Lecture no. 5

 

Held on: Monday, August 7, 2000

 

Notes Prepared by: Tushar Gupta and Sumit Sharma

 

 

Signal Noise Model

 

Observing X_i  at a time ti ( i = 1….n )

 

Model :   X_i = f(t_i) + Î_i   à ( signal + noise )

 

Problem : We need to forecast X for t* however Î_i     ( noise ) has no deterministic                  component.

 

Estimate problem :

 

What is f?

 

So we assume a parameterized model:

 

ie.  f(t) =  a + bt   and so  Î_i  = X_i – f(t)

 

There are 2 ways to estimate a and b : 

 

1. Least squares method:

 

minimize the square of the Î (noise or error) to get a value of a and b.

 

a,b = arg min å[ X_i - f_a,b(t_i) ]²   (  i=1..n)

 

2. Least absolute deviations:

 

a,b = arg min å | X_i - f_a,b(t_i) |             (  i=1..n)

 

à forecast at t* is f_a,b(t_i *)

 

It is also good to find out a forecast of Π , like this if we know Î_i   then we know Î_{i +1} .

 

Forecast at t_n+1 à f_a,b(t_n+1) + Î_n+1

 

If we know Î from (1….n) then we can forecast for Î_n+1.

 

Î_i   + g(t_i) + h_i     ( ie. g should be captured in f where x_i = f(t_i) +Î_i   )

 

so X_i = f(t_i) + g(t_i) + h

 

We can assume Î_n+1to be a function of  ( Î₁….Î_n).

 

So we come to the …….

 

 

AUTO-REGRESSIVE MODEL

 

Î_n+1 =  F₁Î_n + F₂Î_n-1 + F_pÎ_n-(p-1) + NOISE   

 

This is the AR(p) model….

 

So in general :

 

Î_i+1 =  F₁Î_i + F₂Î_i-1 + F_pÎ_i-(p-1) + NOISE   

 

For example :

 

AR(1) à  Î_i+1= FÎ_i + h_i+1

 

If the above is correct then à  X_i = f(ti) + Î_i   

 

What is F ?

 

If Î is positively correlated then :

Large Î_i   à large Î_i+1

Small Î_i   à small Î_i+1

and  the above is vice versa for a negative correlation!

 

F > 0 for a positive correlation ( low frequency ( red noise))

F < 0 for a negative correlation ( high frequency (blue noise))

F = 0 is for white noise.

 

Definition of Correlation :

 

r    =                  å ( y_i – y)(z_i – z)                                             -1 < r < 1

 

            (1/nå (y_i – y)²)^1/2  (1/nå (y_i – y)²)^1/2

 

Estimate of F in AR(1) is the PMR between (Î₁ … Î_n-1) and (Î₂… Î_n)

ie. subsequent pairs of observations

 

Assume åÎ_i   = 0  ( i=1..n-1)     & åÎ_i   = 0  (i=1..n)

 

Then F estimated by :

 

 F =             å Î_i   Î_{i +1}( i = 1..n-1)   

              

            (å (Î_i  ²)^1/2  (åÎ_i  ²)^1/2   ( i = 1 .. n)