International Journal of Modern Research in Engineering & Management (IJMREM) ||Volume|| 2 ||Issue|| 6 ||Pages|| 25-36 || June 2019 || ISSN: 2581-4540
The Predictive Capacity of the Holt-Winters Model Using Different Initialization Methods. An aplicación to the Agricultural Sector 1,
Juan Manuel Izar-Landeta 2, Carmen Berenice Ynzunza-Cortés, 3,Raúl Ignacio Hernández-Molinar Researcher, Universidad del Centro de México Researcher, Universidad Tecnológica de Querétaro Program Coordinator, Universidad Autónoma de San Luis Potosí
----------------------------------------------------ABSTRACT-----------------------------------------------------This work introduces the application of the Holt-Winters (HW) Model to forecast the consumption of electrical energy using three different initialization methods. The model is used with historical data associated to electricity consumption of a produce farm located in the state of San Luis Potosi, Mexico. Three smoothing constants are employed, which are: α to consider the variable level; β for its tendency; and γ, for seasonality. The idea is to propose an evaluation method that minimizes the prediction error of forecasting technique. To get it, the first step consists in verify that HW model is feasible, that means that the variable presents tendency and seasonality. Then the results are estimated using three initialization methods, and a comparison of them is realized. The findings show that the best option has been the second one, since it has a lower error value when the method considers the initial smoothing constants and their optimal values. The obtained results prove that the application of this type of forecasting methods can contribute to improve the decisions making in order to save in consuming electrical energy for irrigation and agricultural activities as well as increasing economic benefits.
KEYWORDS: Electrical Power Consumption, Electrical Power Forecast, Holt-Winters Model, Seasonality, Trend. ----------------------------------------------------------------------------------------------------------------------------- ---------Date of Submission: Date, 18 July 2019 Date of Publication: 29. July 2019 ----------------------------------------------------------------------------------------------------------------------------- ----------
I.
INTRODUCTION
Globalization, market changes and other factors have had a significant impact on competitiveness and agricultural sector profitability in all countries of the world, affecting small producers in developing nations to a greater extent. The activities associated with the agricultural crop production process are affected by a number of factors: environmental, social and financial, which have had a significant effect on expected profits, productivity and profitability.The agricultural sector in Mexico consists mostly of small family producers, oriented towards the production of commodities, fruits and vegetables, which depend on the income generated by the field for their livelihood and financing. Decrease in subsidies for agricultural activity, price regulation and increased agricultural inputs and associated indirect costs, such as electricity consumption, are having a significant impact on the sector's revenues which requires from agricultural producers to establish business strategies and new policy schemes. Companies or individuals engaged in agricultural activities use electricity for their irrigation water pumping processes or cooling systems for the conservation of vegetables or fruits, so the forecast of electricity demand assumes an important role in the profitability of the sector, especially by the use of pumping systems that consume energy the most. Similarly, changes in electricity demand come as a result of weather conditions and hourly rates, which can cause costs to fluctuate significantly. Also, because of the type of equipment used and the habits of use or consumption, which for decades had been favored by government subsidies granted to the agricultural electricity tariffs. Agricultural pumping in the country is one of the sectors that consume the greatest amounts of electricity, being the fourth in importance. On the other hand, the cost of generation and electricity tariffs are under the control of the government, subject to monthly adjustments and have shown an upward trend in recent years, in addition to the signs about the effect on the product market prices of the granted grant, and recent changes in the subsidies and the trend towards eliminating a preferential tariff quota for those using water pumping and pumping systems for agricultural use, which has led to the decline in the profitability of the sector. Based in the previous information, the objective of this research, focuses on the application of the HW methodology as a tool to plan electricity consumption and related decision-making to improve the operational efficiency of this type of companies, their competitiveness and profitability,
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The Predictive Capacity of the Holt-Winters Model Using‌ considering that in organizations, many of the decisions are made from forecast models, which are of the utmost use and very useful, since from them it is possible to give certainty to operations related to production processes, such as planning and control, assuming of course, that in all these models is present an error value in the prediction that can have a significant impact on actions, especially those related to demand and supply of articles and/or services in production, logistics and distribution environments. In this study, a series of energy consumption data from a small orange-producing agricultural farm in the San Luis Potosi orange producer region has been taken. The HW model has been applied with three different initialization methods for obtaining the starting values of the level, trend and seasonal factors, to determine which the best of the three methods is. In all cases, the values of the smoothing constants that lead to the minimization of the forecast error have been optimized.
II.
LITERATURE REVIEW
Forecast models are a valuable tool for operational management and their use has been extended to various fields. This is because it allows the assessment of the behavior or seasonality patterns of multiple variables, including prediction systems electricity demand [1], [2].Several methodologies have been proposed seeking to improve methods and minimize error in prediction, by considering, for example, not only the structure of the data series, but also the linear or nonlinear effects of the analysis variables [3] through various models, such as the univariates of exponential smoothing and double seasonality [4], [5]; the integrated self-regressive models of the moving average (Aritma), dynamic regression or hybrid approaches that seek to improve predictions, in this case energy prices [6], [7], [8], among others. The HW methodology has been applied during the last six decades to numerous cases to estimate the forecast variables that show characteristics of trend and seasonality [9]. Others [10] assert that HW technique is still a good method for estimating forecasts when the variable shows seasonality and trend. Among its benefits are: its simplicity, that require small space for data storage, easily programmable and easy adaptation to changes in trend and seasonality patterns, as are new data of the predicted variable. HW methodology produces very good forecasts with a minimal amount of calculations to identify and apply the model [11], [12]. However, there are researchers who point out that the HW does not provide a good estimation of prediction intervals [13], [14], [15]. Other scholars emphasize the importance of properly choose initial conditions to apply the HW Model [16]. The reason for this is that some academics have incorporated to the technique, other features that allow it to address: (1) the management of atypical data, (2) patterns of seasonality and (3) the obtaining of good confidence intervals for the variable to predict. For example, one academic has extended the application of HW to deal with double and triple patterns of seasonality [17] and others have provided better methods for estimating confidence intervals [18]. Similarly, other authors obtained via maximum likelihood the demand confidence intervals, the initial parameter values and smoothing constants [19], while others have applied the multiplicative version of the HW model to obtain the predicted variable intervals [13]. Other author [20] raised the normalization of the seasonal factors for additive models, concluding that this is not necessary, since a small change in the smoothing constants accomplishes the same effect. In addition, he suggests a more simple and efficient technique to make other normalization process of these factors. In this sense, another academic has proposed new rules for the estimation of the initial conditions applying the HW multiplicative model which derive from the initial conditions combined with the methodology of the moving average-weighted, which generates best values of the predicted variable [21].Others have applied a model of purchasing management to supply inventory using HW to forecast demand, in order to achieve different objective service levels, which has resulted in lower costs of the capital invested in inventory, as well as reducing depletion of stocks [22].Other scholars [19] have used the HW additive model and others the multiplicative model to forecast the air transportation demand for passengers with very good results [15]. Similarly, other academics have used it successfully to forecast the traffic of tourists in 3 Serbian cities [23]. Other authors [24] have employed the HW methodology with cushioned trend, along with a model of neural networks for the forecast of sales. Others have used it to predict univariate time series with atypical data [25]. Other scholars [26] have used HW with ARIMA for forecasting goods demand with seasonality and trend with irregular components. Using the HW methodology, others have forecasted the demand for products from a Mexican cosmetics company that handles more than 36000 items, thus having achieved significant improvements in the management of their inventories [27]. Similarly, other academics using spreadsheet have developed forecasts with HW for sales or demand data for goods and by nonlinear programming have optimized the initial values of the model components. They point out that there is no empirical evidence as to whether any of the initialization techniques are better than the others [28]. On the other hand, other authors have successfully predicted with a multiplicative HW model, the cost of electricity in the National Health Service (NHS), whose data are fitted to a Weibull distribution function [29]. Similarly, others have applied this methodology in conjunction with multiple regression to forecast the heat load for an energy company in Slovenia [30].
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The Predictive Capacity of the Holt-Winters Model Using‌ III.
METHODOLOGY
To carry out the development of this research, data were used on the electricity consumption of the agricultural farm located in the middle area of San Luis Potosi, which is considered to be one of the regions with the highest production of citrus fruits in the country. The prediction model applied was the multiplicative Holt-Winters model using the energy consumption data set from 2015 to 2018. The Holt-Winters model, also known as triple exponential smoothing, uses three smoothing constants, one by the variable level, one by its tendency and one more for seasonality. This technique applies in cases where the variable to be predicted behaves trendy and seasonally, i.e. its values vary cyclically over time, being higher in some periods and smaller in others. For this reason, the first recommendation is to observe how the variable has behaved in the past and if it shows seasonal and trendy behavior, then the Holt-Winters (HW) methodology is advisable. This technique is divided into two steps: additive and multiplicative, which is presented in this study, considering that the difference between the two steps is the way in calculating the forecast of the variable. The model uses three smoothing constants, which are: Îą, to include the variable level; β, to consider its tendency; as well as Îł, to include seasonality; these values must be between zero and one. If they approach to zero, more weight is given to the former data, or if their value is close to one, to give more weight to the most recent data. It is usually suggested to use values of the 3 constants between 0.1 and 0.4. (1) to get the level in the period t is [31]: đ?‘Œđ?‘Ą (1) đ??żđ?‘Ą = đ?›ź + (1 − đ?›ź)(đ??żđ?‘Ąâˆ’1 + đ?‘‡đ?‘Ąâˆ’1 ) đ?‘†đ?‘Ąâˆ’đ?‘? Being: đ??żđ?‘Ą = Estimated level for the period t đ??żđ?‘Ąâˆ’1 = Estimated level for the period t-1 đ?‘‡đ?‘Ąâˆ’1 = Value of the trend in the period t-1 đ?‘Œđ?‘Ą = Actual value of the predicted variable in the period t St-p = Seasonal factor from the previous period t Îą = Smoothing Constant for Level (2) for calculating the trend is [31]: đ?‘‡đ?‘Ą = đ?›˝(đ??żđ?‘Ą − đ??żđ?‘Ąâˆ’1 ) + (1 − đ?›˝)đ?‘‡đ?‘Ąâˆ’1 Being β the smoothing constant for the trend. The seasonal factor St, is obtained with (3) [31]: đ?‘Œ đ?‘†đ?‘Ą = đ?›ž đ?‘Ą + (1 − đ?›ž)đ?‘†đ?‘Ąâˆ’đ?‘?
(2)
(3)
đ??żđ?‘Ą
Being Îł the smoothing constant for the seasonality. Applying (4) finally you get the smoothed forecast of the variable đ?‘ŒĚ‚đ?‘Ą , [31]: đ?‘ŒĚ‚đ?‘Ą = (đ??żđ?‘Ąâˆ’1 + đ?‘?đ?‘‡đ?‘Ąâˆ’1 )đ?‘†đ?‘Ąâˆ’đ?‘? (4) To be able in applying these equations, the value of the smoothing constants must be considered, as well as initial estimates of the level and trend values in the previous immediate period, as well as the seasonal factors of a previous season. For this, three different approximations are used, in order to compare them with each other, to see which of them the best one is. To assess the goodness of each approximation, the expected value of the mean absolute percentage error (MAPE) is estimated, which can be defined using (5) [31]: đ?‘€đ??´đ?‘ƒđ??¸ =
Ě‚đ?‘Ą đ?‘Œđ?‘Ą −đ?‘Œ ∑đ?‘› | 1| đ?‘Œđ?‘Ą
(5)
đ?‘›
Being Yt the actual value of the variable, đ?‘ŒĚ‚đ?‘Ą is its predicted value and n the number of data. For the initial estimation of the parameters, this work includes the following three options: (a) the simplest, which assigns the last real value to the variable to be predicted at the zero level that corresponds the zero period, with trend zero, and all the seasonal factors equal to one; (b) an intermediate option that consists in making a linear regression to the complete data set, placing the y-intercept as the value of the level of the zero period, the trend equal to the value of the slope of the regression equation and the seasonal factors are obtained from the average of the previous years based on the respective available information; and (c) the most elaborated option, which implies the application of (6) to (10).
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The Predictive Capacity of the Holt-Winters Model Using‌ (6) is useful to estimate the initial tendency value T 0: Ě…Ě…Ě…Ě… Ě…Ě…Ě…Ě… đ??ˇ −đ??ˇ
đ?‘‡0 = 2 1 (6) đ?‘› Ě…Ě…Ě…1 are the average previous demands for the year 2 and year 1 respectively, n is the number of Where Ě…Ě…Ě… đ??ˇ2 đ?‘Ś đ??ˇ anual seasons. Then, the initial value for the demand level L0 is obtained: đ?‘›+1 đ??ż0 = Ě…Ě…Ě… đ??ˇ1 − đ?‘‡0 2 All the terms have been defined previously.
(7)
Then the subsequent values of L can be calculated with (8): đ??żđ?‘Ą = đ??ż0 + đ?‘›đ?‘‡0 (8) That is, each next L is incremented by the initial value of trend T 0. Seasonal factors for previous years are then obtained by ratio of the demand for the period and the value of the smoothed forecast: đ?‘Œđ?‘Ą đ?‘†đ?‘Ą = (9) đ??żđ?‘Ą Then using the previous years, the average value of the seasonal factors đ?‘†đ?‘Ą is obtained and in case their sum is not n, they should be normalized with (10): đ?‘› (10) đ?‘†đ?‘Ąâˆ’đ?‘› = đ?‘†đ?‘Ą đ?‘†đ?‘˘đ?‘š Being St-n the normalized seasonal factor and the sum is related to the non-standard factors. After this transformation, it is possible to define the initial values of the demand level, trend and previous normalized seasonal factors. Although there are other techniques to initialize the HW model, these three are presented in this work, in order to make a comparison and assess which of them has the best fit for the case of the energy cost data consumed in the produce farm. Based on the results, the optimization of the forecast adjustment will be made using Excel's Solver, looking for the values of Îą, β y Îł that minimize the sum of the data errors, as measured by the MAPE indicator. This is made using a nonlinear programming approach.
IV.
APPLICATION TO ENERGY CONSUMPTION IN THE PRODUCE FARM
To apply the HM model, the quarterly energy consumption costs of an orange produce farm in the state of San Luis PotosĂ, in Mexico, are registered, and data are shown in Table 1: Table 1. Quarterly energy consumption of the produce farm Year 2015
2016
2017
2018
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Quarter 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
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Consumption, $ 7,576.00 10,027.00 6,708.00 3,414.00 10,539.00 7,725.00 4,914.00 2,284.00 10,557.00 11,674.00 5,166.00 3,335.00 9,100.00 11,435.00 7,697.00 2,568.00
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The Predictive Capacity of the Holt-Winters Model Using‌ Based on this information, the variable to be analyzed is evaluated. In Fig. 1, the energy consumption behavior of the produce farm can be seen. It is possible to observe that presents a tendency and seasonality; so the HoltWinters model is appropriate for estimating forecasts. The suggested values of the smoothing constants for a first estimation process with each initialization option are: ι = 0.1 β = 0.3 γ = 0.2 14,000.00
Consumption, $/Quarter
12,000.00 10,000.00 8,000.00 6,000.00 4,000.00 2,000.00 0.00 0
2
4
6
8
10
12
14
16
Quarter
Figure 1. graph of data Therefore, it is possible to proceed to estimate the predicted values with each of the options described above. For the case of the first option establish that L0 is set as the first quarter consumption value, in this case $7,576.00, the initial trend T0 in zero, and the seasonal factors for each quarter equal to one. Then, (1) can then be applied to obtain L1: đ?‘Œ1 đ??ż1 = đ?›ź + (1 − đ?›ź)(đ??ż0 + đ?‘‡0 ) đ?‘†1−đ?‘? 7576 đ??ż1 = (0.1) + (1 − 0.1)(7576 + 0) = 7576 1 With (2) the tendency T1 is calculated: đ?‘‡1 = đ?›˝(đ??ż1 − đ??ż0 ) + (1 − đ?›˝)đ?‘‡0 đ?‘‡1 = (0.3)(7576 − 7576) + (1 − 0.3)(0) = 0 And (3) is useful to obtain the new seasonal factor S1 đ?‘Œ1 đ?‘†1 = đ?›ž + (1 − đ?›ž)đ?‘†1−đ?‘? đ??ż1 7576 đ?‘†1 = (0.2) + (1 − 0.2)(1) = 1 7576 Finally, for this first period, using (4), the value of the forecast is obtained: đ?‘ŒĚ‚1 = (đ??ż0 + đ?‘‡0 )đ?‘†1−đ?‘? = (7576 + 0)(1) = 7576 We can observe that this first value is equal to the actual consumption, so the MAPE indicator is zero. If you continue with this sequence of calculations, the results presented in Table 2, are generated:
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The Predictive Capacity of the Holt-Winters Model Using‌ Table 2. Results from the first initialization option YEAR
đ?‘ŒĚ‚
MAPE
0.00 0.00 73.53 37.93
S 1 1 1 1 1 1.0564 0.9725
7,576.00 7,576.00 7,894.63
0 0.244 0.177
2015 2015 2015
T -3 -2 -1 0 1 2 3
Y
L
T
7,576.00 10,027.00 6,708.00
7,576.00 7,576.00 7,821.10 7,775.97
2015
4
3,414.00
7,373.91
-94.07
0.8926
7,813.90
1.289
2016
5
10,539.00
7,605.76
3.71
1.0771
7,279.84
0.309
2016
6
7,725.00
7,579.77
-5.20
1.0490
8,038.71
0.041
2016
7
4,914.00
7,322.39
-80.85
0.9122
7,366.51
0.499
2016
8
2,284.00
6,773.27
-221.33
0.7815
6,463.78
1.830
2017
9
10,557.00
6,876.84
-123.86
1.1687
7,057.30
0.332
2017
10
11,674.00
7,190.60
7.42
1.1639
7,083.60
0.393
2017
11
5,166.00
7,044.51
-38.63
0.8765
6,566.35
0.271
2017
12
3,335.00
6,732.03
-120.79
0.7243
5,475.23
0.642
2018
13
9,100.00
6,728.74
-85.54
1.2055
7,726.80
0.151
2018
14
11,435.00
6,961.38
9.92
1.2596
7,731.82
0.324
2018
15
7,697.00
7,152.35
64.23
0.9164
6,110.08
0.206
2018
16
2,568.00
6,849.48
-45.90
0.6544
5,226.93
1.035
The mean value for the MAPE indicator of the 16 calculated points results in 0.484, which is a high value, since it corresponds to a 48% of mean error. If you get the forecast of the quarter 17, which is the first of the year 2019, this result in a consumption of $8,201.52, while the real value has been $8,765, taking an error of 6.4%. Applying the Excel Solver to get the values of the smoothing constants that minimizes the average MAPE indicator, is possible to obtain the following: ι=0 β = 0.2977 γ = 1, MAPE = 0.284
Consumtion, $/Quarter
The value of the forecast for the first quarter of 2019 is $9,100.00, with an error with respect to the real value, of 3.8% The graph of both approaches for this first option is shown in Fig. 2: 14,000.00 12,000.00 10,000.00 8,000.00 6,000.00 4,000.00 2,000.00 0.00 0
2
4
6
8
10
12
14
16
18
Quarter Real
Initial
Optimal
Figure 2. graph of the first option
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The Predictive Capacity of the Holt-Winters Model Using‌ It is possible to observe in the graph that the biggest mistakes between the real consumption and those predicted for the case of optimal values for the constants of smoothing, are found in the first quarters. While in the case of the initial values of these constants, errors have been high throughout the entire chart. If you consider the second initialization option in the HW model, it is important to make a linear regression using the 16 quarters in order to obtain the regression equation, and it will provide the values of L0 y T0. Based on the regression model (Fig. 3), it is shown that even when the linear model is not enough, since the R-squared value has been 0.0064, which was to be expected given that the data does not follow a linear trend, the equation has a negative slope of -54.28, which would be the initial trend value T0 and the y intercept 7631.3 can be used as the initial value of the demand level L0. 14,000.00 y = -54.278x + 7631.3 R² = 0.0064
Consumption, $/Querter
12,000.00 10,000.00 8,000.00 6,000.00 4,000.00 2,000.00 0.00 0
2
4
6
8
10
12
14
16
18
Quarter
Figure 3. linear regressiĂłn model Then to estimation of the seasonal factors is realized; in the case of the second option, they are obtained by dividing the average demand of the 4 years of each quarter over the average demand of 4 years. This is illustrated in the case of the first quarter. The average demand in the first quarter (D1) for the period 2015 to 2018 is: 7576 + 10539 + 10557 + 9100 đ??ˇ1 = = 9443.00 4 Following with this method, the second quarter average demand is 10,215.25, the third quarter is 6,121.25 and the fourth quarter is 2,900.25, while the average demand of 4 years is 28,679.75. Then the seasonal factor of each quarter to initialize the calculations is simply the ratio that is generated dividing the quarter average consumption by the annual average consumption and multiplied by 4, and the defined factors are: S1 = 1.3170, S2 = 1.4247, S3 = 0.8537 and S4 = 0.4045 The sum of these seasonal factors is approximated to 3.9999, and the factors already are normalized. Using these initial values, it is possible to calculate L1 with (1): đ?‘Œ1 đ??ż1 = đ?›ź + (1 − đ?›ź)(đ??ż0 + đ?‘‡0 ) đ?‘†1−đ?‘? 7576 đ??ż1 = (0.1) + (1 − 0.1)(7631.3 − 54.278) = 7394.55 1.317 The first quarter trend T1 is calculated using (2): đ?‘‡1 = đ?›˝(đ??ż1 − đ??ż0 ) + (1 − đ?›˝)đ?‘‡0 = (0.3)(7394.55 − 7631.3) + (1 − 0.3)(−54.278) = −109.02
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The Predictive Capacity of the Holt-Winters Model Using‌ The new seasonal factor S1 is obtained applying (3): đ?‘Œ1 đ?‘†1 = đ?›ž + (1 − đ?›ž)đ?‘†1−đ?‘? đ??ż1 7576 = (0.2) + (1 − 0.2)(1.317) = 1.2585 7394.55 Then the forecast value is: đ?‘ŒĚ‚1 = (đ??ż0 + đ?‘‡0 )đ?‘†1−đ?‘? = (7631.3 − 54.278)(1.317) = 9979.14 If we continue with the calculations for the following periods, the results shown in Table 3 are obtained. Table 3. Results obtained in the initialization second option đ?‘ŒĚ‚
MAPE
-54.28 -109.02 -116.45 -95.06 -55.48 -20.17 -75.34 -116.74 -156.13 -106.36
S 1.3170 1.4247 0.8537 0.4045 1.2585 1.4160 0.8689 0.4177 1.2950 1.3501 0.8376 0.4029 1.3531
9,979.14 10,379.95 6,099.37 2,880.27 9,057.64 10,328.96 6,113.13 2,832.57 8,408.48
0.317 0.035 0.091 0.156 0.141 0.337 0.244 0.240 0.204
6,762.11
-43.53
1.4253
8,846.55
0.242
5,166.00
6,663.46
-60.06
0.8252
5,627.66
0.089
12
3,335.00
6,770.81
-9.84
0.4208
2,660.50
0.202
2018
13
9,100.00
6,757.42
-10.90
1.3518
9,147.99
0.005
2018
14
11,435.00
6,874.14
27.38
1.4730
9,616.02
0.159
2018
15
7,697.00
7,144.16
100.17
0.8756
5,694.83
0.260
2018
16
2,568.00
7,130.12
65.91
0.4087
3,048.63
0.187
YEAR
Y
L
T
2015 2015 2015 2015 2016 2016 2016 2016 2017
t -3 -2 -1 0 1 2 3 4 5 6 7 8 9
7,576.00 10,027.00 6,708.00 3,414.00 10,539.00 7,725.00 4,914.00 2,284.00 10,557.00
7,631.30 7,394.55 7,260.76 7,215.60 7,252.49 7,314.71 7,110.65 6,897.31 6,649.26 6,659.04
2017
10
11,674.00
2017
11
2017
The mean value of the MAPE indicator is equal to 0.182, with the initial values of the smoothing constants. Using the Excel Solver it is possible to obtain their optimal values: ι=0 β = 0.2025 γ = 0, MAPE = 0.158
Consumo, $/trimestre
Figure 4 shows the graph of the approximation. 15,000.00 10,000.00 5,000.00 0.00 0
2
4
6
8
10
12
14
16
18
Trimestre Real
Initial
Optimal
Figure 4. graph of the second option
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The Predictive Capacity of the Holt-Winters Model Using‌ Based on the results given in Fig. 4, It is noted that the lines are closer in comparison with the first initialization option (see Fig. 2), and the optimal values of the smoothing constants have a better approach than the initial values for the smoothing constants. In the case of the first quarter forecast of 2019, using the initial values of the smoothing constants is $9,727.46, which has an error of 11%, whereas with the optimal values has resulted in $8,835.37. This final result is a very good approximation, since the error compared with the actual value has been less than 1%. Finally, for initialization third option, it starts by calculating the average demands Ě…Ě…Ě… đ??ˇ2 đ?‘Ś Ě…Ě…Ě… đ??ˇ1 for the years 2018 and 2017 obtaining $7,700 and $7,683, respectively. The initial trend T 0 is also calculated: 7700 − 7683 = 4.25 4 The initial value for the level L0 is calculated using (7): 5 đ??ż0 = 7683 − (4.25) = 7672.375 2 đ?‘‡0 =
Using (8), it is possible to obtain the corresponding L values. We can note that the values are increasing based on the initial trend value T0. We obtain L1 =7676.625, L2 = 7680.875, L3 = 7685.125 and L4 = 7689.375. Then the seasonal factors are estimated considering (9): đ?‘Œ1 7576 đ?‘†1 = = = 0.9869 đ??ż1 7676.625 đ?‘Œ2 10027 = = 1.3055 đ??ż2 7680.875 đ?‘Œ3 6708 đ?‘†3 = = = 0.8729 đ??ż3 7685.125 đ?‘Œ4 3414 đ?‘†4 = = = 0.4440 đ??ż4 7689.375 We can note that the sum is not 4, but 3.6093. This suggests the application of a normalization process applying (10). Based on this we obtain: S1 = 1.0938, S2 = 1.4468, S3 = 0.9674 and S4 = 0.4921 đ?‘†2 =
The result in the sum is 4.0001 given the numerical approximation. Considering these seasonal factors, initials values of level and the corresponding trend, it is possible apply (1) to (5) in order to obtain the forecasts and the MAPE indicator. These calculations can be summarized in the case of the initial values for the smoothing constants in Table 4. The Table shows the MAPE indicator presents certain variation in the interval from 0.06 to 0.48, with an average value of 0.2165. This provokes that the mean error is greater than 21% when we compare the forecasted and real values. Using the Excel Solver to make an optimization process, it possible to obtain the following approximation: ι = 0.0688 β = 0.2356 γ = 0.3712, MAPE = 0.208 The graph of these two approximations is shown in Fig. 5. We can note that the bigger errors appeared when the maximum values of energy consumption have occurred, and the results related to initial values of smoothing constants and optimal values are very close. Table 4. Result of the third initialization option
Year
2015 2015 2015 2015 2016 2016 2016
t -3 -2 -1 0 1 2 3 4 5 6 7
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Y
L
T
7,576.00 10,027.00 6,708.00 3,414.00 10,539.00 7,725.00 4,914.00
7,672.38 7,601.62 7,518.08 7,425.64 7,328.09 7,515.77 7,314.93 7,049.83
4.25 -18.25 -37.84 -54.22 -67.22 9.25 -53.78 -117.17 IJMREM
S 1.0938 1.4468 0.9674 0.4921 1.0743 1.4242 0.9546 0.4868 1.1399 1.3506 0.9031
Yforecast
MAPE
8,396.35 10,971.68 7,236.16 3,627.22 7,800.57 10,717.06 6,931.26
0.108 0.094 0.079 0.062 0.260 0.387 0.411 Page 33
The Predictive Capacity of the Holt-Winters Model Using… 2016 2017 2017 2017 2017 2018 2018 2018 2018
8 9 10 11 12 13 14 15 16
2,284.00 10,557.00 11,674.00 5,166.00 3,335.00 9,100.00 11,435.00 7,697.00 2,568.00
6,708.55 6,797.85 6,890.38 6,733.93 6,719.55 6,739.17 6,839.27 7,038.99 6,944.31
-184.41 -102.29 -43.85 -77.63 -58.65 -35.17 5.41 63.70 16.19
0.4576 1.2225 1.4193 0.8759 0.4653 1.2481 1.4698 0.9194 0.4462
3,375.01 7,436.96 9,042.78 6,182.84 3,045.63 8,143.15 9,514.98 5,995.13 3,304.93
0.478 0.296 0.225 0.197 0.087 0.105 0.168 0.221 0.287
Consumption, $/Quarter
14,000.00 12,000.00 10,000.00 8,000.00 6,000.00 4,000.00 2,000.00 0.00 0
2
4
6
8
10
12
14
16
18
Quarter Real
Initial
Optimal
Figure 5. graph of third option We can observe that the forecast for the first quarter of 2019, with the initial values of the smoothing constants has been $8,687.32, with an error less than 1%, whereas with the optimal values has resulted in $9,266.01, which is not a best approximation, since it has 5.7% of error respecting to the actual value. Table 5 summarizes the results achieved with the three initialization techniques applied. Table 5. Summary of results for the three options Concept MAPE Indicador Forecast: Quarter 17
α, β, γ Values Initials Optimals Initials
1 0.484 0.284 $8,201.52 6.4% error $9,100.00 3.8% error α= 0 β= 0.2977 γ=1
Optimals
Optimal values of α, β, γ
V.
Option 2 0.182 0.158 $9,727.46 11% error $8,835.37 0.8% error α= 0 β= 0.2025 γ=0
3 0.217 0.208 $8,687.32 0.9% error $9,266.01 5.7% error α= 0.0688 β= 0.2356 γ=0.3712
CONCLUSIONS
The first option employs as initialization factors the initial trend T0 in the origin (zero), seasonal factors of each quarter with a value of one and the initial level as the last value of the predicted variable. This has been the worst approximation, however, the forecast for the first quarter of the year 17 (2019) has had a good approximation, both with the initial values of the smoothing constants, as with the optimal values. In this first
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The Predictive Capacity of the Holt-Winters Model Using… option, we observe a big difference between the best approximation with respect to the initial values for the smoothing constants, since in this case the average MAPE indicator was 48.4% and the optimal values behind 20 percentage points, reaching almost half (28.4%).The second option starts from the linear regression model considering the 16 quarters for estimating the trend, the initial level and the average demand for seasonal factors. This option has proved to be the best one, since it gives lower values of the MAPE indicator, when we compare their initial and optimal values of the smoothing constants. However, it is important to mention that for this option a good forecast of the quarter 17 has not been obtained when the initial values of the smoothing constants are used, because the 11% error, but when we consider the case of the optimal values, the results are positive, since the error has been less than 1%. This option does not imply so many calculations to obtain the initialization values for the HW model, in comparison with the third option which required a greater number of mathematical operations to get the estimators. With regard to the third option, the values of trend, seasonality and demand were estimated using as reference the average demand of consumption in the years 2017 and 2018. This has been the initialization method with which it has obtained the best forecast of energy consumption for the quarter 17, using the initial values of the smoothing constants. Also, it is observed that this third option generated small differences between the MAPE indicator of the initial calculation in respecting with the optimal values of the smoothing constants, and the change of the values of these constants. Based on the results, the application of the second option is suggested when the multiplicative Holt-Winters needs to be applied for forecasting. The method considers realizing a regression analysis of the data, placing the y intercept as the initial value of the level of the variable to predict and the slope as the initial trend, in order to obtain the starting seasonal factors, considering the historical average data values. This seems to be an appropriated alternative, as it makes fewer calculations to generate the estimates for the initial values and provides an acceptable approximation of energy consumption that usually has high variability. In the case of the orange produce farm, even when it seems to have a forecast with an average error of 16%, it is possible to suggest that is a useful approximation, since in agricultural activities fluctuations in costs can vary drastically from one period to the next. Similarly, by having an estimating process on the consumption of their pumping systems, could contribute to more precisely determine their energy needs, to select a better rate and implement actions to save energy and minimize costs. In this sense, the method can be employed as a core element when empirical modelling is required.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
A. Lawrence, and D. Aigner, Modeling and forecasting time of day and seasonal electricity demands, Journal of Econometrics, 9(1), 1979, 59-78. I. Matsukawa, Household respond to optional Peak Load pricing of electricity, Journal of Regulatory Economics, 20(3), 2001, 249-267. J. R. Cancelo, A. Espasa, and R. Gafe, Forecasting the electricity load from one day to one week ahead for the Spanish system operator, International Journal of Forecasting, 24, 2008, 558-602. J. Taylor, L. De Menezes, and P. McSharry, Comparison of univariate methods for forecasting electricity demand up to a day ahead, International Journal of Forecasting, 22(1), 2006, 1-16. B. Lanzilotta, and T. Rosá, Modelos de predicción de Energía Eléctrica con datos diarios para Uruguay, CINVE http://aladee.org/elaee2013/, 2012. S. Botero, and J. Cano, Análisis de series de tiempo para la predicción de los precios de la energía en la bolsa de Colombia, Cuadernos de Economía, 27(48), 2008, 173–208. S. Medina, and J. García, Predicción de demanda de energía en Colombia mediante un sistema de inferencia difuso neuronal, Revista Energética, 33, 2005, 15–24. F. Nogales, J. Contreras, A. Conejo, and R. Espínola, Forecasting Next-Day Electricity Prices by Time Series Models, IEEE Transactions on Power Systems, 17(2), 2002, 342-348. P. R. Winters, Forecasting sales by exponentially weighted moving averages, Management Science, 6, 1960, 324-342. P. Goodwin, The Holt-Winters Approach to Exponential Smoothing: 50 Years Old and Going Strong, Foresight, 2010, 30-33. S. Makridakis, S. C. Wheelwright, and R. J. Hyndman, Forecasting Methods and Applications, 3a ed., (New York: Wiley, 1998). S. Makridakis, and M. Hibon, The M3-competition: Results, conclusions and implications, International Journal of Forecasting, 16, 2000, 451-476. J. K. Ord, A. B. Koehler, and R. D. Snyder, Estimation and Prediction for a Class of Dynamic Nonlinear Statistical Models, Journal of the American Statistical Association, 92(440), 1997, 1621-1629. A. B. Koehler, R. D. Snyder, and J. K. Ord, Forecasting Models and prediction intervals for multiplicative Holt-Winters method, International Journal of Forecasting, 17(2), 2001, 269-286.
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The Predictive Capacity of the Holt-Winters Model Using… 15. H. Grubb, and A. Mason, Long lead-time forecasting of UK air passengers by Holt-Winters methods with damped trend, International Journal of Forecasting, 17(1), 2001, 71-82. 16. E. Vercher, A. Corberán-Vallet, J. V. Segura, and J. D. Bermúdez, Initial Conditions Estimation for Improving Forecast Accuracy in Exponential Smoothing, Top, 20(2), 2012, 517-533. 17. J. W. Taylor, Triple seasonal methods for short-term electricity demand forecasting, European Journal of Operational Research, 204(1), 2010, 139-152. 18. R. J. Hyndman, A. B. Koehler, J. K. Ord, and R. D. Snyder, Prediction intervals for exponential smoothing using two new classes of state space models, Journal of Forecasting, 24(1), 2005, 17-37. 19. J. D. Bermúdez, J. V. Segura, and E. Vercher, Holt-Winters Forecasting: An Alternative Formulation Applied to UK Air Passengers Data, Journal of Applied Statistics, 34(9), 2007, 1075-1090. 20. E. McKenzie, Renormalization of seasonal in Winters´ forecasting systems: Is it necessary?, Operations Research, 34(1), 1986, 174-176. 21. S. Hansun, New Estimation Rules for Unknown Parameters on Holt-Winters Multiplicative Method, Journal of Mathematical & Fundamental Sciences, 49(2), 2017, 127-135. 22. J. A. Arango-Marín, J. A. Giraldo-García, and O. D. Castrillón-Gómez, Gestión de compras e inventarios a partir de pronósticos Holt-Winters y diferenciación de nivel de servicio por clasificación ABC, Scientia et Technica, 18(4), 2013, 743-747. 23. N. Papić-Blagojević, A. Vujko, and T. Gajić, Comparative analysis of exponential smoothing models to tourists´ arrivals in Serbia, Economics of Agriculture, 63(3), 2016, 835-845. 24. A. Kotsialos, M. Papageorgiou, and A. Poulimenos, Long-term sales forecasting using Holt-Winters and neural network methods, Journal of Forecasting, 24(5), 2005, 353-368. 25. S. Gelper, R. Fried, and C. Croux, Robust forecasting with exponential and Holt-Winters smoothing, Journal of Forecasting, 29(3), 2010, 285-300. 26. R. Gamberini, F. Lolli, B. Rimini, and F. Sgarbossa, Forecasting of Sporadic Demand Patterns with Seasonality and Trend Components: An Empirical Comparison between Holt-Winters and S(ARIMA) Methods, Mathematical Problems in Engineering, 2010, 2010, 1-14. 27. L. L. Mira-Segura, A. Trejo-Martínez, and D. López-Cruz, Aplicación de Holt-Winters para pronósticos de inventarios, Ciencia UANL, 21(90), 2018, 38-42. 28. J. V. Segura, and E. Vercher, A Spreadsheet Modeling Approach to the Holt-Winters Optimal Forecasting, European Journal of Operational Research, 131(2), 2001, 375-388. 29. R. J. Kirkham, A. H. Boussabaine, and M. P. Kirkham, Stochastic time series forecasting of electricity costs in an NHS acute care hospital building, for use in whole life cycle costing, Engineering Construction & Architectural Management, 9(1), 2002, 38-52. 30. L. Ferbar-Tratar, and E. Strmčnik, The comparison of Holt-WInters method and multiple regression method: A case study, Energy, 109, 2016, 266-276. 31. J. E. Hanke, and D. W. Wichern, Pronósticos en los Negocios, 9a ed., (México: Pearson Educación, 2010).
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