Analysis of ARMA processes

Question 1

time series interpretation

Answer 1

many different inputs concur to create a measurable output, but
- the inputs are not measurable
and/or
- each of them has a small influence on the output

Question 2

how to model a time series

Answer 2

white noise signal is used as a fictitious, un-measurable input for a Mathematical model that gives the output

Question 3

interpretation of the output

Answer 3

given N values of the output
{y(1), y(2), … , y(N)}
the interpretation is that these numbers are a finite realization of a stationary stochastic process

Question 4

def mean value of a ssp

Answer 4

ssp y(t)
my = E[y(t)]

Question 5

def covariance function of a ssp

Answer 5

ssp y(t)
γy(τ) = E[(y(t)-my)*(y(t-τ)-my)]
τ = 0, +-1, +-2,…

Question 6

properties of the covariance function

Answer 6

1) non-negativity:
γy(0) > = 0 (variance)

2) variance prevalence:
|γy(τ)| < = γy(0) for any τ

3) symmetry
γy(τ) = γy(-τ) (even function)

Question 7

def spectrum of a ssp

Answer 7

spectrum of a ssp y(t):
discrete Fourier transform of γy(τ):
Γy(ω) = Σ(τ=-inf,+inf) γy(τ)*e^-jωτ

Question 8

properties of the spectrum

Answer 8

• real function of ω
• non-negative function of ω
• even function of ω
• periodic function, with period 2pi

Question 9

how to obtain the covariance function from the spectrum

Answer 9

inverse discrete Fourier transform

γy(τ) = 1/2pi * int(-pi,pi) Γy(ω)*e^jωτ dω

Question 10

def moving average process

Answer 10

A ssp is a moving average process of order n (MA(n)) if
y(t) = c0e(t) + c1e(t-1) + … + cn*e(t-n)
where
e ~ WN(0, λ^2)
c0, c1, … , cn are the coefficients
n is the order

Question 11

def auto-regressive process

Answer 11

A ssp is an auto-regressive process of order m (AR(m)) if:
y(t) = a1y(t-1) + a2y(t-2) + … + amy(t-m) + c0e(t)
where
e ~ WN(0, λ^2)
a1, a2,…, am, c0 are the coefficients
m is the order

Question 12

def ARMA process

Answer 12

A ssp is said an Auto Regressive Moving Average process of orders (m,n) (ARMA(m,n)) if
y(t) = a1y(t-1) + a2y(t-2) + … + amy(t-m) + c0e(t) + c1e(t-1) + … + cne(t-n)
where
e ~ WN(0, λ^2)
a1, a2,…, am, c0, c1, … , cn are the coefficients
(m,n) are the orders

Question 13

result on stationarity of a stochastic process

Answer 13

if y(t) is the output of a system with transfer function W(z), when the input is e(t):
y(t) is stationary if and only if
- e(t) is stationary
- W(z) is asymptotically stable

Question 14

general rules on stationarity of MA(n) and AR(m) processes

Answer 14

MA(n) processes are characterized by

• n general zeros
• n poles in the origin
• > MA(n) processes are Always stationary

AR(m) processes are characterized by

• m general poles
• m zeros in the origin
• > stationarity is not a priori guaranteed and it should be checked

Question 15

result on the computation of the spectrum of a ssp

Answer 15

if v(t) is a ssp, W(z) is an as. stable system, the spectrum of the output y(t) is given byu:
Γy(ω) = Γv(ω) * |W(e^jω)|^2 for any ω app R

Question 16

Different equivalent descriptions of ARMA processes

Answer 16

1. time domain (definition)
   y(t) = a1y(t-1) + a2y(t-2) + … + amy(t-m) + c0e(t) + c1e(t-1) + … + cne(t-n)
   e ~ WN(0, λ^2)
2. transfer function
   y(t) = C(z) / A(z) * e(t)
3. probabilistic domain
   my,
   γy(τ) τ = 0, +-1, +-2,…
4. frequency domain
   my,
   Γy(ω) ω app R

Question 17

def stochastic process

Answer 17

A stochastic process (SP) is an infinite sequence of random variables, all defined on the same probabilistic space.

Question 18

def mean value of a stochastic process

Answer 18

m(t) = E[x(t,s)] = int(Π) x(t,s)pdf(s)ds
where Π is the probabilistic space
pdf(s) is the probability distribution of the random variables

Question 19

def covariance function of a stochastic process

Answer 19

γ(t1,t2) = E[ ( x(t1,s)-m(t1) ) * ( x(t2,s)-m(t2) ) ]

Question 20

def stationary stochastic process

Answer 20

A stochastic process is said to be stationary (in a wide-sense) if

1) m(t) = m for any t (the mean is a constant)
2) γ(t1,t2) depends only on τ=t1-t2 - > γ(τ)

Question 21

def white noise

Answer 21

A SSP x(t) is said to be a white noise if

1) E[x(t)] = μ
2) γ(0) = λ^2
3) γ(τ) = 0 for any τ != 0