Data Mining on Time Series: An Illustration Using Fast-Food Restaurant Franchise Data, by Liu, L.-M, Bhattacharyya, S., Sclove, S., Chen, R., Lattyak, W. (2001) is pending publicationin Computational Statistics and Data Analysis. Given the widespread use of modern information technology, a large number of time series may be collected during normal business operations. We use a fast-food restaurant franchiseas a case to illustrate how data mining can be applied to such time series, and help the franchise reap the benefits of such an effort. Time series data mining at both the store level and corporate level are discussed. Box-Jenkins seasonal ARIMA models are employed to analyze and forecast the time series. Instead of a traditional manual approach of Box-Jenkins modeling, an automatic time series modeling procedure is employed to analyze a large number of highly periodic time series. In addition, an automatic outlier detection and adjustment procedure is used for both model estimation and forecasting. The improvementin forecast performance due to outlier adjustment is demonstrated. Adjustment of forecasts based on stored historical estimates of like-events is also discussed. Outlier detection also leads to information that can be used not only for better inventory management and planning, but also to identify potential sales opportunities. To illustrate the feasibility and simplicity of the above automatic procedures for time series data mining, the SCA Statistical System is employed to perform the related analysis.

Keywords: Automatic time series modeling, Automatic outlier detection, Outliers, Forecasting, Expert system, Knowledge discovery

A Multiple-Input Transfer Function Model of Okun's Misery Index: An Empirical Test of the Maharishi Effect by Cavanaugh, K., King, K.D., Ertuna, C., was a presented paper at the 1989 meeting of the American Statistical Association. This paper analyzes the time seriesbehavior of Okun's economic "misery index" of inflation and unemployment, defined as the sumof the inflation rate and unemployment rate. We identify and estimate a multiple-inputtransfer function model of the monthly misery index for the U.S. during the period 1979 to1988 using Liu's (1985) Linear Transfer Function (LTF) method supplemented by the use of Akaike Information Criterion (AIC) to provide an objective criterion of model selection.Maximum likelihood estimates of the model are used to further test a novel hypothesis suggested by a new field-theoretic pardadigm of consciousness and socioeconomic behaviorproposed by Maharishi Mahesh Yogi. The results of this study suggest that the substantialimprovement in this index of economic performance and economic quality of life over this period was significantly influenced by the collective practice of a subjective technologyof consciousness, the Transcendental Meditation and TM-Sidhi program. Controlling for theeffects of monetary growth, growth in crude materials prices, and the rate of change ofindustrial production, the null hypothesis of no effect of the TM and TM-Sidhi group on the misery index must be decisively rejected for these data, thus lending strong supportto the Maharishi effect hypothesis.

Keywords: Multiple time series, Inflation, Unemployment, AIC, TranscendentalMeditation Program

An Approach To Detection and Treatment Of Outliers in Multiple Time Seriesby Pankratz, A. was presented in as a poster session at the 1992 meeting of the AmericanStatistical Association. This paper offers a first approach to the detection of outliersin multiple time series by examining the case of transfer function plus autocorrelated disturbance, with one imput described by an ARIMA process. Special attention is given to effects on the output series of outliers arising in the input series. These effects mayor may not be passed through the transfer function to the output. I extend a single-seriesmethod to include multivariate least squares estimates to detect certain multiple seriesoutliers. I illustrate using a real data set.

Keywords: ARIMA, Dynamic regression, Intervention, Multiple time series, Multivariate restricted least squares, outliers, transfer function.

A (Relatively) Simple Sectoral Employment Level Forecasting Model For Small Regional Economies by Weller, B. was presented at the 36th North American Meetings of the Regional Science Association. This paper illustrates the use of a small multi-input/multi-equation transfer function model to generate a set of internallyconsistent regional employment as well as employment in two major sub-sectors of theeconomy, specifically manufacturing and non-manufacturing. In addition to providinginternally consistent forecasts, the model is relatively easy to develop since it usesreadily available national aggregates as drivers. Values for the driver variables are, themselves, forecast using univariate ARIMA models. Thus, the model is self-contained, requiring no user supplied forecasts of the input variables. Further, the model is moreintuitively appealing than pure time series approaches since it incorporates some aspectsof causality. Post-fit tests of forecast accuracy, spanning intervals of widely varyingdegress of instability, indicate the multiple-equation transfer function model outperformsunivariate sectoral forecasting models over horizons ranging from one through twelvemonths. Altough most pronounced during periods of continuity or relative stability, theaccuracy advantage of te multi-equation model is apprarent regardless of whether theforecast interval is highly unstable or relatively stable.

Keywords: Employment forecasting, Leading indicators, Multi-equation transfer function models, Regional forecasting.

A Statistical Assessment of the Effect of theCar Inspection/Maintenance Program on Ambient CO Air Quality in Phoenix, Arizona by Tiao, G.C., Liu, L.-M., Hudak, G. appeared in Environmental Science & Technology inJuly 1989. As mandated by Clean Air Act Ammendments, several states have implementedcar inspection and maintenance (I/M) programs to reduce automobile emission. This paperinvestigates such a program on the ambient CO concentrations in Phoenix, AZ. Empiricalmodels are formulated to describe the observed concentrations as a function of trafficand meteorological variables. These models help determine the existence of any effects due to the I/M program after adjusting for changes in traffic and meteorologicalconditions. In particular, by comparing our empirical results with emission factorsderived from MOBILE3 analysis, we find some evidence to support the hypothesis that theI/M program has reduced ambient CO levels.

Water treatment control using the joint estimation outlier detection methodby Christine Wright and David Booth appeared in Environmental Modeling and Assessment 6:77-82, 2001. In many industries, it is important to determine when the process is out-of-control, (i.e., when significant adverse process changes occur). The idea is to discover theseadverse process changes while they are still relatively minor, before substandard productor significant pollution is produced. One example of an important chemical process controlproblem is wastewater data. The objective of the research was to determine if out-of-controlobservations (i.e., abnormal states) could be detected by JE in the period when they firstoccurred. The process control method reported herein can be used for any compound for whichan analytical chemical detection method exists. The method that we consider, Joint Estimation(JE), has the potential to be extremely important to both general pollution control and statistical process control.

Effectiveness of Joint Estimation When the Outlier Is the Last Observationin an Autocorrelated Short Time Series by Wright, C., Hu, M.Y., and Booth, D.E., appeared in Decision Sciences (1999), Volume 30, Number 3. The Effectiveness of the jointestimation (JE) outlier detection method as a process control technique for short autocorrelated time series is investigated and compared with exponentially weightedmoving average (EWMA). The research goal is to determine the effectiveness of themethod for detecting out-of-control observations when they are the last observation in ashort autocorrelated time series. This is an important problem because detecting anoutlier in the period when it occurs, rather than several periods after it occurs, willpreclude the production of more defective units. Two cases are investigated: short simulated time series when normality is assumed, and short real time series whenthe assumption is violated. The results show that JE is effective for short time series,particularly for autoregressive series when normality is violated. Joint estimation isalso effective for moving average series under the normality assumption and lesseffective when the assumption is violated. In all cases, JE is found to be more effectivethan EWMA.

Asymptotic Least Squares Estimation Efficiency Considerations And Applications by Kodde, D.A., Palm, F.C., and Pfann, G.A., appeared in Journal of Applied Econometrics (March 1989). This paper is concerned with the large sample efficiency of the asymptotic least squares (ALS) estimatorsintroduced by Gourieroux, Monfort, and Trognon (1982, 1985). We show how the efficiency of these estimatorsis affected when additional information is incorporated into the estimation procedure. The relationshipbetween ALS and maximum likelihood is discussed. It is shown that ALS can be used to obtain asymptoticallyefficient estimates for a large range of econometric models. Many results from the literature on estimationare special cases of the framework adopted in this paper.

An application of ALS to a dynamic rational expectations factor demand model in the manufacturing sector is the Netherlands demonstrates the potential ofthe method in the estimation of the parameters in models which are subject to non-linear cross-equation restrictions.

A Time Series Analysis of Gonorrhea Surveillance Data by Schnell, D., Zaidi, A., and Reynolds, G., appeared in Statistics In Medicine, Vol. 8, Page 343-352 (September 1989). Gonorrhea is the most frequently reported communicable disease in the United States. In response to rapidly rising ratesin the late 1960s, the Public Health Service instituted a gonorrhea control programme. An important component of theprogramme is the screening of women for gonococcal infections. We use a time series intervention model to estimate theinitial increase in reporting of cases in women associated with the control programme. From the middle 1970s to the middle1980s, a regular seasonal pattern in the data is conspicuous. We use a second time series model to quantify the seasonalvariation during this period and to construct forecasts.

Back-end Manager: An Interface Between A Knowledge-based Front End And Its Application Subsystems by Prat, A., Lores, J., Fletcher, P., and Catot, J.M.,appeared in Seccion de Tecnicas Cuantitativa de Gestion, Vol. 3, No. 4 (December 1990). Front Ends for Open and Closed User Systems (FOCUS) is an ESPRIT/2 (no. 2620) project aimed at designing toolsand techniques for the construction of knowledge-based front ends (KBFEs) for open-user systems (reusable softwarecomponents, libraries, etc.) and closed user-systems (free-standing software, packages, etc.). An important partof the project involves the establishment of an architecture for KBFEs and the specification of the KBFE/back-end interface. This paper describes the properties and related issues of such an interface, known as the back-end manager (BEM), and its relationship to the proposed KBFE architecture.

Keywords: Knowledge-based front end, interface separability, back-end manager, user interface.

Combining Forecasts With Contemporaneously Correlated Errors Using Benchmarking by Pankratz, A.,Dept. of Economics and Management, DePauw University (June 1989). Combining forecasts from different sources often improves forecast accuracy. Benchmarking methods have beenproposed for combining forecasts that are stated across different time spans. For example, a set of monthlyforecasts may be adjusted to reflect an annual forecast for the same series obtained from another source. Previous authors presenting benchmarking methods have taken the forecast errors from the different sources to be independent, but it could easily happen that these errors are correlated. This paper gives a benchmarkingmethod for combining forecasts which allows for contemporaneous correlation among the forecast errors from thedifferent sources. The solution is found by adapting generalized least squares estimation of linear model parametersin the presence of extraneous information to the problem of combining forecasts. It is shown that combinedforecasts based on benchmarking methods may be interpreted as modified weighted averages. A vector-ARMA model-basedapproach is used for implmentation; however, the results may be adapted to other extrapolative model forecasts. An empirical example is given.

Keywords: ARIMA models, composite forecasts, distributed lag regression, dynamic regression, extraneousinformation, extrapolative forecasts, generalized least squares, mixed estimation, transfer function, vector ARMA models.

Death Rated & Real Wages: An Analysis of Granger Causality with Post Sample Data & Different Forecast Horizons by Hagnell, M. and Salomonsson, A.,Dept. of Statistics, University of Lund, (June 1989). For yearly Swedish data 1751-1850 we investigated whether an index of the real wages was a Granger cause of the death ratein ages 25-50 years. The investigation was done by comparing the post sample forecasting performance of on one hand a univariate ARIMA model for the death rate and on the other hand a bivariate time series model, where the death rate wasexplained by the real wages index. As bivariate time series model both a transfer function and a distrubuted-lag model was used. Different post sample periods were used and the effect of outliers in the death rate was taken into consideration. Inthe evaluation of forecasting performance we used multi-step ahead forecasts. This approach proved valuable in supportingthe evidence for the hypothesis of causality. On the whole, the analysis showed that the real wages index was a Grangercause of the death rate, although the evidence was weaker when the effect of outliers was eliminated.

Linear Combination of Restrictions and Forecasts in Time Series Analysis by Guerrero, V.M. and Pena, D.,Journal of Forecasting, Vol. 19, Page 103-122, 2000. An important tool in time series analysis is that of combining information in an optimal way. Here we establish a basic combiningrule of linear predictors and show that such problems as forecast updating, missing value estimation, restricted forecasting with bindingconstraints, analysis of outliers and temporal disaggregation can be viewed as problems of optimal linear combination of restrictionsand forecasts. A compatibility test statistic is also provided as a companion tool to check that the linear restrictions arecompatible with the forecasts generated from the historical data.

Keywords: Compatibility testing; disaggregation; missing data; outliers; restricted forecasts.

Measuring Intervention Effects on Multiple Time Series Subjected to Linear Restrictions: A Banking Example by Guerrero, V.M., Pena, D., and Poncela, P.,Journal of Business & Economic Statistics, Vol. 16, No. 4, October 1998.We consider the problem of estimating the effects of an intervention on a time series vector subjected to a linear constraint. Minimum variance linear and unbiased estimators are provided for two different formulations of the problem - (1) when amultivariate intervention analysis is carried out and an adjustment is needed to fulfill the restriction and (2) when a univariate intervention analysis was performed on the aggregate series obtained from the linear constraint, previous tothe multivariate analysis, and the results of both analyses are required to be made compatible with each other. A bankingexample that motivates this work illustrates our solutions.

Keywords: Accounting constraint; Linear estimators; Multivariate intervention; Restricted estimation; VARMA models.

Dynamic Relationship Analysis of US Gasoline and Crude Oil Prices by Liu, L.-M.,Journal of Forecasting, Vol. 10, Page 521-547, 1991.This paper studies the dynamic relationships between U.S. gasoline prices, crude oil prices, and the stock of gasoline. Usingmonthly data between January 1973 and December 1987, we find that the US gasoline price is mainly influenced by the price ofcrude oil. The stock of gasoline has little or no influence on the price of gasoline during the period before the second energy crisis, and seems to have some influence during the period after. We also find that the dynamic relationship between the prices of gasoline and crude oil changes over time, shifting from a longer lag response to a shorter lag response. Box-Jenkins ARIMAand transfer function models are employed in this study. These models are estimated using estimation procedure with and without outlier adjustment. For model estimation with outlier adjustment, an iterative procedure for the joint estimation of modelparameters and outlier effects is employed. The forecasting performance of these models is carefully examined. For the purpose of illustration, we also analyze these time series using classical white-noise regression models. The results show the importanceof using appropriate time-series methods in modeling and forecasting when the data is serially correlated. This paper also demonstrates the problems of time-series modeling when outliers are present.

Keywords: Transfer function model; ARIMA models; Linear transfer function method; Outlier detection; White-noise regressionmodels; Gasoline prices; Crude oil price, Stock of gasoline; Energy crisis.

Dynamic Structural Analysis And Forecasting Of Residential Electricity Consumption by Liu, L.-M., Harris, J.L.,International Journal of Forecasting, Vol. 9, Page 437-455, 1993.This paper studies the dynamic relationships between electricity consumption and several potentially relevant variables,such as weather, price, and consumer income. Monthly data from January 1969 to December 1990 for all-electric residencesin the southeast United States are used for this study. Because of the nature of the annual weather cycle, several of thesetime series are highly seasonal. Multiple-input transfer function models are employed to analyze the data for their dynamic structure and to evaluate future levels of electricity consumption. The linear transfer function (LTF) method is employed in the identification of transfer function models for structural analysis and forecasting. A major finding is that priceplays a major role in explaining conservation behavior by electricity consumers. This result has important implications for forecasting the consumption of electric energy. This paper also demonstrates the appropriate construction of models for economictime series with strong seasonality.

Keywords: Transfer function models; ARIMA models; Seasonality; Electricity consumption; Electricity prices; Energy conservation; Weather conditions.

Forecasting Time Series With Outliers by Chen, C. and Liu, L.-M.,Journal of Forecasting, Vol. 12, Page 13-35, 1993.Time-series data are often contaminated with outliers due to the influenceof unusual and non-repetitive events. Forecast accuracy in such situationsis reduced due to (1) a carry-over effect of the outlier on the point forecastand (2) a bias in the estimates of model parameters. Hilimer (1984) andLedolter (1989) studied the effect of additive outliers on forecasts. It wasfound that forecast intervals are quite sensitive to additive outliers, butthat point forecasts are largely unaffected unless the outlier occurs near theforecast origin. In such a situation the carry-over effect of the outlier canbe quite substantial. In this study, we investigate the issues of forecastingwhen outliers occur near or at the forecast origin. We propose a strategywhich first estimates the model parameters and outlier effects using theprocedure of Chen and Liu (1993) to reduce the bias in the parameterestimates, and then uses a lower critical value to detect outliers near theforecast origin in the forecasting stage. One aspect of this study is on thecarry-over effects of outliers on forecasts. Four types of outliers areconsidered: innovational outlier, additive outlier, temporary change, andlevel shift. The effects due to a misidentification of an outlier type areexamined. The performance of the outlier detection procedure is studiedfor cases where outliers are near the end of the series. In such cases, wedemonstrate that statistical procedures may not be able to effectivelydetermine the outlier types due to insufficient information. Some strategiesare recommended to reduce potential difficulties caused by incorrectlydetected outlier types. These findings may serve as a justification forforecasting in conjunction with judgment. Two real examples areemployed to illustrate the issues discussed.