Understanding Time Series Analysis Components and Models
Explore the components - Trend, Seasonality, Cyclic, and Irregular - in time series analysis, along with additive and multiplicative models. Learn how to choose between these models and methods of isolating trends. Dive into practical examples to comprehend the concepts.
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Presentation Transcript
TIME SERIES ANALYSIS Tides Sunspots Dow Jones Industrial Averages
FOUR COMPONENTS: Trend (T) The long-term progression in the data (upward, downward, or stable). Seasonality (S) Regular patterns repeating over specific periods (e.g., monthly, quarterly). Cyclic (C) Fluctuations not of fixed period, usually tied to economic or business cycles. Irregular/Random (R) Random noise or residuals that cannot be explained by other components.
TIME SERIES ANALYSIS Time Secular Trend Short time Oscillation Cyclical Fluctuations Seasonal fluctuations Irregular fluctuations
MODELS Additive Model Multiplicative Model
MULTIPLICATIVE MODEL ? = ? ? ? ? Y Observed value T Trend C Cyclical fluctuation S Seasonal fluctuation I Irregular fluctuation
ADDITIVE MODEL ? = ? + ? + ? + ? Y Observed value T Trend C Cyclical fluctuation S Seasonal fluctuation I Irregular fluctuation
CHOOSING BETWEEN ADDITIVE AND MULTIPLICATIVE Characteristic Additive Model Multiplicative Model Seasonal Variation Constant over time Varies with trend Suitable for Linear trends and stable seasonality Exponential growth or increasing seasonal effects Example Temperature Sales revenue
METHODS OF ISOLATING TRENDS Free hand method Semi average method Method of moving averages Method of least squares
t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Year Year 1 Quarter 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 Sales(1000s) 4.8 4.1 6 6.5 5.8 5.2 6.8 7.4 6 5.6 7.5 7.8 6.3 5.9 8 8.4 Year 2 Year 3 Year 4
TIME SERIES ANALYSIS Consider a data of number of disk access for 50 database quires. Number of disk access for 50 database quries 140 120 100 80 60 40 20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 4849 50
AUTOREGRESSIVE MODEL Predict the variable as a linear regression of the immediate past value ? ? = ?0+ ?1? ? 1
? ? = ?0+ ?1? ? 1 + ?2? ? 2 + + ??? ? ? ? ? = ?0+ ?1? ? 1 + ?2? ? 2 + +??? ? ? + ? ?
? ? = ?0+ ?1? ? 1 ?0= ? ? ? ? 12 ? ? 1 ? ? ? ? 1 ? ? ? 12 ? ? 1 2 ?1=? ? ? ? ? 1 ? ? ? ? 1 ? ? ? 12 ? ? 1 2
? = 2 ? ? = ?0+ ?1? ? 1 + ?2? ? 2 + e ? ? ? ?2 ??? = ?=3 ? 2 ??? = ? ? ?0+ ?1? ? 1 + ?2? ? 2 ?=3 ???? ??0 ???? ??1 ???? ??1 = 0 = 0 = 0
? 2 ?0+ ?1? ? 1 + ?2? ? 2 = 0 ?=3 ? 2? ? 1 ?0+ ?1? ? 1 + ?2? ? 2 = 0 ?=3 ? 2? ? 2 ?0+ ?1? ? 1 + ?2? ? 2 = 0 ?=3
? 2 ? ? 1 ? ? 2 ? ? ?0 ?1 ?2 ? ? 12 ? ? 1 ? ? 1 ? ? 2 ? ? ? ? 1 = ? ? 22 ? ? 2 ? ? 1 ? ? 2 ? ? ? ? 2 1 ? 2 ? ? 1 ? ? 2 ? ? ?0 ?1 ?2 ? ? 12 ? ? 1 ? ? 1 ? ? 2 ? ? ? ? 1 = ? ? 22 ? ? 2 ? ? 1 ? ? 2 ? ? ? ? 2
