ARMAX processes

Question 1

definition of ARMAX model

Answer 1

y(t) is the output of an ARMAX(m,n,p+k) model if:
y(t) = a1y(t-1) + a2y(t-2) + … + amy(t-m) + e(t) + c1e(t-1) + … + cne(t-n) + b0u(t-k) + b1(t-k-1) + … + bpu(t-k-p)
where k is the instrinsic delay of the input

Question 2

k-steps predictor from data of an ARMAX process

Answer 2

y(t) = B(z)/A(z)u(t-k) + C(z)/A(z)e(t)

y^(t|t-k) = B(z)*E(z)/C(z) * u(t-k) + R~(z)/C(z) * y(t-k)

Question 3

prediction error for an ARMAX process

Answer 3

ε(t) = E(z)*e(t)

Question 4

1-step predictor from data for an ARMAX process

Answer 4

y^(t|t-1) = B(z)/C(z)*u(t-1) + (C(z)-A(z))/C(z) * y(t)

Question 5

how to decide when using ARMA or ARMAX models

Answer 5

1. there is one input that is much more important that the other, used as a control variable - > ARMAX
2. the output is defined by many inputs, none of the inputs is particularly important (individually), and they are not measurable - > ARMA
3. Cases in the middle:
   - if there is a control variable - > ARMAX
   - if there is any variable more important than the other, and it is measurable - > ARMAX
   - if not - > ARMA

In general, measuring a variable adds complexity to the problem, but it gives better results in prediction performances

Question 6

On the optimality of the PEM

Answer 6

2 situations can happen:

1) S app m(θ) - > θ_ = θo
if N is big, prediction error with the identified model should be a white noise

2) S !app m(θ) - > θ_ ! = θo
the witheness test on the prediction error fails for any value of N
the model class should be modified to include S

Question 7

Optimality check

Answer 7

If the system is included in the model classs, then the prediction error must be a white noise, equal to the noise of the real system.
From this result, after the optimization, a whiteness test on the prediction error with the identified parameters can be performed:
- if this error is white, then the model identified is optimal
- if the error is not white, the model is not optimal and the system is not included in the selected model class: the model class can be extended increasing the order of the models, or a non-linear model class can be used