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Part II – TIME SERIES ANALYSIS C1 Introduction to TSA "If we could first know where we are, then whither we are tending, we could then decide what to do and how to do it." Abraham Lincoln, 1809-1865 © Angel A. Juan & Carles Serrat - UPC 2007/2008 2.1.1: Motivation An essential aspect of managing any organization is planning for the future. The long-run success of an organization is closely related to how well management is able to anticipate the future and develop appropriate strategies. The purpose of this part is to introduce several methods which make use of existent historical data to produce forecasts (predictions). Historical data form a time series, i.e.: a set of observations on a variable measured at successive points in time or over successive periods of time. Examples: daily temperature measurements, weekly sales figures, monthly stock market prices, quarterly profits, yearly power-consumption data, etc, Two main goals of time series analysis: 1. Identifying the nature of the phenomenon represented by the sequence of observations, and 2. Forecasting (predicting future values of the time series variable) 2.1.2: Classification of Forecasting Methods Forecasting methods can be classified as quantitative or qualitative. Quantitative forecasting methods can be used when: 1. Past information about the variable being forecast is available, 2. The information can be quantified, and 3. It can be assumed that the pattern of the past will continue into the future In such cases, a forecast can be developed using a time series method or a causal method: Time series methods: The historical data are restricted to past values of the variable. The objective is to discover a pattern in the historical data and then extrapolate the pattern into the future. Ex.: trend analysis, classical decomposition, moving averages, exponential smoothing, ARIMA. Causal forecasting methods: Based on the assumption that the variable we are forecasting has a cause-effect relationship with one or more other variables (e.g.: the sales volume can be influenced by advertising expenditures). Ex.: regression analysis. Qualitative methods generally involve the use of expert judgment to develop forecasts. Time Series Methods Causal Forecasting Methods 2.1.3: General Aspects of Time Series In TSA it is assumed that the data consist of a systematic pattern (usually a set of identifiable components) and random noise (error) which usually makes the pattern difficult to identify. Most TSA techniques involve some form of filtering out noise in order to make the pattern more salient. Time series patterns can be usually described in terms of two basic classes of components: Trend: Represents a general systematic component that changes over time and does not repeat. Seasonality: It repeats itself in systematic intervals over time. Those two general classes of time series components may coexist in real-life data. In this example, the amplitude of the seasonal changes remains approximately constant. This pattern is called additive seasonality. In this example, the amplitude of the seasonal changes increases with the overall trend. This pattern is called multiplicative seasonality. seasonality 2.1.4: Plotting a Time Series (Minitab) File: RIVERC.MTW Stat > Time Series > Time Series Plot… • C1 Hour: lists the time of day (in military format) of each measurement. • C2 Temp: gives the temperature measurement in ºC. The plot indicates a periodical pattern beginning with temperatures at 2 p.m. around 42 degrees, which rise for the next 8 hours and then fall until they reach a low of about 37 degrees at 7 a.m. This suggests that your time series model might include a seasonal component. 45,0 42,5 Temp The river data are fairly well-behaved, as evidenced by the time series plot. If you were to draw a horizontal line at the mean of the series, you would see that the data oscillate with the same cyclical pattern. A time series like this, whose mean and variance do not change with time, is called stationary. Conversely, a time series that appears to wander and does not systematically fluctuate around a single value is called nonstationary. Time Series Plot of Temp 40,0 37,5 35,0 1400 2200 700 1600 100 1000 1900 Hour 400 1300 2200 700 2.1.5: Classification of Time Series Methods Simple forecasting and smoothing methods: Based on the idea that reliable forecasts can be achieved by modeling patterns in the data that are usually visible in a time series plot. Your choice of method should be based upon whether the patterns are static (constant in time) or dynamic (changes in time), the nature of the trend and seasonal components, and how far ahead that you wish to forecast. Simple Forecasting Methods Smoothing Methods Generally easy and quick to apply. ARIMA methods: Also make use of patterns in the data, but these patterns may not be easily visible in a plot of the data. Instead, ARIMA modeling uses differencing and the autocorrelation and partial autocorrelation functions to help identify an acceptable model. ARIMA stands for Autoregressive Integrated Moving Average, which represent the filtering steps taken in constructing the ARIMA model until only random noise remains. Fitting a model is an iterative approach that may not lend itself to application speed and volume. ARIMA Methods References for Time Series Analysis Anderson, D.; Sweeney, D.; Williams, T. (2004): Statistics for Business and Economics. South-Western College Pub. Berk, K.; Carey, P. (2003): Data Analysis with Microsoft Excel. Duxbury Press. Hanke, J.; Wichern, D. (2004): Business Forecasting. Prentice Hall. McKenzie, J.; Goldman, R. (1998): The Student Edition of MINITAB for Windows. Addison Wesley Longman. Mendenhall, W.; Sincich, T. (2003): A Second Course in Statistics: Regression Analysis. Prentice Hall. StatSoft (2007): Electronic Statistics Textbook. Available at http://www.statsoft.com/textbook/stathome.html