This book is downloadable from http://www.irisa.fr/sisthem/kniga/. Many monitoring problems can be stated as the problem of detecting a change in the parameters of a static or dynamic stochastic system. The main goal of this book is to describe a unified framework for the design and the performance analysis of the algorithms for solving these change detection problems. Also the book contains the key mathematical background necessary for this purpose. Finally links with the analytical redundancy approach to fault detection in linear systems are established. We call abrupt change any change in the parameters of the system that occurs either instantaneously or at least very fast with respect to the sampling period of the measurements. Abrupt changes by no means refer to changes with large magnitude; on the contrary, in most applications the main problem is to detect small changes. Moreover, in some applications, the early warning of small - and not necessarily fast - changes is of crucial interest in order to avoid the economic or even catastrophic consequences that can result from an accumulation of such small changes. For example, small faults arising in the sensors of a navigation system can result, through the underlying integration, in serious errors in the estimated position of the plane. Another example is the early warning of small deviations from the normal operating conditions of an industrial process. The early detection of slight changes in the state of the process allows to plan in a more adequate manner the periods during which the process should be inspected and possibly repaired, and thus to reduce the exploitation costs.

Detection of abrupt changes: theory and application

https://www.robjhyndman.com/papers/ijf25.pdf

25 years of time series forecasting

This paper is a critical review of exponential smoothing since the original work by Brown and Holt in the 1950s. Exponential smoothing is based on a pragmatic approach to forecasting which is shared in this review. The aim is to develop state-of-the-art guidelines for application of the exponential smoothing methodology. The first part of the paper discusses the class of relatively simple models which rely on the Holt-Winters procedure for seasonal adjustment of the data. Next, we review general exponential smoothing (GES), which uses Fourier functions of time to model seasonality. The research is reviewed according to the following questions. What are the useful properties of these models? What parameters should be used? How should the models be initialized? After the review of model-building, we turn to problems in the maintenance of forecasting systems based on exponential smoothing. Topics in the maintenance area include the use of quality control models to detect bias in the forecast errors, adaptive parameters to improve the response to structural changes in the time series, and two-stage forecasting, whereby we use a model of the errors or some other model of the data to improve our initial forecasts. Some of the major conclusions: the parameter ranges and starting values typically used in practice are arbitrary and may detract from accuracy. The empirical evidence favours Holt's model for trends over that of Brown. A linear trend should be damped at long horizons. The empirical evidence favours the Holt-Winters approach to seasonal data over GES. It is difficult to justify GES in standard form–the equivalent ARIMA model is simpler and more efficient. The cumulative sum of the errors appears to be the most practical forecast monitoring device. There is no evidence that adaptive parameters improve forecast accuracy. In fact, the reverse may be true.

Exponential smoothing: The state of the art

1. Introduction to Forecasting. 1.1 The Nature and uses of Forecasts. 1.2 Some Examples of Time Series. 1.3 The Forecasting Process. 1.4 Resources for Forecasting. 2. Statistics Background for Forecasting. 2.1 Introduction. 2.2 Graphical Displays. 2.3 Numerical Description of Time Series Data. 2.4 Use of Data Transformations and Adjustments. 2.5 General Approach to Time Series Analysis and Forecasting. 2.6 Evaluating and Monitoring Forecasting Model Performance. 3. Regression Analysis and Forecasting. 3.1 Introduction. 3.2 Least Squares Estimation in Linear Regression Models. 3.3 Statistical Inference in Linear Regression. 3.4 Prediction of New Observations. 3.5 Model Adequacy Checking. 3.6 Variable Selection Methods in Regression. 3.7 Generalized and Weighted Least Squares. 3.8 Regression Models for General Time Series Data. 4. Exponential Smoothing Methods. 4.1 Introduction. 4.2 First-Order Exponential Smoothing. 4.3 Modeling Time series Data. 4.4 Second-Order Exponential Smoothing. 4.5 Higher-Order Exponential Smoothing. 4.6 Forecasting. 4.7 Exponential Smoothing for Seasonal Data. 4.8 Exponential Smoothers and ARIMA Models. 5. Autoregressive Integrated Moving Average (ARIMA) Models. 5.1 Introduction. 5.2 Linear Models for Stationary Time Series. 5.3 Finite Order Moving Average (MA) Processes. 5.4 Finite Order Autoregressive Processes. 5.5 Mixed Autoregressive-Moving Average (ARMA) Processes. 5.6 Non-stationary Processes. 5.7 Time Series Model Building . 5.8 Forecasting ARIMA Processes . 5.9 Seasonal Processes. 5.10 Final Comments. 6. Transfer Function and Intervention Models. 6.1 Introduction. 6.2 Transfer Function Models. 6.3 Transfer Function-Noise Models. 6.4 Cross Correlation Function. 6.5 Model Specification. 6.6 Forecasting with Transfer Function-Noise Models. 6.7 Intervention Analysis. 7. Survey of Other Forecasting Methods. 7.1 Multivariate Time Series Models and Forecasting. 7.2 State Space Models. 7.3 ARCH and GARCH Models. 7.4 Direct Forecasting of Percentiles. 7.5 Combining Forecasts to Improve Prediction Performance. 7.6 Aggregation and Disaggregation of Forecasts. 7.7 Neural Networks and Forecasting. 7.8 Some Comments on Practical Implementation and use of Statistical Forecasting Techniques. Bibliography. Appendix. Appendix A Statistical Tables. Table A.1 Cumulative Normal Distribution. Table A.2 Percentage Points of the Chi-Square Distribution. Table A.3 Percentage Points of the t Distribution. Table A.4 Percentage Points of the F Distribution. Table A.5 Critical Values of the Durbin-Watson Statistic. Appendix B Data Sets for Exercises. Table B.1 Market Yield on U.S. Treasury Securities at 10-year Constant Maturity. Table B.2 Pharmaceutical Product Sales. Table B.3 Chemical Process Viscosity. Table B.4 U.S Production of Blue and Gorgonzola Cheeses. Table B.5 U.S. Beverage Manufacturer Product Shipments, Unadjusted. Table B.6 Global Mean Surface Air Temperature Anomaly and Global CO22 Concentration. Table B.7 Whole Foods Market Stock Price, Daily Closing Adjusted for Splits. Table B.8 Unemployment Rate - Full-Time Labor Force, Not Seasonally Adjusted. Table B.9 International Sunspot Numbers. Table B.10 United Kingdom Airline Miles Flown. Table B.11 Champagne Sales. Table B.12 Chemical Process Yield, with Operating Temperature (Uncontrolled). Table B.13 U.S. Production of Ice Cream and Frozen Yogurt. Table B.14 Atmospheric CO2 Concentrations at Mauna Loa Observatory. Table B.15 U.S. National Violent Crime Rate. Table B.16 U.S. Gross Domestic Product. Table B.17 U.S. Total Energy Consumption. Table B.18 U.S. Coal Production. Table B.19 Arizona Drowning Rate, Children 1-4 Years Old. Table B.20 U.S. Internal Revenue Tax Refunds. Index.

/pdf/introduction-to-time-series-analysis-and-forecasting-3ovs13b87u.pdf

Introduction to Time Series Analysis and Forecasting

A comprehensive account of economic size distributions around the world and throughout the years In the course of the past 100 years, economists and applied statisticians have developed a remarkably diverse variety of income distribution models, yet no single resource convincingly accounts for all of these models, analyzing their strengths and weaknesses, similarities and differences. Statistical Size Distributions in Economics and Actuarial Sciences is the first collection to systematically investigate a wide variety of parametric models that deal with income, wealth, and related notions. Christian Kleiber and Samuel Kotz survey, compliment, compare, and unify all of the disparate models of income distribution, highlighting at times a lack of coordination between them that can result in unnecessary duplication. Considering models from eight languages and all continents, the authors discuss the social and economic implications of each as well as distributions of size of loss in actuarial applications. Specific models covered include: Pareto distributions Lognormal distributions Gamma-type size distributions Beta-type size distributions Miscellaneous size distributions Three appendices provide brief biographies of some of the leading players along with the basic properties of each of the distributions. Actuaries, economists, market researchers, social scientists, and physicists interested in econophysics will find Statistical Size Distributions in Economics and Actuarial Sciences to be a truly one-of-a-kind addition to the professional literature.

Statistical Size Distributions in Economics and Actuarial Sciences

Use is often made of the Wiener and Ornstein-Uhlenbeck (O.U.) processes in various applications of stochastic processes to problems of engineering interest. These applications frequently involve the presence of barriers. Although mathematical methods for solving Kolmogorov's forward equation for the above processes have previously been discussed ([1], [2]), many solutions for problems with two barriers do not seem to be available in the literature. Instead, one finds solutions for unrestricted processes or simulation used in place of analytical solutions in various applications ([3], [4], [5]). In this paper, solutions of Kolmogorov's forward equations in the presence of constant absorbing and/or reflecting barriers are obtained by means of separation of variables. This enables one to obtain expressions for the probability density functions for first passage times when absorbing barriers are present. The solution for the O.U. process is used to obtain a result of Breiman's [6] concerning first passage times.

Solutions for some diffusion processes with two barriers

This paper presents expressions for the variance of the forecast error for arbitrary lead times for both the additive and multiplicative Holt-Winters seasonal forecasting models. It is shown that even when the smoothing constants are chosen to have values between zero and one, when the period is greater than four, the variance may not be finite for some values of the smoothing constants. In addition, the regions where the variance becomes infinite are almost the same for both models. These results are of importance for practitioners, who may choose values for the smoothing constants arbitrarily, or by searching on the unit cube for values which minimize the sum of the squared errors when fitting the model to a data set. It is also shown that the variance of the forecast error for the multiplicative model is nonstationary and periodic.

Computing the variance of the forecast error for the holt‐winters seasonal models

It is argued that plots of the hazard rate for the lognormal random variable which have appeared in some recent literature are incorrect and/or misleading; the hazard rate always begins at zero, rises to a maximum, then decreases very slowly to zero. An equation for the location of the maximum of the hazard rate is derived. The maximum lies in a finite interval for all positive values of the standard deviation of the associated normal random variable. Approximations that can be used to compute the hazard rate for parameter values outside of the usual range in the tables associated with the normal (Gaussian) random variable are presented. >

On the hazard rate of the lognormal distribution

A note on a comparison of exponential smoothing methods for forecasting seasonal series

A discrete exponential model has been proposed for use as a learning curve. This model describes different learning behavior than the classical or a modified version of the power-form model and it provides some practical advantages. Tests for selection between the models are given. Methods of parameter estimation for the exponential model based on least-squared errors, maximum likelihood and mean likelihood criteria are shown for individual cycle time data. Estimation methods are also given for data on the number of completed cycles during a fixed time interval. Monte Carlo simulation experiments were conducted to test the past description and future forecast quality of this model using these estimation methods. The classical power-form model was also included in these experiments. Estimation precision is described for both models. Other features of the exponential model are shown in contrast to the power-form models.

Arnold L. Sweet

Papers

Solutions for some diffusion processes with two barriers

Computing the variance of the forecast error for the holt‐winters seasonal models

On the hazard rate of the lognormal distribution

A note on a comparison of exponential smoothing methods for forecasting seasonal series

Parameter Estimation Methods for Discrete Exponential Learning Curves