Prerequisites
Basic notions of set theory, logic, calculus, function maximization.
Course contents
System Identification deals with methodologies that enable the construction of mathematical models of systems and signals based on experimental data. In presence of complex systems whose behavior can be hardly reduced to known "laws of nature", the use of identification techniques is often the only way to obtain models to be used in the context of forecasting, simulation, and control. The methods presented in the course are widely used in heterogeneous fields such as automation, biomedical engineering, econometry, hydrology, geophysics and telecommunications. Some basic notions of probability, estimation theory and stochastic processes are recalled. The main properties (stability, input-output description in the time and frequenct domains) of linear discrete-time systems are introduced. In the context of parametric estimation, the issues of model validation and model complexity are extensively discussed. Neural based identification is also illustrated and discussed, pointing out pros and cons with respect to standard approaches. The study of dynamic systems addresses three main topics: the optimal prediction of stationary stochastic processes (Wiener filtering), the identification of linear discrete-time systems, and spectral estimation (both nonparametric and maximum-entropy).
Probability: basic notions
probability notion;
independence, conditional probability, total probability and Bayes theorems;
Bernoulli trials, Poisson events;
the notion of random variable (R.V.), cumulative distribution function, probability density function, functions on one R.V.;
mode, median, moments of a R.V.;
joint random variables: distribution, density, moments, independence, incorrelation, functions of random variables;
Law of Lrge Numbers, Gaussian R.V., Central Limit Theorem.
Statistics: basic notions
notion of estimator; properties of estimators;
sample moments and their main properties;
confidence interval for the sample mean, Student's t.
Identification of linear-in-parameter models:
the least squares method, normal equations, identifiability;
Best Linear Unbiased Estimator: estimator, variance of parameters;
validation and choice of complexity: chi-square test, F-test, FPE, AIC, and MDL criteria.
Reccomended or required readings
Lecture notes (http://sisdin.unipv.it/labsisdin/teaching/teaching.php).
M. Bramanti. Calcolo delle probabilità e statistica. Esculapio.
A. Papoulis. Probability, Random Variables, and Stochastic Processes. McGraw-Hill.
L. Ljung. System Identification: Theory for the User. Prentice-Hall.
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