Lecture N0. 6

 

Held on: Wednesday, August 9, 2000

 

Notes Prepared by: Abhishek Bapna & Aditya Gandhi

 

Review

Input Sequence of observations

X1,X2.....Xn

Objective - To forecast Xn+1, Xn+2

Step 1

Signal Extraction

Fit Model

Find parameter estimates that minimize say least squares

i.e. minimize

Step 2

Obtain "residuals"

Fit a time series model to AR(p) model

Autoregression process of order p

Exogenous (from outside) Indogenous (from inside)

Assume that the signal has been filtered out (been subtracted/ has been detrended)

This means that

 

i.e if

then g(t) should be zero or else the estimate for f(t) is wrong.

 

Why do we need to estimate

From AR(p) and assuming linear trends for f(t)

if all parameters are estimated

_En is already available from Xi - f(ti)

 

Similarly using Xn+1 we can find the value of Xn+2

As the n+2th terms forecast is based on a previous forecast other than data thus possibility of inaccuracy is higher.

This can be understood as a spread, which increases as we move further away

Forecasting strategy:

• Use historical data if available else replace with forecast

 

How do we estimate value of ?'s?

For AR(p)

Where n is structured

To fine the E's we can use least squares of E's i.e. mininise

Or minimize

This estimates almost exactly the product moment correlation

Choice of p-

Diagnostic

Autocorrelation function (ACF)

The correlation coefficient r relates the two random quantities Y and Z and is given as

r_k = correlation between E and E lagged by k

i.e we consider observations (E₁, E_k+1), (E₂, E_k+2) etc.

say E's have some process

for ex.- for sales the data is more likely to be periodic with similar trends every 12 months.

Typical ACF for periodic/ seasonal series

This somewhat resembles a dampened sinusoid

One can write a program to find ACF.

Standard packages also are available.

Base on when the value of ACF dies or shrinks to 0 can help in deciding p.

The expression for E can also look like this -

Note: we may not be using all the terms and just some part of the data.

This can also be interpreted as a variable time gap situation.

 

• Use only those parameters that are useful in prediction

This is known as the Parasimonious choice of p

 

AIC (Akaike Information Criteria)

This incorporates a factor for p also in it and our minimisation function changes to

More the p's better fit possible but can lead to junk parameters as p goes up

Thus there is an optimal value of p