AR(p) Autoregressive Model (Time Series Models for Forecasting)

AR(p) Autoregressive Model (Time Series Models for Forecasting)

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  AR(p) Model (Autoregressive Modelof order ‘p’) Presented by: A V Chandana
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  Introduction An Autoregressive (AR)model is a time series model where the current value of a variable depends on its past values. General form is : Today’s value = combination of previous values + random error This is called AR(p) because the model uses p previous observations.
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  Z Current value ofthe time series at time t. Z ,Z​ ,...,Z ​ These are past values of the series. ϕ ,ϕ ​ ,...,ϕ​ These are AR coefficients (parameters). They tell us how strongly past values influence the present value. p (Order of the Model) The number p tells how many past values are used. a This is called the white noise error term. It represents random factors we cannot predict. t t-1 t-2 t-p 1 2 p t
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  Derivation of AutoregressiveModel of Order p (AR(p)) Let Zt​be a stationary time series such that E(Zₜ) = μ, Var(Zₜ) = σ ² Let {at} be a white noise process associated with the time series Zt which represents the random error. Define the centered series Z =Z −μ so that the mean of Z​ becomes zero. The autoregressive model of order p is defined by where ϕ ,ϕ ,…,ϕ​ are constants called autoregressive parameters, and at​is white noise. Now introduce the backshift operator B defined as BZ =Z Then BZ =Z and B Z =Z Z ~ t t ~ t ......................(1) 1 2 p t-1 t t t-1 t 2 t-2
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  From equation (2), Z=[ϕ(B)] a Let ψ(B)=[ϕ(B)] Then Z =ψ(B)a Thus the AR(p) model can be expressed as a linear filter of the white noise process. There are p+2 unknown parameters in the AR(p) model: μ,σ ,ϕ ,ϕ ,…,ϕ using the backshift operator, equation (1) becomes Z =ϕ BZ +ϕ B Z +⋯+ϕ B Z +a Taking Z​ common, (1−ϕ B−ϕ B−⋯−ϕ B )Z =a Let ϕ(B)=1−ϕ B−ϕ B −⋯−ϕ B which is called the Autoregressive operator. Hence the AR(p) model can be written as ϕ(B)Z =a ..........................(2) t t t 1 2 2 p t t p ~ ~ ~ ~ ~ t 1 2 2 p p ~ t t 1 2 2 p p t t t ~ t -1 -1 ~ t t Z 2 1 2 p
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  Stationarity AR(p) is stationaryif the roots of ϕ(B)=0 lie outside the unit circle. Variance Autocovariance This shows how values at different time lags relate. Autocorrelation Properties
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  Applications Stock Market prediction Weather Forecasting Electricity Demand Forecasting Sales Forecasting Traffic Flow prediction Economic Analysis
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  Simple and easyto understand. Useful for short-term forecasting. Uses past data effectively. Flexible by adjusting order p. Requires stationary data. Cannot predict sudden changes. Difficult to choose correct order p. Not good for long-term forecasting. ADVANTAGES DISADVANTAGES
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  Example:Temperature Prediction Suppose wewant to predict today's temperature. We observe that: today's temperature depends on yesterday yesterday depends on the day before So we model: 𝑇𝑒𝑚𝑝 =0.6𝑇𝑒𝑚𝑝 +0.3𝑇𝑒𝑚𝑝 +𝑎 ​ This is an AR(2) model. Meaning: Factor Influence Yesterday's temperature 60% Temperature 2 days ago 30% Random weather variation 𝑎𝑡 Meteorologists use such models for short-term forecasting. t-1 t-2 t t
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