next up previous [pdf]

Next: The common criteria Up: Choosing the Operator Length Previous: Choosing the Operator Length

A mean square error criterion

One theoretically sound procedure for choosing a truncation point is based on a mean square error criterion (, ). Suppose that $ P_M(f)$ is our $ M$-th estimate of the true spectrum $ P(f)$. Then we might wish to mimimize the square error

$\displaystyle E[P_M(f)-P(f)]^2 = {Var}[P_M(f)] + B^2[P_M(f)],$ (A-32)

where $ E$ is expectation, $ Var$ is variance

$\displaystyle Var(P_M) = E(P_M^2) - E^2(P_M),$ (A-33)

and $ B$ is bias

$\displaystyle B[P_M(f)] = P(f)- E[P_M(f)].$ (A-34)

In general, as $ M$ increases, the bias decreases while the variance increases. Thus, (32) will have a minimum for some value of $ M$.

This criterion is not of practical value unless it is possible to obtain reasonably good estimates of the variance and bias of $ P_M$. This problem is not easily solved, but satisfactory approximate solutions can probably be found. However, this approach will NOT be pursued here.

This notation has been introduced to help the reader understand why one should expect such an optimum operator length to exist. Spectral estimates are nearly always designed to decrease the bias as $ M$ increases. (MESA is clearly designed this way.) However, when the bias is small, the variance is a measure of the ragged oscillations $ P_M(f)$ makes around $ P(f)$. Since most people prefer to study a smooth spectrum, a balance between variance and bias is our goal.


next up previous [pdf]

Next: The common criteria Up: Choosing the Operator Length Previous: Choosing the Operator Length

2009-04-13