ABSTRACT
The sum result of our analysis in §7-6 and §7-7 above is that in continuous time, the instantaneous measurement error may be described by the following stochastic differential equation:
lim t→0
x(t) = lim t→0
[μt +z(t)] = dx(t) = μdt + dz(t)
where x(t) is the accumulated measurement error up to and including time t, dx(t) = x(t + dt) − x(t) is the measurement error that arises over the instantaneous period from time t until time t + dt, z(t) is the accumulated stochastic (or total unexpected) component of the measurement error up to and including time t, dz(t) = z(t + dt) − z(t) is the stochastic (or unexpected) component of the measurement error that arises over the instantaneous period from time t until time t + dt, and μ is the rate at which the accumulated measurement error is expected to drift upwards (if μ is positive) or downwards (if μ is negative), stated on a per unit time basis. Moreover, we can integrate through the expression for dx(t) given above and thereby show that the accumulated measurement error up to and including time t can be stated as follows:
x(t) = t∫
dx(s) = t∫
[μds+ dz(s)] = μt + z(t)
where z(t) = ∫ t0 dz(s). Taking expectations through this expression shows E0[z(t)] =
E0[dz(s)] = 0
that is, the accumulated stochastic (or unexpected) component of the measurement error has an expected value of zero. Moreover, it will be recalled from §7-7 that the variance of the stochastic component of the measurement error over the instantaneous period from time t until time t + dt will be Vart[dz(t)] = σ 2dt. Hence, we can apply Wiener’s Theorem (as in §7-5) with the weighting function f (t) = 1 to determine the variance of the accumulated stochastic component of the measurement error, namely,
Var0[z(t)] = Var0 ⎡⎣ t∫
f (s)dz(s)
⎤⎦= Var0 ⎡⎣ t∫
dz(s)
⎤⎦= σ 2 t∫ 0
ds = σ 2t
These results mean that the accumulated measurement error x(t) at time t will have an expected value E0[x(t)] = E0[μt + z(t)] = μt and a variance Var0[x(t)] = Var0[μt + z(t)] = Var0[z(t)] = σ 2t. Note that the variance for the accumulatedmeasurement error obtained here using Wiener’s Theorem is the same as the variance based on the discrete-time development of the accumulated measurement error in §7-2 above. Moreover, since
lim t→0
z( jt) = t∫
dz(s) = z(t)
it follows that z(t) is the sum of a large number n of small stochastic error terms and will therefore satisfy the requirements for the application of the Central Limit Theorem. This in
turn will mean that the probability distribution for the accumulated measurement error x(t) will asymptotically approach that of a normal distribution with a mean E0[x(t)] = μt and a variance Var0[x(t)] = σ 2t.
