20), lwd=2, col=3 )
lines( ksmooth( time(Zt), Zt, "normal", bandwidth=40), lwd=2, col=4 )
lines( ksmooth( time(Zt), Zt, "normal", bandwidth=30), lwd=2, col=5 )
# Lowess Smoother
plot(Zt)
lines(lowess(Zt, f=0.10), lwd=2, col=4)
lines(lowess(Zt, f=0.20), lwd=2, col=2)
lines(lowess(Zt, f=0.15), lwd=2, col=3)
# Periodigram
x<- periodogram(Zt)
#top Frequency can be found when maximizing one column and looking at the other value for spec/freq
Frequency<- data.frame(x$spec, x$freq)
Frequency
## x.spec x.freq
## 1 0.20255824 0.003333333
## 2 0.25452626 0.006666667
## 3 0.26854215 0.010000000
## 4 5.37018603 0.013333333
## 5 0.99040423 0.016666667
## 6 164.33505092 0.020000000
## 7 0.62759237 0.023333333
## 8 0.03598642 0.026666667
## 9 2.44591867 0.030000000
## 10 0.26165310 0.033333333
## 11 0.65127321 0.036666667
## 12 0.07980522 0.040000000
## 13 6.56363554 0.043333333
## 14 0.70915077 0.046666667
## 15 6.01736680 0.050000000
## 16 0.25298745 0.053333333
## 17 0.46450510 0.056666667
## 18 1.26491137 0.060000000
## 19 3.50172438 0.063333333
## 20 117.74043331 0.066666667
## 21 1.61139230 0.070000000
## 22 2.02382388 0.073333333
## 23 0.42356794 0.076666667
## 24 3.34268052 0.080000000
## 25 1.72408596 0.083333333
## 26 0.06779503 0.086666667
## 27 3.63017852 0.090000000
## 28 11.14589629 0.093333333
## 29 0.60438176 0.096666667
## 30 3.71913910 0.100000000
## 31 0.21259289 0.103333333
## 32 0.76574870 0.106666667
## 33 0.49489378 0.110000000
## 34 1.06280948 0.113333333
## 35 1.94527502 0.116666667
## 36 0.47986371 0.120000000
## 37 1.25834316 0.123333333
## 38 0.16977255 0.126666667
## 39 1.11108710 0.130000000
## 40 3.28380718 0.133333333
## 41 0.50672785 0.136666667
## 42 1.00780606 0.140000000
## 43 2.43465741 0.143333333
## 44 0.81215826 0.146666667
## 45 1.37706132 0.150000000
## 46 2.59975056 0.153333333
## 47 1.40968066 0.156666667
## 48 3.07790763 0.160000000
## 49 1.75884484 0.163333333
## 50 1.03283235 0.166666667
## 51 0.63514440 0.170000000
## 52 1.49140891 0.173333333
## 53 0.63101256 0.176666667
## 54 4.06429451 0.180000000
## 55 0.03391974 0.183333333
## 56 1.55927815 0.186666667
## 57 0.04504360 0.190000000
## 58 5.98253920 0.193333333
## 59 3.15387145 0.196666667
## 60 0.17881130 0.200000000
## 61 3.51127032 0.203333333
## 62 1.93446046 0.206666667
## 63 3.56341457 0.210000000
## 64 0.89570982 0.213333333
## 65 0.97135822 0.216666667
## 66 1.89606101 0.220000000
## 67 1.53777254 0.223333333
## 68 0.33044328 0.226666667
## 69 4.94103395 0.230000000
## 70 2.53213370 0.233333333
## 71 0.09987796 0.236666667
## 72 4.12093376 0.240000000
## 73 0.77499529 0.243333333
## 74 0.20244869 0.246666667
## 75 2.25339012 0.250000000
## 76 1.36813265 0.253333333
## 77 1.26078132 0.256666667
## 78 1.04100097 0.260000000
## 79 0.07764672 0.263333333
## 80 0.41189295 0.266666667
## 81 0.22492400 0.270000000
## 82 0.88580828 0.273333333
## 83 0.07159542 0.276666667
## 84 5.82667658 0.280000000
## 85 0.85726516 0.283333333
## 86 0.27148612 0.286666667
## 87 1.49870107 0.290000000
## 88 0.13149972 0.293333333
## 89 7.39686358 0.296666667
## 90 2.33163056 0.300000000
## 91 1.12054066 0.303333333
## 92 1.28751632 0.306666667
## 93 0.50346745 0.310000000
## 94 1.59956013 0.313333333
## 95 2.21993547 0.316666667
## 96 2.44850495 0.320000000
## 97 6.06709875 0.323333333
## 98 0.19215809 0.326666667
## 99 1.35701898 0.330000000
## 100 1.92156124 0.333333333
## 101 1.86588108 0.336666667
## 102 0.72632262 0.340000000
## 103 3.76824330 0.343333333
## 104 2.75099041 0.346666667
## 105 2.29659133 0.350000000
## 106 3.15283634 0.353333333
## 107 6.65848092 0.356666667
## 108 1.99952470 0.360000000
## 109 0.21221501 0.363333333
## 110 2.76663559 0.366666667
## 111 1.30448941 0.370000000
## 112 2.95594247 0.373333333
## 113 2.50903996 0.376666667
## 114 4.15324331 0.380000000
## 115 1.25836633 0.383333333
## 116 6.40866057 0.386666667
## 117 0.15061650 0.390000000
## 118 2.11115528 0.393333333
## 119 0.65894932 0.396666667
## 120 1.29576178 0.400000000
## 121 1.97039493 0.403333333
## 122 0.37163845 0.406666667
## 123 2.62904471 0.410000000
## 124 2.18195086 0.413333333
## 125 1.43819830 0.416666667
## 126 0.67075843 0.420000000
## 127 0.51379734 0.423333333
## 128 3.14035020 0.426666667
## 129 3.02232327 0.430000000
## 130 4.74266612 0.433333333
## 131 0.95566846 0.436666667
## 132 5.28159926 0.440000000
## 133 0.44949748 0.443333333
## 134 3.02726337 0.446666667
## 135 3.21092214 0.450000000
## 136 0.79687780 0.453333333
## 137 0.32895795 0.456666667
## 138 4.62653869 0.460000000
## 139 1.64201100 0.463333333
## 140 2.71396894 0.466666667
## 141 1.94250310 0.470000000
## 142 5.86290057 0.473333333
## 143 6.06907042 0.476666667
## 144 1.13212401 0.480000000
## 145 3.62671066 0.483333333
## 146 1.40205783 0.486666667
## 147 3.83018383 0.490000000
## 148 2.11962808 0.493333333
## 149 3.51395091 0.496666667
## 150 0.08091168 0.500000000