Heuristics approach for the Holt-Winters multiplicative method with new initial values

Abstract

When trend and seasonality are detected, the Holt-Winters multiplicative approach is one of the most commonly used methods for forecasting time series data. Choosing the proper initial values for level, trend, and seasonality plays a vital role in this method. In this paper, a new and efficient procedure to choose the initial values for the Holt-Winters multiplicative method is developed. A total of 12 types of agricultural satellite backscatter values are used for analysis, estimated, and compared with the existing Hansun and Holt-Winters methods and the proposed initial setting method with the best smoothing constants. According to the analysis of the mean absolute percentage error, symmetric mean absolute percentage error, Theil-U statistics, and root mean squared error, the proposed approaches outperformed the existing methods in this experiment.

Keywords

Weighted moving average time series forecasting initial conditions hansun method Holt-Winters method

1. Introduction

Time series analysis is an important and necessary activity in science and engineering. Time series forecasting serves to make a model and predict experimental future observations based on theoretically observed values at a present state. In many time series applications, one of the emerging ideas behind the researcher is the challenge of setting and estimating parameters, which are internally involved when using the maximum likelihood method of estimation. The method is useful for estimating any parameter by using standard probability distributions that can be employed in a mathematical model and gives the predicted observations in standard applications in environmental and biological science, ecology, business, and engineering. One of the major methods to implement time series models with a wide variety of big data in agricultural and environmental studies is exponential smoothing techniques (EST), which is easy to understand and easy to implement for a reliable forecast in a large variety of different applications.

At present, the analysis made possible by using different data analytic source languages is available to implement advanced forecasting model building techniques, usually using the R language and Python, to select and analyze robust time series models. And segregate the different comparisons by using performance metrics, which are mean absolute percentage (MAP), symmetric mean absolute percentage error (SMAP), Theil-U, root mean squared error (RMSE), Squared R, and Theil-U measures. The primary goal is to identify and determine the seasonality of the Holt-Winters technique in order to assign appropriate weight to large amounts of data. The method is classified into different techniques for time series based on trends and seasonality. In the case of stationary time series data, single exponential smoothing is a perfectly suitable approach for forecasting. But simple exponential smoothing (SES) is not providing proper results for the non-stationary time series data. Most of the agricultural applications have changed over time, so they show a trend. The trend may always move upward and downward.

2. Review of literature

Michael and Makowski (2013) studied the comparison of statistical models for wheat yield time series data analysis. They investigated different statistical models to forecast wheat production at different scales and found that the Holt model performed better when compared with others. In the early phases of rice production in India, Kumari et al. (2014) separated the forecasting abilities of statistical models. They studied different ways of exponentially smoothing models to forecast rice production. It was observed that the Holt model performed comparatively well with other ESMs. Pathmanathan et al. (2015) attempted tea export forecasting from India with the utilization of exponential smoothing technique methodologies and also observed that the Holt linear trend procedures are perfect for forecasting the export of tea from India up to 2020.

Khayati (2015) studied vegetable crop production forecasting in Tunisia. He observed that the Holt methodology for prediction of different vegetables like potatoes, peppers, and artichokes. The present study depends on the Holt-Winters method, which affects the selection of smoothing parameters and their initial values, which are the major issues. For initial values and smoothing parameters, they used maximum likelihood methods (Ord et al., 1997; Broze & Melard, 1990; Hyndman et al., 2002; Hanzak, 2008; Osman & King, 2015). The improvement of forecasting accuracy and the selection of initial values are critical for forecasting (Vercher et al., 2012). In seasonal model building, the adopted methods are multiplicative methods for nonlinear seasonality and additive methods for linear seasonality.

In recent research, Hansun (2017) made new assumptions for the initial values and got good results, and Dhali et al. (2019) have discussed selecting the proper smoothing constants for seasonal time series data while using the Holt-exponential Winters smoothing approach to predict demand for a company’s toy manufacturing. Setiawan et al. (2020) gave a real-world example to show the methods for choosing constants and calculating the forecast values. Wongoutong (2021) has modified the initial values for Holt’s linear multiplicative methods by using Hansun’s proposed methods.

In the present study, the focus is on optimizing the initial values for the Holt-Winters technique based on the principle of weighted moving averages. The method can be compared favorably with Hansun’s method.

3. Methodology

3.1 Holt-Winters multiplicative technique

The Holt-Winters method is suitable for seasonality data that can also predict the long-term forecast. It consists of two assumptions, which are multiplicative and additive effect assumptions. A multiplicative effect occurs when the size of the seasonal variation increases as the data set’s level increases. When the absolute size of the variations is independent of each other, this is referred to as an “additive effect”. The generic expressions of the involved quantities are below.

$\displaystyle\text{Smoothing level}\ L_{t}=\alpha\frac{Y_{t}}{S_{t-s}}+\left({% 1-\alpha}\right)\left({L_{t-1}+b_{t-1}}\right)$ (1) $\displaystyle\text{Smoothing trend}\ b_{t}=\beta\left({L_{t}-L_{t-1}}\right)+% \left({1-\beta}\right)b_{t-1},$ (2) $\displaystyle\text{Smoothing seasonal}\ S_{t}=\gamma\frac{Y_{t}}{L_{t}}+\left(% {1-\gamma}\right)S_{t-s},$ (3) $\displaystyle\text{Forecast}\ F_{t+m}=\left({L_{t}+b_{t}m}\right)S_{t-s+m},$ (4)

where $Y_{t}$ is the observed values at time $t$ , $s$ is the length of the seasonality, $L_{t}$ is the overall estimate level value, $b_{t}$ is the trend component, $S_{t}$ is the seasonal component, and $F_{t+m}$ is the estimate for $m$ in the next period.

To estimate the initial values of the seasonal indices, one needs to take at least one complete season’s worth of data. The initialized trend, level, and seasonality values are found in the following equations:

$\displaystyle\text{Initial seasonal factors}\ S_{1}=\frac{Y_{1}}{L_{s}},S_{2}=% \frac{Y_{2}}{L_{s}},\ldots,S_{s}=\frac{Y_{s}}{L_{s}}.$ (5) $\displaystyle\text{Initializing trend}\ T_{s}=\frac{1}{s}\left[{\frac{Y_{s+1}-% Y_{1}}{s}+\frac{Y_{s+2}-Y_{2}}{s}+\ldots\frac{Y_{s+s}-Y_{s}}{s}}\right]=\frac{% 1}{s^{2}}\left[{\mathop{\sum}\limits_{i=1}^{s}\left({Y_{s+i}-Y_{i}}\right)}% \right].$ (6) $\displaystyle\text{Initial level}\ L_{s}=\textit{Average}\ \left({Y_{1},Y_{2},% Y_{3},\ldots Y_{s}}\right)=\frac{Y_{1}+Y_{2}+Y_{3}+\ldots+Y_{s}}{s}.$ (7)

3.2 Performance metrics

By comparing actual and predicted values, we can determine the performance of the forecast methods. We have the following performance metrics to analyze the accuracy of the forecast values: The following formulas calculate the MAPE, SMAPE, RMSE, and Theil-U statistics, which are all calculated by the following formulas:

$\displaystyle\textit{MAPE}=\frac{100\%}{n}\sum_{t=1}^{n}\left[\left|\frac{Y_{t% }-\hat{Y}_{t}}{y_{t}}\right|\right]Y_{t}\quad\textit{RMSE}=\sqrt{\frac{\sum_{t% =1}^{n}(Y_{t}-\hat{Y}_{t})^{2}}{n},}$ $\displaystyle\text{sMAPE}=\frac{100\%}{n}\sum_{t=1}^{n}\frac{\left|Y_{t}-\hat{% Y}_{t}\right|}{(|Y_{t}|+|\hat{Y}_{t}|)/2^{\prime}}\quad\text{Theil-U}=\sqrt{% \frac{\sum_{t=1}^{n-1}(\hat{Y}_{t+1}-Y_{t+1})^{2}}{\sum^{n-1}_{t=1}(\frac{Y_{t% +1}-Y_{t}}{Y_{t}})}}$

where $\hat{Y}_{t}$ is the forecasted value at time $t$ and $n$ is the number of data.

3.3 Hansun proposed estimation for initial value

Hansun (2017) suggested the new initial values setting for the Holt-Winters multiplicative method using the basic principles of a weighted moving average to enable more appropriate values for the recent data for Level ( $L_{s}$ ) and trend ( $T_{s}$ ). The initial seasonality is the same as the Holt-Winters method. Hansun proposed the new initial values as like $L_{s}^{H}$ and $T_{s}^{H}$ as follows:

$\displaystyle L_{s}^{H}=\frac{sY_{s}+\left({s-1}\right)Y_{s-1}+\ldots+\left({s% -m-2}\right)Y_{s-m+2}+\left({s-m+1}\right)Y_{s-m+1}}{s+\left({s-1}\right)+% \ldots+\left({s-m+2}\right)+\left({s-m+1}\right)},$ (8) $\displaystyle T_{s}^{H}=\frac{1}{s^{2}}\left[\frac{2sY_{2s}+\left({2s-1}\right% )Y_{2s-1}+\ldots+\left({s+2}\right)Y_{s+2}+\left({s+1}\right)Y_{s+1}}{s+\left(% {s-1}\right)+\ldots+\left({s+2}\right)+\left({s+1}\right)}\right.\left.-\frac{% sY_{s}+\left({s-1}\right)Y_{s-1}+\ldots+2Y_{2}+Y_{1}}{s+\left({s-1}\right)+% \ldots+2+1}\right].$ (9)

Compared with the Holt-Winters method, the Hansun method performs better and gives good results. All the information’s are given in Hansun (2017).

4. The proposed model

It is important to note that the moving average approach, with some slight alterations, is used to find appropriate beginning values, which improves the Holt-Winters multiplicative method. We propose to choose the initial values for seasonality as in the Holt-Winters method and taking the level $L_{s}^{\textit{New}}$ is proposed Model Eq. (10) and choose the trend $T_{s}^{\textit{New}}$ as a major modification to Hansun’s proposed model, as follows:

$\displaystyle\text{Initial level}\ L_{s}^{\textit{New}}=\frac{Y_{s}}{S_{s}}=% \frac{Y_{s}-Y_{s-1}}{S_{s}-S_{s-1}},$ (10) $\displaystyle\text{Initializing trend}\ T_{s}^{\textit{New}}=\frac{1}{s^{2}}% \left[\frac{2sY_{2s}+\left({2s-1}\right)Y_{2s-1}+\ldots+\left({s+2}\right)Y_{s% +2}+\left({s+1}\right)Y_{s+1}}{s+\left({s-1}\right)+\ldots+\left({s+2}\right)+% \left({s+1}\right)}\right.$ (11) $\displaystyle\hskip 102.429921pt\left.-\frac{Y_{s}-Y_{s-1}}{S_{s}-S_{s-1}}\right]$ $\displaystyle\text{Initial seasonal values}\ S_{1}=\frac{Y_{1}}{L_{s}},S_{2}=% \frac{Y_{2}}{L_{s}},\ldots,S_{s}=\frac{Y_{s}}{L_{s}}.$ (12)

The steps involving choosing the initial values are:

Choose the smoothing initial values of level as in Eq. (10) of the proposed model.

Select the smoothing initial trend taken from (Equation 9) by subtracting $L_{s}^{\textit{New}}$ Eq. (10).

Calculate the smoothing seasonality values as the Holt-Winters multiplicative method Eq. (5).

5. Experimental data analysis

5.1 Dataset

The various agricultural time series data sets have been chosen for the study to determine the performance of the new initial value setting. The data were collected during the Kharif season as a short-term crop and the Rabi season as a long-term crop and analyzed using the Holt-Winters multiplicative method. The data are available on the MOSDOC website.

5.2 Data analysis

In this study, we introduce alternative formulas for determining the initial values of Holt’s Winter Multiplicative technique, which incorporate the weighted Moving average (WMA) model with Hansun’s proposed initial value approach to test the level of validity of the proposed revised HW Multiplicative method.

In general, the Holt-Winters multiplicative method had three smoothing constants between 0 and 1. We use the trial-and-error method to determine the optimum parameter, which gives the least MAE values. In this experiment, we use Python to find the values with the fewest errors. Table 1 presents the best parameters for the dataset.

Table 1
Best parameters for the dataset

Dataset	DS-01	DS-02	DS-03	DS-04	DS-05	DS-06	DS-07	DS-08	DS-09	DS-10	DS-11	DS-12
$\alpha$	0.96	0.96	0.83	0.94	1	0.81	0.91	0.94	0.82	0.97	1	1
$\beta$	0	0.1	0.2	0.2	0.2	0.2	0.2	0.1	0.1	0.1	0.1	0.1
$\gamma$	0.5	0.5	0.5	0	0	0.5	0.5	0.5	0	0	0	0.5
MAE	0.073	0.036	0.064	0.038	0.044	0.053	0.056	0.038	0.046	0.051	0.074	0.082

By using the best parameter values for each data set, the forecasting was able to derive the MAPE, RMSE, SMAPE, and Theil-U statistics, which are shown in Tables 2 and 3. From there, it is easy to discuss the proposed initial value model that can give the lowest error factors.

Table 2

Comparison between the Holt-Winters (HW), Hansun and the new approach with RMSE and MAPE

Dataset	MAPE			RMSE
	Holt’s Winter	Hansun’s Method	Proposed Method	Holt’s Winter	Hansun’s Method	Proposed Method
DS-01	0.774	0.751	0.385	0.089	0.087	0.044
DS-02	0.895	0.885	0.684	0.107	0.105	0.077
DS-03	0.401	0.524	0.398	0.050	0.066	0.050
DS-04	0.645	0.631	0.452	0.084	0.083	0.061
DS-05	0.667	0.665	0.537	0.079	0.079	0.066
DS-06	0.945	0.977	0.581	0.114	0.117	0.071
DS-07	0.777	0.718	0.412	0.087	0.084	0.048
DS-08	0.816	0.862	0.483	0.102	0.106	0.060
DS-09	0.863	0.888	0.540	0.100	0.105	0.063
DS-10	0.819	0.809	0.806	0.093	0.092	0.091
DS-11	0.924	0.914	0.881	0.107	0.107	0.106
DS-12	0.852	0.850	0.816	0.099	0.098	0.093

Table 3

Comparison between the HW, Hansun and the new approach with SMAPE and Theil-U statistic

Dataset	sMAPE			Theil-U
	Holt’s Winter	Hansun’s Method	Proposed Method	Holt’s Winter	Hansun’s Method	Proposed Method
DS-01	0.774	0.752	0.385	0.203	0.199	0.101
DS-02	0.894	0.883	0.682	0.254	0.248	0.178
DS-03	0.400	0.523	0.397	0.144	0.192	0.144
DS-04	0.643	0.629	0.451	0.204	0.202	0.146
DS-05	0.666	0.663	0.535	0.208	0.207	0.169
DS-06	0.941	0.974	0.580	0.277	0.284	0.173
DS-07	0.774	0.715	0.411	0.229	0.221	0.130
DS-08	0.812	0.857	0.482	0.247	0.259	0.148
DS-09	0.859	0.885	0.539	0.332	0.349	0.209
DS-10	0.817	0.806	0.805	0.305	0.300	0.295
DS-11	0.921	0.911	0.879	0.363	0.360	0.362
DS-12	0.850	0.848	0.814	0.255	0.252	0.237

In the case of fitting the Holt-Winters multiplicative method for forecasting agricultural data, out of various initial value setting procedures, this proposed model gives the best result. From Tables 2 and 3, comparing the performance metrics values of the Holt-Winters and Hansun’s methods with the values of the proposed model, it is easy to conclude that the proposed model gives the minimum error values. We may utilize our proposed forecasting methods to accurately predict diverse seasonal agricultural data with few errors.

Using the RMSE, MAPE, SMAPE, and Theil-U statistics, values estimated by the three methods have been drawn in Fig. 1 via bar charts. From that, it is seen that the proposed initial value setting is performed well and gives the least error value compared with the Holt-Winters and Hansun’s methods.

Figure 1.

Comparable error statistics for the proposed initial value approach.

In the experiment study, three smoothing parameters $\left({\alpha,\beta,\gamma}\right)$ were set from 0 to 1 with 0.1 increments. The results show that, almost often, the proposed method exceeded Holt-Winters and Hansun’s method in terms of MAPE, sMAPE, RMSE, and Theil-U values. In comparison to Holt-Winters and Hansun techniques, the suggested method reduces MAPE, Theil-U, RMSE, and sMAPE by 20% to 25% on average. As a result, the suggested approach is a good substitute for estimating the initial values for the Holt-Winter multiplicative method.

In order to validate all types of datasets, the proposed technique has error values that are significantly lower in the datasets DS01-DS09. The error values for DS10-DS12 are also quite low and significantly coincide. However, when compared to the Holt-Winters and Hansun’s methods, the better values were always improved.

6. Conclusion

In this research, we have given a new formula for the Holt-Winter multiplicative method’s initial value. To give more weight to the most recent data, which are subsequently employed in the Holt-multiplicative Winters technique, the proposed formulas utilize the fundamental idea of the weighted moving average. A promising outcome was obtained from the experiment employing 12 different agricultural satellite backscatter values. For predicting accuracy, the MAPE, MASE, sMAPE, and Theil-U statistics criteria are used.

The proposed Holt-Winters multiplicative initial value approaches yield the lowest MAPE and MASE values. This indicates that, when compared to the Hansun’s approach and the original Holt-Winters multiplicative approach, the new formula for determining the initial values in the procedure produced the minimum error and best accuracy level. Therefore, we can suggest that the proposed method is practiced as an alternative to the Holt-Winters multiplicative approaches for establishing initial values.

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