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Preliminaries of KF – Linear least-mean-squres (minimum variance) estimator

17/05/2011

In this post I shall talk more about the linear least-mean-squares (minimum variance) estimators, which is very important for understanding KF.

A) what is a least-mean-squares estimator?

x is the random variable to be estimated. \hat x is the estimator. If the estimator can minimize the least-mean-squares

E(x-\hat{x})(x-\hat{x})^T

then \hat{x} is called a least-mean-squares estimator. More importantly, E(x-\hat{x})(x-\hat{x})^T actually is the covariance matrix of the estimator error, so a least-mean-squares estimator is also known as minimum-variance estimator. If it is further unbiased, it is well-known as Minimum Variance Unbiased Estimator (MVUE).

B) how to minimize a matrix?

As mentioned, a minimum variance estimator can minimize the error covariance matrix. But how can we minimize a matrix? In fact, we are going to minimize some scalar index of the matrix such as a^TE(x-\hat{x})(x-\hat{x})^Ta with a as an arbitrary vector or the trace of the covariance matrix.  However, the most convenient way to understand is that: estimator \hat{x}^* is optimal in the sense of minimizing covariance iff

E(x-\hat{x})(x-\hat{x})^T -E(x-\hat{x}^*)(x-\hat{x}^*)^T \ge 0

where \hat{x} is another arbitrary estimator. The above equation means the left hand side of the equation is positive semi-definite. And we can often find the optimal estimator by using completion of squares method.

C) what is a linear minimum-variance estimator?

When we give some measurements y, if the estimator is a linear (or generally affine) function of y, say

\hat{x}=Ky

then the estimator is called a linear estimator. Linear estimators are the most widely used estimators.

D) with or without model?

I’m surprised to find out: we can design a linear minimum variance estimator without having a model of x and y. For zero-mean cases, the l.m.v.e is a linear form of y; for non-zero-mean cases, the l.m.v.e is an affine form of y. All needed in the estimator are the statistics characteristics of x, y. See ‘linear estimation’ section 3.2 for details.

Usually we can get a model of x and y. Then the role of the model actually is to establish the relation between the statistics characteristics of x, y. We may simply apply the results obtained without models to the cases with models. The most widely used model is:

y=Hx+v

This actually is the measurement model of a linear state space model. From here we start to see what kalman filter looks like.

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