Minitab Project Report - Practical 2. Time Series Plot of Electricity. Index

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1 Problem : Australian Electricity Production Minitab Project Report - Practical Plot of the Data Australian Electricity Production Data Time Series Plot of Electricity Electricity Since these are monthly data, the period length is. The data indicate clear trend and seasonality. Over the years, the production steadily increases, though it has slowed down slightly during the last four years. The seasonal effects are additive, as they seem to be similar over all periods and the variability does not increase with time. There is a higher production during May to August than the rest of the year; a clear peak is in y. Small Trend Method We assume the additive model Xt = mt + st + Yt, () where E(Yt) =, () st = st-d (the seasonality effect is assumed to be the same for the same seasons), () and Σsk = (the seasonality effects over one period sum to zero). () The small trend method relies on the assumption that the trend is constant over each period and it is estimated by an average of the observations for all seasons in the period. Each seasonal component is estimated by an average of the detrended data over the observations for the given season in all periods.

2 Time Series Plot of Electricity, m_t Variable Electricity m_t Data Comments: The detrended data set (x-m, where m is the mean over the months of a year) is still showing seasonal effects and noise effects, but no trend. The values fluctuate about zero. The seasonal effects (s) show a peak in electricity production in y and the lowest value in February. The deseasonalised data show a clear upward trend in electricity production over the years. The residuals are free of the trend and seasonal effects. The TS looks stationary with mean zero. Note that this analysis is done based on full years. The last year has missing data for the last four months and so the estimates of seasonal effects are affected by the missing values. Hence, this method should only be used for complete date in all periods.

3 Differencing Method Assuming model (), we may remove the trend or seasonal effects by differencing. Differencing by the lag equal to the length of the period (lag = d) may remove the seasonal effect, assuming (). Differencing the deseasonalized data by a small lag may remove the trend. For example, we take lag = if the trend is approximately linear. Deseasonlized data: Electricity production in Australia Deseasonlized and detrended data: Electricity production in Australia diff diffdiff Month Year Month Year Comments: The deseasonalised data show some non-constant trend. The residuals obtained by additional differencing with lag of the differenced data with lag are scattered about zero. However, there are some bursts, which increase the variance of the residuals; the variance may not be constant. So the residuals may not be stationary. Minitab Decomposition Method Time Series Decomposition Plot for Electricity Additive Model Variable Actual Fits Trend Electricity Accuracy Measures MAPE MAD MSD

4 Component Analysis for Electricity Additive Model Original Data Detrended Data Data Detr. Data - Seas. Adj. Data Seasonally Adjusted Data Seas. Adj. and Detr. Data Seasonally Adj. and Detrended Data - Seasonal Analysis for Electricity Additive Model Seasonal Indices Detrended Data, by Seasonal Period - - Percent Variation, by Seasonal Period Residuals, by Seasonal Period - Comments: This method uses a linear trend approximation, which, for this data set, does not completely remove the trend from the data, as seen in the Detrended Data plot. The deseasonalised data indicate an upward trend with a small bend around index, which is year. The residuals still carry the trend features. The seasonal indices show that the maximum production is in y and the minimum in February. The highest seasonal variation is in August, although the variation by seasonal period is not large.

5 Comparison of the Methods Summary for N(N) Summary for RESI A -Squared. P-Value <. A -Squared. P-Value <. -. StDev. V ariance. Skew ness -. Kurtosis. N. StDev. V ariance. Skew ness. Kurtosis. N - - Minimum -. st Q uartile rd Q uartile. Maximum. % C onfidence Interv al for Minimum -. st Q uartile rd Q uartile. Maximum. % C onfidence Interv al for % Confidence Intervals % C onfidence Interv al for StDev.. % Confidence Intervals % C onfidence Interv al for StDev The histogram of the residuals for the Small Trend Method has a shape closest to a normal distribution. The residuals have the smallest variance and the test does not reject the null hypothesis of normality. Hence, this method seems to produce the best result, removing the trend and seasonality most successfully.

6 Australian Beer Production Data Beer Production in Australia,. - Aug. Beer Production Month Year Since these are monthly data, the period length is. This data set shows possible seasonal effects and a small decreasing trend. The beer production is highest in November and December, the summer months in Australia. Comparison of the Methods Summary for Diff_ Summary for RESI A -Squared. A -Squared. P-V alue. P-V alue.. StDev. V ariance. -. StDev. V ariance. Skew ness -. Kurtosis -. N Skew ness. Kurtosis. N Minimum -. st Q uartile -. Minimum -. st Q uartile rd Q uartile. Maximum rd Q uartile. Maximum. % C onfidence Interv al for % C onfidence Interv al for % Confidence Intervals % C onfidence Interv al for StDev.. % Confidence Intervals % C onfidence Interv al for StDev For the Australian beer production data, there is no clear difference between the methods, when compared with respect to their residuals. The Small Trend Method gives residuals with the smallest range and the largest p-value for the Anderson- Darling test for normality, however the residuals are obtained from the four full year data only.