Required Packages:

Task 1:

Introduction:

The dataset provided includes monthly averages of the amount of horizontal solar radiation reaching the ground at a location over the globe, starting from January 1960. The task is to analyses and forecast and find the best model for series.
This data was first imported, checked the class of the data and head of it.
Then converted each column to time series using the ts function separately.
Then I checked the class of the data and then plot the figure (Fig 1.0) of the model to visualize the trend of the model.
Then checked the correlation between dependent and independent variables.
Then after plotting the solar column (Fig 1.1) first it was observed that:
1. There is no trend.
2. There is a seasonality effect.
3. Series has a Moving average behaviour.
4. There in no intervention point.
5. There is changing in variance.
Then when plot for precipitation column (Fig 1.2) observed that:
1. There is no trend.
2. seasonality effects.
3. Series is Moving average.
4. There in no intervention point.
5. There is changing in variance.
Then plotted all the plots (Fig 1.3) in the same frame but the scales will be problematic, and we won’t be able to see the series clearly.
Then scaled the data so can clearly see all the plots (Fig 1.4), in this plot blue is solar and red is precipitation

solprep <- read_csv("C:/Users/hassa/Downloads/RMIT/3rd Semester/Forecasting/Assignment 2/data1.csv")

## Parsed with column specification:
## cols(
##   solar = col_double(),
##   ppt = col_double()
## )

class(solprep)

## [1] "spec_tbl_df" "tbl_df"      "tbl"         "data.frame"

head(solprep)

## # 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

solar = ts(solprep$solar,start=c(1960,1),frequency = 12) 
ppt = ts(solprep$ppt,start=c(1960,1),frequency = 12) 

class(solar)

## [1] "ts"

class(ppt)

## [1] "ts"

solarppt = ts.intersect(solar , ppt) 
plot(solarppt , yax.flip=T, main = "Fig 1.0")

cor(solprep)

##            solar        ppt
## solar  1.0000000 -0.4540277
## ppt   -0.4540277  1.0000000

# Analysing solar time series:

plot(solar, main = "Time series plot for Solar radiations Fig 1.1")
points(y=solar,x=time(solar), pch=as.vector(season(solar)))

# Analysing prep time series:

plot(ppt, main = "Time series plot for precipitation Fig 1.2")
points(y=ppt,x=time(ppt), pch=as.vector(season(ppt)))

plot(solarppt, plot.type="s",col = c("blue", "red"), main = "Measurements of solar radiations and 
     precipitation Fig 1.3")
legend("topleft",lty=1, text.width = 1, col=c("blue","red"), c("Solar", "Precipitation"))

data.scaled = scale(solarppt)
plot(data.scaled, plot.type="s",col = c("blue", "red"), main = "Measurements of solar radiations and 
     precipitation Fig 1.4")
legend("topleft",lty=1, text.width = 1, col=c("blue","red"), c("Solar", "Precipitation"))

Analysis:

First the X-12 decomposition was done for both variables to reverify seasonality.
For the solar series (Fig 2.0) the seasonally adjusted plot and the original show a significant deviation from each other indicating the presence of seasonality, and the seasonally adjusted plot and the trend plot almost overlap each other. And then plot the seasonal factors by period and SI ratio (Fig 2.1).
For precipitation series (Fig 2.2) the seasonally adjusted plot and the original show a significant deviation from each other indicating the presence of seasonality, and the seasonally adjusted plot and the trend plot almost overlap each other. And then plot the seasonal factors by period and SI ratio (Fig 2.3).
Now for checking the stationarity and seasonality by using the ACF and PACF plot.
Observed that in the solar series wave like pattern confirms seasonality in ACF (Fig 2.4) and in PACF plot (Fig 2.5) observe high significant lags.
And for the precipitation series also ACF confirms Wave like pattern confirms seasonality in ACF (Fig 2.6) and in PACF plot (Fig 2.7) observe high significant lags.
Then did the adf test on both the series and the test confirms that both the series are stationary.
Now as seasonality was confirmed moving forward for the STL decomposition.
Then apply STL on solar series (Fig 2.8) and observed the difference between the original data, trend, seasonality and the remainders.
Then apply STL on precipitation series (Fig 2.9) and observed the difference between the original data, trend, seasonality and the remainders.
Then plot the seasonal sub-series plot of the seasonal component for both the solar (Fig 2.10) and precipitation series (Fig 2.11).
After adjusting the seasonal components of both the series and extract the seasonal component from the output observed the final look of the series of solar (Fig 2.12) and precipitation (Fig 2.13).

# X-12 decomposition:

fits.x12 = x12(solar) 
plot(fits.x12 , sa=TRUE , trend=TRUE, main = "X12 - Solar radiations Fig 2.0")

plotSeasFac(fits.x12, main = "Seasonal Factors by period and SI Ratio Fig 2.1")

fitp.x12 = x12(ppt) 
plot(fitp.x12 , sa=TRUE , trend=TRUE, main = "X12 - Precipitations Fig 2.2")

plotSeasFac(fitp.x12, main = "Seasonal Factors by period and SI Ratio Fig 2.3")

# Checking stationarity and seasonality using ACF and PACF:

par(mfrow=c(1,2))
acf(solar, main="Sample ACF for solar 
    radiations Fig 2.4")
pacf(solar, main="Sample PACF for solar 
radiations Fig 2.5")

par(mfrow=c(1,2))
acf(ppt, main="Sample ACF for 
    ppt Fig 2.6")
pacf(ppt, main="Sample PACF for 
     ppt Fig 2.7")

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: stationary

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: stationary

# Since seasonality has been confirmed we go ahead and use STL decomposition:

fit.solar <- stl(solar, t.window=15, s.window="periodic", robust=TRUE) 
plot(fit.solar, main = "STL on Solar Series Fig 2.8")

fit.ppt <- stl(ppt, t.window=15, s.window="periodic", robust=TRUE) 
plot(fit.ppt, main = "STL on Precipitation Series Fig 2.9")

par(mfrow=c(1,1))
monthplot(fit.solar,choice = "trend", main="Seasonal sub-series plot of the seasonal 
          component of solar series Fig 2.10", ylab="Trend")

monthplot(fit.ppt,choice = "trend", main="Seasonal sub-series plot of the seasonal 
          component of precipitation series Fig 2.11", ylab="Trend")

#Adjusting seasonal component:

# Solar:

par(mfrow=c(1,1))
fit.solar.seasonal = fit.solar$time.series[,"seasonal"]
solar.seasonally.adjusted = solar - fit.solar.seasonal
plot(solar.seasonally.adjusted, ylab='Solar radiations ',xlab='Time',type='o', main = "Seasonally adjusted solar radiations series 
     Fig 2.12")

# Precipitation:

fit.ppt.seasonal = fit.ppt$time.series[,"seasonal"]
ppt.seasonally.adjusted = solar - fit.ppt.seasonal
plot(ppt.seasonally.adjusted, ylab='Precipitation',xlab='Time',type='o', main = "Seasonally adjusted precipitation series 
     Fig 2.13")

Solar with Precipitation (dLagM Package):

Now to find the suitable lag models of Solar with Precipitation, as Solar here is dependent variable while Precipitation is an independent variable.

dlm:

First for diagnostic checking fitted a finite DLM with a finite lag length and for this there is no need for the time series data as this function looks for vectors.
A function is created which checks for the value of q from 1 to 10 and compare the values if AIC and BIC and from all these values we can easily pick the smallest value of AIC and BIC.
This table shows that q with value 10 has the smallest values of both AIC and BIC.
Then when we modelled with q = 10 and checks its summary none of the values were significant, value of adjusted R-square is also very low, but the p-value is good.
Then checked the residuals (Fig 3.0)they are not random, correlation is prominent, pattern in histogram is almost normalized. According to the ACF and BG test result it can be concluded that the serial correlation left in residuals is highly significant.
Then checked the multicollinearity and all were FALSE which means that there is no multicollinearity between these two variables.
Finally, the MASE value of 1.577996 was observed which is not good.

finiteDLMauto:

Then I used finiteDLMauto function which shows the lag length in “q - k”.
Then again checked the model with q = 10 and still it’s not good.

polyDlm:

Then I tried to fit the polynomial DLM of order 10 and checks its summary the values were insignificant; value of adjusted R-square is also very low but the p-value is good.
Then checked the residuals (Fig 3.1) were not random, correlation is prominent, pattern in histogram was also almost normalized. According to the ACF and BG test result we can conclude that the serial correlation left in residuals is highly significant.
Was not able to do the VIF test because of the aliased coefficients in the model.
Finally, the MASE value of 1.580235 was observed which is not good.
Then I tried to fit the polynomial DLM of order 1 and checks its summary the values were significant; value of adjusted R-square is also very low but the p-value is good.
Then checked the residuals (Fig 3.2) were not random, correlation is prominent, pattern in histogram is almost normalized. According to the ACF and BG test result we can conclude that the serial correlation left in residuals is highly significant.
Then checked the multicollinearity and all were FALSE which means that there is no multicollinearity between these two variables.
Finally, the MASE value of 1.688457 was observed which is not good.

koyckDlm:

Then tried to fit the Koyck model and checks its summary, parameters are significant, value of adjusted R-square is not very good but better than the previous models and the p-value is also good.
Then checked the residuals (Fig 3.3) it was random, wave like pattern indicating the effect of seasonality on the residuals in ACF and the histogram was normalized. This model seems good as compared to all the previous ones.
Then checked the multicollinearity and all were FALSE in this model which means that there is no multicollinearity between these two variables.
Then checked the value of AIC nd BIC and the values of 3946.476 and 3964.439 was observed respectively.
Finally, the MASE value of 1.032483 was observed which is better than all the previous models.

ardlDlm:

Then tries to fit autoregressive distributed lag models, a function is created which checks for the value of q and p from 1 to 15 and compare the values if AIC and BIC and from all these values we can easily pick the smallest value of AIC and BIC.
From the table it was observed that at p = 9 and q = 15 the value of AIC and BIC were the lowest, the when its summary was checked only 2 values were insignificant, value of adjusted R-square is very good and the p-value is good.
Then checked the residuals (Fig 3.4) it was random, no pattern or significant lag in ACF and the histogram was also normalized.
Then checked the multicollinearity and 9 were FALSE and 15 were TRUE in this model which means that there is still some multicollinearity between these two variables.
Finally, the MASE value of 0.3812687 was observed which is the best than all the previous models.

Forecasting:

As the best MASE observed was with ardlm so by taking this model forecasting was done by taking the values given for the next two years prediction on precipitation and with the help of this the amount of solar radiations were predicted for the next 2 years.
Then checked the values of the forecasting done and then plotted it with the whole series and observed that the forecasting shown in the plot does not seems good (Fig 3.5).

# dlm:

for ( i in 1:10){
  modelca1 = dlm( x =as.vector(ppt) , y = as.vector(solar), q = i )
  cat("q = ", i, "AIC = ", AIC(modelca1$model), "BIC = ", BIC(modelca1$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 =  4660.858

modelca2 = dlm( x = as.vector(ppt) , 
                y = as.vector(solar), q = 10 )
summary(modelca2)

## 
## 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 4660.858

checkresiduals(modelca2$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

VIF.modelca2 = vif(modelca2$model)
VIF.modelca2 > 10

##   x.t   x.1   x.2   x.3   x.4   x.5   x.6   x.7   x.8   x.9  x.10 
## FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE

MASE(modelca2)

##              MASE
## modelca2 1.577996

# finiteDLMauto:

finiteDLMauto( x = as.vector(ppt) , y = as.vector(solar), q.min = 1, q.max = 10, 
               model.type = "dlm", error.type ="AIC", trace = TRUE)

##    q - k    MASE      AIC      BIC R.Adj.Sq Ljung-Box
## 10    10 1.57800 4602.658 4660.858  0.29615         0
## 9      9 1.59312 4615.084 4668.827  0.28882         0
## 8      8 1.60481 4625.986 4675.267  0.28251         0
## 7      7 1.60704 4632.716 4677.532  0.28096         0
## 6      6 1.60753 4637.489 4677.837  0.28214         0
## 5      5 1.61385 4644.622 4680.499  0.28072         0
## 4      4 1.64636 4663.600 4695.003  0.26577         0
## 3      3 1.66270 4688.551 4715.478  0.24326         0
## 2      2 1.67597 4712.649 4735.095  0.22173         0
## 1      1 1.68846 4728.713 4746.676  0.21036         0

finiteDLMauto( x = as.vector(ppt) , y = as.vector(solar), q.min = 1, q.max = 10, 
               model.type = "dlm", error.type ="BIC", trace = TRUE)

##    q - k    MASE      AIC      BIC R.Adj.Sq Ljung-Box
## 10    10 1.57800 4602.658 4660.858  0.29615         0
## 9      9 1.59312 4615.084 4668.827  0.28882         0
## 8      8 1.60481 4625.986 4675.267  0.28251         0
## 7      7 1.60704 4632.716 4677.532  0.28096         0
## 6      6 1.60753 4637.489 4677.837  0.28214         0
## 5      5 1.61385 4644.622 4680.499  0.28072         0
## 4      4 1.64636 4663.600 4695.003  0.26577         0
## 3      3 1.66270 4688.551 4715.478  0.24326         0
## 2      2 1.67597 4712.649 4735.095  0.22173         0
## 1      1 1.68846 4728.713 4746.676  0.21036         0

# polyDlm:

modelca2 = polyDlm( x =as.vector(ppt) ,
                    y = as.vector(solar) , 
                    q = 10 , k = 10 , show.beta = FALSE)
summary(modelca2)

## 
## Call:
## "Y ~ (Intercept) + X.t"
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -19.0521  -5.4604  -0.7981   4.0705  30.8643 
## 
## Coefficients: (1 not defined because of singularities)
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.901e+01  1.094e+00  17.382  < 2e-16 ***
## z.t0        -7.349e+00  1.897e+00  -3.875 0.000118 ***
## z.t1         8.893e+00  9.385e+01   0.095 0.924533    
## z.t2         3.090e+00  2.181e+02   0.014 0.988702    
## z.t3        -1.004e+01  1.974e+02  -0.051 0.959477    
## z.t4         6.545e+00  9.302e+01   0.070 0.943921    
## z.t5        -2.069e+00  2.537e+01  -0.082 0.935048    
## z.t6         3.641e-01  4.157e+00   0.088 0.930227    
## z.t7        -3.648e-02  4.032e-01  -0.090 0.927934    
## z.t8         1.950e-03  2.132e-02   0.091 0.927177    
## z.t9        -4.323e-05  4.736e-04  -0.091 0.927293    
## z.t10               NA         NA      NA       NA    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.25 on 639 degrees of freedom
## Multiple R-squared:  0.3079, Adjusted R-squared:  0.297 
## F-statistic: 28.42 on 10 and 639 DF,  p-value: < 2.2e-16

checkresiduals(modelca2$model)

## 
##  Breusch-Godfrey test for serial correlation of order up to 15
## 
## data:  Residuals
## LM test = 587.98, df = 15, p-value < 2.2e-16

MASE(modelca2)

##              MASE
## modelca2 1.580235

modelca3 = polyDlm( x =as.vector(ppt) ,
                    y = as.vector(solar) , 
                    q = 1 , k = 1 , show.beta = FALSE)
summary(modelca3)

## 
## Call:
## "Y ~ (Intercept) + X.t"
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -17.816  -5.736  -0.742   4.717  32.283 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  23.2467     0.5839  39.814  < 2e-16 ***
## z.t0        -15.7626     1.5425 -10.219  < 2e-16 ***
## z.t1         19.8764     2.9067   6.838 1.84e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.715 on 656 degrees of freedom
## Multiple R-squared:  0.2128, Adjusted R-squared:  0.2104 
## F-statistic: 88.65 on 2 and 656 DF,  p-value: < 2.2e-16

checkresiduals(modelca3$model)

## 
##  Breusch-Godfrey test for serial correlation of order up to 10
## 
## data:  Residuals
## LM test = 568.25, df = 10, p-value < 2.2e-16

VIF.modelca3 = vif(modelca3$model)
VIF.modelca3 > 10

##  z.t0  z.t1 
## FALSE FALSE

MASE(modelca3)

##              MASE
## modelca3 1.688457

# koyckDlm:

modelca4 = koyckDlm(x = as.vector(ppt) , y = as.vector(solar))
summary(modelca4)

## 
## 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 
## 
##                              alpha     beta       phi
## Geometric coefficients:  -154.0203 5.346844 0.9854613

checkresiduals(modelca4$model)

VIF.modelca4 = vif(modelca4$model)
VIF.modelca4 > 10

##   Y.1   X.t 
## FALSE FALSE

attr(modelca4$model,"class") = "lm"
AIC(modelca4$model)

## [1] 3946.476

BIC(modelca4$model)

## [1] 3964.439

MASE(modelca4)

##              MASE
## modelca4 1.032483

# ardlDlm:

for (i in 1:15){
  for(j in 1:15){
    modelca5 = ardlDlm(x =as.vector(ppt) ,
                       y = as.vector(solar), p = i , q = j )
    cat("p = ", i, "q = ", j, "AIC = ", AIC(modelca5$model), "BIC = ", BIC(modelca5$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 =  1 q =  11 AIC =  2979.302 BIC =  3046.433 
## p =  1 q =  12 AIC =  2942.604 BIC =  3014.186 
## p =  1 q =  13 AIC =  2929.206 BIC =  3005.235 
## p =  1 q =  14 AIC =  2926.209 BIC =  3006.683 
## p =  1 q =  15 AIC =  2919.972 BIC =  3004.888 
## 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 =  2 q =  11 AIC =  2977.955 BIC =  3049.562 
## p =  2 q =  12 AIC =  2942.125 BIC =  3018.181 
## p =  2 q =  13 AIC =  2928.617 BIC =  3009.12 
## p =  2 q =  14 AIC =  2924.793 BIC =  3009.738 
## p =  2 q =  15 AIC =  2918.361 BIC =  3007.746 
## 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 =  3 q =  11 AIC =  2978.439 BIC =  3054.521 
## p =  3 q =  12 AIC =  2943.273 BIC =  3023.803 
## p =  3 q =  13 AIC =  2929.094 BIC =  3014.068 
## p =  3 q =  14 AIC =  2925.218 BIC =  3014.634 
## p =  3 q =  15 AIC =  2918.579 BIC =  3012.433 
## 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 =  4 q =  11 AIC =  2969.385 BIC =  3049.943 
## p =  4 q =  12 AIC =  2932.294 BIC =  3017.298 
## p =  4 q =  13 AIC =  2917.646 BIC =  3007.093 
## p =  4 q =  14 AIC =  2914.65 BIC =  3008.537 
## p =  4 q =  15 AIC =  2908.197 BIC =  3006.521 
## 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 =  5 q =  11 AIC =  2970.343 BIC =  3055.376 
## p =  5 q =  12 AIC =  2933.501 BIC =  3022.979 
## p =  5 q =  13 AIC =  2918.19 BIC =  3012.109 
## p =  5 q =  14 AIC =  2915.232 BIC =  3013.59 
## p =  5 q =  15 AIC =  2908.636 BIC =  3011.429 
## 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 =  6 q =  11 AIC =  2972.342 BIC =  3061.85 
## p =  6 q =  12 AIC =  2934.744 BIC =  3028.696 
## p =  6 q =  13 AIC =  2919.101 BIC =  3017.493 
## p =  6 q =  14 AIC =  2916.365 BIC =  3019.194 
## p =  6 q =  15 AIC =  2909.475 BIC =  3016.737 
## 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 =  7 q =  11 AIC =  2960.241 BIC =  3054.225 
## p =  7 q =  12 AIC =  2921.752 BIC =  3020.177 
## p =  7 q =  13 AIC =  2908.528 BIC =  3011.392 
## p =  7 q =  14 AIC =  2905.445 BIC =  3012.744 
## p =  7 q =  15 AIC =  2898.172 BIC =  3009.903 
## 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 =  8 q =  11 AIC =  2959.055 BIC =  3057.515 
## p =  8 q =  12 AIC =  2921.816 BIC =  3024.715 
## p =  8 q =  13 AIC =  2908.703 BIC =  3016.039 
## p =  8 q =  14 AIC =  2905.932 BIC =  3017.702 
## p =  8 q =  15 AIC =  2898.381 BIC =  3014.582 
## 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 =  9 q =  11 AIC =  2947.033 BIC =  3049.968 
## p =  9 q =  12 AIC =  2913.567 BIC =  3020.94 
## p =  9 q =  13 AIC =  2899.978 BIC =  3011.787 
## p =  9 q =  14 AIC =  2897.119 BIC =  3013.359 
## p =  9 q =  15 AIC =  2891.053 BIC =  3011.723 
## 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 =  3065.571 
## p =  10 q =  11 AIC =  2946.44 BIC =  3053.851 
## p =  10 q =  12 AIC =  2913.952 BIC =  3025.8 
## p =  10 q =  13 AIC =  2899.906 BIC =  3016.187 
## p =  10 q =  14 AIC =  2897.117 BIC =  3017.829 
## p =  10 q =  15 AIC =  2891.07 BIC =  3016.209 
## p =  11 q =  1 AIC =  3477.139 BIC =  3544.27 
## p =  11 q =  2 AIC =  3162.828 BIC =  3234.434 
## p =  11 q =  3 AIC =  3071.052 BIC =  3147.135 
## p =  11 q =  4 AIC =  3072.21 BIC =  3152.768 
## p =  11 q =  5 AIC =  3043.91 BIC =  3128.944 
## p =  11 q =  6 AIC =  2978.876 BIC =  3068.385 
## p =  11 q =  7 AIC =  2966.892 BIC =  3060.876 
## p =  11 q =  8 AIC =  2968.47 BIC =  3066.929 
## p =  11 q =  9 AIC =  2970.38 BIC =  3073.315 
## p =  11 q =  10 AIC =  2960.572 BIC =  3067.982 
## p =  11 q =  11 AIC =  2948.052 BIC =  3059.937 
## p =  11 q =  12 AIC =  2915.641 BIC =  3031.962 
## p =  11 q =  13 AIC =  2901.696 BIC =  3022.449 
## p =  11 q =  14 AIC =  2898.937 BIC =  3024.119 
## p =  11 q =  15 AIC =  2892.818 BIC =  3022.426 
## p =  12 q =  1 AIC =  3472.271 BIC =  3543.853 
## p =  12 q =  2 AIC =  3160.003 BIC =  3236.06 
## p =  12 q =  3 AIC =  3068.088 BIC =  3148.618 
## p =  12 q =  4 AIC =  3069.332 BIC =  3154.336 
## p =  12 q =  5 AIC =  3041.35 BIC =  3130.828 
## p =  12 q =  6 AIC =  2974.581 BIC =  3068.533 
## p =  12 q =  7 AIC =  2963.3 BIC =  3061.726 
## p =  12 q =  8 AIC =  2964.786 BIC =  3067.685 
## p =  12 q =  9 AIC =  2966.735 BIC =  3074.108 
## p =  12 q =  10 AIC =  2957.638 BIC =  3069.486 
## p =  12 q =  11 AIC =  2946.207 BIC =  3062.529 
## p =  12 q =  12 AIC =  2917.586 BIC =  3038.381 
## p =  12 q =  13 AIC =  2903.694 BIC =  3028.92 
## p =  12 q =  14 AIC =  2900.937 BIC =  3030.59 
## p =  12 q =  15 AIC =  2894.784 BIC =  3028.861 
## p =  13 q =  1 AIC =  3466.396 BIC =  3542.425 
## p =  13 q =  2 AIC =  3149.639 BIC =  3230.141 
## p =  13 q =  3 AIC =  3054.07 BIC =  3139.045 
## p =  13 q =  4 AIC =  3055.592 BIC =  3145.039 
## p =  13 q =  5 AIC =  3029.941 BIC =  3123.86 
## p =  13 q =  6 AIC =  2963.045 BIC =  3061.437 
## p =  13 q =  7 AIC =  2948.98 BIC =  3051.844 
## p =  13 q =  8 AIC =  2950.463 BIC =  3057.799 
## p =  13 q =  9 AIC =  2952.384 BIC =  3064.193 
## p =  13 q =  10 AIC =  2942.543 BIC =  3058.824 
## p =  13 q =  11 AIC =  2930.864 BIC =  3051.617 
## p =  13 q =  12 AIC =  2902.893 BIC =  3028.119 
## p =  13 q =  13 AIC =  2904.307 BIC =  3034.005 
## p =  13 q =  14 AIC =  2901.428 BIC =  3035.552 
## p =  13 q =  15 AIC =  2895.384 BIC =  3033.931 
## p =  14 q =  1 AIC =  3457.014 BIC =  3537.489 
## p =  14 q =  2 AIC =  3142.399 BIC =  3227.344 
## p =  14 q =  3 AIC =  3048.7 BIC =  3138.116 
## p =  14 q =  4 AIC =  3050.305 BIC =  3144.192 
## p =  14 q =  5 AIC =  3025.438 BIC =  3123.796 
## p =  14 q =  6 AIC =  2959.487 BIC =  3062.315 
## p =  14 q =  7 AIC =  2945.846 BIC =  3053.145 
## p =  14 q =  8 AIC =  2947.343 BIC =  3059.113 
## p =  14 q =  9 AIC =  2949.294 BIC =  3065.534 
## p =  14 q =  10 AIC =  2938.922 BIC =  3059.633 
## p =  14 q =  11 AIC =  2926.974 BIC =  3052.156 
## p =  14 q =  12 AIC =  2900.118 BIC =  3029.771 
## p =  14 q =  13 AIC =  2901.711 BIC =  3035.835 
## p =  14 q =  14 AIC =  2903.258 BIC =  3041.852 
## p =  14 q =  15 AIC =  2897.289 BIC =  3040.305 
## p =  15 q =  1 AIC =  3443.884 BIC =  3528.8 
## p =  15 q =  2 AIC =  3131.919 BIC =  3221.304 
## p =  15 q =  3 AIC =  3042.451 BIC =  3136.305 
## p =  15 q =  4 AIC =  3044.249 BIC =  3142.573 
## p =  15 q =  5 AIC =  3020.493 BIC =  3123.286 
## p =  15 q =  6 AIC =  2954.704 BIC =  3061.966 
## p =  15 q =  7 AIC =  2940.973 BIC =  3052.705 
## p =  15 q =  8 AIC =  2942.547 BIC =  3058.747 
## p =  15 q =  9 AIC =  2944.503 BIC =  3065.173 
## p =  15 q =  10 AIC =  2934.634 BIC =  3059.773 
## p =  15 q =  11 AIC =  2923.348 BIC =  3052.956 
## p =  15 q =  12 AIC =  2896.573 BIC =  3030.651 
## p =  15 q =  13 AIC =  2898.068 BIC =  3036.614 
## p =  15 q =  14 AIC =  2899.728 BIC =  3042.744 
## p =  15 q =  15 AIC =  2899.254 BIC =  3046.74

modelca6 = ardlDlm(x =as.vector(ppt) ,
                 y = as.vector(solar), p = 9 , q = 15 )
summary(modelca6)

## 
## Time series regression with "ts" data:
## Start = 16, End = 660
## 
## Call:
## dynlm(formula = as.formula(model.text), data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.9250  -1.0273  -0.1059   1.0047  17.7120 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.72860    0.54818   4.978 8.36e-07 ***
## X.t         -0.25000    0.52374  -0.477  0.63330    
## X.1          0.06512    0.71130   0.092  0.92708    
## X.2          0.84196    0.72275   1.165  0.24449    
## X.3          0.91264    0.72134   1.265  0.20627    
## X.4         -1.06564    0.71401  -1.492  0.13609    
## X.5          0.22576    0.71440   0.316  0.75210    
## X.6         -2.56092    0.71233  -3.595  0.00035 ***
## X.7          2.11687    0.71688   2.953  0.00327 ** 
## X.8          0.81572    0.71332   1.144  0.25326    
## X.9         -1.57762    0.52537  -3.003  0.00278 ** 
## Y.1          1.09456    0.03984  27.472  < 2e-16 ***
## Y.2          0.12209    0.05904   2.068  0.03907 *  
## Y.3         -0.17339    0.05826  -2.976  0.00303 ** 
## Y.4         -0.15757    0.05745  -2.743  0.00627 ** 
## Y.5         -0.19002    0.05761  -3.298  0.00103 ** 
## Y.6          0.10409    0.05818   1.789  0.07408 .  
## Y.7          0.08006    0.05807   1.379  0.16852    
## Y.8         -0.10972    0.05777  -1.899  0.05799 .  
## Y.9          0.12455    0.05809   2.144  0.03242 *  
## Y.10         0.06356    0.05821   1.092  0.27527    
## Y.11         0.11699    0.05791   2.020  0.04380 *  
## Y.12        -0.16714    0.05792  -2.886  0.00404 ** 
## Y.13        -0.06223    0.05791  -1.075  0.28299    
## Y.14        -0.04757    0.05787  -0.822  0.41141    
## Y.15         0.06061    0.03877   1.563  0.11847    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.228 on 619 degrees of freedom
## Multiple R-squared:  0.9508, Adjusted R-squared:  0.9488 
## F-statistic: 478.8 on 25 and 619 DF,  p-value: < 2.2e-16

checkresiduals(modelca6$model)

## 
##  Breusch-Godfrey test for serial correlation of order up to 29
## 
## data:  Residuals
## LM test = 55.251, df = 29, p-value = 0.002316

VIF.modelca6 = vif(modelca6$model)
VIF.modelca6 > 10

##        X.t  L(X.t, 1)  L(X.t, 2)  L(X.t, 3)  L(X.t, 4)  L(X.t, 5) 
##      FALSE      FALSE      FALSE      FALSE      FALSE      FALSE 
##  L(X.t, 6)  L(X.t, 7)  L(X.t, 8)  L(X.t, 9)  L(y.t, 1)  L(y.t, 2) 
##      FALSE      FALSE      FALSE      FALSE       TRUE       TRUE 
##  L(y.t, 3)  L(y.t, 4)  L(y.t, 5)  L(y.t, 6)  L(y.t, 7)  L(y.t, 8) 
##       TRUE       TRUE       TRUE       TRUE       TRUE       TRUE 
##  L(y.t, 9) L(y.t, 10) L(y.t, 11) L(y.t, 12) L(y.t, 13) L(y.t, 14) 
##       TRUE       TRUE       TRUE       TRUE       TRUE       TRUE 
## L(y.t, 15) 
##       TRUE

MASE(modelca6)

##               MASE
## modelca6 0.3812687

# Forecasting:

forecast.ardldlm = dLagM::forecast(model=modelca6, x = c(0.189009998,0.697262522,0.595213491,0.487388526,0.261677017,0.808606651,0.94186202,0.905636325,1.059964682,0.341438784,0.525805322,0.602471062, 0.109860632,0.781464707,0.69685501,0.502413906,0.649385609,0.745960773,0.663047123,0.533770112,0.61542621,0.54606508,0.142673325,0.013650407), h=24)$forecasts

forecast.ardldlm

##  [1]  5.6685145  5.5469002  7.6026670 10.9141865 18.1481627 20.3932993
##  [7] 21.8324065 16.8962240 14.6455820 13.2075047  8.3822395  6.4341929
## [13]  6.2354723 11.4372455 12.6418302  8.3207547  4.2004823  0.9825388
## [19]  9.3370308 17.6871819 23.2888147 25.7715376 16.7694623 14.4553104

plot(ts(c(as.vector(solar), forecast.ardldlm),start=c(1960,1),frequency = 12), ylab = "Solar Radiation", main = "Forecasting of the amount
     of Solar Radiation for the next 2 years Fig 3.5")

Solar (Exponential Smoothing Package):

As it is observed that there is no trend and there is seasonality, the models which includes seasonality components are (Additive or Multiplicative) and might include the error term so the candidate models are:
1. No Trend, Additive Seasonality.
2. No Trend, Additive Seasonality, Damped.
3. No Trend, Multiplicative Seasonality.
4. No Trend, Multiplicative Seasonality, Exponential.
5. No Trend, Multiplicative Seasonality, Damped.
Then by using the first candidate make a model checks its summary and residuals.
Then checked its summary and observed the values of AIC, AICc and BIC are 5434.708, 5435.662 and 5511.076 respectively. The MASE is about 0.24716 and the p-values is also good.
Then checked the residuals (Fig 4.0) they are random, correlation is prominent, pattern in histogram was also almost normalized but ACF has some patterns.
Then by using the second candidate make a model checks its summary and residuals.
Then checked its summary and observed the values of AIC, AICc and BIC are 5428.422, 5429.489 and 5509.282 respectively. The MASE is about 0.2461797 which is slightly better than the previous model and the p-values is also good.
Then checked the residuals (Fig 4.1) they are random, correlation is prominent, pattern in histogram was also almost normalized but ACF has some patterns.
Then by using the third candidate make a model checks its summary and residuals.
Then checked its summary and observed the values of AIC, AICc and BIC are 6648.746, 6649.699 and 6725.114 respectively. The MASE is about 0.2233077 which is smaller than all the previous model and the p-values is also good.
Then checked the residuals (Fig 4.2) they are random, correlation is prominent, pattern in histogram was also almost normalized and ACF also seems fine.
Then by using the fourth candidate make a model checks its summary and residuals.
Then checked its summary and observed the values of AIC, AICc and BIC are 6584.208, 6585.161 and 6660.576 respectively. The MASE is about 0.2320404 which is slightly higher than the previous model and the p-values is ok.
Then checked the residuals (Fig 4.3) they are random, correlation is prominent, pattern in histogram was also almost normalized and ACF also seems fine.
Then by using the fifth candidate make a model checks its summary and residuals.
Then checked its summary and observed the values of AIC, AICc and BIC are 6366.701, 6367.768 and 6447.561 respectively. The MASE is about 0.2035619 which is slightly higher than the previous model and the p-values is ok.
Then checked the residuals (Fig 4.4) they are random, correlation is prominent, pattern in histogram was also almost normalized and ACF also seems fine.
The third, fourth and fifth candidate models gives the best result according to MASE but taking other aspects into consideration like p-value, AIC, AICc, BIC and residual analysis the fifth model of holt’s winter with multiplicative seasonality is the best one.
Then just for testing purposes an automatic ets test was run and it gave us the model “ETS(A,Ad,A)”, so this model was fitted on the solar series and then analysed.
- Then checked its summary and observed the values of AIC, AICc and BIC are 5428.422, 5429.489 and 5509.282 respectively. The MASE is about 0.2461797 which is slightly higher than the model selected, and the p-values is also good.
Then checked the residuals (Fig 4.5) they are random, correlation is prominent, pattern in histogram was also almost normalized but ACF has some patterns.
Then the forecasting was plotted for the best model and found out that this candidate model shows the beter prediction then ardlm as shown in (Fig 4.6).
So finally, the fifth model with the lowest MASE was selected and forecasting was done with the help of that model.
Forecasting was done to calculate the amount of solar radiations for the next two years and the following plot was observed (Fig 4.8), and this model in the exponential smoothing package shows some great forecasting as compared to the one done in dLagM model.
So, at last this forecasting was selected to be the best forecast.

fit1.hw <- hw(solar,seasonal="additive", h=5*frequency(solar))
summary(fit1.hw)

## 
## 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
## Training set -0.1179476 2.328736 1.504489 -2.230359 12.68101 0.24716
##                   ACF1
## Training set 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 46.44828

checkresiduals(fit1.hw)

## 
##  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

fit2.hw <- hw(solar,seasonal="additive",damped = TRUE, h=5*frequency(solar))
summary(fit2.hw)

## 
## 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 40.79252

checkresiduals(fit2.hw)

## 
##  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

fit3.hw <- hw(solar,seasonal="multiplicative", h=2*frequency(solar))
summary(fit3.hw)

## 
## Forecast method: Holt-Winters' multiplicative method
## 
## Model Information:
## Holt-Winters' multiplicative method 
## 
## Call:
##  hw(y = solar, h = 2 * 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       5.610335   2.75534413  8.465326   1.2440033  9.976666
## Feb 2015       6.512806   2.03809262 10.987520  -0.3306776 13.356290
## Mar 2015       8.786120   1.33667747 16.235562  -2.6068190 20.179058
## Apr 2015      10.192785  -0.02672876 20.412298  -5.4366123 25.822181
## May 2015      12.512597  -1.96157245 26.986766  -9.6237347 34.648928
## Jun 2015      14.061609  -4.41048190 32.533701 -14.1890164 42.312235
## Jul 2015      14.502088  -6.89909775 35.903274 -18.2282011 47.232377
## Aug 2015      12.945620  -8.35024040 34.241481 -19.6235881 45.514829
## Sep 2015      10.352210  -8.52365174 29.228072 -18.5159294 39.220350
## Oct 2015       7.418279  -7.51119146 22.347750 -15.4143760 30.250935
## Nov 2015       4.989730  -6.05900058 16.038460 -11.9078451 21.887305
## Dec 2015       3.871776  -5.53876625 13.282319 -10.5204066 18.263960
## Jan 2016       4.199400  -7.01753622 15.416336 -12.9554236 21.354224
## Feb 2016       4.838892  -9.30305408 18.980837 -16.7893479 26.467131
## Mar 2016       6.477174 -14.21845674 27.172805 -25.1740619 38.128410
## Apr 2016       7.452633 -18.56883703 33.474103 -32.3437711 47.249037
## May 2016       9.069749 -25.52978297 43.669280 -43.8456686 61.985166
## Jun 2016      10.099485 -31.99838423 52.197355 -54.2836502 74.482621
## Jul 2016      10.315199 -36.68017471 57.310572 -61.5580226 82.188420
## Aug 2016       9.113765 -36.29212001 54.519651 -60.3285438 78.556075
## Sep 2016       7.208700 -32.09306283 46.510463 -52.8981594 67.315559
## Oct 2016       5.105869 -25.38360449 35.595343 -41.5237568 51.735495
## Nov 2016       3.391945 -18.81647619 25.600367 -30.5729044 37.356795
## Dec 2016       2.597256 -16.07148862 21.266001 -25.9541251 31.148637

checkresiduals(fit3.hw)

## 
##  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

fit4.hw <- hw(solar,seasonal="multiplicative",exponential = TRUE, h=5*frequency(solar))
summary(fit4.hw)

## 
## 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
## Training set 0.06388152 2.208727 1.412454 -1.303792 11.28449 0.2320404
##                  ACF1
## Training set 0.153871
## 
## Forecasts:
##          Point Forecast      Lo 80     Hi 80       Lo 95    Hi 95
## Jan 2015       5.797870 2.97914127  8.639807 1.644427474 10.08164
## Feb 2015       7.513359 3.34951727 12.086812 1.776075505 15.14153
## Mar 2015      10.758212 4.35349718 18.537524 2.106769334 24.63747
## Apr 2015      12.694087 4.57249648 22.654960 2.278843226 31.88680
## May 2015      15.340074 4.94165089 29.167861 2.507785659 41.06949
## Jun 2015      17.239431 5.11339134 33.674137 2.569511030 50.43154
## Jul 2015      18.004700 4.87967149 35.945475 2.368986841 57.32599
## Aug 2015      16.206596 4.13374510 33.623020 1.848377916 55.02534
## Sep 2015      13.011256 3.03797566 26.719663 1.384463300 46.37224
## Oct 2015       9.215869 1.95869287 19.842040 0.778253176 33.66363
## Nov 2015       6.214762 1.23796550 13.835791 0.484541733 24.02569
## Dec 2015       4.565667 0.79942799 10.072896 0.318601385 18.19733
## Jan 2016       5.220661 0.85191836 11.844437 0.333097263 22.00930
## Feb 2016       6.765364 0.97461353 15.679979 0.411623568 30.40340
## Mar 2016       9.687175 1.32055366 22.405436 0.490075887 46.21142
## Apr 2016      11.430324 1.49942805 25.309714 0.545389487 54.52494
## May 2016      13.812889 1.72334790 32.375938 0.633684274 66.54740
## Jun 2016      15.523155 1.76130552 37.165623 0.683339891 77.38735
## Jul 2016      16.212237 1.73688573 39.112202 0.667969779 77.79028
## Aug 2016      14.593143 1.42714888 35.783835 0.526816671 73.99590
## Sep 2016      11.715916 1.09720777 28.867406 0.386589873 62.97202
## Oct 2016       8.298381 0.73432376 20.764577 0.254772202 44.81023
## Nov 2016       5.596049 0.47498064 13.634956 0.165502522 31.96281
## Dec 2016       4.111130 0.30693173 10.103351 0.102762210 23.08154
## Jan 2017       4.700916 0.34077620 11.455586 0.108451855 27.93883
## Feb 2017       6.091836 0.41153565 14.623204 0.142476360 34.24187
## Mar 2017       8.722765 0.57272432 21.361564 0.171126426 51.21713
## Apr 2017      10.292374 0.62521372 25.330971 0.203461389 61.93930
## May 2017      12.437743 0.69422891 31.462603 0.220746702 72.89489
## Jun 2017      13.977742 0.72734228 34.815583 0.221942090 83.10217
## Jul 2017      14.598223 0.73967972 35.475739 0.215652517 90.10927
## Aug 2017      13.140319 0.61010909 31.536671 0.175293660 84.00275
## Sep 2017      10.549535 0.46031504 25.634413 0.137920006 67.83462
## Oct 2017       7.472233 0.30713038 18.215372 0.096019860 49.37026
## Nov 2017       5.038933 0.19667979 11.903813 0.056634514 33.79673
## Dec 2017       3.701845 0.13325885  8.771328 0.034893897 26.50091
## Jan 2018       4.232915 0.14176887 10.110233 0.042735728 29.03712
## Feb 2018       5.485362 0.17374491 13.180913 0.042337983 39.63338
## Mar 2018       7.854368 0.23010252 18.916882 0.058274558 52.68137
## Apr 2018       9.267714 0.26941164 21.272417 0.069594807 67.23293
## May 2018      11.199499 0.30989087 26.881461 0.083207249 81.01326
## Jun 2018      12.586184 0.31792254 29.445596 0.086571893 91.66146
## Jul 2018      13.144893 0.31260736 31.058934 0.082098956 95.09699
## Aug 2018      11.832130 0.27412365 29.193981 0.067111573 95.12446
## Sep 2018       9.499273 0.19752408 22.514504 0.049742958 72.99732
## Oct 2018       6.728333 0.13708672 16.501488 0.035693856 49.00958
## Nov 2018       4.537281 0.08776856 10.737959 0.020983370 30.66318
## Dec 2018       3.333307 0.06055159  7.870714 0.013673368 23.02917
## Jan 2019       3.811506 0.06522347  9.044785 0.014436128 28.36450
## Feb 2019       4.939265 0.08028840 11.336514 0.019510520 36.49698
## Mar 2019       7.072424 0.10623340 16.277128 0.025812970 54.43265
## Apr 2019       8.345064 0.12275074 19.162108 0.028530524 61.27708
## May 2019      10.084530 0.14091459 22.351285 0.029835468 75.21518
## Jun 2019      11.333163 0.14788676 24.899355 0.029638167 91.86197
## Jul 2019      11.836249 0.14232035 26.717738 0.027589269 90.80075
## Aug 2019      10.654179 0.12651664 23.945552 0.025190493 82.86509
## Sep 2019       8.553570 0.08925397 19.128946 0.020093198 66.01332
## Oct 2019       6.058491 0.06521343 13.737042 0.012525610 48.66728
## Nov 2019       4.085570 0.04060032  9.375168 0.008598106 33.17750
## Dec 2019       3.001459 0.02816107  6.415085 0.005961340 23.82144

checkresiduals(fit4.hw)

## 
##  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: 24

fit5.hw <- hw(solar,seasonal="multiplicative",damped = TRUE, h=5*frequency(solar))
summary(fit5.hw)

## 
## 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  71.382889

checkresiduals(fit5.hw)

## 
##  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

fit.auto.bonds = ets(solar,model="ZZZ",ic="bic")
fit.auto.bonds$method

## [1] "ETS(A,Ad,A)"

fit6 = ets(solar)
summary(fit6)

## ETS(A,Ad,A) 
## 
## Call:
##  ets(y = solar) 
## 
##   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 0.1700724

checkresiduals(fit6)

## 
##  Ljung-Box test
## 
## data:  Residuals from ETS(A,Ad,A)
## Q* = 210.76, df = 7, p-value < 2.2e-16
## 
## Model df: 17.   Total lags used: 24

plot(forecast(fit5.hw$model), ylab="Solar",plot.conf=FALSE, type="l", xlab="Year", main = "Forecasting for next two years Fig 4.6")

Conclusion:

After running all the models in dLagM and exponential smoothing, ardlm comes up with the best model in dLagM and the third, fourth and fifth models come up with the best MASE so at last by comparing all the models and by analysing from all the angles the fifth model (Fig 4.6) which is No Trend, Multiplicative Seasonality, Damped was selected as the best model among all and forecasting was done.

Task 2:

Introduction:

The following data is about the quarterly Residential Property Price Index (PPI) in Melbourne and quarterly population change over the previous quarter in Victoria between September 2003 and December 2016.
This data was first imported, checked the class of the data and head of it.
Then converted each column to time series using the ts function separately.
Then joined the series of price and change where they were common in frequency and then plotted them (Fig 5.0). Both the graphs show that there is a clear trend while the change series also have seasonality in it.
Then checked the CCF (Fig 5.1) and observed that there is a clear pattern showing that there is seasonality in the series.
Then difference was taken for both the series and then again joined.
And finally, by using the prewhiten function checked the CCF (Fig 5.2) of the series and observed that no significant lag now left in the series.
So, this makes it clear that there is now no correlation between them.

PPI <- read_csv("data2.csv")

## Parsed with column specification:
## cols(
##   Quarter = col_character(),
##   price = col_double(),
##   change = col_double()
## )

class(PPI)

## [1] "spec_tbl_df" "tbl_df"      "tbl"         "data.frame"

head(PPI)

## # 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

Price_ts <- ts(PPI$price,start=c(2003,3),frequency = 4)
Price_ts

##       Qtr1  Qtr2  Qtr3  Qtr4
## 2003              60.7  62.1
## 2004  60.8  60.9  60.9  62.4
## 2005  62.5  63.2  63.1  64.0
## 2006  65.4  67.2  68.1  69.5
## 2007  71.0  76.0  79.5  84.7
## 2008  85.7  85.5  83.0  82.3
## 2009  82.5  86.9  92.5  98.6
## 2010 103.3 106.2 104.3 105.7
## 2011 104.7 103.5 101.3 100.0
## 2012  99.4  99.3  98.6 100.4
## 2013 100.8 102.7 105.9 109.7
## 2014 110.7 112.1 113.1 115.2
## 2015 115.9 120.8 124.3 126.3
## 2016 127.3 130.7 132.9 140.0

Change_ts <- ts(PPI$change,start=c(2003,3),frequency = 4)
Change_ts

##       Qtr1  Qtr2  Qtr3  Qtr4
## 2003             14017 12350
## 2004 17894  9079 16210 13788
## 2005 21195 10904 16995 16962
## 2006 25004 13059 22327 20372
## 2007 30109 19448 24993 20988
## 2008 33497 23375 30174 26736
## 2009 34387 24262 26940 20375
## 2010 25923 15929 17609 17001
## 2011 24667 17439 27892 27378
## 2012 34569 25773 29453 29174
## 2013 35704 28048 32942 29061
## 2014 37877 26282 33154 31115
## 2015 38857 27872 32948 31683
## 2016 43701 37949 32275 32703

Joint = ts.intersect(Price_ts, Change_ts)
plot(Joint, yax.flip=T, main = "Joint Fig 5.0")

ccf(as.vector(Joint[,1]), as.vector(Joint[,2]),ylab='CCF', main = "Sample CCF between Fig 5.1")

me.dif=ts.intersect(diff(diff(Price_ts,4)),diff(diff(Change_ts,4)))
prewhiten(as.vector(me.dif[,1]),as.vector(me.dif[,2]),ylab='CCF', main = "Sample CFF after prewhitening Fig 5.2")

Conclusion:

By analysing the data and then by applying pre-whitening, and having a look at the CCF plot, now there is no correlation between them.

Assignment 2

Syed Hassan Afsar

16 September 2019

Required Packages:

Task 1:

Introduction:

Analysis:

Solar with Precipitation (dLagM Package):

dlm:

finiteDLMauto:

polyDlm:

koyckDlm:

ardlDlm:

Forecasting:

Solar (Exponential Smoothing Package):

Conclusion:

Task 2:

Introduction:

Conclusion: