CHAPTER TIME SERIES ANALYSIS Time series data Numerical data obtained at regular time intervals The time intervals can be annually, quarterly, daily, hourly, etc Example Annual sales of a firm for successive years Year 2000 2001 2002 2003 2004 Sales 75.3 74.2 78.5 79.7 80.2 Example Number of registered journeys for a Home Removals firm: Qtr Qtr Qtr Qtr Year 73 90 121 98 Year 69 92 145 107 Year 86 111 157 122 Year 88 109 159 131 Time series cycle General pattern which broadly repeats Regularly occur but may vary in length Often measured peak to peak or trough to trough Cycle Sales Year Standard time series models There are various types of model that can be used to describe time series data Two main models: Additive model: y=t+s+r Multiplicative model y=txSxR y: is a given time series value wher e t: is the trend component s: is the seasonal component r: is the residual component Time series components Time-Series Trend Component Seasonal Component Residual Component Trend component The underlying, long-term tendency of the data series (overall upward or downward movement) Some techniques are used for extracting a trend from a given time series: semiaverage, regression (least square method), moving average… Trend Component Example Sales trend d r a Upw Time Trend Component Trend can be upward or downward Trend can be linear or non-linear Sales Sales Time Downward linear trend Time Upward nonlinear trend 10 Q1 Q2 Q3 Q4 Year -50.06 108.21 -93.52 24.75 Year -36.98 111.29 -80.44 32.83 Year -43.90 99.37 -92.36 Totals 318.8 -266.32 130.9 106.29 -88.77 Averages 443.65 -0.03 Adjustments Adjusted average -43.65 106.2 s1 s2 88.77 s3 Sum 20.91 78.49 26.16 0.03 26.16 s4 0.00 48 Year Qtr y 2000 180 340 140 260 200 350 160 275 200 345 155 270 2001 2002 t (y - t) 230.0 -50.06 231.7 108.21 233.5 -93.52 235.2 24.75 236.9 -36.98 238.7 111.29 240.4 -80.44 242.1 243.9 245.6 247.3 249.0 s y-s -43.65 223.65 106.26 233.74 -88.77 228.77 26.16 233.84 -43.65 243.6 243.7 106.26 -88.77 248.77 248.8 243.90 32.83 26.16 -43.90 -43.65 99.37 106.26 -92.36 -88.77 238.7 243.77 20.91 26.16 49 243.84 Forecasting for quarters of 2003 Quarter 1, 2003:x = 13 test (Q1,2003) = 228.33 + 1.73 ×13 = 250.82 yest (Q1,2003) =test (Q1,2003) + s1 = 250.82 − 43.65 = 207.17 Quarter 2, 2003: x = 14 test (Q 2,2003) = 228.33 + 1.73 ×14 = 252.55 yest (Q 2,2003) = test (Q 2,2003) + s2 = 252.55 + 106.26 = 358.81 50 Forecasting for quarters of 2003 Quarter 3, 2003:x = 15 test (Q 3,2003) = 228.33 + 1.73 ×15 = 254.28 yest (Q 3,2003)=test (Q 3,2003) + s3 = 254.28 − 88.77 = 165.51 Quarter 4, 2003: x = 16 test (Q 4,2003) = 228.33 + 1.73 ×16 = 256.01 yest (Q 4,2003) = test (Q 4,2003) + s4 = 256.01 + 26.16 = 282.17 51 Example The following data represents the number of registered guests (in thousands) of a hotel in Sapa Qtr Qtr Qtr Qtr 2000 180 340 140 260 2001 200 350 160 275 2002 200 345 155 270 Forecast the number of guests for quarters of 2003 using moving average method to 52 Year Qtr y 2000 180 340 140 260 200 350 160 275 200 345 155 270 2001 2002 Moving totals Moving average 920 940 230 235 950 970 237.5 985 246.25 985 246.2 245 980 975 970 242.5 243.7 5242.5 Centered moving average 232.5 236.25 240 244.375 246.25 245.625 244.375 243.125 53 Year Qtr y 2000 180 340 2001 2002 t (y - t) 140 232.5 -92.50 260 23.75 200 236.2 240 350 160 275 200 345 155 270 244.37 246.2 245.62 244.37 -40 105.625 -86.25 29.375 -44.375 243.125 101.875 54 Q1 Q2 Year Year -40 105.625 Q3 Q4 -92.50 23.75 -86.25 29.37 Year -44.375 101.87 Totals -84.375 207.5 -178.75 53.125 Averages -42.1875 -89.375 Adjustments 103.7 1.25 26.562 Adjusted average 42.1875 105.00 -89.375 s1 s2 s3 26.562 s4 Sum -1.25 0.00 55 Year 2000 2001 2002 Qtr y 180 340 140 260 200 350 160 275 200 345 t (y - t) s y-s -42.1875 222.1875 105.00 232.5 236.25 -92.50 23.75 240 244.37 -40 105.625 246.25 245.62 -86.25 29.375 244.37 -44.375 243.12 101.875 235 -89.375 229.375 26.5625 233.4375 -42.1875 242.1875 105.00 245 -89.375 249.375 26.5625 248.4375 -42.1875 242.1875 105.00 240 -89.375 244.375 26.5625 243.437556 Forecasting for quarters of 2003 Range of trend values: R = 243.125 − 232.5 = 10.625 Average change per time period: = 10.625 ÷ = 1.518( approx) 57 Forecasting for quarters of 2003 Quarter 1, 2003: test (Q1,2003) = 243.125 + 1.518 × = 247.679 yest (Q1,2003)=test (Q1,2003) + s1 = 247.679 − 42.1875 = 205.4915 Quarter 2, 2003: test (Q 2,2003) = 243.125 + 1.518 × = 249.197 yest (Q 2,2003) = test (Q 2,2003) + s2 = 249.197 + 105 = 354.197 58 Forecasting for quarters of 2003 Quarter 3, 2003: test (Q 3,2003) = 243.125 + 1.518 × = 250.715 yest (Q 3,2003) =test (Q 3,2003) + s3 = 250.715 − 89.375 = 161.34 Quarter 4, 2003: test (Q 4,2003) = 243.125 + 1.518 × = 252.233 yest (Q 4,2003) = test (Q 4,2003) + s4 = 252.233 + 26.5625 = 278.7955 59 Forecasting for quarters of 2003 Proportional multiplier of trend values: pm = 243.125 / 232.5 = 1.046 Geometric mean multiplier of the period: gmm = 1.046 = 1.0064 60 Forecasting for quarters of 2003 Quarter 1, 2003: test (Q1,2003) = 243.125 ×1.0064 = 247.826 yest (Q1,2003)=test (Q1,2003) + s1 = 247.826 − 42.1875 = 205.6384 Quarter 2, 2003: test (Q 2,2003) = 243.125 ×1.0064 = 249.413 yest (Q 2,2003) = test (Q 2,2003) + s2 = 249.413 + 105 = 354.413 61 Forecasting for quarters of 2003 Quarter 3, 2003: test (Q 3,2003) = 243.125 ×1.0064 = 251.0103 yest (Q 3,2003)=test (Q 3,2003) + s3 = 251.0103 − 89.375 = 161.635 Quarter 4, 2003: test (Q 4,2003) = 243.125 ×1.0064 = 252.6178 yest (Q 4,2003) = test (Q 4,2003) + s4 = 252.6178 + 26.5625 = 279.1803 62 ... 200 350 160 2 75 200 3 45 155 270 2001 2002 t (y - t) 230.0 -50 .06 231.7 108.21 233 .5 -93 .52 2 35. 2 24. 75 236.9 -36.98 238.7 111.29 240.4 -80.44 242.1 243.9 2 45. 6 247.3 249.0 s y-s -43. 65 223. 65 106.26... 231.7 3 140 4 260 200 350 160 2 75 200 10 3 45 11 155 12 270 2001 2002 233 .5 2 35. 2 236.9 238.7 1240.4 242.1 243.9 02 45. 6 247.3 6249.0 ∑ x = 78 ∑ y = 28 75 ∑ xy = 189 35 x ∑ = 650 The regression for... t (y - t) 2000 180 -50 .06 340 230.0 231.7 140 260 200 350 160 2 75 200 3 45 155 270 2001 2002 233 .5 2 35. 2 236.9 238.7 240.4 242.1 243.9 2 45. 6 247.3 249.0 108.21 -93 .52 24. 75 -36.98 111.29 -80.44