TIME SERIES FORECASTING BASED ON THE COMBINED USE OF VARIOUS EXTRAPOLATION METHODS

V. Z. Yampolsky, V. A. Silich

(Presented by the scientific-technical seminar of the Department of Automated Control Systems and the Control Laboratory).

In the practice of forecasting system parameters on the basis of time series, methods of isolated extrapolation are widely used, and, in particular, methods of blind extrapolation. These include:

1) the moving averages method [1];
2) exponential smoothing methods [2];
3) forecasting on the basis of Laguerre polynomials [2], Lagrange polynomials [3];
4) forecasting on the basis of the method of quasi-optimal filtering of random processes [2].

Forecasts obtained on the basis of extrapolation methods have rather low reliability, and when one of the methods is used, significant errors may occur in the forecast values of the parameters. It is obvious that the joint use of various methods in order to obtain a certain compromise forecast vector \(\vec X_0\) \((X_0=(x_1^0,\ldots,x_\theta^0)\)—a vector whose components are compromise forecast values of the system parameters \(x_1,\ldots,x_\theta\)) will make it possible to guard against “large” forecast errors.

The procedure for deciding on the compromise value of the vector \(\vec X_0\) for \(\chi\) periods is proposed to be regarded as a game with nature under conditions of uncertainty.

Let \(\vec X_i=(x_1^i,\ldots,x_\theta^i)\), \(i=\overline{1,s}\), be the value of the forecast vector \(\vec X=(x_1,\ldots,x_\theta)\) computed by the \(i\)-th forecasting algorithm \((i=\overline{1,s}\), where \(s\) is the number of forecasting algorithms \(P_1,\ldots,P_s)\).

The problem consists in finding the compromise value of the vector \(\vec X_0\) by determining the best mixture of vectors of the form

\[ \vec X_0=\lambda_1\vec X_1+\lambda_2\vec X_2+\ldots+\lambda_s\vec X_s \tag{1} \]

with

\[ \sum_{i=1}^{s}\lambda_i=1,\quad \lambda_i\geq 0,\quad i=\overline{1,s}. \tag{2} \]

Here \(\lambda_i\) is a certain weight coefficient with which the vector should be present in the mixture \(\vec X_0\).

First of all, let us introduce a measure for measuring the magnitude of the discrepancy between the forecast vectors \(X_i,\ i=\overline{1,s}\):

\[ g_{kl}=\sum_{j=1}^{\theta}\left|\frac{x_j^l-x_j^k}{x_j^l}\right|,\quad k,l=\overline{1,s}. \tag{3} \]

The value \(g_{kl}\) quantitatively expresses the inconsistency of the forecast \(\vec X_k\) with the forecast \(\vec X_l\), obtained by the forecasting algorithms \(P_k\) and \(P_l\), respectively, and is the sum of the ratios, component by component, of the difference between the vectors \(\vec X_k\) and \(\vec X_l\) to the components of the vector \(\vec X_k\). The value \(g_{kl}\) is the smallest, that is, equal to zero, in all cases when \(k=l\).

Formula (3) is recommended for estimating the characteristic \(g_{kl}\) for the following reasons:

a) \(g_{kl}\) is a dimensionless quantity, which makes it possible to avoid difficulties due to different units and scales of measurement of the parameters;

b) \(g_{kl}\) makes it possible to take into account deviations both upward and downward, since the numerator takes the absolute value of the difference between the vectors. Using the introduced measure (1), we construct the square loss matrix \(G^*=(-g_{kl})\) (Table).

Table

Each row of the matrix \(G^*\) corresponds to one of the \(s\) possible values of the forecast vector obtained by means of different forecasting algorithms, and each column corresponds to one of the forecasting algorithms.

The diagonal elements of the matrix are equal to 0. The remaining elements of the matrix are less than zero or equal to zero in the case when the forecast vectors coincide for different forecasting algorithms. Further, the matrix \(G^*\) is considered as the payoff matrix in a two-person game.

The possible strategies of one of the players (nature) are the different forecasting algorithms \(P=\{P_1,\ldots,P_s\}\) for the values of time series, while those of the second are the different forecast vectors \(\vec X=\{\vec X_1,\ldots,\vec X_s\}\), computed by the algorithms \(P_1,\ldots,P_s\). It is necessary to determine such an optimal strategy \(\vec X_0\) of the first player (i.e., to determine \(\lambda_1,\ldots,\lambda_s\)) that would correspond to minimal deviations of the vector \(\vec X_0\) from the vectors \(\vec X_1,\ldots,\vec X_s\). From the structure of the matrix \(G^*\) it follows that it has no saddle point, because

\[ \max_{k}\min_{l}(-g_{kl})<\min_{l}\max_{k}(-g_{kl})=0. \]

The solution of the game specified by the matrix \(G^*\) is determined in the form of mixed strategies and can be found by one of the known methods [4]. The considered problem of determining the compromise vector can be solved by methods of linear programming. The optimization problem then takes the form: find

\[ v \longrightarrow \max \tag{4} \]

under the conditions:

\[ \left. \begin{aligned} &-\lambda_1 g_{11}-\ldots-\lambda_s g_{s1}\ge v;\\ &\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ &\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ &-\lambda_1 g_{1s}-\ldots-\lambda_s g_{ss}\ge v; \end{aligned} \right\} \tag{5} \]

\[ \sum_{i=1}^{s}\lambda_i=1; \tag{6} \]

\[ \lambda_i\ge 0;\qquad i=\overline{1,s}. \tag{7} \]

Here \(g_{kl}\) \((k,l=\overline{1,s})\) represent the elements of the game matrix \(G^*\), and the variables are \(\lambda_1,\ldots,\lambda_s\), which constitute the components of the optimal strategy being determined, while \(v\) is the value of the game.

The requirement of maximizing the right-hand side of (5) is identical to the requirement, for the compromise forecast, to minimize the maximum deviation with respect to all vectors \(\vec X_i\) \((i=\overline{1,s})\). When determining the value of the compromise forecast vector for \(x\) periods, the elements \(g_{kl}\) of the game matrix \(G^*\) are computed by the formula

\[ g_{kl}^{x}=\sum_{\nu=1}^{x}\sum_{j=1}^{\theta}\frac{\left|x_j^{l\nu}-x_j^{k\nu}\right|}{x_j^{l\nu}}, \tag{8} \]

where \(x_j^{l\nu}\) is the value of the \(j\)-th parameter of the vector \(\vec X\), obtained by the \(l\)-th forecasting algorithm \(P_l\) for the \(\nu\)-th forecasting period \((j=\overline{1,\theta},\ l=\overline{1,s};\ \nu=\overline{1,x})\).

The approach considered in this article to solving the forecasting problem on the basis of time series should be regarded as an approximate solution that makes it possible to prevent large values of forecast errors arising from the application of each of the methods separately under conditions of substantial uncertainty. Finding the described compromise forecast is a substantially more correct procedure than the simple averaging of forecast results obtained by individual methods, as used in forecasting practice.

REFERENCES

1. Yu. I. Ryzhikov. Inventory Control. Moscow, “Nauka,” 1969.
2. A. D. Smirnov. Modeling and Forecasting Socialist Reproduction. Moscow, “Ekonomika,” 1970.
3. E. A. Galperin. Long-Term Forecasting Program. In: Digital Computing Technology and Programming, issue 5, Moscow, “Sov. radio,” 1969.
4. R. D. Luce, H. Raiffa. Games and Decisions. Moscow, IL, 1961.