MIDA1 Model Identification and Data Analysis
Course information
Complete course name: 051588 - MODEL IDENTIFICATION AND DATA ANALYSIS - 1ST MODULE (BITTANTI SE...
Stochastic processes and weak description
A signal is a function of time, usually symbolized $v(t,s)$. In a noisy signal, the exact value o...
Stationary stochastic processes (SSP)
A stochastic process $v(t,s)$ is stationary if: $m_v(t)=m_v$ $\forall{t}$: the mean function is ...
Shift operators
The shift operators are: $z^{-1}$ backward shift operator $z^1$ forward shift operator Given ...
Model classes
White noise
A white noise is a sequence of uncorrelated random variables with the same mean and the same vari...
Moving average processes
Given a zero mean white noise $e(t) \sim WN(0,\lambda^2)$ we define the moving average process o...
Auto regressive processes
AR processes Given a zero mean white noise $e(t) \sim WN(0, \lambda^2)$ then the stochastic proce...
Transfer function (digital filter)
A transfer function $W(z)=\frac{C(z)}{A(z)}$ is an operator that given a process $v(t)$ outputs t...
When ARMA is well defined?
Given the steady state output of the recursive equation defined by the transfer function $W(z)$ f...
Non steady state solutions
If we consider a stationary stochastic process $v(t)$ and an assintotically stable transfer func...
Weak (wide sense) characterization of ARMA
Given the ARMA process: $$y(t)=\frac{c_0+c_1z^{-1}+...+c_nz^{-n}}{1-a_1z^{-1}-...-a_mz^{-m}}e(t)=...
Non zero mean ARMA
Consider the ARMA process $y(t)=\frac{C(z)}{A(z)}e(t)$ where is a non zero mean white noise$e(t) ...
Auto regressive processes with exogenous input
With ARMA we model time-series: we analize the output of a system without observing any input. In...
Frequency Domain Analysis
Frequency domain and spectrum
The frequency domain is another way to obtain the weak (wide sense) characterization of a station...
Spectrum antitrasformation and relation with covariance
There exists an antitrasformation: $$\gamma_y(\tau)=\frac{1}{2\pi}\int_{-\pi}^{+\pi}{\Gamma_y(\om...
Wiener–Khinchin theorem
Spectrum of ARMA processes
Given the $ARMA$ process $y(t)=\frac{C(z)}{A(z)}e(t)$ with $e(t) \sim WN(0,\lambda^2)$ the spectr...
Model prediction
Linear optimal prediction
Starting from an $ARMA$ process $y(t)=W(z)e(t)=\frac{C(z)}{A(z)}e(t)$, wehere $e(t) \sim WN(0,\la...
Linear predictors from noise
Starting from an $ARMA$ process $y(t)=W(z)e(t)=\frac{C(z)}{A(z)}e(t)$, wehere $e(t) \sim WN(0,\la...
Long k-step division method
The steady state solution of an $ARMA$ process can be obtained with a long k-step division of $C(...
Linear predictor from output
In the real world we cannot measur the white noise, the only aviable information is the values of...
Prediction of non zero mean ARMA
Starting from an $ARMA$ process $y(t)=W(z)e(t)=\frac{C(z)}{A(z)}e(t)$, wehere $e(t) \sim WN(\mu,\...
ARMAX predictors
Starting from an $ARMAX$ process $y(t)=\frac{B(z)}{A(z)}u(t-d)+\frac{C(z)}{A(z)}e(t)$, wehere $e(...