Abstract :
In his section “Orthogonal Projections”, Kalman purports to derive his filter by orthogonally projecting a solution x(τ) onto noisy measurement data y(t) = x(t) + n(t); effectively assuming x̂ = Σaiyi, minimizing E[(xi − aiyi)2], and determining ai. Conversely, under his later rubric “Solution of the Wiener Problem” he instead does the reverse by projecting data onto a prescribed solution in the form of his state equation. This is virtually equivalent to projecting data onto a polynomial; assuming x̂ = Σbjτj, minimizing E[(yi − Σbjtij)2], and determining bj. Exceptionally subtle and cryptic, but enormously consequential; projecting a solution onto data - as Kalman purports, but fails, to do - offers superior position as well as derivative accuracy through minimization of the true MSE - comprising both variance and bias. As Kalman does, projecting data onto a prescribed solution (e.g., a state equation or polynomial) minimizes only the curve fitting variance in the constrained special case of merely producing an “average curve” (an “average state equation” in Kalman´s filter) - yielding sub-optimal accuracy. Moreover, a very disturbing consequence of Kalman´s misapplication of orthogonal projection is that the simple analysis herein strongly suggests that Kalman´s Filter is truly optimum only in the absence of the specific additive statistical measurement noise it is explicitly designed to filter out.
