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A Practical Two-Stage Decomposition Approach to Filtering Endogenous Irregularities in Seasonal Series
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Forecast combinations have flourished remarkably in the forecasting community and, in recent years, have become part of the mainstream of forecasting research and activities. Combining multiple forecasts produced from single (target) series is now widely used to improve accuracy through the integration of information gleaned from different sources, thereby mitigating the risk of identifying a single "best" forecast. Combination schemes have evolved from simple combination methods without estimation, to sophisticated methods involving time-varying weights, nonlinear combinations, correlations among components, and cross-learning. They include combining point forecasts and combining probabilistic forecasts. This paper provides an up-to-date review of the extensive literature on forecast combinations, together with reference to available open-source software implementations. We discuss the potential and limitations of various methods and highlight how these ideas have developed over time. Some important issues concerning the utility of forecast combinations are also surveyed. Finally, we conclude with current research gaps and potential insights for future research.
Forecast combinations have been widely applied in the last few decades to improve forecasting. Estimating optimal weights that can outperform simple averages is not always an easy task. In recent years, the idea of using time series features for forecast combinations has flourished. Although this idea has been proved to be beneficial in several forecasting competitions, it may not be practical in many situations. For example, the task of selecting appropriate features to build forecasting models is often challenging. Even if there was an acceptable way to define the features, existing features are estimated based on the historical patterns, which are likely to change in the future. Other times, the estimation of the features is infeasible due to limited historical data. In this work, we suggest a change of focus from the historical data to the produced forecasts to extract features. We use out-of-sample forecasts to obtain weights for forecast combinations by amplifying the diversity of the pool of methods being combined. A rich set of time series is used to evaluate the performance of the proposed method. Experimental results show that our diversity-based forecast combination framework not only simplifies the modeling process but also achieves superior forecasting performance in terms of both point forecasts and prediction intervals. The value of our proposition lies on its simplicity, transparency, and computational efficiency, elements that are important from both an optimization and a decision analysis perspective.
Ensemble forecasting is a modeling approach that combines data sources, models of different types, with alternative assumptions, using distinct pattern recognition methods. The aim is to use all available information in predictions, without the limiting and arbitrary choices and dependencies resulting from a single statistical or machine learning approach or a single functional form, or results from a limited data source. Uncertainties are systematically accounted for. Outputs of ensemble models can be presented as a range of possibilities, to indicate the amount of uncertainty in modeling. We review methods and applications of ensemble models both within and outside of transport research. The review finds that ensemble forecasting generally improves forecast accuracy, robustness in many fields, particularly in weather forecasting where the method originated. We note that ensemble methods are highly siloed across different disciplines, and both the knowledge and application of ensemble forecasting are lacking in transport. In this paper we review and synthesize methods of ensemble forecasting with a unifying framework, categorizing ensemble methods into two broad and not mutually exclusive categories, namely combining models, and combining data; this framework further extends to ensembles of ensembles. We apply ensemble forecasting to transport related cases, which shows the potential of ensemble models in improving forecast accuracy and reliability. This paper sheds light on the apparatus of ensemble forecasting, which we hope contributes to the better understanding and wider adoption of ensemble models.
Sugar price forecasting has attracted extensive attention from policymakers due to its significant impact on people’s daily lives and markets. In this paper, we present a novel hybrid deep learning model that utilizes the merit of a time series decomposition technology empirical mode decomposition (EMD) and a hyperparameter optimization algorithm Tree of Parzen Estimators (TPEs) for sugar price forecasting. The effectiveness of the proposed model was implemented in a case study with the price of London Sugar Futures. Two experiments are conducted to verify the superiority of the EMD and TPE. Moreover, the specific effects of EMD and TPE are analyzed by the DM test and improvement percentage. Finally, empirical results demonstrate that the proposed hybrid model outperforms other models.
The stock market prediction patterns are seen as an important activity and it is more effective. Hence, stock prices will lead to lucrative profits from sound taking decisions. Because of the stagnant and noisy data, stock market-related forecasts are a major challenge for investors. Therefore, forecasting the stock market is a major challenge for investors to use their money to make more profit. Stock market predictions use mathematical strategies and learning tools. This paper provides a complete overview of 30 research papers recommending methods that include calculation methods, ML algorithms, performance parameters, and outstanding journals. The studies are selected based on research questions. Hence, these selected studies are helping to find the ML techniques along with their dataset for stock market prediction. Most widely ANN and NN techniques are used to achieve precise predictions of the stock market. While much amount of work is done, the latest stock market-related prediction methodology has many limitations. It can be assumed in this study that stock market forecasting is an integrated process and distinctive parameters for forecast the stock market should be considered more accurate.
The aim of this paper is to present a comprehensive literature review on the application of decomposition methods of time series forecasting in power systems. A comprehensive search is conducted, relevant publications are identified and summarised in terms of the aim of the forecast, decomposition forecasting method used, and a comparison is made with other forecasting techniques. Moreover, publications are analysed by number of publications per year, number of papers by the decomposition method used, and number of publications by area of application of the method in power systems. It is shown that the number of publications per annum grew substantially after 2014 due to studies on the application of decomposition methods in forecasting distributed generation output. Of the methods of forecasting, most publications have used multiplicative decomposition, a lower number has used additive decomposition, with the balance of the papers only using decomposition of the time series as an initial phase towards forecasting using other techniques. The analysis of the papers also shows that decomposition methods have been used in power systems mainly for load, price and distributed generation forecasting. The review contributes to literature by providing an extensive overview of the papers and a repository of references to researchers interested in decomposition forecasting techniques.
This study aimed to predict seasonal time series data using the Holt-Winters
exponential smoothing additive model. The data used in this study is data
on the number of passengers departing at Hasanudin Airport in 2009-2019,
the source of the data obtained from the o�cial website of the Central Statis-
tics Agency. The results showed that the Holt-Winters exponential smoothing
method on the passenger's number at Hasanudin Airport from 2009 to 2019
contained trend patterns and seasonal patterns by �rst determining the initial
values and smoothing parameters minimize forecasting errors.
Daily SKU demand forecasting is a challenging task as it usually involves predicting irregular series that are characterized by intermittency and erraticness. This is particularly true when forecasting at low cross-sectional levels, such as at a store or warehouse level, or dealing with slow-moving items. Yet, accurate forecasts are necessary for supporting inventory holding and replenishment decisions. This task is typically addressed by utilizing well-established statistical methods, such as the Croston's method and its variants. More recently, Machine Learning (ML) methods have been proposed as an alternative to statistical ones, but their superiority remains under question. This paper sheds some light in that direction by comparing the forecasting performance of various ML methods, trained both in a series-by-series and a cross-learning fashion, to that of statistical methods using a large set of real daily SKU demand data. Our results indicate that some ML methods do provide better forecasts, both in terms of accuracy and bias. Cross-learning across multiple SKUs has also proven to be beneficial for some of the ML methods.
Forecasting the short-term metro ridership is an important issue for operation management of metro systems. However, it cannot be solved well by the single long short-term memory (LSTM) neural network alone for the irregular fluctuation caused by various factors. This paper proposes a hybrid algorithm (STL-LSTM) which combines the addition mode of Seasonal-Trend decomposition based on Loess (STL) and the LSTM neural network to mitigate the influences of irregular fluctuation and improve the performance of short-term metro ridership prediction. First, the original series is decomposed into three sub-series by the addition mode of STL. Then, the LSTM neural network is employed to predict each decomposed series. Finally, all the predicted outputs are merged as the overall output. The results show that the STL-LSTM model can achieve higher accuracy than the single LSTM model, support vector regression (SVR), and the EMD-LSTM model which combines the empirical mode decomposition and the LSTM neural network.
This article introduces this JBR Special Issue on simple versus complex methods in forecasting. Simplicity in forecasting requires that (1) method, (2) representation of cumulative knowledge, (3) relationships in models, and (4) relationships among models, forecasts, and decisions are all sufficiently uncomplicated as to be easily understood by decision-makers. Our review of studies comparing simple and complex methods - including those in this special issue - found 97 comparisons in 32 papers. None of the papers provide a balance of evidence that complexity improves forecast accuracy. Complexity increases forecast error by 27 percent on average in the 25 papers with quantitative comparisons. The finding is consistent with prior research to identify valid forecasting methods: all 22 previously identified evidence-based forecasting procedures are simple. Nevertheless, complexity remains popular among researchers, forecasters, and clients. Some evidence suggests that the popularity of complexity may be due to incentives: (1) researchers are rewarded for publishing in highly ranked journals, which favor complexity; (2) forecasters can use complex methods to provide forecasts that support decision-makers’ plans; and (3) forecasters’ clients may be reassured by incomprehensibility. Clients who prefer accuracy should accept forecasts only from simple evidence-based procedures. They can rate the simplicity of forecasters’ procedures using the questionnaire at simple-forecasting.com.
This article proposes a unifying theory, or the Golden Rule, of forecasting. The Golden Rule of Forecasting is to be conservative. A conservative forecast is consistent with cumulative knowledge about the present and the past. To be conservative, forecasters must seek out and use all knowledge relevant to the problem, including knowledge of methods validated for the situation. Twenty-eight guidelines are logically deduced from the Golden Rule. A review of evidence identified 105 papers with experimental comparisons; 102 support the guidelines. Ignoring a single guideline increased forecast error by more than two-fifths on average. Ignoring the Golden Rule is likely to harm accuracy most when the situation is uncertain and complex, and when bias is likely. Non-experts who use the Golden Rule can identify dubious forecasts quickly and inexpensively. To date, ignorance of research findings, bias, sophisticated statistical procedures, and the proliferation of big data, have led forecasters to violate the Golden Rule. As a result, despite major advances in evidence-based forecasting methods, forecasting practice in many fields has failed to improve over the past half-century.
From its foundation, operational research (OR) has made many substantial contributions to practical forecasting in organizations. Equally, researchers in other disciplines have influenced forecasting practice. Since the last survey articles in JORS, forecasting has developed as a discipline with its own journals. While the effect of this increased specialization has been a narrowing of the scope of OR's interest in forecasting, research from an OR perspective remains vigorous. OR has been more receptive than other disciplines to the specialist research published in the forecasting journals, capitalizing on some of their key findings. In this paper, we identify the particular topics of OR interest over the past 25 years. After a brief summary of the current research in forecasting methods, we examine those topic areas that have grabbed the attention of OR researchers: computationally intensive methods and applications in operations and marketing. Applications in operations have proved particularly important, including the management of inventories and the effects of sharing forecast information across the supply chain. The second area of application is marketing, including customer relationship management using data mining and computer-intensive methods. The paper concludes by arguing that the unique contribution that OR can continue to make to forecasting is through developing models that link the effectiveness of new forecasting methods to the organizational context in which the models will be applied. The benefits of examining the system rather than its separate components are likely to be substantial.
This paper describes a numerically efficient Machine Learning approach for solving the farm manager’s land allocation problem in a manner that requires neither a distributional assumption on crop returns nor an expected utility specification. The method is based on the idea that a farm manager seeks the allocation of crops that minimizes the chance of realizing a harvest return below some pre-determined target rate of return, a notion supported by results from financial portfolio theory, behavioral economics, and historical land allocation research. The method also includes a way to connect a farm manager’s choice of target rate to their risk attitudes. Corn-soybean return data from Illinois is used to demonstrate the approach for an array of shortfall values; the optimally fitted model matches well to observed planting behavior. The proposed approach is not inherently limited to land allocation decisions; future researchers may discover that it is useful in other types of shortfall and non-shortfall decision processes.
The recent advances in Big Data have opened up the opportunity to develop competitive Global Forecasting Models (GFM) that simultaneously learn from many time series. Although, the concept of series relatedness has been heavily exploited with GFMs to explain their superiority over local statistical benchmarks, this concept remains largely under-investigated in an empirical setting. Hence, this study attempts to explore the factors that affect GFM performance, by simulating a number of datasets having controllable characteristics. The factors being controlled are along the homogeneity/heterogeneity of series, the complexity of patterns in the series, the complexity of forecasting models, and the lengths/number of series. We simulate time series from simple Data Generating Processes (DGP), such as Auto Regressive (AR), Seasonal AR and Fourier Terms to complex DGPs, such as Chaotic Logistic Map, Self-Exciting Threshold Auto-Regressive and Mackey-Glass Equations. We perform experiments on these datasets using Recurrent Neural Networks (RNN), Feed-Forward Neural Networks, Pooled Regression models and Light Gradient Boosting Models (LGBM) built as GFMs, and compare their performance against standard statistical forecasting techniques. Our experiments demonstrate that with respect to GFM performance, relatedness is closely associated with other factors such as the availability of data, complexity of data and the complexity of the forecasting technique used. Also, techniques such as RNNs and LGBMs having complex non-linear modelling capabilities, when built as GFMs are competitive methods under challenging forecasting scenarios such as short series, heterogeneous series and having minimal prior knowledge of the data patterns.
Decomposition methods are extensively used for processing the complex patterns of tourism demand data. Given tourism demand data’s intrinsic complexity, it is critical to theoretically understand how different decomposition methods provide solutions. However, a comprehensive comparison of decomposition methods in tourism demand forecasting is still lacking. Hence, this study systematically investigates the forecasting performance of decomposition methods in tourism demand. Nine popular decomposition methods and six forecasting methods are employed, and their forecasting performance is compared. With Hong Kong visitor arrivals from eight major sources as a sample, three main conclusions are obtained from empirical results. First, all the decomposition methods generally outperform benchmark at all horizons, in both the level and directional forecasting. Second, decomposition methods can be divided into four categories based on forecasting accuracy. Finally, variational mode decomposition method is consistently superior to other eight decomposition methods and can provide the best forecasts in all cases.
Inaccurate forecasts of demand during promotions diminish the already meager profit margins of retailers. No forecasting method described in the literature can accurately account for the combination of seasonal sales variations and promotion-induced sales peaks over forecasting horizons of several weeks or months. We address this research gap by developing a forecasting method for seasonal, frequently promoted products that generates accurate predictions, can handle a large number of sales series, and requires minimal training data. In our method's first stage, we forecast the seasonal sales cycle by fitting a harmonic regression model to a decomposed training set, which excludes promotional and holiday sales, and then extrapolate that model to a testing set. In the second stage, we integrate the resulting seasonal forecast into a multiplicative demand function that accounts for consumer stockpiling and captures promotional and holiday sales uplifts. The final model is then fitted using ridge regression. We use sales data from a grocery retailing chain to compare the forecasting accuracy of our method with popular seasonal and promotion demand forecasting models at multiple aggregation levels for both short and long forecasting horizons. The significantly more accurate forecasts generated by our model attest to the merit of the approach developed here.
Oil price forecasting has received a prodigious attention by scholars and policymakers due to its significant effect on various economic sectors and markets. Incentivized by this issue, the author proposes a novel hybrid model for crude oil price forecasting whose focus is on improving the accuracy of prediction taking into consideration the characteristics existing in the oil price time series. In so doing, the author constitutes a hybrid model consisting of complete ensemble empirical mode decomposition, support vector machine, particle swarm optimization, and Markov-switching generalized autoregressive conditional heteroskedasticity to capture the nonlinearity and volatility of the time series more effectively. Mean absolute error, root mean square error, and mean absolute percentage error tests are used to measure forecasting errors. Results robustness and forecasting quality of the proposed hybrid model compared with counterparts are also investigated by Diebold-Mariano test. Finally, empirical results demonstrate that the proposed hybrid model outperforms other models.
Demand forecasting is an important task for retailers as it is required for various operational decisions. One key challenge is to forecast demand on special days that are subject to vastly different demand patterns than on regular days. We present the case of a bakery chain with an emphasis on special calendar days, for which we address the problem of forecasting the daily demand for different product categories at the store level. Such forecasts are an input for production and ordering decisions. We treat the forecasting problem as a supervised machine learning task and provide an evaluation of different methods, including artificial neural networks and gradient-boosted decision trees. In particular, we outline and discuss the possibility of formulating a classification instead of a regression problem. An empirical comparison with established approaches reveals the superiority of machine learning methods, while classification-based approaches outperform regression-based approaches. We also found that machine learning methods not only provide more accurate forecasts but are also more suitable for applications in a large-scale demand forecasting scenario that often occurs in the retail industry.
The demand for a particular product or service is typically associated with different uncertainties that can make them volatile and challenging to predict. Demand unpredictability is one of the managers’ concerns in the supply chain that can cause large forecasting errors, issues in the upstream supply chain and impose unnecessary costs. We investigate 843 real demand time series with different values of coefficient of variations (CoV) where promotion causes volatility over the entire demand series. In such a case, forecasting demand for different CoV require different models to capture the underlying behavior of demand series and pose significant challenges due to very different and diverse demand behavior. We decompose demand into baseline and promotional demand and propose a hybrid model to forecast demand. Our results indicate that our proposed hybrid model generates robust and accurate forecast and robust inventory performance across series with different levels of volatilities. We stress the necessity of decomposition for volatile demand series. We also model demand series with a number of well known statistical and machine learning (ML) models to investigate their forecast accuracy and inventory performance empirically. We found that ARIMA with covariate (ARIMAX) works well to forecast volatile demand series, but exponential smoothing with covariate (ETSX) has a poor performance. Support vector regression (SVR) and dynamic linear regression (DLR) models generate robust forecasts across different categories of demands with different CoV values. In terms of inventory performance, ARIMAX and combination models have superior performance to the other presented models. The hybrid algorithm also depicts robust performance across different series with different CoVs and has low inventory costs.
This paper reviews the research literature on forecasting retail demand. We begin by introducing the forecasting problems that retailers face, from the strategic to the operational, as sales are aggregated over products to stores and to the company overall. Aggregated forecasting supports strategic decisions on location. Product-level forecasts usually relate to operational decisions at the store level. The factors that influence demand, and in particular promotional information, add considerable complexity, so that forecasters potentially face the dimensionality problem of too many variables and too little data. The paper goes on to evaluate evidence on the comparative forecasting accuracy. Although causal models outperform simple benchmarks, adequate evidence on machine learning methods has not yet accumulated. Methods for forecasting new products are examined separately, with little evidence being found on the effectiveness of the various approaches. The paper concludes by describing company forecasting practices, offering conclusions as to both research gaps and the barriers to improved practice.
Effective and stable load forecasting is necessary and of great importance in ensuring a reliable supply of electricity and the security of the power system. However, due to such factors as cyclicity and seasonality, electric load series show complex nonlinearity characteristics. As a result, obtaining the desired forecasting accuracy becomes highly difficult and challenging. To address this problem, based on the “divide and conquer” idea, we developed a novel decomposition‐ensemble model for short‐term load forecasting (STLF) by integrating singular spectrum analysis (SSA), a support vector machine, the Autoregressive Integrated Moving Average model and the cuckoo search algorithm. To effectively tackle nonlinearity characteristics and later improve forecasting performance, an SSA-based decomposition and reconstruction strategy was introduced into the proposed model and performed based on the pre-analysis of hidden characteristics of the data. Specifically, the decomposed modes that reflect the inner data characteristics were analyzed and selected to establish specific individual predictors. Finally, the cuckoo search algorithm was employed to generate the ensemble result. To verify the performance of the proposed model, half-hourly load data from New South Wales and hourly load data from Singapore were used as illustrative cases. The experimental results demonstrate that the proposed decomposition-ensemble model can provide more accurate electric power forecasting compared with the eight models discussed.
Research into time series forecasting for call center management suggests that a forecast based on the simple Seasonal Moving Average (SMA) method outperforms more sophisticated approaches at long horizons where capacity planning decisions are made. However in the short to medium term where decisions concerning the scheduling of agents are required, the SMA method is usually outperformed. This study is the first systematic evaluation of the SMA method across averages of different lengths using call arrival data sampled at different frequencies from 5 min to 1 h. A hybrid method which combines the strengths of the SMA method and nonlinear data-driven artificial neural networks (ANNs) is proposed to improve short-term accuracy without deteriorating long-term performance. Results of forecasting the intraday call arrivals to banks in the US, UK and Israel indicate that the proposed method outperforms standard benchmarks, and leads to improvements in forecasting accuracy across all horizons.
In the article a simple neural model with local learning for forecasting time series with multiple seasonal cycles is presented. This model uses patterns of the time series seasonal cycles: input ones representing cycles preceding the forecast moment and forecast ones representing the forecasted cycles. Patterns simplify the forecasting problem especially when a time series exhibits nonstationarity, heteroscedasticity, trend and many seasonal cycles. The artificial neural network learns using the training sample selected from the neighborhood of the query pattern. As a result the target function is approximated locally which leads to a reduction in problem complexity and enables the use of simpler models. The effectiveness of the proposed approach is illustrated through applications to electrical load forecasting and compared with ARIMA and exponential smoothing approaches. In a day ahead load forecasting simulations indicate the best results for the one-neuron network.
Forecasting special events such as conflicts and epidemics is challenging because of their nature and the limited amount of historical information from which a reference base can be built. This study evaluates the performances of structured analogies, the Delphi method and interaction groups in forecasting the impact of such events. The empirical evidence reveals that the use of structured analogies leads to an average forecasting accuracy improvement of 8.4% compared to unaided judgment. This improvement in accuracy is greater when the use of structured analogies is accompanied by an increase in the level of expertise, the use of more analogies, the relevance of these analogies, and the introduction of pooling analogies through interaction within experts. Furthermore, the results from group judgmental forecasting approaches were very promising; the Delphi method and interaction groups improved accuracy by 27.0% and 54.4%, respectively.
A number of procedures for forecasting a time series from its own current and past values are surveyed. Forecasting performances of three methods--Box-Jenkins, Holt-Winters and stepwise autoregression--are compared over a large sample of economic time series. The possibility of combining individual forecasts in the production of an overall forecast is explored, and we present empirical results which indicate that such a procedure can frequently be profitable.
This article presents the results of a survey to determine the degree of familiarity and usage, accuracy obtained, and evaluation of different forecasting techniques. It was found that regression analysis, subjective techniques, exponential smoothing, and moving average were well known and used for specific situations. Accuracy was relatively high for aggregate short range forecasts, but decreased for longer range and product level forecasts.
This paper is a re-examination of the benefits and limitations of decomposition and combination techniques in the area of forecasting, and also a contribution to the field, offering a new forecasting method. The new method is based on the disaggregation of time series components through the STL decomposition procedure, the extrapolation of linear combinations of the disaggregated sub-series, and the reaggregation of the extrapolations to obtain estimates for the global series. Applying the forecasting method to data from the NN3 and M1 Competition series, the results suggest that it can perform well relative to four other standard statistical techniques from the literature, namely the ARIMA, Theta, Holt-Winters' and Holt's Damped Trend methods. The relative advantages of the new method are then investigated further relative to a simple combination of the four statistical methods and a Classical Decomposition forecasting method. The strength of the method lies in its ability to predict long lead times with relatively high levels of accuracy, and to perform consistently well for a wide range of time series, irrespective of the characteristics, underlying structure and level of noise of the data.
This paper introduces five new univariate exponentially weighted methods for forecasting intraday time series that contain both intraweek and intraday seasonal cycles. Applications of relevance include forecasting volumes of call centre arrivals, transportation, e-mail traffic and electricity loads. The first method that we develop extends an exponential smoothing formulation that has been used for daily sales data, and which involves smoothing the total weekly volume and its split across the periods of the week. Two new methods are proposed that use discount weighted regression (DWR). The first uses DWR to estimate the time-varying parameters of a model with trigonometric terms. The second introduces DWR splines. We also consider a time-varying spline that uses exponential smoothing. The final new method presented here involves the use of singular value decomposition followed by exponential smoothing. Empirical results are provided using a series of intraday call centre arrivals.
This paper concerns the forecasting of seasonal intraday time series. An extension of Holt-Winters exponential smoothing has been proposed that smoothes an intraday cycle and an intraweek cycle. A recently proposed exponential smoothing method involves smoothing a different intraday cycle for each distinct type of day of the week. Similar days are allocated identical intraday cycles. A limitation is that the method allows only whole days to be treated as identical. We introduce an exponential smoothing formulation that allows parts of different days of the week to be treated as identical. The result is a method that involves the smoothing and initialisation of fewer terms than the other two exponential smoothing methods. We evaluate forecasting up to a day ahead using two empirical studies. For electricity load data, the new method compares well with a range of alternatives. The second study involves a series of arrivals at a call centre that is open for a shorter duration at the weekends than on weekdays. By contrast with the previously proposed exponential smoothing methods, our new method can model in a straightforward way this situation, where the number of periods on each day of the week is not the same.
A study of business cycles defined as sequences of expansions and contractions in the level of general economic activity does not require trend estimation and elimination, but a study of growth cycles defined as sequences of high and low growth phases does. Major cyclical slowdowns and booms deserve to be analyzed along with classical recessions and expansions, but the needed time series decomposition presents difficult problems, mainly because trends and cycles influence each other. We compare cyclical movements in levels, deviations from trend, and smoothed growth rates of the principal measures of aggregate economic activity - the quarterly real GDP and the monthly U.S. Coincident Index - using the phase average trend (PAT). Then we compare alternative trend estimates, deterministic and stochastic, linear and nonlinear, and the corresponding estimates of 'cyclical components,' that is, series of deviations from these trends. We discuss how these measures differ in terms of the patterns, timing, amplitudes, and smoothness of the resulting estimates of U.S. growth cycles in the post-World War II period. The results of PAT show great similarity to the results obtained with the H-P and band-pass filtering methods, but in matters of detail PAT is often superior.
A new approach is proposed for forecasting a time series with multiple seasonal patterns. A state space model is developed for the series using the innovations approach which enables us to develop explicit models for both additive and multiplicative seasonality. Parameter estimates may be obtained using methods from exponential smoothing. The proposed model is used to examine hourly and daily patterns in hourly data for both utility loads and traffic flows. Our formulation provides a model for several existing seasonal methods and also provides new options, which result in superior forecasting performance over a range of prediction horizons. In particular, seasonal components can be updated more frequently than once during a seasonal cycle. The approach is likely to be useful in a wide range of applications involving both high and low frequency data, and it handles missing values in a straightforward manner.