Symmetric Seasonality of Time Series in Interval Prediction for Financial Management of the Branch

Abstract

The paper examines the task of managing the finances of a company with branches when funds are saved on the central company account, from which payments for the expenses of the branches are made. The dynamics of these expenses may have similar dynamics, which makes it possible to build a single model for the entire group. This article is devoted to the construction of theoretical concepts of the nonlinear dynamics approach and the formalization of criteria for combining time series into a single model. We introduce the concept of series with the same type of symmetrical seasonality, based on phase portraits, which allows formalizing the similarity criterion based on symmetry transformations. Considering time series that are recognized as similar, we bypass nonstationarity by considering the series included in the group as realizations of a random process. Finally, the use of new concepts allows solving an important practical problem, reducing the analysis to grouping by seasonal similarity and statistical characteristics of deviations when symmetry transformations are violated.

1. Introduction

The financial management of company branches is an important task. Determining the optimal amount of funds reserved for each branch is a strategic decision of the bank for its branches [1,2,3,4]. The existence of branches of companies and banks is recognized as one of the most effective channels for sales and the provision of services, as well as the development of relationships with customers. At the same time, each branch needs to have certain available funds [5]. A certain amount of funds should be saved and ready to pay current payments, but holding too much cash eliminates the opportunity to generate additional profits. All this determines the importance of forecasting the range of funds needed by the branch every day.

The paper discusses the task of managing the finances of a company with branches when funds are saved on the central account of the company, from which payments for the expenses of the branches are made. When a branch needs to pay its expenses, it applies to the central office, where it indicates the required amount. Next, the application is sent for execution, where the funds are transferred to the accounts of the payment recipients. Income and expenses on branch accounts arise unevenly; the company’s general account must always have a balance of available funds to avoid the possibility of a gap. At the same time, the excess of free cash should not be very large; therefore, measures are necessary to manage the size of the daily balance. Free cash is directed toward investing in various financial instruments. Thus, the task comes down to the task of forecasting expenses on branch accounts, while forecasting models, based on the priority of the tasks of uninterrupted provision of financial obligations and management of financial resources, require maximum reliability and forecasting of the minimum balance on accounts. At the same time, the expenses of the branches are quite similar in nature, having seasonal components at the level of the month, week, and holidays. It is proposed to use a dynamic approach to build a unified cost forecasting model for all similar branches.

Similar tasks occur in the management of bank branches, cash management at ATMs, and supply inventory management based on forecasting.

Several new financial management models have been developed for bank branches. It must be considered that there are two ways for cash to reach a branch: the branch requests cash from the central office and its own deposits. The branch’s cash levels are periodically adjusted, and, if necessary, the branch requests a certain amount. Funds management procedures in bank branches mainly consist of the processing of historical data; however, currently, calculations of amounts are mainly based on the experience of branch managers. In [6], an algorithm is constructed to optimize the funds of branches, considering the weekly request for funds from the center. In [7], an intelligent system for monitoring cash levels in a branch network was developed using a theoretical model based on Markov random fields (MRFs). The system provides alerts at certain security thresholds in the event of a shortage of funds at any point in the branch network. In addition, the construction of the model provides a unified view of the group of branches, which ensures that branches with similar monetary characteristics will be subject to similar management. However, the authors of [7] do not consider dynamic models. The authors of [8] modeled cash logistics in bank branches as an optimization problem and developed machine learning models to estimate the uncertainty of cash demand. The paper [9] presents cash strategies for bank branches using deep learning techniques. It is proposed to predict, in the form of an upper and lower limit, the level of cash collections and customer requirements for cash withdrawals. The machine learning model was applied to predict the upper and lower bounds of spending for 14 consecutive days using the previous 30 days of data collection in a sliding time window.

A similar formulation is used in the problem of loading ATMs [10]. In [11], changes in the set of denominations over time are considered depending on the forecast of withdrawals from an ATM. It is necessary to note that the models are constructed in the form of evolutionary equations. There is an analysis of the forecast values of branch parameters in a similar formulation in logistics problems [12,13], where demand forecasting is important. There are a large number of examples of financial management based on forecasting parameters of a branch and macroeconomics: the impact of interstate banking operations in the USA and deregulation [14], the management of digital currencies by the central bank [15,16], the management of research and development in China [17], the management of cash transfers to Syrian refugees in Turkey [18], and other applied research.

A tool for forecasting branch expenses is the use of dynamic forecasting models with a seasonal component [19]. Further, the dynamics of the expenses of branches as a time series will be considered. At the same time, it is necessary to create branches with similar dynamics of expenses. The grouping of several time series allows one to reduce the dimension of the problem [20] by creating a single model for a group of series. Time series can be considered the result of the evolution of a discrete dynamic system, resulting in a solution to an unknown evolutionary model [21]. The use of dynamic models underlies various modern economic time series models [22,23]. Such models are used if the next values are obtained based on the regression dependencies of previous values [24,25,26]. However, the use of an econometric approach faces several issues, such as: (1) absence of external control when constructing a regression equation based on a time series; (2) filter of emissions, but they are a reaction of the system, for example, to a macroeconomic factor; (3) existence of several output parameters while a single parameter is being observed; (4) the question of checking stationarity in the general case, etc.

This article proposes a nonlinear dynamics approach for the specific problem of constructing an interval prediction for several series of branch expenses that are similar in their seasonal component, which has practical value in reducing the dimensionality of forecast models. Considering several time series of the same type of economic indicators, we combine them as having symmetrical seasonality. The phrase “of the same type” means that related or similar processes occur or that transformations of the symmetries of phase trajectories can be formally established.

The developed nonlinear dynamic approach consists of considering two or more processes simultaneously and checking for symmetrical seasonality. For each time series, a phase portrait is built. If the phase trajectories are in a certain sense symmetrical under the conditions of given ranges of deviations, then for such time series, it is possible to construct a prediction interval model that would not smooth out the oscillations that are an internal property of both dynamic systems. The developed approach allows for solving some problems with regression econometric models: (1) The use of two or more series allows to assume there is a lack of control (or stable control), to which the systems (which generate the series) react in the same way; (2) Statistical methods can solve the problem of stationarity in time series; after normalization (tension/compression transformation), the series can be represented as an implementation of a single random process, which makes it possible to determine stationarity in fact from these realizations. The solution is being sought in the form of a boundary estimate of the forecast interval [27], since the irreducible level of funds reserved for the expenses of each branch is important for the problem.

The contribution of this paper is the following:

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The property of uniformity of time series is defined as the satisfaction of given symmetry transformations to phase portraits within the boundaries of symmetry violation.

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A technique has been formulated for constructing one interval forecast for a group of time series with a symmetrical seasonal component.

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The construction of a unified model of series of the same type is based on combining intervals, subject to the fulfillment of specified criteria for values going beyond the boundaries, which allows making the assumption that series are considered as different implementations of a single random process, which ultimately makes it possible to operate random processes without the assumption of stationarity.

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An example is given in which the approach allows reducing the dimension of the analyzed data; 21 rows are replaced by one predictive forecast.

The paper is organized as follows: Section 2 presents interdisciplinary research related to the problem under consideration; Section 3 presents methods and results; and Section 4 presents the conclusion.

2. Background

A certain approach has been formed in the theory of time series to the analysis of time series with seasonal components [28,29]. Seasonality [30] is a periodically recurring phenomenon (data outliers, jumps, and dips) observed in time series. Repeated oscillations are found in many dynamic behaviors of time series. In business and economic time series [31,32], in the sense of repeating deviations in a dynamic sense, this is the most studied object since seasonality is easily distinguished and explained by factors of weekends and holidays and changing seasons. Seasonality can be a reaction of similar financial instruments to the economic behavior of agents and participants in economic relations and macroeconomic and political changes. Seasonal components are also observed in technical objects [33], in the traffic of various networks [34], including computer ones [35], overflowing flows during working hours, and becoming free at night.

Due to the wide distribution of seasonal time series, solving the problem of their accurate forecasting is important for effective solutions in financial management, marketing, production, accounting, personnel management, and other business sectors. Modeling and forecasting seasonal time series [36,37] has long been large-scale research of great practical importance.

Researchers faced with seasonal series try to implement decomposition, which consists of dividing the original seasonal time series into three components: seasonal, trend, and stochastic [28]. There are approaches to splitting into two components—the main factor model and noise [38]. At the same time, a seasonal time series, as a rule, is non-stationary, and a large number of methods are aimed at obtaining a stationary time series from historical data. The purpose of seasonal time series decomposition is to remove seasonal variations, build forecasts for the filtered data [39], and then add the previously extracted seasonal component to the resulting model. However, seasonally adjusting the data for further analysis is controversial [40]. Some empirical studies show that seasonal variations are not always constant, and, at least in some time series, seasonal components and non-seasonal components cannot be considered independent and therefore separable. The difficulty in distinguishing seasonal from non-seasonal variations leads to the development of seasonal models—periodic models that explicitly take seasonal variations into account. From the point of view of dynamic systems, the phase portraits of the seasonal components of time series are close to limit cycles [37,41].

Often, there is a question of the stationarity of time series [42]. In the Box–Jenkins model of time series, decomposition is used to achieve stationarity on the average. However, splitting is not always the right way to deal with a trend, and linear detrending may be more appropriate. Depending on the nature of the nonstationarity, time series can be modeled in different ways [43]. In practice, however, it is often difficult to determine whether a given series is stationary by trend or stationery by subtracting some of its components.

Artificial neural networks are widely used for seasonal series forecasting [44,45]. As a flexible modeling tool, neural networks can model any type with high accuracy. With neural networks, no specific assumptions are required for the model. Its underlying relationships are determined solely through data mining. Since neural networks are a universal function of approximators, they are used to model seasonal and trend changes [46,47,48]. However, neural networks are highly dependent on the size of the training sample. This does not work for short-term series.

A limitation of econometric models is that the form of the model must be predetermined without knowing the true basis of the data generation process. In addition, the essentially linear relationship assumed in these models limits their ability to model complex nonlinear problems often encountered.

For forecasting purposes, accuracy is not always important. Often, it is required to use robust models that are quite reliable, at the expense of accuracy. Prediction intervals belong to this class. The point is to find such boundaries around historical data so that the predictive model with the found boundaries is accurate to a certain extent. As an example of such intervals, confidence intervals around stochastic models can be used, given inaccuracies in data measurements. In turn, the most effective tool here is neural networks that build a forecast, and the width of the interval is determined based on the optimization problem of the given criteria. An overview of the criteria is given in the article [49]. It can be shown that intervals in financial time series absorb local extremes, become insensitive to them, but still capture global changes [50], including seasonal ones.

The following approach was developed within this research:

In the following, several similar time series will be considered. These are series with economic data that have a seasonal component. By time series of the same type, the following time series will be meant: (1) obtained by observing similar processes (that is, the dynamics of expenses of related or similar companies, the dynamics of related currencies, and the dynamics of computer traffic in two branches of one company); (2) with decentralization and decomposition of a complex, multiply-connected system onto a simply connected one, or, more formally, from the point of view of nonlinear dynamics, guarantee the topological equivalence of the phase portrait. In other words, with scaled normalization of time series, when constructing phase portraits, there is a symmetric transformation (a combination of rotation and translation transformations) that transforms one phase portrait into another, possibly under conditions of weak symmetry violation. It is necessary to verify that the group includes series of the same type, which is carried out on the basis of nonlinear dynamics and the theory of symmetries. This combination of several series of one type allows for the hypothesis that they can be considered as several implementations of one random process. In turn, this makes it possible to isolate a truly non-stationary function of the average value from the series. This approach will not allow accurate forecasts, but it can:

(1)

Qualitatively predict behavior;

(2)

Provide the type of behavior for robust models, such as interval forecasts; and

(3)

Using several scaled non-stationary series of the same type to obtain their average value and other characteristics of a quasi-random process.

As an application of the approach, the problem of dimensionality reduction in the analysis of many series, when one robust forecast is sufficient for the whole group, is considered.

3. Main Results

There is a single account at the central office from which payments for branch expenses are made. It is necessary to always have a balance of available funds in the company’s general account to avoid the possibility of a cash gap. It is necessary to forecast the required minimum balance reserved for branch expenses. At the same time, the expenses of the branches are of a fairly uniform nature. Namely, it is assumed that there are several time series (expenses of branches) that potentially claim to be of the same type, for which it is necessary to construct a single interval forecast. The series are taken, the seasonal component of which is justified by equivalent factors; therefore, in order not to analyze them separately, it is necessary to build a sufficiently reliable interval forecast to decide.

Here, an example will be considered using real data on spending funds from the accounts of 21 branches of a company within a year.

We will develop an approach based on the similarity of seasonal components. At first, two series without loss of generality were taken. It was assumed that there is a transformation that converts the invariant characteristic of the system generating the time series into a similar characteristic of another system. Note that if it is talking about different partial solutions of one differential equation, then Lie groups would be obtained [51]. It is proposed to use phase trajectories as a characteristic of a nonlinear dynamic system because they are quite easy to construct based on observed discrete data. If such a transformation exists, two series would be considered to be series of the same type with seasonal symmetry.

For a practically oriented and intuitive interpretation by applied researchers, we will limit ourselves to the desired transformation as a combination of translation (shift) and stretching/compression transformations.

Definition 1.

Let two time series $X^1=\{x_1^1,x_2^1,\dots ,x_t^1,\dots ,x_n^1\}$ and $X^2=\{x_1^2,x_2^2,\dots ,x_t^2,\dots ,x_n^2\}$ of the same dimension n of discrete time t be given. Phase trajectories were constructed in coordinates x, $∆x/∆t$.

The curve connecting the phase points is denoted by $A_1$ and $A_2$.

Let there be a transformation taking

$$\Phi :A_1\stackrel{}{\to }{A}_2.$$

If Φ consists of a combination of transformations {stretching/compression, shifting}, the difference between the phase portraits at each point of the phase space does not exceed ε, then such series will be called the same type with seasonal symmetry.

$$\forall x_i^1,x_i^2,(j=\overline{1,n}):x_i^1-x_i^2<\mathrm{\epsilon }.$$

Definition 2.

If there is a transformation $\Phi$ satisfying Definition 1, then there can be a weak symmetry breaking such that each $x_i^1$ in the phase trajectory $A_1$ can be transformed in δ-surroundings of the corresponding $(\mathrm{\delta }\to \mathrm{\epsilon })$ of $x_i^2$ in the phase trajectory.

Based on the introduced Definition 2 here, as a criterion for the weakness of deviations in symmetric transformation $\Phi$, the average value of pairwise deviations of trajectory points with the corresponding numbers is taken:

$$d=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left|\right|x_i^1-x_i^2\left|\right|}.$$

Example 1.

Figure 1 shows the time series of spending funds from the company’s account by its branches 3 and 7.

Phase portraits are shown in Figure 2.

The scaling factor for account 7 on the X₁ and X₂ axes is [1.6, 1.5]. Transfer coefficients for account 7 along the X₁ and X₂ axes: [0.2, 0.1]. Characteristic d = 0.16.

Remark 1.

Here, it is possible to apply the theory of hidden attractors [52] or the reconstruction of dynamical systems using time series [53], but this is beyond the scope of this study.

To construct a prediction interval, such a deviation satisfies the requirements of the decision-making system.

To build an interval model, the criterion will be used. Let ${\tilde{x}}_i$—prediction value of time series ($i=\overline{1,N}$), N is the data sample size.

PICP [27,49] is a valuation index for prediction intervals indicating the probability that future values will fall within the lower and upper bounds:

$$\mathrm{PICP}=\frac{1}{N}{\displaystyle \sum _{i=1}^N{a}_i},$$

where a_i is a binary variable defined by the formula:

$$a_i=\left\{\begin{array}{l}1,{\tilde{x}}_i\in [{\tilde{x}}_i-{\mathrm{\epsilon }}_1,{\tilde{x}}_i+{\mathrm{\epsilon }}_2],\\ 0,{\tilde{x}}_i\notin [{\tilde{x}}_i-{\mathrm{\epsilon }}_1,{\tilde{x}}_i+{\mathrm{\epsilon }}_2].\end{array}\right.$$

Here, ${\mathrm{\epsilon }}_1,{\mathrm{\epsilon }}_2$ are the lower and upper limits. To ensure prediction interval accuracy, the PICP is typically required to exceed a predetermined confidence level.

Thus, the prediction interval allows setting a given level of confidence.

An important consequence of the introduced concept of seasonal symmetries is a reduction in the dimension of models for a group of series, allowing one to work with non-stationary time series.

The following hypothesis is formulated.

Hypothesis 1.

Time series of the same type in the introduced sense (seasonal symmetry) can be considered as different implementations of the same random process.

As a rule, the application of the theory of random processes to time series modeling has several significant problems: it is impossible to repeat tests, short time series, etc. For applied problems of analyzing a large number of economic or financial indicators, problems of distribution of limited resources, etc., under the conditions of the formulated hypothesis, it is possible to estimate the moments of a random process based on several series, i.e., we have several implementations of a discrete random process X(t).

Using the hypothesis, for a group of symmetric seasonal series, each discrete time ti corresponds to several points (corresponding to the number of series of the same type), which we accept as the implementation of a random variable xi.

The introduced hypothesis allows avoiding tests for stationarity, considering stationarity in a broad or narrow sense, based on observed data, and not based on assumptions about stationarity.

Further, an application example is given.

The company under consideration has 21 branches, the expenses of which represent the possibility of constructing one prediction interval based on the PICP index < 0.85, applied only to the lower bound (to forecast balances in the accounts of branches).

Many branch expense series are similar in some sense, for example, because they have receipts from the parent company and expenses associated with specific calendar dates. It is proposed to consider several series of the same type after their normalization as implementations of random processes, which will eliminate the study of stationarity based on preliminary hypotheses.

Under the assumption of the implementation of a single random process, a lower bound would be constructed.

Figure 3 shows the time series of 21 branches. We will divide the data into two parts: a sample for constructing an interval forecast of the model (rows 1–11) and a test sample for assessing the quality of the model according to the PICP criterion. In Figure 4, the red line is the checkmate expectation of a non-stationary process, and the black line is the interval prediction intervals for historical data. The model is built around the average based on the assumption that PICP = 0.85, with a minimum interval width. Figure 5 shows the test sample. The value of exceeding the interval is demonstrated by several series, with the PICP value on the test sample equal to 0.91.

Based on the above, an algorithm for constructing one interval model for a group of time series is formulated.

• Selection of time series that claim to be uniform in the sense of symmetrical seasonality.
• Construction of phase trajectories.
• Application of scaling and shift transformations, normalization.
• Calculation of the deviations of each point of the phase trajectories.
• Checking the number of points that go beyond the boundary.
• The series is included in the group of the same type if the quantitative parameters satisfy the specified threshold deviation values.
• Construction of the average value for the group as a function of the expectation value of the non-stationary series.
• Construction of prediction intervals under given conditions and restrictions.

The problem of dimensionality reduction by analyzing and predicting the behavior of several related or similar time series is attractive for its solution; such heuristic approaches are used in many companies and large analytical systems on an intuitive level. This result tries to theoretically substantiate this approach based on several theories—nonlinear dynamic systems, interval methods, the theory of time series forecasting, the theory of random processes, and the theory of symmetries. The approaches used made it possible to formalize the concepts of similar time series, calling them series with symmetric seasonality, and the formalization also defined evaluation criteria and areas of applicability.

4. Conclusions

This paper formulates the problem of constructing a prediction interval for a group of time series that have the property of seasonal symmetry. Formal criteria for seasonal symmetry are introduced, which are represented by a form of series whose phase porters may contain a given error by means of symmetry transformations. For series with seasonal symmetry (if this is true within the meaning of the problem), a hypothesis has been introduced that each of the series combined into a group is a realization of a random process. Under these conditions, a method for constructing the average under conditions of nonstationarity was obtained.

Examples of applications based on real data on financial expenditures from the accounts of company branches are given. The problem of forecasting the lower limit of balances on branch accounts is considered. Real data for one year from 21 branches was considered. Taking 11 time series for modeling, a prediction interval boundary was constructed. For the test part consisting of 10 time series, 91% of the data did not exceed the constructed lower limit. This result shows that in real conditions, it is possible to construct prediction intervals under conditions of nonstationarity, provided that the seasonal components of the belt series are symmetrically similar.

The further development of this research consists of automating the identification of symmetries and introducing the concepts of weakly broken symmetries. As well as expanding the types of transformations for grouping time series, which will make it possible to determine not only seasonality but also other types of dependencies.