Time Demand Temperature
Min. :2012-01-01 00:00:00 Min. :2858 Min. : 1.50
1st Qu.:2012-09-30 22:52:30 1st Qu.:3969 1st Qu.:12.30
Median :2013-07-01 22:45:00 Median :4635 Median :15.40
Mean :2013-07-01 22:45:00 Mean :4665 Mean :16.27
3rd Qu.:2014-04-01 23:37:30 3rd Qu.:5244 3rd Qu.:19.40
Max. :2014-12-31 23:30:00 Max. :9345 Max. :43.20
Date Holiday
Min. :2012-01-01 Mode :logical
1st Qu.:2012-09-30 FALSE:51120
Median :2013-07-01 TRUE :1488
Mean :2013-07-01
3rd Qu.:2014-04-01
Max. :2014-12-31
The original dataset contains 48 half-hourly observations per day. I aggregated these observations into daily total electricity demand so that the overall annual pattern could be examined more clearly. Daily maximum temperature and holiday status were also retained for use as external predictors in the dynamic regression analysis.
Date Demand Temperature Holiday
Min. :2013-01-01 Min. :168.2 Min. :10.20 Mode :logical
1st Qu.:2013-04-02 1st Qu.:207.3 1st Qu.:16.30 FALSE:355
Median :2013-07-02 Median :224.0 Median :19.80 TRUE :10
Mean :2013-07-02 Mean :223.2 Mean :21.13
3rd Qu.:2013-10-01 3rd Qu.:241.8 3rd Qu.:24.40
Max. :2013-12-31 Max. :311.4 Max. :40.60
Day_Type
Length:365
Class :character
Mode :character
#Check for the missing data or NAvic_elec_2013|>summarise(Missing_Demand=sum(is.na(Demand)),Missing_Temperature=sum(is.na(Temperature)),Missing_Holiday=sum(is.na(Holiday)) )
The time-series plot shows several large spikes in electricity demand, particularly near the beginning and end of the year. Because these periods correspond to the Australian summer, the higher demand may be partly associated with increased air-conditioning use. However, electricity demand may also rise during colder periods because of heating, so temperature should be examined as a potentially nonlinear external predictor.
Original Series ACF and PACF
vic_elec_2013 |>gg_tsdisplay( Demand,plot_type ="partial",lag_max =36 ) +labs(title ="Daily Electricity Demand with ACF and PACF")
Although the original time-series plot appears to show higher electricity demand during the beginning and end of the year, this should not be interpreted as monthly seasonality. ACF analysis reveals strong autocorrelation at multiples of seven days rather than thirty days, indicating that the dominant seasonal pattern is weekly. The peaks observed during the summer months are more likely driven by changes in temperature than by a repeating monthly cycle. Therefore, a 7-day seasonal difference is more appropriate than a 30-day difference, while temperature can be incorporated later as an external predictor in the ARIMAX model.
Warning: Removed 1 row containing missing values or values outside the scale range
(`geom_line()`).
Warning: Removed 1 row containing missing values or values outside the scale range
(`geom_point()`).
After applying the first difference, the series appears to fluctuate around a relatively constant mean with no obvious long-term trend, suggesting that the trend component has been largely removed. However, the ACF still shows strong significant spikes at lags 7, 14, 21, and 28, indicating that a clear weekly seasonal pattern remains.Therefore, an additional seasonal difference with a lag of 7 days may be required before fitting an ARIMA model.
Second Difference
vic_elec_2013 |>gg_tsdisplay(difference(Demand, lag =7),plot_type ="partial" )
Warning: Removed 7 rows containing missing values or values outside the scale range
(`geom_line()`).
Warning: Removed 7 rows containing missing values or values outside the scale range
(`geom_point()`).
After applying a seasonal difference with a lag of 7 days, the weekly seasonal pattern has been substantially reduced. The differenced series fluctuates around a relatively constant mean without any obvious long-term trend, suggesting that the seasonal non-stationarity has largely been removed.
# A mable: 1 x 2
Manual_ARIMA Auto_ARIMA
<model> <model>
1 <ARIMA(1,0,1)(0,1,0)[7]> <ARIMA(2,0,0)(2,1,2)[7]>
report(arima_models)
Warning in report.mdl_df(arima_models): Model reporting is only supported for
individual models, so a glance will be shown. To see the report for a specific
model, use `select()` and `filter()` to identify a single model.
The manually specified ARIMA(1,0,1)(0,1,0)[7] model was chosen based on the ACF and PACF plots, which showed strong lag-1 autocorrelation and partial autocorrelation after seasonal differencing. However, the automatically selected ARIMA model achieved substantially lower AIC, AICc, and residual variance, indicating a much better fit to the data. This demonstrates that while ACF and PACF are useful for proposing candidate models, automatic model selection can identify more appropriate ARIMA structures by evaluating a wider range of parameter combinations.
The winning Accuracy results of the model prediction was Auto ARIMA. All the accuracy scores of Auto Arima was higher than manual one. It was expected to have such results because of its metric that was already represented as AIC, AICc and BIC score.
Part 2. Dynamic Regression
vic_elec_2013|>ggplot(aes(x= Temperature, y= Demand))+geom_point()+geom_smooth(method="lm", formula = y~x+I(x^2))+labs(title ="Electricity Demand and Temperature",subtitle ="Quadratic relationship",x ="Daily Maximum Temperature",y ="Daily Electricity Demand (GWh)" )
As the temperature was going extreme value to 10 degree and 40 degree, the demand of the electricity has increased, so we can assume that there is a correlation of demand between temperature. Also a quadratic temperature term was included because electricity demand may increase during both extremely hot and relatively cold conditions.
Series: Demand
Model: TSLM
Residuals:
Min 1Q Median 3Q Max
-54.257 -14.096 1.035 18.842 75.347
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.155e+02 2.500e+00 86.193 < 2e-16 ***
I(Temperature^2) 1.589e-02 4.403e-03 3.608 0.000352 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 24.96 on 363 degrees of freedom
Multiple R-squared: 0.03462, Adjusted R-squared: 0.03196
F-statistic: 13.02 on 1 and 363 DF, p-value: 0.00035197
Same as the expected relationship from the graph, there is a positive relationship of temperature and Demand. The hypothesis that the temperature is causing demand increase.
The residual diagnostics indicate that the dynamic regression model does not fully account for the autocorrelation in electricity demand. Therefore, an ARIMAX model is used to combine external predictors with errors, capturing the remaining temporal dependence and improving forecast performance.
ARIMAX model
models <- train |>model(TSLM =TSLM( Demand ~I(Temperature^2)+ Temperature ),ARIMAX =ARIMA( Demand ~I(Temperature^2)+ Temperature ) )forecast_models<- models|>forecast(new_data = test)forecast_models
# A tibble: 2 × 10
.model .type ME RMSE MAE MPE MAPE MASE RMSSE ACF1
<chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 ARIMAX Test -5.52 19.4 13.8 -3.36 7.00 NaN NaN 0.710
2 TSLM Test -14.5 27.8 22.1 -8.15 11.4 NaN NaN 0.434
The ARIMAX model achieved lower RMSE and MAE than the dynamic regression model. By incorporating ARIMA errors, the model successfully accounted for the residual autocorrelation that remained after fitting the regression model. As a result, ARIMAX produced more accurate forecasts while retaining the influence of external predictors such as temperature and day type.
