Solution to Exercise 2.3.  No Title  Solution to Exercise 2.1.

Solution to Exercise 2.2.

Problem Statement: Are the following valid autocorrelation functions?

(a) $\rho_1$ = .8, $\rho_2$ = .55, $\rho_k$ = for k > 2.

(a) $\rho_1$ = .8, $\rho_2$ = .28, $\rho_k$ = for k > 2.

Solution: In order for a given sequence to be a valid autocorrelation function, we must of course have $\rho_0$ = 1. We assume this is intended, as the value of $\rho_0$ is not explicitly stated. Furthermore, as in section 2.1.3, all of the correlation matrices

$$\left[ \begin{array}{ccccc}1 & \rho_1 & \rho_2 & \ldots &... ...\ldots & 1\end{array} \right] \; = \; \mbox{\boldmath{P}}_n$$ must be nonnegative definite (this is a correction to what's in the book - they say positive definite) for all n = 2, 3, $\ldots$. This business with positive definiteness vs. nonnegative definiteness is discussed in the notes below.

On p. 28 we see the conditions
Actually, all the < should be $\le$ (because it is only necessary that the determinants that were evaluated to produce these be nonnegative, not positive). In fact, these are necessary but not sufficient for the given function to be a valid autocorrelation function since they only verify positivity of the determinants of $\mbox{\boldmath{P}}_n$ for n = 1, 2, and 3, not all n. We can apply these conditions to the given potential autocorrelation functions, and if any of the conditions fails, then we know the given functions are not valid autocorrelation functions. However, even if all three conditions are verified, then we cannot be sure that the given function is an autocorrelation function because we don't know about the sign of the determinant of $\mbox{\boldmath{P}}_n$ for n > 3.

In both cases, clearly (1) and (2) hold, so there only remains to check (3). For the putative autocorrelation in (a),

$$\frac{.55 - .8^2}{1-.8^2} \; = \; -.25 ,$$ so that one is OK by (3), and for (b), the Matlab computations are

» format long
» (.28-.8^2)/(1-.8^2)

ans =

  -1.00000000000000

It looks like it fails the condition as stated in the book, but doesn't fail it according to the amended statement as above (and mentioned in class).

So at this point, we don't really know more than when we started. It is quite impossible to check the conditions that the determinant of $\mbox{\boldmath{P}}_n$ is nonnegative for all n. However, as mentioned in class, $\gamma_k$ is a valid autocovariance if and only if  

$$g(f) \; = \; \gamma_0 \, + \, 2 \sum_{k=1}^{\infty} \gamma_k cos(2 \pi f k)$$ (1)

is a nonnegative function. This is sort of alluded to in the book. About midway down p. 28, ``As will be shown in Section 2.2.3, all of these conditions can be brought together in the definition of the spectrum.'' However, all I can find in Section 2.2.3 is the following statement in the first paragraph on p. 40, ``Conversely, since the $\{ \gamma_k \}$ from a positive definite sequence, it follows from Herglotz's theorem ... that a unique function p(f) exists such that ...'' It doesn't actually state that nonnegativeness of the Fourier series transform as in (4) is equivalent to the nonnegative definiteness property.

Anyway, applying this fancier condition to the first alleged autocorrelation function, the corresponding spectrum (or spectral density function) is

$$g(f) \; = \; 1 \, + \, 1.6 \cos (2 \pi f ) \, + \, 1.1 \cos (4 \pi f) .$$ The plot of this function can be found at this link. This function clearly goes negative, so we do not have a valid autocorrelation function. For the spectral density in part(b): $$g(f) \; = \; 1 \, + \, 1.6 \cos(2 \pi f) \, + \, .56 \cos (4 \pi f) ,$$ the corresponding plot is here. We see this one also goes negative in places, so we do not have a valid autocorrelation function for this one either.

Comments: I don't expect you to understand necessarily what is going on with this problem. The take home message is that it is nontrivial to determine if a given function is an autocorrelation function unless it is computed from a process known to be stationary.

Also, regarding the requirement of (strict) positive definiteness of all $\mbox{\boldmath{P}}_n$ vs. nonnegative definiteness, the latter is all that is needed, but if one of the determinants is , that says there is some linear combination of finitely many values of the process which is almost surely a constant ( if the process mean is ). This would mean that the process if perfectly predictable given a large enough block of past observations. This never occurs in practice, so the condition as stated in the text is ``safe'' in practice but not mathematically correct.

3/10/1999