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Unformatted text preview: CSE 6740 Lecture 11 How Do I Treat Temporal Data? I (Time Series Analysis) Alexander Gray [email protected] Georgia Institute of Technology CSE 6740 Lecture 11 – p. 1/3 4 Today 1. Time Series 2. Univariate Linear Methods 3. Extensions CSE 6740 Lecture 11 – p. 2/3 4 Time Series General concepts in time series analysis. CSE 6740 Lecture 11 – p. 3/3 4 Time Series Time series are not IID. Each data point is somehow dependent on previous ones. CSE 6740 Lecture 11 – p. 4/3 4 Parts of a Time Series Some sources of variation: Seasonality. For example, unemployment is typically high in the winter and lower in summer. Trend. A longterm change in the mean level. Example of a trend: X t = m t + ǫ t (1) where m t = β + βt. (2) and ǫ t is zeromean noise. CSE 6740 Lecture 11 – p. 5/3 4 Parts of a Time Series Seasonality can be modeled in different ways, including: X t + m t + S t + ǫ t additive (3) X t + m t S t + ǫ t multiplicative (4) X t m t S t ǫ t multiplicative (5) Note that the third model can be made into the first one with a logarithmic transformation. CSE 6740 Lecture 11 – p. 6/3 4 Stationarity A time series is said to be stationary , roughly speaking, if there is no systematic change in mean (no trend) or variance and if strictly periodic variations have been removed. In other words, the properties of one section of the data look much like that of any other section. (Actually, only models are stationary, not data.) CSE 6740 Lecture 11 – p. 7/3 4 Differencing Differencing is a preprocessing operation which is effective for removing a trend. It is performed on the time series { x 1 ,... ,x N } to obtain a new time series { y 2 ,... ,y N } by y t = x t − x t 1 ≡ ∇ x t . (6) Occasionally secondorder differencing is required: ∇ 2 x t = ∇ x t −∇ x t 1 = x t − 2 x t 1 + x t 2 . (7) A seasonal effect can be removed with seasonal differencing, e.g.: ∇ 12 x t = ∇ x t −∇ x t 12 . (8) CSE 6740 Lecture 11 – p. 8/3 4 Stochastic Process A stochastic process or random process is a sequence of random variables X 1 ,... ,X t ,... ,X N ordered in time. The mean function is defined by μ ( t ) = E ( X t ) . (9) The variance function is defined by σ 2 ( t ) = V ( X t ) . (10) The autocovariance function is defined by γ ( t 1 ,t 2 ) = E { ( X t 1 − μ t 1 )( X t 2 − μ t 2 ) } , (11) which generalizes the variance function when t 1 negationslash = t 2 . CSE 6740 Lecture 11 – p. 9/3 4 Stationary Process A time series is said to be strictly stationary if the joint distribution of X 1 ,... ,X N is the same as the joint distribution of X 1+ τ ,... ,X N + τ for all t 1 ,... ,t N and τ . In other words, shifting the time origin by an amount τ has no effect on the joint distributions, which thus must depend only on the intervals between t 1 ,... ,t N ....
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This note was uploaded on 04/03/2010 for the course CSE 6740 taught by Professor Staff during the Fall '08 term at Georgia Institute of Technology.
 Fall '08
 Staff

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