10 Forecasting

Let \(X_1,X_2,..............,X_n\) be an observed time series. The problem is to estimate further values \(X_{t+k}\), \(\hspace {0.2cm} k \)is called lead time. The forecast \(X_{n+k}\) at time \(n\) at \(k\) steps ahead can be denoted by \(\widehat {x}(n,k)\). Various methods for forecasting exist. No single method is universally available.
Forecasting is of the form of extrapolation condition of a characteristic of a time series.

Forecasting Methods: Are broadly based on

1.: Subjective
-judgment
-intuition
-business knowledge
or any other relevant information.
2.: Univariate
based on a model fitted only the passed observation of \(\{X_t\}\) so that \(\widehat {x}(n,k)\) depends only on \(X_t,X_t,........,X_n\)
3.: Multivariate
Used for non-seasonal time series showing no trends. For time series with trends or seasonal patterns these need to be removed to produce stationarity.

Suppose \(X_1,X_2,.......,X_n\) is a non-seasonal time series with no trend. We take \(X_{n+1}\) as a waited a sum of passed observations \[\widehat {x}(n,1)= C_0X_n+C_1X_{n-1}+C_2X_{n-2}+\cdots \cdots \] then \(\{C_i\}\) are the weights (Geometric weights) are used so that
\(C_i=\alpha (1-\alpha )^i, \hspace {0.5cm}i=1,2,....\) where \(\alpha \) is such that \(0<\alpha <1.\)

The \(\hspace {0.3cm}\widehat {x}(n,1)=C_0X_n+C_1X_{n-1}+C_2X_{n-2}+\cdots \cdots \)

becomes \(\hspace {0.3cm}\widehat {x}(n,1)=\alpha X_n+\alpha (1-\alpha )X_{n-1}+\alpha (1-\alpha )^2X_{n-2}+\cdots \cdots \cdots \)
or

\[\widehat {x}(n,1)=\alpha X_n+(1-\alpha )\Big [\alpha X_{n-1}+\alpha (1-\alpha )X_{n-2}+\alpha (1-\alpha )^2X_{n-3}+\cdots \cdots \Big ]\] Therefore \(\widehat {x}(n,1)=\alpha X_n+(1-\alpha )\widehat {x}(n-1,1)\).

If \(\widehat {x}(1,1)=X_1\), the equation can be used recursively to compute forecasting, this is called exponential smoothing.

We can also write \begin {align*} \widehat {x}(n,1) &=\alpha X_n+(1-\alpha )\widehat {x}(n-1,1)\\ &=\alpha \Big [X_n-\widehat {x}(n-1,1)\Big ] +\widehat {x}(n-1,1)\\ &=\alpha e_n +\widehat {x}(n-1,1) \end {align*}

where \(e\) is the prediction error at time n. The values of \(\alpha \) depend on the time series properties, usually the values of \(\alpha \) are taken \(0.1\leq \alpha \leq 0.3\).

10.1 The Partial Autocorrelation Function

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