A wavelet smoothing method to improve conditional sales forecasting

This article proposes a wavelet smoothing method to improve conditional forecasts generated from linear regression sales response models. The method is applied to the forecasted values of the predictors to remove forecast errors and thereby improve the overall forecasting performance of the models. Eight empirical studies are presented in which the purpose was to forecast detergent sales in the Netherlands, and wavelet smoothing was compared with a moving average and a band-pass filter. All methods were found to improve forecasts. Wavelet smoothing provided the best results when applied on highly volatile marketing time series. In contrast, it was less effective when applied on highly aggregated and smooth time series. An advantage of wavelets is that they are flexible enough to allow for data characteristics like abrupt changes, spikes and cyclical changes that are usually associated with price changes and promotions.     

Short-term electricity demand forecasting using double seasonal exponential smoothing

This paper considers univariate online electricity demand forecasting for lead times from a half-hour-ahead to a day-ahead. A time series of demand recorded at half-hourly intervals contains more than one seasonal pattern. A within-day seasonal cycle is apparent from the similarity of the demand profile from one day to the next, and a within-week seasonal cycle is evident when one compares the demand on the corresponding day of adjacent weeks. There is strong appeal in using a forecasting method that is able to capture both seasonalities. The multiplicative seasonal ARIMA model has been adapted for this purpose. In this paper, we adapt the Holt–Winters exponential smoothing formulation so that it can accommodate two seasonalities. We correct for residual autocorrelation using a simple autoregressive model. The forecasts produced by the new double seasonal Holt–Winters method outperform those from traditional Holt–Winters and from a well-specified multiplicative double seasonal ARIMA model.     

Practical exponential smoothing credibility

Since some time so-called credibility estimators with geometric weights are of some practical importance. As alternatives one can use so-called exponential smoothing credibility estimators. In the present paper the second ones are prepared for practical application.     

On the brown method of exponential smoothing

In the literature the Brown Method is often recommended for forecasting with the smoothing constant α = 0.1 or α = 0.2. We describe an experiment for checking the recommendation, the results of which indicate that it has severe drawbacks. An alternative is suggested.     

The admissible parameter space for exponential smoothing models

We discuss the admissible parameter space for some state space models, including the models that underly exponential smoothing methods. We find that the usual parameter restrictions (requiring all smoothing parameters to lie between 0 and 1) do not always lead to stable models. We also find that all seasonal exponential smoothing methods are unstable as the underlying state space models are neither reachable nor observable. This instability does not affect the forecasts, but does corrupt the state estimates. The problem can be overcome with a simple normalizing procedure. Finally we show that the admissible parameter space of a seasonal exponential smoothing model is much larger than that for a basic structural model, leading to better forecasts from the exponential smoothing model when there is a rapidly changing seasonal pattern.     

Initial conditions estimation for improving forecast accuracy in exponential smoothing

In this paper we analyze the importance of initial conditions in exponential smoothing models on forecast errors and prediction intervals. We work with certain exponential smoothing models, namely Holt’s additive linear and Gardner’s damped trend. We study some probability properties of those models, showing the influence of the initial conditions on the forecast, which highlights the importance of obtaining accurate estimates of initial conditions. Using the linear heteroscedastic modeling approach, we show how to obtain the joint estimation of initial conditions and smoothing parameters through maximum likelihood via box-constrained nonlinear optimization. Point-wise forecasts of future values and prediction intervals are computed under normality assumptions on the stochastic component. We also propose an alternative formulation of prediction intervals in order to obtain an empirical coverage closer to their nominal values; that formulation adds an additional term to the standard formulas for the estimation of the error variance. We illustrate the proposed approach by using the yearly data time-series from the M3-Competition.     

The development of efficient portfolios in Japan with particular emphasis on sales and earnings forecasting

In this study, we show that earnings forecasting creates an index-tracking portfolio that dominates the historical model trade-off curve. We find that using Toyo Keizai earnings forecasts improves geometric means by over 300 basis points compared to the historical model. Weighted latent root regression is used in this study to create portfolios that have outperformed the Japanese market in backtest and in real-time performance.     

Pullback exponential attractors for evolution processes with the difference of 2 solutions lacking smoothing property

In this paper, we construct the pullback exponential attractors for evolution processes in which the difference of 2 solutions lacks the smoothing property. To do this, by the uniform squeezing property of the corresponding discrete process, we add the points to the pullback attractor such that every new set of it has the finite fractal dimension and pullback exponentially attracts every bounded subset of the phase space. As the applications, we establish the existence of pullback exponential attractors for non‐autonomous reaction‐diffusion equation without any restriction on the growing order of nonlinear term and non‐autonomous strongly damped wave equation in with critical nonlinearity.     

Multi-item lot-sizing with joint set-up costs

We consider a multi-item lot-sizing problem with joint set-up costs and constant capacities. Apart from the usual per unit production and storage costs for each item, a set-up cost is incurred for each batch of production, where a batch consists of up to C units of any mix of the items. In addition, an upper bound on the number of batches may be imposed. Under widely applicable conditions on the storage costs, namely that the production and storage costs are nonspeculative, and for any two items the one that has a higher storage cost in one period has a higher storage cost in every period, we show that there is a tight linear program with O(mT ²) constraints and variables that solves the joint set-up multi-item lot-sizing problem, where m is the number of items and T is the number of time periods. This establishes that under the above storage cost conditions this problem is polynomially solvable. For the problem with backlogging, a similar linear programming result is described for the uncapacitated case under very restrictive conditions on the storage and backlogging costs. Computational results are presented to test the effectiveness of using these tight linear programs in strengthening the basic mixed integer programming formulations of the joint set-up problem both when the storage cost conditions are satisfied, and also when they are violated.     

Quadratic interval innovation diffusion models for new product sales forecasting

An appropriate sales forecasting method is vital to the success of a business firm. The logistic model and the Gompertz model are usually adopted to forecast the growth trends and the potential market volume of innovative products. All of these models rely on statistics to explain the relationships between dependent and independent variables, and use crisp parameters. However, fuzzy relationships are more appropriate for describing the relationships between dependent and independent variables; these relationships require less data than traditional models to generate reasonable estimates of parameters. Therefore, we have combined fuzzy regression with the logistic and Gompertz models to develop a quadratic-interval Gompertz model and a quadratic-interval logistic model, and we applied the models to three cases. Our practical application of the two models shows that they are appropriate tools that can reveal the best and worst possible sales volume outcomes.     

Bestimmung statistisch gesicherter Koeffizienten bei der exponentiellen ausgleichung (exponential smoothing)

Zusammenfassung Zur Vorhersage von Zeitreihen, hauptsächlich bei der Lagerhaltung, verwendet man neuerdings die Methode der exponentiellen Ausgleichung. Eine Schwierigkeit ist dabei die Wahl des Vorhersagemodells.Die exponentielle Ausgleichung liefert die gleichen Vorhersagegleichungen wie eine Regressionsanalyse mit exponentiell verteilten Gewichten.Die schrittweise Regressionsanalyse führt schrittweise zu den am besten statistisch gesicherten Koeffizienten, wobei bei jedem Schritt das Minimum der Quadratsumme der Abweichungen gesucht wird. Er werden dabei nur die Koeffizienten in die Vorhersagegleichung aufgenommen, die eine vorgegebene statistische Sicherheitsbedingung erfüllen. Verlieren im Verlauf der Rechnung Koeffizienten durch starke Interkorrelation ihre statistische Sicherheit, so werden sie aus der Vorhersagegleichung eliminiert.Neue Beobachtungen werden laufend berücksichtigt und die neuen Koeffizienten berechnet. Umfangreiche Rechnungen sind nur dann nötig, wenn aufgrund der statistischen Sicherheitsbedingung Koeffizienten aus der Vorhersagegleichung eliminiert oder neu in sie aufgenommen werden müssen.

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Summary In order to forecast time series, mainly for Inventory Control, recently the method of Exponential Smoothing is used. But it is difficult to select a good forecasting model.Exponential Smoothing leads to the same forecasting equations as the Regression Analysis with exponentially distributed weights.Stepwise regression analysis leads step by step to the most significant coefficients, where the minimum of the sum of squared residuals is searched at each step. Only these coefficients which satisfy a stated significance condition are taken into forecast equation. If during computation by intercorrelation coefficients loose their significance, they are eliminated from forecast equation.New observations are currently taken up, and the new coefficients are computed. Extensive calculations are only necessary, if the coefficients—because of significance condition—must be eliminated from or taken into forecast equation.

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Vortrag auf der Jahrestagung 1963 der DGU in Braunschweig

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Vorgel. v.:J. Nitsche     

Tactical sales forecasting using a very large set of macroeconomic indicators

Tactical forecasting in supply chain management supports planning for inventory, scheduling production, and raw material purchase, amongst other functions. It typically refers to forecasts up to 12 months ahead. Traditional forecasting models take into account univariate information extrapolating from the past, but cannot anticipate macroeconomic events, such as steep increases or declines in national economic activity. In practice this is countered by using managerial expert judgement, which is well known to suffer from various biases, is expensive and not scalable. This paper evaluates multiple approaches to improve tactical sales forecasting using macro-economic leading indicators. The proposed statistical forecast selects automatically both the type of leading indicators, as well as the order of the lead for each of the selected indicators. However as the future values of the leading indicators are unknown an additional uncertainty is introduced. This uncertainty is controlled in our methodology by restricting inputs to an unconditional forecasting setup. We compare this with the conditional setup, where future indicator values are assumed to be known and assess the theoretical loss of forecast accuracy. We also evaluate purely statistical model building against judgement aided models, where potential leading indicators are pre-filtered by experts, quantifying the accuracy-cost trade-off. The proposed framework improves on forecasting accuracy over established time series benchmarks, while providing useful insights about the key leading indicators. We evaluate the proposed approach on a real case study and find 18.8% accuracy gains over the current forecasting process.     

An application of support vector machines to sales forecasting under promotions

This paper deals with sales forecasting of a given commodity in a retail store of large distribution. For many years statistical methods such as ARIMA and Exponential Smoothing have been used to this aim. However the statistical methods could fail if high irregularity of sales are present, as happens for instance in case of promotions, because they are not well suited to model the nonlinear behaviors of the sales process. In recent years new methods based on machine learning are being employed for forecasting applications. A preliminary investigation indicates that methods based on the support vector machine (SVM) are more promising than other machine learning methods for the case considered. The paper assesses the application of SVM to sales forecasting under promotion impacts, compares SVM with other statistical methods, and tackles two real case studies.     

A novel signal extraction approach for filtering and forecasting noisy exponential series

The coefficients of Linear Recurrent Relations (LRR) play a pivotal role in many forecasting techniques. Precise and closed form of the LRR coefficients enables one to achieve more accurate forecasts. On account to the fact that, in real-world situations, a time series data is contaminated with noise, extracting the noiseless series is of great importance. This paper seeks to obtain a closed form, with less noise level, of LRR coefficients for noisy exponential time series. Improving the filtering performance through employing noiseless eigenvectors of the covariance matrix is another novelty of this study. Our simulation results confirm that the proposed approach enhances filtering and forecasting results.     

On the identification of sales forecasting models in the presence of promotions

Shorter product life cycles and aggressive marketing, among other factors, have increased the complexity of sales forecasting. Forecasts are often produced using a Forecasting Support System that integrates univariate statistical forecasting with managerial judgment. Forecasting sales under promotional activity is one of the main reasons to use expert judgment. Alternatively, one can replace expert adjustments by regression models whose exogenous inputs are promotion features (price, display, etc). However, these regression models may have large dimensionality as well as multicollinearity issues. We propose a novel promotional model that overcomes these limitations. It combines Principal Component Analysis to reduce the dimensionality of the problem and automatically identifies the demand dynamics. For items with limited history, the proposed model is capable of providing promotional forecasts by selectively pooling information across established products. The performance of the model is compared against forecasts provided by experts and statistical benchmarks, on weekly data; outperforming both substantially.     

A sales forecasting system based on fuzzy neural network with initial weights generated by genetic algorithm

Sales forecasting is highly complex due to the influence of internal and external environments. However, reliable prediction of sales can improve the quality of business strategy. Recently, artificial neural networks (ANNs) have been applied for sales forecasting due to their promising performance in the areas of control and pattern recognition. However, further improvement is still necessary since unique circumstances such as promotion can cause sudden changes in sales patterns. Thus, the present study utilizes the proposed fuzzy neural network with initial weights generated by genetic algorithm (GFNN) for the sake of learning fuzzy IF–THEN rules for promotion obtained from marketing experts. The result from GFNN is further integrated with an ANN forecast using the time series data and the promotion length from another ANN. Model evaluation results for a convenience store (CVS) company indicate that the proposed system can perform more accurately than the conventional statistical method and a single ANN.