This is equivalent to the following orthogonality conditions:
Solving gives us
then
Gaussian random variables are the hardest predict because nonlinearity should improve our error, but it does not in the Gaussian case. This means if affine estimation works well, we shouldn’t try and find better non-linear estimators.
Least Squares
Theorem 2 (Gauss Markov Theorem)
Recursive Least Squares
Applying Theorem 1 and solving the normal equation, we see
By applying the Sherman-Morrison-Woodbury identity, we can see that
Theorem 3 (Recursive Least Squares Update)
Applying the Sherman-Morrison-Woodbury identity,
Theorem 4 (Recursive Least Squares Downdate)
The best least squares estimate using all but the kth observation can be found by updating the best least squares estimate using all data points using
In Linear Estimation, we are trying to estimate a random variable X using an observation Y with a linear function of Y. If Y is finite dimensional, then we can say X^(Y)=WY where W is some matrix. Using Theorem 1 and the orthogonality principle, we know that
⟨X−WY,Y⟩=0⇔RXY=WRY
This is known as the Normal Equation. If RY is invertible, then we can apply the inverse to find W. Otherwise, we can apply the pseudoinverse RY† to find W, which may not be unique. If we want to measure the quality of the estimation, since X=X+(X−X^),
If we allow ourselves to consider an affine function for estimation X^(Y)=WY+b, then this is equivalent to instead finding an estimator
X^(Y′)=WY′ where Y′=[Y1]
⟨X−X^,Y⟩
⟨X−X^,1⟩
X^(Y)=W(Y−μY)+μx where WΣY=ΣXY.
ΣY and ΣXY are the auto-covariance and cross-covariance respectively. Recall that if
[XY]∼N([μXμY],[ΣXΣYXΣXYΣY])
X∣Y∼N(μX+ΣXYΣY−1(Y−μY),ΣX−ΣXYΣY−1ΣYX)
Thus in the Joint Gaussian case, the mean of the conditional distribution is the best affine estimator of X using Y, and the covariance is the estimation error. This has two interpretations.
Under the Gaussian assumption, the best nonlinear estimator E[X∣Y]is affine
The theory of linear estimation is very closely connected with the theory behind least squares in linear algebra. In least squares, we have a deterministic x and assume nothing else about it, meaning we are looking for an unbiased estimator. Theorem 2 tells us how to find the best linear unbiased estimator in a linear setting.
Suppose that Y=Hx+Z and Z is zero-mean with ⟨Z,Z⟩=I, H is full-column rank, then xb^=(H∗H)−1H∗Yis the best linear unbiased estimator.
Suppose we extend the least squares setup to allow a stochastic, but fixed, X where ⟨X,X⟩=Π0. At each timestep, we receive observations of X such that Yi=hi∗X+Vi where ⟨Vi,Vj⟩=δ[i,j] and ⟨X,V⟩. Define
Suppose we want to do this in an online fashion where at each timestep i, we only use the current hi,Yi and our previous estimate Xi−1. Let Pi=(Π0−1+Hi∗Hi)−1. Then
Pi−1=Π0+∑k=0ihkhk∗=Pi−1−1+hihi∗.
Pi=Pi−1=Pi−11+hi∗P−1hihihi∗Pi−1
The best least squares estimate using i+1 data points can be found by updating the best least squares estimate using i data points using
Notice that this formula scales an innovation in order to improve the current estimate of X.
Just as we could compute a recursive update, we can also compute a “downdate” where we forget a particular observation. More concretely, we want to use X^i to find X^i∣k, the best linear estimator of X using Y0,Y1,⋯,Yk−1,Yk+1,⋯,Yi. Defining Pi∣k=(Π0−1+Hi∣k∗Hi∣k)−1,