S3800545 Assignment 2
Shubham Chougule
Assignment 2 Forecasting
TASK 1
In this task we will compare all Distributed lag models,Dynamic lag models and exponential smoothing and corresponding state-space models to check the best fitting model to the series and best fit is checked using the accuracy(MASE) of model and then make the forecast of next 2 years.
Importing dataset for Task 1
Solar_data <- read_csv("/Users/shubhamchougule/Downloads/data1 (1).csv")## Parsed with column specification:
## cols(
## solar = col_double(),
## ppt = col_double()
## )#Checking the dataset
head(Solar_data)## # A tibble: 6 x 2
## solar ppt
## <dbl> <dbl>
## 1 5.05 1.33
## 2 6.42 0.921
## 3 10.8 0.947
## 4 16.9 0.615
## 5 24.0 0.544
## 6 26.3 0.703#Checking the class of dataset
class(Solar_data)## [1] "spec_tbl_df" "tbl_df" "tbl" "data.frame"Changing the dataset to time series
#Converting dataset to time series starting from Jan 1960
Solar<-ts(Solar_data$solar,start = 1960,frequency = 12)
PPT<-ts(Solar_data$ppt,start = 1960,frequency = 12)Checking the existence of nonstationarity in series
For making the prediction we need to make the dataset stationary so that each point should be independent of one another. And stationarity of a series is checked using ACF and PACF plots also by conducting an ADF test. Non-stationarity can be removed by applying transformation and differencing to the series. Series can be well understood when it is decomposed using a different techniques.
Plot the time series dataset
#Plotting Solar series and check
plot(Solar,ylab="Monthly Average Solar Radiations",xlab="YEAR",main="Solar Radiations")
- In solar series we do not see any trend.
- We see seasonality in the series.
- No intervention point is observed.
- Moving average can be seen.
- Changing varince can be seen.
#Plotting precipitation series and check
plot(PPT,ylab="Monthly precipitation series measured",xlab="YEAR",main="Precipitation Series")
- In precipitation series we do not see any trend.
- We see seasonality in the series.
- No intervention point is observed.
- Moving average can be seen.
- Changing varince can be seen.
ACF and PACF plots
#ACF and PACF plots
par(mfrow=c(2,2))
acf(Solar,main="ACF of Horizontal solar radiation")
acf(PPT,main="ACF of Precipitation series")
pacf(Solar,main="PACF of Horizontal solar radiation")
pacf(PPT,main="PACF of Precipitation series")
- ACF plot of Solar radiation shows seasonal pattern and PACF plot shows significant lags.
- ACF plot of Precipitation series shows seasonal pattern and PACF plot shows significant lags.
ADF test
Non-stationary can be check by adf test.
Assumptions are as follows
NULL hypothesis:
H0:Non-stationary
Alternative hypothesis:
HA:stationary
#ADF test for Solar Series
adf.test(Solar)## Warning in adf.test(Solar): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: Solar
## Dickey-Fuller = -4.7661, Lag order = 8, p-value = 0.01
## alternative hypothesis: stationaryAugmented Dickey-Fuller Test states that solar series is stationary as we get p-value less than 5% hence rejecting null hypothesis.
#ADF test for PPT series
adf.test(PPT)## Warning in adf.test(PPT): p-value smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: PPT
## Dickey-Fuller = -6.7438, Lag order = 8, p-value = 0.01
## alternative hypothesis: stationaryAugmented Dickey-Fuller Test states that precipitation series is stationary as we get p-value less than 5% hence rejecting null hypothesis.
Decomposition
The individual effect of the existing components and past effects can be analyzed by using the decomposition technique. For decomposition, we will use the X12-ARIMA decomposition method as we will check STL decomposition.
#Decomposition function giving output decomposed series, SI ratio plot and STL plot.
decompose <- function(x){
DECOM = x12(x)
plot(DECOM, sa=TRUE , trend=TRUE , forecast = TRUE)
plotSeasFac(DECOM)
stldec=stl(x, t.window=15, s.window="periodic", robust=TRUE)
plot(stldec)
}
#decomposition
decompose(Solar)
We can observe seasonally adjusted series does not follows original series in the same pattern. which states that there is seasonal effect affecting the series. The trend looks following the original series hence can say that trend has no effect on the series. We can also see the forecast.
We See that seasonal factors are clustered around the mean for most of the months.
STL first graph shows the original series, the seasonal pattern are same says seasonality. The trend graph shows no any upward or downward fall and hence do not see a trend.
Distributed Lag Model
To check amoung the Distributed Lag Model which best fits the data based on the AIC and MASE scores. The model we are going to fit are ;
- Finite Distributed-Lag Model,
- Polynomial Distributed Lags
- Koyck Distributed Lag Model
- Autoregressive Distributed Lag Mode
Finite Distributed-Lag Model
#selection of "q" value on basis of AIC and BIC for fitting of DLM model
for(i in 1:10){
model1 = dlm( x = as.vector(PPT) , y = as.vector(Solar), q = i )
cat("q = ", i, "AIC = ", AIC(model1$model), "BIC = ", BIC(model1$model),"\n")
}## q = 1 AIC = 4728.713 BIC = 4746.676
## q = 2 AIC = 4712.649 BIC = 4735.095
## q = 3 AIC = 4688.551 BIC = 4715.478
## q = 4 AIC = 4663.6 BIC = 4695.003
## q = 5 AIC = 4644.622 BIC = 4680.499
## q = 6 AIC = 4637.489 BIC = 4677.837
## q = 7 AIC = 4632.716 BIC = 4677.532
## q = 8 AIC = 4625.986 BIC = 4675.267
## q = 9 AIC = 4615.084 BIC = 4668.827
## q = 10 AIC = 4602.658 BIC = 4660.858Lowest value of AIC and BIC is observed when q=10 hence using this vales for DLM model.
model1<- dlm(x = as.vector(PPT) , y = as.vector(Solar), q = 10)
summary(model1)##
## Call:
## lm(formula = model.formula, data = design)
##
## Residuals:
## Min 1Q Median 3Q Max
## -18.9353 -5.4124 -0.7911 4.0184 30.8900
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 19.0105 1.0942 17.374 < 2e-16 ***
## x.t -7.3843 1.8995 -3.887 0.000112 ***
## x.1 -0.4763 2.5395 -0.188 0.851288
## x.2 -0.1324 2.5734 -0.051 0.958980
## x.3 1.7902 2.5781 0.694 0.487691
## x.4 1.9686 2.5808 0.763 0.445877
## x.5 3.4928 2.5807 1.353 0.176402
## x.6 0.5243 2.5787 0.203 0.838943
## x.7 1.6762 2.5797 0.650 0.516088
## x.8 0.9282 2.5673 0.362 0.717817
## x.9 0.3754 2.5338 0.148 0.882272
## x.10 -5.3798 1.8760 -2.868 0.004272 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.256 on 638 degrees of freedom
## Multiple R-squared: 0.3081, Adjusted R-squared: 0.2962
## F-statistic: 25.82 on 11 and 638 DF, p-value: < 2.2e-16
##
## AIC and BIC values for the model:
## AIC BIC
## 1 4602.658 4660.858checkresiduals(model1$model)
##
## Breusch-Godfrey test for serial correlation of order up to 15
##
## data: Residuals
## LM test = 588.43, df = 15, p-value < 2.2e-16- The summary express that one of the lag is significant.
- Adj R-squared value is very low
- In Breusch-Godfrey test we see value is less than 5% states that there is a serial correlation. * From the ACF plot we conclude that residuals are highly significant.Hence violate general assumptions.
vif(model1$model)#states multicollinerity## x.t x.1 x.2 x.3 x.4 x.5 x.6 x.7
## 4.244594 7.665259 7.910115 7.952633 7.957841 7.941836 7.911213 7.901898
## x.8 x.9 x.10
## 7.847965 7.653512 4.228221We see all the values are less than 10 hence there does not exist multicollinerity.
Polynomial Distributed Lag Model
model2 = polyDlm(x = as.vector(PPT) , y = as.vector(Solar) , q = 2 , k = 2 , show.beta = TRUE)## Estimates and t-tests for beta coefficients:
## Estimate Std. Error t value P(>|t|)
## beta.0 -12.90 1.76 -7.37 5.32e-13
## beta.1 -2.59 2.56 -1.01 3.12e-01
## beta.2 5.83 1.75 3.33 9.06e-04summary(model2)##
## Call:
## "Y ~ (Intercept) + X.t"
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.481 -5.773 -0.921 4.576 31.726
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 22.3441 0.6374 35.054 < 2e-16 ***
## z.t0 -12.9460 1.7577 -7.366 5.33e-13 ***
## z.t1 11.3215 8.0564 1.405 0.160
## z.t2 -0.9659 4.0021 -0.241 0.809
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.65 on 654 degrees of freedom
## Multiple R-squared: 0.2253, Adjusted R-squared: 0.2217
## F-statistic: 63.39 on 3 and 654 DF, p-value: < 2.2e-16checkresiduals(model2$model)
##
## Breusch-Godfrey test for serial correlation of order up to 10
##
## data: Residuals
## LM test = 577.49, df = 10, p-value < 2.2e-16- The summary express that one of the lag is significant.
- Adj R-squared value is very low
- In Breusch-Godfrey test we see value is less than 5% states that there is a serial correlation. * From the ACF plot we conclude that residuals are highly significant.Hence violate general assumptions.
vif(model2$model)## z.t0 z.t1 z.t2
## 23.71744 583.13654 412.20072Above all the values are higher than 10 and there exist of multicollinerity.
Koyck Distributed Lag Model
model3 = koyckDlm(x = as.vector(PPT) , y = as.vector(Solar))
summary(model3,diagnostics=TRUE)##
## Call:
## "Y ~ (Intercept) + Y.1 + X.t"
##
## Residuals:
## Min 1Q Median 3Q Max
## -13.0926 -3.5961 0.3176 3.6103 14.8399
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.23925 0.76549 -2.925 0.00356 **
## Y.1 0.98546 0.02424 40.650 < 2e-16 ***
## X.t 5.34684 0.84383 6.336 4.37e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.814 on 656 degrees of freedom
## Multiple R-Squared: 0.7598, Adjusted R-squared: 0.7591
## Wald test: 1104 on 2 and 656 DF, p-value: < 2.2e-16
##
## Diagnostic tests:
## df1 df2 statistic p-value
## Weak instruments 1 656 710.7209 1.191744e-106
## Wu-Hausman 1 655 146.8017 1.248856e-30
##
## alpha beta phi
## Geometric coefficients: -154.0203 5.346844 0.9854613checkresiduals(model3$model)
- The summary express that all of the lags are significant.
- Adj R-squared value also good.
- In Breusch-Godfrey test we see value is less than 5% states that there is a serial correlation. * From the ACF plot we conclude that residuals are highly significant.Hence violate general assumptions.
We will check the multicollinerity.
vif(model3$model)## Y.1 X.t
## 1.605001 1.605001Here we have values less than 10 confirming of no multicollinerity.
Autorregressive Distributed Lag Model
#Checking p and q vales on basis fo AIC and BIC
for (i in 1:10){
for(j in 1:10){
model4 = ardlDlm(x = as.vector(PPT) , y = as.vector(Solar), p = i , q = j )
cat("p = ", i, "q = ", j, "AIC = ", AIC(model4$model), "BIC = ", BIC(model4$model),"\n")
}
}## p = 1 q = 1 AIC = 3712.311 BIC = 3734.765
## p = 1 q = 2 AIC = 3239.416 BIC = 3266.352
## p = 1 q = 3 AIC = 3143.522 BIC = 3174.936
## p = 1 q = 4 AIC = 3138.399 BIC = 3174.288
## p = 1 q = 5 AIC = 3100.283 BIC = 3140.644
## p = 1 q = 6 AIC = 3033.2 BIC = 3078.031
## p = 1 q = 7 AIC = 3014.566 BIC = 3063.864
## p = 1 q = 8 AIC = 3011.894 BIC = 3065.654
## p = 1 q = 9 AIC = 3006.442 BIC = 3064.663
## p = 1 q = 10 AIC = 2996.457 BIC = 3059.135
## p = 2 q = 1 AIC = 3639.223 BIC = 3666.159
## p = 2 q = 2 AIC = 3229.051 BIC = 3260.476
## p = 2 q = 3 AIC = 3137.634 BIC = 3173.535
## p = 2 q = 4 AIC = 3132.962 BIC = 3173.337
## p = 2 q = 5 AIC = 3097.288 BIC = 3142.134
## p = 2 q = 6 AIC = 3031.898 BIC = 3081.212
## p = 2 q = 7 AIC = 3013.616 BIC = 3067.395
## p = 2 q = 8 AIC = 3010.8 BIC = 3069.04
## p = 2 q = 9 AIC = 3005.437 BIC = 3068.136
## p = 2 q = 10 AIC = 2995.768 BIC = 3062.922
## p = 3 q = 1 AIC = 3608.793 BIC = 3640.207
## p = 3 q = 2 AIC = 3226.623 BIC = 3262.524
## p = 3 q = 3 AIC = 3139.409 BIC = 3179.798
## p = 3 q = 4 AIC = 3134.777 BIC = 3179.638
## p = 3 q = 5 AIC = 3098.808 BIC = 3148.139
## p = 3 q = 6 AIC = 3032.418 BIC = 3086.216
## p = 3 q = 7 AIC = 3013.636 BIC = 3071.897
## p = 3 q = 8 AIC = 3010.775 BIC = 3073.495
## p = 3 q = 9 AIC = 3005.95 BIC = 3073.127
## p = 3 q = 10 AIC = 2996.59 BIC = 3068.221
## p = 4 q = 1 AIC = 3602.664 BIC = 3638.553
## p = 4 q = 2 AIC = 3224.285 BIC = 3264.66
## p = 4 q = 3 AIC = 3131.289 BIC = 3176.15
## p = 4 q = 4 AIC = 3131.424 BIC = 3180.772
## p = 4 q = 5 AIC = 3096.024 BIC = 3149.839
## p = 4 q = 6 AIC = 3027.07 BIC = 3085.35
## p = 4 q = 7 AIC = 3006.43 BIC = 3069.173
## p = 4 q = 8 AIC = 3003.603 BIC = 3070.804
## p = 4 q = 9 AIC = 2998.61 BIC = 3070.266
## p = 4 q = 10 AIC = 2986.78 BIC = 3062.889
## p = 5 q = 1 AIC = 3599.402 BIC = 3639.764
## p = 5 q = 2 AIC = 3221.853 BIC = 3266.699
## p = 5 q = 3 AIC = 3127.103 BIC = 3176.434
## p = 5 q = 4 AIC = 3127.868 BIC = 3181.684
## p = 5 q = 5 AIC = 3097.877 BIC = 3156.177
## p = 5 q = 6 AIC = 3029.06 BIC = 3091.824
## p = 5 q = 7 AIC = 3008.224 BIC = 3075.448
## p = 5 q = 8 AIC = 3005.42 BIC = 3077.101
## p = 5 q = 9 AIC = 3000.385 BIC = 3076.52
## p = 5 q = 10 AIC = 2988.572 BIC = 3069.158
## p = 6 q = 1 AIC = 3586.116 BIC = 3630.948
## p = 6 q = 2 AIC = 3214.331 BIC = 3263.645
## p = 6 q = 3 AIC = 3119.154 BIC = 3172.952
## p = 6 q = 4 AIC = 3120.48 BIC = 3178.76
## p = 6 q = 5 AIC = 3094.149 BIC = 3156.912
## p = 6 q = 6 AIC = 3030.976 BIC = 3098.223
## p = 6 q = 7 AIC = 3010.203 BIC = 3081.908
## p = 6 q = 8 AIC = 3007.386 BIC = 3083.546
## p = 6 q = 9 AIC = 3002.341 BIC = 3082.954
## p = 6 q = 10 AIC = 2990.569 BIC = 3075.631
## p = 7 q = 1 AIC = 3575.799 BIC = 3625.097
## p = 7 q = 2 AIC = 3212.372 BIC = 3266.151
## p = 7 q = 3 AIC = 3117.291 BIC = 3175.551
## p = 7 q = 4 AIC = 3118.61 BIC = 3181.352
## p = 7 q = 5 AIC = 3091.439 BIC = 3158.663
## p = 7 q = 6 AIC = 3022.352 BIC = 3094.058
## p = 7 q = 7 AIC = 3002.413 BIC = 3078.6
## p = 7 q = 8 AIC = 2998.638 BIC = 3079.278
## p = 7 q = 9 AIC = 2994.189 BIC = 3079.28
## p = 7 q = 10 AIC = 2980.152 BIC = 3069.691
## p = 8 q = 1 AIC = 3538.751 BIC = 3592.512
## p = 8 q = 2 AIC = 3180.78 BIC = 3239.02
## p = 8 q = 3 AIC = 3093.999 BIC = 3156.719
## p = 8 q = 4 AIC = 3095.328 BIC = 3162.529
## p = 8 q = 5 AIC = 3068.742 BIC = 3140.423
## p = 8 q = 6 AIC = 3006.28 BIC = 3082.441
## p = 8 q = 7 AIC = 2991.824 BIC = 3072.465
## p = 8 q = 8 AIC = 2993.801 BIC = 3078.922
## p = 8 q = 9 AIC = 2990.093 BIC = 3079.664
## p = 8 q = 10 AIC = 2977.316 BIC = 3071.333
## p = 9 q = 1 AIC = 3503.723 BIC = 3561.943
## p = 9 q = 2 AIC = 3168.825 BIC = 3231.525
## p = 9 q = 3 AIC = 3074.896 BIC = 3142.074
## p = 9 q = 4 AIC = 3075.884 BIC = 3147.54
## p = 9 q = 5 AIC = 3048.439 BIC = 3124.574
## p = 9 q = 6 AIC = 2983.773 BIC = 3064.386
## p = 9 q = 7 AIC = 2972.197 BIC = 3057.289
## p = 9 q = 8 AIC = 2973.907 BIC = 3063.477
## p = 9 q = 9 AIC = 2975.583 BIC = 3069.631
## p = 9 q = 10 AIC = 2963.123 BIC = 3061.616
## p = 10 q = 1 AIC = 3486.25 BIC = 3548.927
## p = 10 q = 2 AIC = 3165.653 BIC = 3232.807
## p = 10 q = 3 AIC = 3072.759 BIC = 3144.39
## p = 10 q = 4 AIC = 3073.91 BIC = 3150.018
## p = 10 q = 5 AIC = 3046.099 BIC = 3126.684
## p = 10 q = 6 AIC = 2980.553 BIC = 3065.615
## p = 10 q = 7 AIC = 2968.654 BIC = 3058.193
## p = 10 q = 8 AIC = 2970.239 BIC = 3064.256
## p = 10 q = 9 AIC = 2972.143 BIC = 3070.636
## p = 10 q = 10 AIC = 2962.601 BIC = 3065.571We choose p=1 and q=5 because for this p and q values AIC and BIC are the least using same in the model.
model4 = ardlDlm(x = as.vector(PPT) , y = as.vector(Solar), p = 9 , q = 7 )
summary(model4)##
## Time series regression with "ts" data:
## Start = 10, End = 660
##
## Call:
## dynlm(formula = as.formula(model.text), data = data, start = 1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -13.2222 -1.2151 -0.1015 0.9864 18.0325
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.54522 0.47727 3.238 0.001268 **
## X.t 0.03524 0.53121 0.066 0.947126
## X.1 0.52602 0.72140 0.729 0.466169
## X.2 0.93915 0.72929 1.288 0.198300
## X.3 0.88547 0.73212 1.209 0.226938
## X.4 -1.07437 0.73290 -1.466 0.143168
## X.5 0.05236 0.73419 0.071 0.943168
## X.6 -2.22446 0.73342 -3.033 0.002520 **
## X.7 2.42465 0.73506 3.299 0.001026 **
## X.8 0.79521 0.73160 1.087 0.277480
## X.9 -2.14299 0.52580 -4.076 5.17e-05 ***
## Y.1 1.14197 0.03892 29.345 < 2e-16 ***
## Y.2 0.09977 0.05946 1.678 0.093864 .
## Y.3 -0.24990 0.05843 -4.277 2.19e-05 ***
## Y.4 -0.18084 0.05849 -3.092 0.002077 **
## Y.5 -0.18410 0.05821 -3.163 0.001637 **
## Y.6 0.13933 0.05869 2.374 0.017889 *
## Y.7 0.14187 0.03884 3.652 0.000281 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.337 on 633 degrees of freedom
## Multiple R-squared: 0.9451, Adjusted R-squared: 0.9436
## F-statistic: 640.5 on 17 and 633 DF, p-value: < 2.2e-16checkresiduals(model4$model)
##
## Breusch-Godfrey test for serial correlation of order up to 21
##
## data: Residuals
## LM test = 86.329, df = 21, p-value = 6.885e-10- The summary express that almost all of the lags are significant.
- Adj R-squared value is very high.
- In Breusch-Godfrey test we see value is less than 5% states that there is a serial correlation. * From the ACF plot we conclude that residuals are not highly significant.Hence see histogram is normalized.
We will check the multicollinerity.
vif(model4$model)## X.t L(X.t, 1) L(X.t, 2) L(X.t, 3) L(X.t, 4) L(X.t, 5) L(X.t, 6) L(X.t, 7)
## 4.192098 7.760879 7.946151 8.013082 8.016774 8.024377 7.990709 8.031374
## L(X.t, 8) L(X.t, 9) L(y.t, 1) L(y.t, 2) L(y.t, 3) L(y.t, 4) L(y.t, 5) L(y.t, 6)
## 7.975689 4.151762 17.405503 40.570447 39.180158 39.291729 38.935939 39.584029
## L(y.t, 7)
## 17.352708Here we have some values are less than 10 some are greater than 10 confirming of some existance of multicollinerity in the series.
Comparision of DLM models
DLM models based on the MASE value. Lower the MASE value better the forecast. From the values we see that Autorregressive Distributed Lag Model has the lowest MASE value and hence could be used for prediction.
mas =MASE(model1, model2, model3, model4)
mas## n MASE
## model1 650 1.5779955
## model2 658 1.6759667
## model3 659 1.0324829
## model4 651 0.4038376Dynamic Lag Models
Dynamic Lag models are created using Y lag, step function,Intevention point, trend and seasonality. Best model is selected by adding and removing different points. Best model with lowest AIC and accuracy(MASE) values will be selected.
From the observations we can see that * Almost all of the models lags are significant. * Have high Adj. R-square value. * Most of models AIC lags are significant. * Histogram is normalized.
#Considerations for Dynlm models
Y.t=Solar
X.t=PPT#Model 1
dynlm1 =dynlm(Y.t~ L(Y.t , k = 1 )+ trend(Y.t)+ season(Y.t))
summary(dynlm1)##
## Time series regression with "ts" data:
## Start = 1960(2), End = 2014(12)
##
## Call:
## dynlm(formula = Y.t ~ L(Y.t, k = 1) + trend(Y.t) + season(Y.t))
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.8501 -0.9906 -0.1119 0.8325 16.8276
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.2105921 0.3640843 3.325 0.000934 ***
## L(Y.t, k = 1) 0.9323820 0.0142258 65.542 < 2e-16 ***
## trend(Y.t) -0.0006813 0.0056406 -0.121 0.903905
## season(Y.t)Feb 1.9262673 0.4397952 4.380 1.39e-05 ***
## season(Y.t)Mar 4.8864291 0.4420720 11.053 < 2e-16 ***
## season(Y.t)Apr 3.7894322 0.4563152 8.304 5.90e-16 ***
## season(Y.t)May 5.1503064 0.4741868 10.861 < 2e-16 ***
## season(Y.t)Jun 3.3685316 0.5051379 6.669 5.56e-11 ***
## season(Y.t)Jul 1.6296826 0.5265113 3.095 0.002052 **
## season(Y.t)Aug -2.1600321 0.5340317 -4.045 5.87e-05 ***
## season(Y.t)Sep -4.7002552 0.5116903 -9.186 < 2e-16 ***
## season(Y.t)Oct -5.8121501 0.4778130 -12.164 < 2e-16 ***
## season(Y.t)Nov -5.0968011 0.4512694 -11.294 < 2e-16 ***
## season(Y.t)Dec -2.8325126 0.4408651 -6.425 2.56e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.295 on 645 degrees of freedom
## Multiple R-squared: 0.9463, Adjusted R-squared: 0.9452
## F-statistic: 874.6 on 13 and 645 DF, p-value: < 2.2e-16checkresiduals(dynlm1)
##
## Breusch-Godfrey test for serial correlation of order up to 24
##
## data: Residuals
## LM test = 132.86, df = 24, p-value < 2.2e-16dynlm2=dynlm(Y.t~ L(Y.t , k = 2)+X.t+ trend(Y.t)+ season(Y.t))
summary(dynlm2)##
## Time series regression with "ts" data:
## Start = 1960(3), End = 2014(12)
##
## Call:
## dynlm(formula = Y.t ~ L(Y.t, k = 2) + X.t + trend(Y.t) + season(Y.t))
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.1731 -1.5067 -0.2347 1.2121 16.5670
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.182851 0.660844 0.277 0.78210
## L(Y.t, k = 2) 0.840701 0.021317 39.438 < 2e-16 ***
## X.t -0.307827 0.497183 -0.619 0.53604
## trend(Y.t) -0.001325 0.008470 -0.156 0.87575
## season(Y.t)Feb 4.511375 0.664449 6.790 2.56e-11 ***
## season(Y.t)Mar 9.252790 0.661321 13.991 < 2e-16 ***
## season(Y.t)Apr 10.985633 0.660987 16.620 < 2e-16 ***
## season(Y.t)May 11.444854 0.679797 16.836 < 2e-16 ***
## season(Y.t)Jun 10.976631 0.724089 15.159 < 2e-16 ***
## season(Y.t)Jul 7.679590 0.781420 9.828 < 2e-16 ***
## season(Y.t)Aug 2.343898 0.811719 2.888 0.00401 **
## season(Y.t)Sep -3.679474 0.809213 -4.547 6.50e-06 ***
## season(Y.t)Oct -7.154745 0.748818 -9.555 < 2e-16 ***
## season(Y.t)Nov -7.569350 0.697556 -10.851 < 2e-16 ***
## season(Y.t)Dec -4.765167 0.668226 -7.131 2.69e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.435 on 643 degrees of freedom
## Multiple R-squared: 0.8799, Adjusted R-squared: 0.8772
## F-statistic: 336.4 on 14 and 643 DF, p-value: < 2.2e-16checkresiduals(dynlm2)
##
## Breusch-Godfrey test for serial correlation of order up to 24
##
## data: Residuals
## LM test = 421.48, df = 24, p-value < 2.2e-16dynlm3 =dynlm(Y.t~ L(Y.t , k = 1 )+X.t+ season(Y.t))
summary(dynlm3)##
## Time series regression with "ts" data:
## Start = 1960(2), End = 2014(12)
##
## Call:
## dynlm(formula = Y.t ~ L(Y.t, k = 1) + X.t + season(Y.t))
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.7834 -1.0052 -0.0964 0.8024 16.8758
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.36184 0.40650 3.350 0.000855 ***
## L(Y.t, k = 1) 0.93192 0.01423 65.471 < 2e-16 ***
## X.t -0.23642 0.33142 -0.713 0.475877
## season(Y.t)Feb 1.90792 0.44040 4.332 1.71e-05 ***
## season(Y.t)Mar 4.86290 0.44315 10.973 < 2e-16 ***
## season(Y.t)Apr 3.76530 0.45742 8.232 1.02e-15 ***
## season(Y.t)May 5.10305 0.47866 10.661 < 2e-16 ***
## season(Y.t)Jun 3.27202 0.52284 6.258 7.10e-10 ***
## season(Y.t)Jul 1.50367 0.55526 2.708 0.006947 **
## season(Y.t)Aug -2.29117 0.56467 -4.058 5.57e-05 ***
## season(Y.t)Sep -4.81495 0.53621 -8.980 < 2e-16 ***
## season(Y.t)Oct -5.86148 0.48261 -12.145 < 2e-16 ***
## season(Y.t)Nov -5.10509 0.45124 -11.313 < 2e-16 ***
## season(Y.t)Dec -2.80858 0.44199 -6.354 3.96e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.294 on 645 degrees of freedom
## Multiple R-squared: 0.9464, Adjusted R-squared: 0.9453
## F-statistic: 875.3 on 13 and 645 DF, p-value: < 2.2e-16checkresiduals(dynlm3)
##
## Breusch-Godfrey test for serial correlation of order up to 24
##
## data: Residuals
## LM test = 134.42, df = 24, p-value < 2.2e-16dynlm4=dynlm(Y.t~ L(Y.t , k = 1 )+X.t+trend(Y.t))
summary(dynlm4)##
## Time series regression with "ts" data:
## Start = 1960(2), End = 2014(12)
##
## Call:
## dynlm(formula = Y.t ~ L(Y.t, k = 1) + X.t + trend(Y.t))
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.4925 -3.7427 0.2951 3.4114 16.6215
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.476695 0.634917 3.901 0.000106 ***
## L(Y.t, k = 1) 0.881095 0.020313 43.376 < 2e-16 ***
## X.t -0.572126 0.563644 -1.015 0.310458
## trend(Y.t) -0.003331 0.010958 -0.304 0.761238
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.456 on 655 degrees of freedom
## Multiple R-squared: 0.7945, Adjusted R-squared: 0.7936
## F-statistic: 844.3 on 3 and 655 DF, p-value: < 2.2e-16checkresiduals(dynlm4)
##
## Breusch-Godfrey test for serial correlation of order up to 24
##
## data: Residuals
## LM test = 493.18, df = 24, p-value < 2.2e-16dynlm5 =dynlm(Y.t~ L(Y.t , k = 1 )+L(Y.t , k = 2) +X.t +season(Y.t))
summary(dynlm5)##
## Time series regression with "ts" data:
## Start = 1960(3), End = 2014(12)
##
## Call:
## dynlm(formula = Y.t ~ L(Y.t, k = 1) + L(Y.t, k = 2) + X.t + season(Y.t))
##
## Residuals:
## Min 1Q Median 3Q Max
## -13.1353 -0.9959 -0.0915 0.8252 17.0059
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.97079 0.41260 4.777 2.21e-06 ***
## L(Y.t, k = 1) 1.13172 0.03849 29.400 < 2e-16 ***
## L(Y.t, k = 2) -0.21447 0.03850 -5.571 3.72e-08 ***
## X.t -0.24527 0.32442 -0.756 0.44990
## season(Y.t)Feb 1.29901 0.44752 2.903 0.00383 **
## season(Y.t)Mar 3.86196 0.46925 8.230 1.04e-15 ***
## season(Y.t)Apr 2.24539 0.52418 4.284 2.12e-05 ***
## season(Y.t)May 3.95247 0.51194 7.721 4.45e-14 ***
## season(Y.t)Jun 1.95757 0.56368 3.473 0.00055 ***
## season(Y.t)Jul 0.68494 0.56305 1.216 0.22425
## season(Y.t)Aug -2.68154 0.55699 -4.814 1.84e-06 ***
## season(Y.t)Sep -4.42079 0.52910 -8.355 4.02e-16 ***
## season(Y.t)Oct -5.03914 0.49435 -10.194 < 2e-16 ***
## season(Y.t)Nov -4.20687 0.46974 -8.956 < 2e-16 ***
## season(Y.t)Dec -2.22125 0.44494 -4.992 7.70e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.244 on 643 degrees of freedom
## Multiple R-squared: 0.9488, Adjusted R-squared: 0.9476
## F-statistic: 850.2 on 14 and 643 DF, p-value: < 2.2e-16checkresiduals(dynlm5)
##
## Breusch-Godfrey test for serial correlation of order up to 24
##
## data: Residuals
## LM test = 109.78, df = 24, p-value = 6.162e-13dynlm6 =dynlm(Y.t~ L(Y.t , k = 1 )+ L(Y.t , k = 2 )+L(Y.t , k = 3 )+X.t +season(Y.t))
summary(dynlm6)##
## Time series regression with "ts" data:
## Start = 1960(4), End = 2014(12)
##
## Call:
## dynlm(formula = Y.t ~ L(Y.t, k = 1) + L(Y.t, k = 2) + L(Y.t,
## k = 3) + X.t + season(Y.t))
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.8910 -0.9667 -0.1398 0.8091 17.1517
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.54381 0.44657 7.936 9.37e-15 ***
## L(Y.t, k = 1) 1.06952 0.03780 28.293 < 2e-16 ***
## L(Y.t, k = 2) 0.11225 0.05653 1.986 0.0475 *
## L(Y.t, k = 3) -0.28856 0.03784 -7.626 8.78e-14 ***
## X.t -0.33681 0.31173 -1.080 0.2803
## season(Y.t)Feb 0.66860 0.43705 1.530 0.1266
## season(Y.t)Mar 2.51605 0.48637 5.173 3.08e-07 ***
## season(Y.t)Apr 0.57076 0.54877 1.040 0.2987
## season(Y.t)May 1.50248 0.58710 2.559 0.0107 *
## season(Y.t)Jun 0.07092 0.59503 0.119 0.9052
## season(Y.t)Jul -1.54179 0.61454 -2.509 0.0124 *
## season(Y.t)Aug -4.34971 0.57777 -7.528 1.75e-13 ***
## season(Y.t)Sep -5.74108 0.53646 -10.702 < 2e-16 ***
## season(Y.t)Oct -5.43685 0.47694 -11.399 < 2e-16 ***
## season(Y.t)Nov -4.08468 0.45070 -9.063 < 2e-16 ***
## season(Y.t)Dec -1.94202 0.42823 -4.535 6.88e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.151 on 641 degrees of freedom
## Multiple R-squared: 0.953, Adjusted R-squared: 0.9519
## F-statistic: 866.3 on 15 and 641 DF, p-value: < 2.2e-16checkresiduals(dynlm6)
##
## Breusch-Godfrey test for serial correlation of order up to 24
##
## data: Residuals
## LM test = 58.344, df = 24, p-value = 0.000109Comparision of Dynamic lag models
Comparision is done based on AIC and MASE value.
aic =AIC(dynlm1, dynlm2, dynlm3, dynlm4, dynlm5,dynlm6)
cbind(Model=c("Dynlm1","Dynlm2","Dynlm3","Dynlm4","Dynlm5","Dynlm6"),
MASE=c(accuracy(dynlm1)[6],accuracy(dynlm2)[6],accuracy(dynlm3)[6],accuracy(dynlm4)[6],
accuracy(dynlm5)[6],accuracy(dynlm6)[6]),aic)## Model MASE df AIC
## dynlm1 Dynlm1 0.3743356 15 2981.018
## dynlm2 Dynlm2 0.5868963 16 3508.219
## dynlm3 Dynlm3 0.3739879 15 2980.514
## dynlm4 Dynlm4 0.9717761 5 3845.485
## dynlm5 Dynlm5 0.3632657 16 2947.658
## dynlm6 Dynlm6 0.3520468 17 2888.991From the resulting values, we can see that dynlm6 has the lowest AIC and MASE values hence dynlm6 is used for forecasting.
Exponential Smoothing Method
In our series we do not see any trend and have a seasonal effect. The holt’s-Winter method will be the best method for our series to deal with the seasonal effects.
Model created using Holt’s-Winter method are:
1.Seasonality=Additive,damped=False 2.Seasonality=Additive,damped=True 3.Seasonality=multiplicative,damped=False 4.Seasonality=multiplicative,damped=True 5.Seasonality=multiplicative,damped=False,Exponential =True
#Holt’s-Winter model 1
hw1 <- hw(Solar,seasonal="additive",h=5*frequency(Solar))
summary(hw1)##
## Forecast method: Holt-Winters' additive method
##
## Model Information:
## Holt-Winters' additive method
##
## Call:
## hw(y = Solar, h = 5 * frequency(Solar), seasonal = "additive")
##
## Smoothing parameters:
## alpha = 0.9993
## beta = 0.0019
## gamma = 1e-04
##
## Initial states:
## l = 11.3306
## b = 0.209
## s = -10.7542 -8.4968 -3.227 3.0768 7.9264 10.9122
## 10.1422 7.3322 2.2353 -1.9528 -7.3626 -9.8316
##
## sigma: 2.3575
##
## AIC AICc BIC
## 5434.708 5435.662 5511.076
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set -0.1179476 2.328736 1.504489 -2.230359 12.68101 0.24716 0.1703355
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Jan 2015 6.127399 3.1061581 9.148641 1.5068095 10.74799
## Feb 2015 8.655602 4.3802222 12.930983 2.1169727 15.19423
## Mar 2015 14.122221 8.8814747 19.362968 6.1071910 22.13725
## Apr 2015 18.366625 12.3095941 24.423656 9.1031957 27.63005
## May 2015 23.522264 16.7439512 30.300578 13.1557290 33.88880
## Jun 2015 26.390991 18.9586817 33.823300 15.0242549 37.75773
## Jul 2015 27.219228 19.1837595 35.254697 14.9300392 39.50842
## Aug 2015 24.290519 15.6920179 32.889020 11.1402463 37.44079
## Sep 2015 19.495874 10.3670365 28.624711 5.5345219 33.45723
## Oct 2015 13.253765 3.6218809 22.885648 -1.4769303 27.98446
## Nov 2015 8.042144 -2.0695730 18.153861 -7.4223926 23.50668
## Dec 2015 5.840110 -4.7313921 16.411612 -10.3276072 22.00783
## Jan 2016 6.819616 -4.1942247 17.833457 -10.0246000 23.66383
## Feb 2016 9.347819 -2.0927719 20.788410 -8.1490551 26.84469
## Mar 2016 14.814438 2.9609172 26.667959 -3.3139578 32.94283
## Apr 2016 19.058842 6.8048110 31.312872 0.3179190 37.79976
## May 2016 24.214481 11.5711780 36.857784 4.8782175 43.55074
## Jun 2016 27.083208 14.0608586 40.105557 7.1672434 46.99917
## Jul 2016 27.911445 14.5194061 41.303485 7.4300887 48.39280
## Aug 2016 24.982736 11.2296053 38.735867 3.9491378 46.01633
## Sep 2016 20.188091 6.0818048 34.294377 -1.3856120 41.76179
## Oct 2016 13.945981 -0.5061083 28.398071 -8.1565824 36.04855
## Nov 2016 8.734361 -6.0566988 23.525420 -13.8866128 31.35533
## Dec 2016 6.532327 -8.5913309 21.655984 -16.5973115 29.66196
## Jan 2017 7.511833 -7.9385282 22.962194 -16.1174555 31.14112
## Feb 2017 10.040036 -5.7313777 25.811450 -14.0802599 34.16033
## Mar 2017 15.506655 -0.5805619 31.593872 -9.0966201 40.10993
## Apr 2017 19.751059 3.3529822 36.149135 -5.3276351 44.82975
## May 2017 24.906698 8.2024281 41.610968 -0.6402783 50.45367
## Jun 2017 27.775425 10.7693726 44.781476 1.7669125 53.78394
## Jul 2017 28.603662 11.3000071 45.907317 2.1400055 55.06732
## Aug 2017 25.674953 8.0776596 43.272246 -1.2377847 52.58769
## Sep 2017 20.880308 2.9931440 38.767471 -6.4757484 48.23636
## Oct 2017 14.638198 -3.5352503 32.811647 -13.1556929 42.43209
## Nov 2017 9.426578 -9.0297392 27.882894 -18.7999231 37.65308
## Dec 2017 7.224543 -11.5113813 25.960468 -21.4295807 35.87867
## Jan 2018 8.204050 -10.8084216 27.216522 -20.8730161 37.28112
## Feb 2018 10.732253 -8.5537326 30.018238 -18.7631166 40.22762
## Mar 2018 16.198872 -3.3577754 35.755519 -13.7104391 46.10818
## Apr 2018 20.443275 0.6186997 40.267851 -9.8757968 50.76235
## May 2018 25.598915 5.5090332 45.688797 -5.1259078 56.32374
## Jun 2018 28.467641 8.1149718 48.820311 -2.6590806 59.59436
## Jul 2018 29.295879 8.6828417 49.908916 -2.2290411 60.82080
## Aug 2018 26.367170 5.4960925 47.238247 -5.5523883 58.28673
## Sep 2018 21.572524 0.4456486 42.699400 -10.7382439 53.88329
## Oct 2018 15.330415 -6.0501007 36.710931 -17.3682620 48.02909
## Nov 2018 10.118794 -11.5132798 31.750868 -22.9646081 43.20220
## Dec 2018 7.916760 -13.9648644 29.798385 -25.5482967 41.38182
## Jan 2019 8.896267 -13.2330166 31.025550 -24.9475516 42.74009
## Feb 2019 11.424470 -10.9505524 33.799492 -22.7951736 45.64411
## Mar 2019 16.891089 -5.7278620 39.510039 -17.7016112 51.48379
## Apr 2019 21.135492 -1.7256366 43.996621 -13.8275871 56.09857
## May 2019 26.291132 3.1895190 49.392744 -9.0397360 61.62200
## Jun 2019 29.159858 5.8194018 52.500314 -6.5362894 64.85601
## Jul 2019 29.988096 6.4103847 53.565807 -6.0709017 66.04709
## Aug 2019 27.059386 3.2459605 50.872812 -9.3601057 63.47888
## Sep 2019 22.264741 -1.7829062 46.312389 -14.5129618 59.04244
## Oct 2019 16.022632 -8.2577885 40.303052 -21.1110666 53.15633
## Nov 2019 10.811011 -13.7007762 35.322798 -26.6765326 48.29855
## Dec 2019 8.608977 -16.1328122 33.350766 -29.2303243 46.44828checkresiduals(hw1)
##
## Ljung-Box test
##
## data: Residuals from Holt-Winters' additive method
## Q* = 205.55, df = 8, p-value < 2.2e-16
##
## Model df: 16. Total lags used: 24#Holt’s-Winter model 2
hw2 <- hw(Solar,seasonal="multiplicative", h=5*frequency(Solar))
summary(hw2)##
## Forecast method: Holt-Winters' multiplicative method
##
## Model Information:
## Holt-Winters' multiplicative method
##
## Call:
## hw(y = Solar, h = 5 * frequency(Solar), seasonal = "multiplicative")
##
## Smoothing parameters:
## alpha = 0.8251
## beta = 1e-04
## gamma = 0.0259
##
## Initial states:
## l = 10.5659
## b = -0.2371
## s = 0.5024 0.6292 0.9319 1.2229 1.4923 1.6309
## 1.5606 1.3672 1.0278 0.8128 0.5106 0.3114
##
## sigma: 0.3971
##
## AIC AICc BIC
## 6648.746 6649.699 6725.114
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.04281136 2.12539 1.359297 -0.4896895 10.87401 0.2233077
## ACF1
## Training set 0.06551473
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Jan 2015 5.61033488 2.75534413 8.465326 1.2440033 9.976666
## Feb 2015 6.51280609 2.03809262 10.987520 -0.3306776 13.356290
## Mar 2015 8.78611971 1.33667747 16.235562 -2.6068190 20.179058
## Apr 2015 10.19278453 -0.02672876 20.412298 -5.4366123 25.822181
## May 2015 12.51259660 -1.96157245 26.986766 -9.6237347 34.648928
## Jun 2015 14.06160944 -4.41048190 32.533701 -14.1890164 42.312235
## Jul 2015 14.50208797 -6.89909775 35.903274 -18.2282011 47.232377
## Aug 2015 12.94562026 -8.35024040 34.241481 -19.6235881 45.514829
## Sep 2015 10.35221007 -8.52365174 29.228072 -18.5159294 39.220350
## Oct 2015 7.41827945 -7.51119146 22.347750 -15.4143760 30.250935
## Nov 2015 4.98972975 -6.05900058 16.038460 -11.9078451 21.887305
## Dec 2015 3.87177644 -5.53876625 13.282319 -10.5204066 18.263960
## Jan 2016 4.19940009 -7.01753622 15.416336 -12.9554236 21.354224
## Feb 2016 4.83889159 -9.30305408 18.980837 -16.7893479 26.467131
## Mar 2016 6.47717405 -14.21845674 27.172805 -25.1740619 38.128410
## Apr 2016 7.45263289 -18.56883703 33.474103 -32.3437711 47.249037
## May 2016 9.06974859 -25.52978297 43.669280 -43.8456686 61.985166
## Jun 2016 10.09948524 -31.99838423 52.197355 -54.2836502 74.482621
## Jul 2016 10.31519859 -36.68017471 57.310572 -61.5580226 82.188420
## Aug 2016 9.11376538 -36.29212001 54.519651 -60.3285438 78.556075
## Sep 2016 7.20870003 -32.09306283 46.510463 -52.8981594 67.315559
## Oct 2016 5.10586905 -25.38360449 35.595343 -41.5237568 51.735495
## Nov 2016 3.39194526 -18.81647619 25.600367 -30.5729044 37.356795
## Dec 2016 2.59725607 -16.07148862 21.266001 -25.9541251 31.148637
## Jan 2017 2.78373153 -19.21776020 24.785223 -30.8646464 36.432109
## Feb 2017 3.15936030 -24.33037133 30.649092 -38.8825562 45.201277
## Mar 2017 4.16047976 -35.76625931 44.087219 -56.9021982 65.223158
## Apr 2017 4.70328433 -45.18192233 54.588491 -71.5895556 80.996124
## May 2017 5.61534367 -60.36391809 71.594605 -95.2912295 106.521917
## Jun 2017 6.12405932 -73.79768601 86.045805 -116.1057024 128.353821
## Jul 2017 6.11425109 -82.77279348 95.001296 -129.8267526 142.055255
## Aug 2017 5.26904280 -80.34316326 90.881249 -125.6635275 136.201613
## Sep 2017 4.05463246 -69.85419810 77.963463 -108.9791696 117.088435
## Oct 2017 2.78569137 -54.42552114 59.996904 -84.7113076 90.282690
## Nov 2017 1.78879320 -39.80734935 43.384936 -61.8270171 65.404604
## Dec 2017 1.31845355 -33.59456540 36.231473 -52.0764012 54.713308
## Jan 2018 1.36330608 -39.72863165 42.455244 -61.4813895 64.208002
## Feb 2018 1.47418475 -49.81449186 52.762861 -76.9650772 79.913447
## Mar 2018 1.83599894 -72.59270008 76.264698 -111.9928734 115.664871
## Apr 2018 1.94469388 -90.98201645 94.871404 -140.1744454 144.063833
## May 2018 2.14932535 -120.68632438 124.984975 -185.7115895 190.010240
## Jun 2018 2.13526664 -146.58736982 150.857903 -225.3163779 229.586911
## Jul 2018 1.89917673 -163.44351234 167.241866 -250.9706448 254.768998
## Aug 2018 1.41138961 -157.79092639 160.613706 -242.0675417 244.890321
## Sep 2018 0.88995573 -136.51563431 138.295546 -209.2537599 211.033671
## Oct 2018 0.45770844 -105.88461019 106.800027 -162.1788332 163.094250
## Nov 2018 0.18024733 -77.12580421 77.486299 -118.0491559 118.409651
## Dec 2018 0.03534794 -64.84280988 64.913506 -99.1872320 99.257928
## Jan 2019 -0.06189953 -76.41489903 76.291100 -116.8337354 116.709936
## Feb 2019 -0.21666264 -95.51037879 95.077054 -145.9558249 145.522500
## Mar 2019 -0.49630645 -138.77793993 137.785327 -211.9798149 210.987202
## Apr 2019 -0.82318361 -173.46750956 171.821142 -264.8598951 263.213528
## May 2019 -1.32836314 -229.53530386 226.878578 -350.3407610 347.684035
## Jun 2019 -1.86695811 -278.16514909 274.431233 -424.4285763 420.694660
## Jul 2019 -2.33009351 -309.50479272 304.844606 -472.1132561 467.453069
## Aug 2019 -2.45925738 -298.22745626 293.308942 -454.7976831 449.879168
## Sep 2019 -2.28538200 -257.56121126 252.990447 -392.6960724 388.125308
## Oct 2019 -1.87811787 -199.44569116 195.689455 -304.0316469 300.275411
## Nov 2019 -1.43371871 -145.05801838 142.190581 -221.0881300 218.220693
## Dec 2019 -1.25208177 -121.78846690 119.284303 -185.5965748 183.092411checkresiduals(hw2)
##
## Ljung-Box test
##
## data: Residuals from Holt-Winters' multiplicative method
## Q* = 38.585, df = 8, p-value = 5.868e-06
##
## Model df: 16. Total lags used: 24#Holt’s-Winter model 3
hw3 <- hw(Solar,seasonal="additive",damped = TRUE, h=5*frequency(Solar))
summary(hw3)##
## Forecast method: Damped Holt-Winters' additive method
##
## Model Information:
## Damped Holt-Winters' additive method
##
## Call:
## hw(y = Solar, h = 5 * frequency(Solar), seasonal = "additive",
##
## Call:
## damped = TRUE)
##
## Smoothing parameters:
## alpha = 0.9999
## beta = 1e-04
## gamma = 1e-04
## phi = 0.9388
##
## Initial states:
## l = 11.154
## b = 0.7632
## s = -10.4919 -8.137 -3.348 2.5794 8.08 11.1219
## 9.9586 6.9916 1.9573 -1.8565 -7.1607 -9.6946
##
## sigma: 2.3446
##
## AIC AICc BIC
## 5428.422 5429.489 5509.282
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -0.01091357 2.314163 1.498521 -1.468083 12.44796 0.2461797
## ACF1
## Training set 0.1700724
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Jan 2015 5.945198 2.9405301 8.949866 1.3499548 10.54044
## Feb 2015 8.479778 4.2305479 12.729009 1.9811413 14.97842
## Mar 2015 13.784309 8.5799016 18.988716 5.8248548 21.74376
## Apr 2015 17.598568 11.5887764 23.608360 8.4073847 26.78975
## May 2015 22.632258 15.9128030 29.351713 12.3557383 32.90878
## Jun 2015 25.599150 18.2380236 32.960277 14.3412786 36.85702
## Jul 2015 26.761775 18.8104968 34.713053 14.6013444 38.92221
## Aug 2015 23.722251 15.2216103 32.222892 10.7216429 36.72286
## Sep 2015 18.222624 9.2059552 27.239293 4.4328191 32.01243
## Oct 2015 12.293259 2.7884705 21.798047 -2.2430606 26.82958
## Nov 2015 7.504219 -2.4648770 17.473315 -7.7421977 22.75064
## Dec 2015 5.150488 -5.2622860 15.563263 -10.7744758 21.07545
## Jan 2016 5.947275 -4.8911752 16.785726 -10.6287044 22.52326
## Feb 2016 8.481728 -2.7662508 19.729707 -8.7205712 25.68403
## Mar 2016 13.786139 2.1429879 25.429291 -4.0205242 31.59280
## Apr 2016 17.600287 5.5749058 29.625668 -0.7909464 35.99152
## May 2016 22.633871 10.2380087 35.029734 3.6760353 41.59171
## Jun 2016 25.600665 12.8450464 38.356283 6.0926298 45.10870
## Jul 2016 26.763197 13.6576670 39.868726 6.7200186 46.80637
## Aug 2016 23.723586 10.2772227 37.169950 3.1591478 44.28802
## Sep 2016 18.223877 4.4450855 32.002669 -2.8489663 39.29672
## Oct 2016 12.294435 -1.8089722 26.397843 -9.2748652 33.86374
## Nov 2016 7.505324 -6.9154138 21.926061 -14.5492911 29.55994
## Dec 2016 5.151525 -9.5797258 19.882776 -17.3779790 27.68103
## Jan 2017 5.948249 -9.0871906 20.983688 -17.0464714 28.94297
## Feb 2017 8.482642 -6.8508988 23.816183 -14.9679850 31.93327
## Mar 2017 13.786997 -1.8389729 29.412968 -10.1108619 37.68486
## Apr 2017 17.601092 1.6880528 33.514132 -6.7358014 41.93799
## May 2017 22.634628 6.4395949 38.829660 -2.1335375 47.40279
## Jun 2017 25.601375 9.1291648 42.073585 0.4093036 50.79345
## Jul 2017 26.763863 10.0190534 43.508673 1.1548866 52.37284
## Aug 2017 23.724212 6.7111602 40.737263 -2.2950052 49.74343
## Sep 2017 18.224465 0.9473269 35.501602 -8.1986374 44.64757
## Oct 2017 12.294987 -5.2422687 29.832242 -14.5259309 39.11590
## Nov 2017 7.505841 -10.2877372 25.299420 -19.7070886 34.71877
## Dec 2017 5.152011 -12.8942567 23.198279 -22.4473739 32.75140
## Jan 2018 5.948705 -12.3468264 24.244236 -22.0318957 33.92931
## Feb 2018 8.483070 -10.0583229 27.024464 -19.8735437 36.83968
## Mar 2018 13.787399 -4.9966434 32.571442 -14.9403150 42.51511
## Apr 2018 17.601470 -1.4221329 36.625072 -11.4926198 46.69556
## May 2018 22.634982 3.3747944 41.895169 -6.8209329 52.09090
## Jun 2018 25.601707 6.1078017 45.095613 -4.2116486 55.41506
## Jul 2018 26.764175 7.0393167 46.489034 -3.4023928 56.93074
## Aug 2018 23.724505 3.7713624 43.677647 -6.7911931 54.24020
## Sep 2018 18.224740 -1.9541074 38.403587 -12.6361439 49.08562
## Oct 2018 12.295245 -8.1068132 32.697304 -18.9070105 43.49750
## Nov 2018 7.506084 -13.1167729 28.128941 -24.0338538 39.04602
## Dec 2018 5.152239 -15.6890799 25.993558 -26.7218076 37.02629
## Jan 2019 5.948919 -15.1086483 27.006486 -26.2558509 38.15369
## Feb 2019 8.483271 -12.7882989 29.754841 -24.0487878 41.01533
## Mar 2019 13.787588 -7.6958556 35.271031 -19.0685035 46.64368
## Apr 2019 17.601647 -4.0916031 39.294896 -15.5753158 50.77861
## May 2019 22.635148 0.7340999 44.536196 -10.8596146 56.12991
## Jun 2019 25.601863 3.4949682 47.708758 -8.2077152 59.41144
## Jul 2019 26.764322 4.4534771 49.075166 -7.3571707 60.88581
## Aug 2019 23.724642 1.2116941 46.237591 -10.7059408 58.15523
## Sep 2019 18.224869 -4.4883862 40.938124 -16.5120571 52.96179
## Oct 2019 12.295366 -10.6164456 35.207178 -22.7452262 47.33596
## Nov 2019 7.506198 -15.6024666 30.614862 -27.8354544 42.84785
## Dec 2019 5.152346 -18.1515090 28.456200 -30.4878245 40.79252checkresiduals(hw3)
##
## Ljung-Box test
##
## data: Residuals from Damped Holt-Winters' additive method
## Q* = 210.76, df = 7, p-value < 2.2e-16
##
## Model df: 17. Total lags used: 24#Holt’s-Winter model 4
hw4 <- hw(Solar,seasonal="multiplicative",damped = TRUE, h=5*frequency(Solar))
summary(hw4)##
## Forecast method: Damped Holt-Winters' multiplicative method
##
## Model Information:
## Damped Holt-Winters' multiplicative method
##
## Call:
## hw(y = Solar, h = 5 * frequency(Solar), seasonal = "multiplicative",
##
## Call:
## damped = TRUE)
##
## Smoothing parameters:
## alpha = 0.805
## beta = 1e-04
## gamma = 0.0124
## phi = 0.94
##
## Initial states:
## l = 9.9033
## b = 0.4703
## s = 0.4341 0.5649 0.8263 1.1716 1.4514 1.6316
## 1.5503 1.3834 1.1045 0.8837 0.5761 0.4221
##
## sigma: 0.3145
##
## AIC AICc BIC
## 6366.701 6367.768 6447.561
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -0.0456208 2.045442 1.239102 -2.172476 10.00296 0.2035619
## ACF1
## Training set 0.04177493
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Jan 2015 5.611616 3.3502090 7.873023 2.1530925 9.070139
## Feb 2015 7.060438 3.3373603 10.783517 1.3664818 12.754395
## Mar 2015 10.515176 3.8550902 17.175263 0.3294535 20.700899
## Apr 2015 12.993446 3.5145037 22.472389 -1.5033454 27.490238
## May 2015 16.284381 2.9386340 29.630128 -4.1261776 36.694939
## Jun 2015 18.261020 1.7242212 34.797820 -7.0298315 43.551872
## Jul 2015 19.031597 0.2101276 37.853067 -9.7533565 47.816551
## Aug 2015 17.105654 -1.2074754 35.418784 -10.9018606 45.113169
## Sep 2015 13.764743 -2.0801991 29.609686 -10.4680049 37.997491
## Oct 2015 9.774733 -2.2585804 21.808047 -8.6286319 28.178099
## Nov 2015 6.581938 -2.0456267 15.209503 -6.6127835 19.776659
## Dec 2015 5.163049 -2.0168711 12.342970 -5.8176914 16.143790
## Jan 2016 5.616270 -2.6571166 13.889656 -7.0367828 18.269322
## Feb 2016 7.066360 -3.9144360 18.047156 -9.7273184 23.860039
## Mar 2016 10.524088 -6.6891829 27.737360 -15.8013383 36.849515
## Apr 2016 13.004566 -9.3405081 35.349641 -21.1692761 47.178409
## May 2016 16.298445 -13.0720043 45.668893 -28.6197807 61.216670
## Jun 2016 18.276925 -16.2140342 52.767885 -34.4724452 71.026296
## Jul 2016 19.048304 -18.5466197 56.643228 -38.4481704 76.544779
## Aug 2016 17.120781 -18.1782620 52.419825 -36.8644468 71.106009
## Sep 2016 13.777000 -15.8651593 43.419159 -31.5567705 59.110770
## Oct 2016 9.783493 -12.1627698 31.729756 -23.7804195 43.347405
## Nov 2016 6.587872 -8.8063951 21.982139 -16.9556278 30.131371
## Dec 2016 5.167730 -7.4021686 17.737629 -14.0562710 24.391731
## Jan 2017 5.621391 -8.6173336 19.860116 -16.1548593 27.397642
## Feb 2017 7.072836 -11.5506968 25.696368 -21.4093994 35.555071
## Mar 2017 10.533777 -18.2831270 39.350681 -33.5378745 54.605428
## Apr 2017 13.016590 -23.9602666 49.993446 -43.5346319 69.567811
## May 2017 16.313574 -31.7864590 64.413606 -57.2490781 89.876225
## Jun 2017 18.293954 -37.6661323 74.254041 -67.2896128 103.877522
## Jul 2017 19.066115 -41.4169669 79.549196 -73.4347761 111.567005
## Aug 2017 17.136842 -39.2196179 73.493302 -69.0529257 103.326610
## Sep 2017 13.789964 -33.2070276 60.786955 -58.0857320 85.665659
## Oct 2017 9.792726 -24.7827962 44.368248 -43.0859719 62.671423
## Nov 2017 6.594106 -17.5190371 30.707248 -30.2837638 43.471975
## Dec 2017 5.172632 -14.4124885 24.757753 -24.7802253 35.125490
## Jan 2018 5.626738 -16.4482035 27.701680 -28.1339719 39.387449
## Feb 2018 7.079578 -21.6650836 35.824241 -36.8815886 51.040745
## Mar 2018 10.543840 -33.7520469 54.839727 -57.2008731 78.288553
## Apr 2018 13.029049 -43.5957293 69.653827 -73.5710763 99.629174
## May 2018 16.329218 -57.0729428 89.731378 -95.9296992 128.588134
## Jun 2018 18.311528 -66.8108540 103.433910 -111.8719210 148.494977
## Jul 2018 19.084459 -72.6443909 110.813310 -121.2027110 159.371630
## Aug 2018 17.153356 -68.0816522 102.388364 -113.2023398 147.509051
## Sep 2018 13.803271 -57.0947408 84.701283 -94.6258813 122.232424
## Oct 2018 9.802188 -42.2336507 61.838027 -69.7797599 89.384137
## Nov 2018 6.600486 -29.6097187 42.810690 -48.7782429 61.979214
## Dec 2018 5.177643 -24.1727779 34.528064 -39.7099521 50.065238
## Jan 2019 5.632196 -27.3848682 38.649259 -44.8630460 56.127437
## Feb 2019 7.086452 -35.8304219 50.003326 -58.5492425 72.722146
## Mar 2019 10.554087 -55.4725310 76.580704 -90.4249112 111.533085
## Apr 2019 13.041723 -71.2326855 97.316131 -115.8448624 141.928308
## May 2019 16.345115 -92.7428126 125.433042 -150.4904731 183.180703
## Jun 2019 18.329369 -108.0080581 144.666797 -174.8870523 211.545791
## Jul 2019 19.103068 -116.8701338 155.076270 -188.8499990 227.056135
## Aug 2019 17.170093 -109.0302062 143.370393 -175.8366093 210.176796
## Sep 2019 13.816749 -91.0420216 118.675520 -146.5509017 174.184400
## Oct 2019 9.811766 -67.0716275 86.695159 -107.7712373 127.394768
## Nov 2019 6.606938 -46.8432208 60.057097 -75.1380259 88.351902
## Dec 2019 5.182708 -38.1032646 48.468680 -61.0174741 71.382889checkresiduals(hw4)
##
## Ljung-Box test
##
## data: Residuals from Damped Holt-Winters' multiplicative method
## Q* = 23.57, df = 7, p-value = 0.001355
##
## Model df: 17. Total lags used: 24#Holt’s-Winter model 5
hw5 <- hw(Solar,seasonal="multiplicative",exponential = TRUE, h=5*frequency(Solar))
summary(hw5)##
## Forecast method: Holt-Winters' multiplicative method with exponential trend
##
## Model Information:
## Holt-Winters' multiplicative method with exponential trend
##
## Call:
## hw(y = Solar, h = 5 * frequency(Solar), seasonal = "multiplicative",
##
## Call:
## exponential = TRUE)
##
## Smoothing parameters:
## alpha = 0.6499
## beta = 1e-04
## gamma = 0.0597
##
## Initial states:
## l = 9.3759
## b = 0.9889
## s = 0.3552 0.4789 0.952 1.0553 1.4608 1.7926
## 1.6746 1.4529 1.1145 0.8672 0.5333 0.2627
##
## sigma: 0.3755
##
## AIC AICc BIC
## 6584.208 6585.161 6660.576
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 0.06388152 2.208727 1.412454 -1.303792 11.28449 0.2320404 0.153871
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Jan 2015 5.797870 3.01983746 8.528547 1.470742117 10.13384
## Feb 2015 7.513359 3.35093186 11.958387 1.669164907 15.05970
## Mar 2015 10.758212 4.34904221 18.297449 2.347809834 24.17894
## Apr 2015 12.694087 4.65323658 22.808540 2.347735124 31.90351
## May 2015 15.340074 4.81477307 28.540756 2.321491741 40.93178
## Jun 2015 17.239431 5.01664783 33.079309 2.453263277 48.88778
## Jul 2015 18.004700 4.82553668 35.341362 2.243021374 55.18168
## Aug 2015 16.206596 4.02605236 32.604057 1.920948848 51.48505
## Sep 2015 13.011256 2.97147232 26.579891 1.292371882 44.98912
## Oct 2015 9.215869 1.95215295 19.034742 0.838474570 31.74227
## Nov 2015 6.214762 1.19746333 12.892833 0.512640464 23.19036
## Dec 2015 4.565667 0.85024036 9.492378 0.348075549 16.87388
## Jan 2016 5.220661 0.83692446 11.235257 0.336724552 20.37616
## Feb 2016 6.765364 1.02974380 14.394190 0.411886002 26.79793
## Mar 2016 9.687175 1.33247389 21.053024 0.506153874 40.56392
## Apr 2016 11.430324 1.53184698 24.613082 0.632220799 47.93269
## May 2016 13.812889 1.62284016 30.816119 0.643394316 61.43697
## Jun 2016 15.523155 1.80159659 34.459063 0.658967237 69.58234
## Jul 2016 16.212237 1.74468503 37.192598 0.682006550 74.79569
## Aug 2016 14.593143 1.43983772 32.651159 0.518788411 71.34345
## Sep 2016 11.715916 1.09106830 26.703472 0.370476001 59.50976
## Oct 2016 8.298381 0.70151135 19.033768 0.211330837 43.55793
## Nov 2016 5.596049 0.45540410 13.253214 0.153663969 30.99148
## Dec 2016 4.111130 0.30766290 9.563693 0.092324269 23.14326
## Jan 2017 4.700916 0.32795798 11.333354 0.105886969 26.25831
## Feb 2017 6.091836 0.37812454 14.774991 0.119789578 36.84959
## Mar 2017 8.722765 0.52468787 20.642138 0.163705449 51.79812
## Apr 2017 10.292374 0.58847686 23.634147 0.169517718 61.38132
## May 2017 12.437743 0.65022471 29.005789 0.164804185 77.67039
## Jun 2017 13.977742 0.68648273 32.448641 0.192749452 86.11528
## Jul 2017 14.598223 0.70112662 33.777933 0.185905463 96.49645
## Aug 2017 13.140319 0.55528477 30.785062 0.156924491 82.45802
## Sep 2017 10.549535 0.43434155 24.481567 0.115703477 70.57232
## Oct 2017 7.472233 0.28964367 17.518987 0.078300908 51.17181
## Nov 2017 5.038933 0.18937631 11.965186 0.045185455 34.23835
## Dec 2017 3.701845 0.12535643 8.689323 0.031861801 24.38203
## Jan 2018 4.232915 0.13714207 9.727921 0.034886237 29.65742
## Feb 2018 5.485362 0.16591746 12.492215 0.045352775 38.90172
## Mar 2018 7.854368 0.22507169 18.415839 0.065025499 55.82959
## Apr 2018 9.267714 0.24633782 21.212573 0.064730761 64.75081
## May 2018 11.199499 0.27552782 25.972435 0.074885645 82.41732
## Jun 2018 12.586184 0.31189185 30.443198 0.076849065 88.66050
## Jul 2018 13.144893 0.31699650 30.783235 0.081303478 94.05588
## Aug 2018 11.832130 0.25088341 27.587627 0.063639693 83.65822
## Sep 2018 9.499273 0.19244843 22.384007 0.049840255 66.73504
## Oct 2018 6.728333 0.13023870 15.584172 0.030674417 49.53922
## Nov 2018 4.537281 0.08535751 10.241058 0.018211279 34.95690
## Dec 2018 3.333307 0.05760253 7.435154 0.014534413 24.76757
## Jan 2019 3.811506 0.06237475 8.345845 0.013607209 29.06785
## Feb 2019 4.939265 0.07983718 10.612912 0.018573250 35.71379
## Mar 2019 7.072424 0.10239469 14.918407 0.023220528 51.75997
## Apr 2019 8.345064 0.11734104 17.562713 0.025365127 64.11096
## May 2019 10.084530 0.12903708 21.939933 0.030918208 74.26760
## Jun 2019 11.333163 0.14366914 23.447829 0.029811663 83.28289
## Jul 2019 11.836249 0.13756828 24.364695 0.032376565 93.12558
## Aug 2019 10.654179 0.11786073 22.658772 0.025664233 78.10276
## Sep 2019 8.553570 0.08645611 17.294352 0.019388425 65.34651
## Oct 2019 6.058491 0.05976678 12.283953 0.012785440 45.32868
## Nov 2019 4.085570 0.03774909 7.939158 0.007373460 31.17401
## Dec 2019 3.001459 0.02612405 5.906397 0.005672383 21.76737checkresiduals(hw5)
##
## Ljung-Box test
##
## data: Residuals from Holt-Winters' multiplicative method with exponential trend
## Q* = 21.246, df = 8, p-value = 0.006521
##
## Model df: 16. Total lags used: 24cbind(Model=c("HW1","HW2","HW3","HW4","HW5"),
MASE=c(accuracy(hw1)[6],accuracy(hw2)[6],accuracy(hw3)[6],
accuracy(hw4)[6],accuracy(hw5)[6]))## Model MASE
## [1,] "HW1" "0.247160046952369"
## [2,] "HW2" "0.223307745945573"
## [3,] "HW3" "0.246179687973231"
## [4,] "HW4" "0.203561894680514"
## [5,] "HW5" "0.23204037862927"By comparing Holt’s-Winter models we can say that:
- RMSE value is least for HW4 and highest for HW1.
- AIC and BIC lowest values can be seen in HW3.
- Mean absolute percentage error is lowest for HW4.
- MASE is least observed in HW4
Hence HW4 is the best fitting model of the Holt’s-Winter method.
State-space model
The state-space model considers error, trend, and seasonality of the series denoted by (E, T, S). Here we will check automatically which model best fits by using the following code.
ssm1 = ets(Solar,model ="ZZZ")
ssm1$method## [1] "ETS(A,Ad,A)"We see that “ETS(A,Ad,A)” best fits the series then checking the summary of “ETS(A,Ad,A)”.
- RMSE is 2.324
- AIC and BIC are 5428.422 & 5509.282.
- MASE is 0.246
summary(ssm1) ## ETS(A,Ad,A)
##
## Call:
## ets(y = Solar, model = "ZZZ")
##
## Smoothing parameters:
## alpha = 0.9999
## beta = 1e-04
## gamma = 1e-04
## phi = 0.9388
##
## Initial states:
## l = 11.154
## b = 0.7632
## s = -10.4919 -8.137 -3.348 2.5794 8.08 11.1219
## 9.9586 6.9916 1.9573 -1.8565 -7.1607 -9.6946
##
## sigma: 2.3446
##
## AIC AICc BIC
## 5428.422 5429.489 5509.282
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -0.01091357 2.314163 1.498521 -1.468083 12.44796 0.2461797
## ACF1
## Training set 0.1700724Forecasting of 2 years
#forecasting dataset
data.x <- read_csv("/Users/shubhamchougule/Downloads/data.x.csv")## Parsed with column specification:
## cols(
## x = col_double()
## )forecast_ts=ts(data.x)
dlmfore = forecast(model4, x = as.vector(forecast_ts) , h = 24)$forecasts
hwforecast=hw4$mean
etsforecast = as.vector(forecast(ssm1,h=24)$mean)
#Dynlm model forecasting
q = 24
n = nrow(dynlm6$model)
landings.frc = array(NA , (n + q))
landings.frc[1:n] = Y.t[1:length(Y.t)]
for (i in 1:q){
months = array(0,11)
months[(i-1)%%12] = 1
data.new = c(1,landings.frc[n-1+i],landings.frc[n-2+i],landings.frc[n-3+i],1,months)
landings.frc[n+i] = as.vector(dynlm6$coefficients) %*% data.new
}dlm = c(Solar,dlmfore)
ets = c(Solar,etsforecast)
hw=c(Solar,hwforecast)
dynlm_forecast=ts(landings.frc[(n+1):(n+q)],start=c(2014,1),frequency = 12)
Dataforecast = ts.intersect(
ts(dlm,start=c(1960,1),frequency = 12),
ts(landings.frc[(n+1):(n+q)],start=c(2015,1),frequency = 12),
ts(hw,start=c(1960,1),frequency = 12),
ts(ets,start=c(1960,1),frequency = 12)
)
ts.plot(Dataforecast,xlim=c(2015,2017),ylim=c(-30,70),
plot.type = c("single"), gpars= list(col=c("red","blue","gray","green")),
main = "Forecasting of next 2 years of Solar Radiations")
legend("bottomleft", col=c("red","blue","gray","green"), lty=1, cex=.65,c("DLM","DYNLM","HW","ETS"))
Conclusion
Based on the MASE value, we can say that HW6 of Holts winter method in Exponential smoothening techniques is the best fitting model to the solar series after comparing different models of the dynamic lag model, distributed lag model, exponential smoothing, and corresponding state-space models.
Task 2
Here we will investigate if there is correlation between quarterly Residential Property Price Index (PPI) in Melbourne and quarterly population change over the previous quarter in Victoria these two series is spurious or not.
#Importing the dataset
task2 <- read_csv("/Users/shubhamchougule/Downloads/data2.csv")## Parsed with column specification:
## cols(
## Quarter = col_character(),
## price = col_double(),
## change = col_double()
## )head(task2)## # A tibble: 6 x 3
## Quarter price change
## <chr> <dbl> <dbl>
## 1 Sep-2003 60.7 14017
## 2 Dec-2003 62.1 12350
## 3 Mar-2004 60.8 17894
## 4 Jun-2004 60.9 9079
## 5 Sep-2004 60.9 16210
## 6 Dec-2004 62.4 13788#Converting dataset to timeseries
task=ts(task2,start = c(2003,3),frequency=4)
price = task[,2]
change = task[,3]
#intersecting both time series
price.joint=ts.intersect(price,change)plot(price.joint,yax.flip=T)
We see that both the series have a upward trend showing moderately correlated.
Check cross correlation
ccf(as.vector(price.joint[,1]),as.vector(price.joint[,2]),
ylab='CCF', main = "Sample CCF between")
We see that lags are highly significant there is no significant evidence for the existence of a spurious correlation and hence to make the series stationary we need to apply differencing. Applying prewhitening.
me.dif=ts.intersect(diff(diff(price,4)),diff(diff(change,4)))
prewhiten(as.vector(me.dif[,1]),as.vector(me.dif[,2]),
ylab='CCF',main="Sample CFF after prewhitening ")
After prewhitening we can see that there are no lags in the plot saying no correlation between price and population change.