Model classes

A white noise is a sequence of uncorrelated random variables with the same mean and the same variance. A stationary stochastic process is called white noise if: $\mathbb{E}[e(t,s)]=\mu$ $\forall t$…

Given a zero mean white noise $e(t) \sim WN(0,\lambda^2)$ we define the moving average process of order $MA(n)$ as: $$y(t)=c_0e(t)+c_1e(t-1)+...+c_ne(t-n)$$ Which is a linear combination of the past…

AR processes Given a zero mean white noise $e(t) \sim WN(0, \lambda^2)$ then the stochastic process $y(t)$ is an auto regressive process if $y(t)$ is stationary and satisfies the recursive equation :…

A transfer function $W(z)=\frac{C(z)}{A(z)}$ is an operator that given a process $v(t)$ outputs the steady state solution of the recursive equation $y(t)$: $$y(t)=W(z)v(t) \iff A(z)y(t)=C(z)v(t)$$ We…

Given the steady state output of the recursive equation defined by the transfer function $W(z)$ fed by the stochastic process $v(t)$: $$y(t)=W(z)v(t)$$ Then $y(t)$ is well defined iff: $v(t)$ is…

If we consider a stationary stochastic process $v(t)$ and an assintotically stable transfer function $W(z)$ then the steady state solution of the recursive equation defined by the transfer function…

Given the ARMA process: $$y(t)=\frac{c_0+c_1z^{-1}+...+c_nz^{-n}}{1-a_1z^{-1}-...-a_mz^{-m}}e(t)=\frac{C(z)}{A(z)}e(t)$$ We can compute the weak (wide sense) characterization for a well defined…

Consider the ARMA process $y(t)=\frac{C(z)}{A(z)}e(t)$ where is a non zero mean white noise$e(t) \sim WN(\mu,\lambda^2)$. Gain theorem If $W(z)=\frac{C(z)}{A(z)}$ is assintotically stable, then:…

With ARMA we model time-series: we analize the output of a system without observing any input. In many application, however, we can observe an input process $u(t)$. $$…