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Unformatted text preview: 28.7 Estimation Theory 501 2. Furthermore, Equation 28.239 states that the η i are normally distributed around the zero point with an unknown, nonzero variance σ 2 . To suppose measurement errors to be normally distributed is quite common and correct in most cases. The white noise 63 in transmission of signals for example is often modeled with Gaussian distributed 64 amplitudes. This second assumption includes, of course, the first one: Being normally distributed with N ( μ = 0 ,σ 2 ) implies a zero expected value of the error. 3. With Equation 28.240 , we assume that the errors η i of the single measurements are stochastically independent. If there existed a connection between them, it would be part of the underlying physical law ϕ and could be incorporated in our measurement device and again be subtracted. Objective: Estimation Assume that we can choose from a, possible infinite large, set of functions (estimators) f ∈ F . f ∈ F ⇒ f : R mapsto→ R (28.241) From this set we want to pick the function f ⋆ ∈ F with that resembles ϕ the best (i. e., better than all other f ∈ F : f negationslash≡ f ⋆ ). ϕ is not necessarily an element of F , so we cannot always presume to find a f ⋆ ≡ ϕ . Each estimator f deviates by the estimation error ε ( f ) (see Definition 28.53 on page 499 ) from the y i-values. The estimation error depends on f and may vary for different estimators. y i = f ( x i ) + ε i ( f ) ∀ i : 0 < i ≤ n (28.242) We consider all f ∈ F to be valid estimators for ϕ and simple look for the one that “fits best”. We now can combine Equation 28.242 with Equation 28.237 : f ( x i ) + ε i ( f ) = y i = ϕ ( x i ) + η i ∀ i : 0 < i ≤ n (28.243) We do not know ϕ and thus, cannot determine the η i . According to the likelihood method, we pick the function f ∈ F that would have most probably produced the outcomes y i . In other words, we have to maximize the likelihood of the occurrence of the ε i ( f ). The likelihood here is defined under the assumption that the true measurement errors η i are normally distributed (see Equation 28.239 ). So what we can do is to determine the ε i in a way that their occurrence is most probable according to the distribution of the random variable that created the η i , N (0 ,σ 2 ). In the best case, the ε ( f ⋆ ) = η i and thus, f ⋆ is equivalent to ϕ ( x i ), at least in for the sample information A available to us. Maximizing the Likelihood Therefore, we can regard the ε i ( f ) as outcomes of independent random experiments, as uncorrelated random variables, and combine them to a multivariate normal distribution. For the ease of notation, we define the ε ( f ) to be the vector containing all the single ε i ( f )- values....
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This document was uploaded on 08/10/2011.
- Spring '11