Time Series Analysis and Forecasting PDF

OPRE 3333 - Unit 5 Time Series Analysis and Forecasting1 Forecasting methods can be classified as qualitative or quantitative. ˆ Qualitative methods: Generally involve the use of expert judgment to develop fore- casts. ˆ Quantitative methods: can be used when...

OPRE 3333 - Unit 5 Time Series Analysis and Forecasting1 Forecasting methods can be classified as qualitative or quantitative. ˆ Qualitative methods: Generally involve the use of expert judgment to develop fore- casts. ˆ Quantitative methods: can be used when – past information about the variable being forecast is available. – the information can be quantified. – it is reasonable to assume that past is prologue (i.e., that the pattern of the past will continue into the future). Time series: A sequence of observations on a variable measured at successive points in time or over successive periods of time. Time series patterns: 1. Horizontal pattern: data fluctuate randomly around a constant mean over time Example (Gasoline Sales Data): Consider the 12 weeks of data shown in the fol- lowing table. These data show the number of gallons of gasoline (in 1,000s) sold by a gasoline distributor in Bennington, Vermont, over the past 12 weeks. 1 Camm, J. D., Cochran, J. J., Fry, M. J., & Ohlmann, J. W. (2020). Business Analytics (4th ed.). Cengage Learning US. OPRE 3333 - Unit 5 2 2. Trend pattern: gradual shifts or movements to relatively higher or lower values over a longer period of time Example: Consider the time series of bicycle sales for a particular manufacturer over the past 10 years, as shown bellow. Example: Consider the sales revenue for a cholesterol drug since the company won FDA approval for the drug 10 years ago, as shown bellow. OPRE 3333 - Unit 5 3 3. Seasonal pattern: recurring patterns over successive periods of time Example: A retailer of snow removal equipment and heavy clothing expects low sales activity in the spring and summer months, with peak sales in the fall and winter months to occur every year. Example: Consider the number of umbrellas sold at a clothing store over the past five years, as shown bellow. Although we generally think of seasonal movement in a time series as occurring over one year, time series data can also exhibit seasonal patterns of less than one year in duration. Example: Daily traffic volume shows within-the-day “seasonal” behavior, with peak levels occurring during rush hours, moderate flow during the rest of the day and early evening, and light flow from midnight to early morning. OPRE 3333 - Unit 5 4 4. Trend & seasonal patterns: both a trend and a seasonal pattern Example: The following time series plot shows quarterly smartphone sales for a par- ticular manufacturer over the past four years. Identifying time series patterns: ˆ The underlying pattern in the time series is an important factor in selecting a forecast- ing method. ˆ A time series plot should be one of the first analytic tools. ˆ We need to use a forecasting method that is capable of handling the pattern exhibited by the time series effectively. Forecast accuracy measures: measures to determine how well a particular forecasting method is able to reproduce the time series data that are already available The key concept associated with measuring forecast accuracy is forecast error. forecast error (et ) = yt − yˆt where yt and yˆt are the actual value and the forecasted values for period t, respectively. Example: Consider the following time series data. A forecasting technique is used to reproduce the time series values. Calculate the forecast error for each period. OPRE 3333 - Unit 5 5 Year Value Forecast Error 1 10 9 2 11 10 3 9 11 1. Mean Absolute Error (MAE): Measure of forecast accuracy that avoids the prob- lem of positive and negative forecast errors offsetting one another. Pn t=k+1| et | M AE = n−k 2. Mean Squared Error (MSE): Measure that avoids the problem of positive and neg- ative errors offsetting each other is obtained by computing the average of the squared forecast errors. Pn 2 t=k+1 et M SE = n−k 3. Mean Absolute Percentage Error (MAPE): Average of the absolute value of percentage forecast errors. Pn et t=k+1 | ( yt )100 | M AP E = n−k OPRE 3333 - Unit 5 6 Forecasting methods: 1. Naı̈ve forecasting method: It uses the most recent data value as the forecast for the next period. Example (Gasoline Sales Data): Using the naı̈ve forecasting method, compute the forecast accuracy measures for the following time series data. Year Time Series Value 1 17 2 21 3 19 4 23 5 18 6 16 7 20 8 18 9 22 10 20 11 15 12 22 2. Average of all past values: It uses the average of all the past data values in the time series as the forecast for the next period. Example (Gasoline Sales Data): Using the average of past values forecasting method, compute the forecast accuracy measures for the Gasoline Sales time series data. 3. Moving averages: It uses the average of the most recent k data values in the time series as the forecast for the next period. P Pt most recent k data values yi yt−k+1 + · + yt−1 + yt ŷt+1 = = i=t−k+1 = k k k where ˆ ŷt+1 = forecast of the time series for period t + 1 ˆ yt = actual value of the time series in period t ˆ k = number of periods of time series data used to generate the forecast OPRE 3333 - Unit 5 7 Parameter k: ˆ If only the most recent values of the time series are considered relevant, a small value of k is preferred. A smaller value of k will track shifts in a time series more quickly. ˆ If a greater number of past values are considered relevant, then we generally opt for a larger value of k. A larger values of k will be more effective in smoothing out random fluctuations. Example (Gasoline Sales Data): Using the moving average forecasting method with k = 3, compute the forecast accuracy measures for the Gasoline Sales time series data. 4. Exponential smoothing: Uses a weighted average of past time series values as a forecast. ŷt+1 = αyt + (1 − α)ŷt where smoothing constant (α) is the weight given to the actual value in period t. OPRE 3333 - Unit 5 8 Parameter α: ˆ Insight into choosing a good value for α can be obtained by rewriting the basic exponential smoothing model: ŷt+1 = αyt + (1 − α)ŷt = αyt + ŷt − αŷt = ŷt + α(yt − ŷt ) = ŷt + αet ˆ If the time series contains substantial random variability, a small value of the smoothing constant is preferred and vice-versa. ˆ Choose the value of α that minimizes the MSE. Example (Gasoline Sales Data): Using the exponential smoothing forecasting method with α = 0.2, compute the forecast accuracy measures for the Gasoline Sales time series data. OPRE 3333 - Unit 5 9 Practice problems: 1. Which of the following states the objective of time series analysis? (a) To study the variation of time with respect to increase in the variable value (b) To analyze the time-dependent environmental factors that affected variable values in the past (c) To use present variable values to study what should have been the ideal past values (d) To uncover a pattern in the time series and then extrapolate the pattern into the future (correct) 2. A set of observations on a variable measured at successive points in time or over successive periods of time constitute a (a) geometric series (b) time invariant set (c) time series (correct) (d) logarithmic series 3. A pattern exists when the data fluctuate randomly around a constant mean over time. (a) vertical (b) seasonal (c) trend (d) horizontal (correct) 4. Using a large value of k in the moving averages method is effective in (a) providing inconsistent forecast values (b) tracking changes in a time series more quickly (c) smoothing out random fluctuations (correct) (d) taking into account the effect of seasonal variations in the time series OPRE 3333 - Unit 5 10 5. Using the four forecasting techniques (naive method, average of all, four-year moving average, exponential smoothing with α = 0.4), compute the following measures of forecast accuracy for the below times series data: (a) Mean absolute error (b) Mean squared error (c) Mean absolute percentage error (d) What is the forecast for year 11? (e) Which forecasting technique is more appropraite in terms of MSE? Year Time Series Value 1 234 2 287 3 255 4 310 5 298 6 250 7 456 8 412 9 525 10 438 6. Using the average of all the historical data as a forecast for the next year, compute the following measures of forecast accuracy for the times series data. (a) Mean forecast error (b) Mean absolute error (c) Mean squared error (d) Mean absolute percentage error (e) What is the forecast for the next year? Year Value 1 10 2 15 3 16 4 14 5 13 OPRE 3333 - Unit 5 11 7. Using a two-year moving average, compute MSE and a forecast for the next year, for the following times series data. Year Value 1 10 2 15 3 16 4 14 5 13 8. Use α = 0.25 to compute the exponential smoothing values for the time series below. Compute MSE and a forecast for the next year. Year Value 1 10 2 15 3 16 4 14 5 13