رگرسیون خطی محلی برای برآورد سری زمانی داده ها

Predicting future performance based on past performance history is a task often undertaken by business process managers. Various statistical and analytical techniques, such as time series and neural network modeling, are available. However, these techniques require the availability of a long time series for the development of a predictive model. Local linear regression (LLR) is an additional nonparametric statistical method that can be used to estimate a time series response variable. The LLR technique does not require a long time series for the development of a predictive model. In fact, the LLR technique can be utilized for prediction once three data points have been collected from the business process. In this work, LLR was evaluated as a tool for predicting future values of process parameters based on historical values. If successful, the LLR technique could be applied in start-up conditions or used as an alternative in some situations to time series modeling. The LLR procedure outperformed traditional time series techniques for the example stationary data sets and had comparable results to the ARIMA model for the example seasonal data set. In addition the LLR technique uses the data that is currently available from a process as its basis for prediction, thus providing a dynamic predictive technique that can continue to function in the presence of process changes.

Business process managers are often interested in predicting the future value of important process parameters based on historical values of those parameters. A multitude of methods is available for making such predictions, including statistical and analytical techniques. Statistical techniques including time series, regression, and nonparametric regression have been used for developing predictive models (Box and Jenkins, 1976; Härdle, 1990). Analytical neural network techniques have also been used extensively for prediction (Chiu et al., 1995; Cook and Chiu, 1997; Hoskins and Himmelblau, 1988; Saad et al., 1998; Gao et al., 1997). These methods require the availability of a long time series that is used for model development and model validation before parameter estimates can be obtained and predictions can be made. The ARIMA approach for time series predictive model development is theoretically and statistically appealing. However, the complexity of these models has often hindered their widespread adoption as a forecasting tool in organizations (Makridakis et al., 1983). The development of neural network models requires large training data sets containing examples of multiple process operating conditions. Extensive network design, training and testing are also typically required for model development. Additionally the performance of the time series and neural network models must be regularly monitored to assure that the model continues to represent the process. If a process change occurs, development of a new model is required. When a process change has occurred, a new data set representing the process must be collected and a new model developed and tested, for both time series and neural network techniques. The ability to develop a predictive model that uses the data that is currently available, does not require a large initial training data set, and responds dynamically to process changes over time is very appealing to many process modelers and managers. Work by Härdle et al. (1997) used nonparametric regression methods for estimating spectral densities, higher order conditional moments or conditional densities for time series data. Tjostheim (1994) used nonparametric regression techniques to test the linearity and independence of time series data. Local linear regression (LLR), the nonparametric procedure presented in this paper, is a technique that meets these specifications. The LLR procedure requires only three data points to obtain an initial prediction and then uses all new data as it becomes available to make future predictions, thus making it a dynamic procedure for predicting time series data. That is, making predictions with the first three measurements from a process, the LLR procedure can dynamically make predictions after each subsequent measurement or observation—rather than waiting until all of the data has been collected. The proposed LLR technique is described and analyzed as follows. First the theory and implementation of the LLR method is described. Then the simulation of various test data series is described. Finally, results are generated using the LLR technique and the performance of the LLR technique is compared to that of traditional time series methods.

This research has shown that the LLR method is preferable in cases where a limited process history is available and in cases of a dynamic process where frequent process changes or shifts occur. For many business processes, frequent process changes or short process runs are standard, as customer demand for the product or service changes or raw material incoming properties vary with changes in suppliers and ambient conditions. If only limited data is available for model development, the ARIMA model is a poor descriptor of the data and the resulting model is typically not optimal. In these cases, the estimates of parameters will be biased with increased variances. The LLR technique is preferable under these various conditions. In addition, the LLR procedure fits seasonal time series data as well as the traditional ARIMA models, demonstrating its feasibility even in the presence of seasonality. If the observed time series data conforms closely to the prescribed ARIMA model, one would expect that the parametric time series approach offers the best results. The LLR technique would be expected to lag behind seasonal or cyclical changes in a time series; however, the LLR technique proved to provide predictions comparable to seasonal ARIMA models. The data-driven bandwidth facilitates a fairly quick reaction to the changes, and the future addition of a functional polynomial term to the LLR methodology would be expected to improve the predictive capability. In the case of the five data sets tested, the LLR method performed as well if not better than the ARIMA method. This superior performance by the LLR method was accomplished with a significantly reduced data set as opposed to that utilized for the ARIMA model development. Consequently, the LLR technique can be viewed as a viable alternative when full data sets are available and process changes are infrequent. A computer automated local linear regression model building technique could be used in place of time series models in instances where statistical model building and monitoring expertise is not available. In addition, the automated LLR technique could be used as a pre-cursor to a predictive modeling technique in start-up situations, while an adequate data set is being collected for the development of other predictive models.