Stochastic processes and weak description
A signal is a function of time, usually symbolized . In a noisy signal, the exact value of the signal is random. Therefore, we will model noisy signals as a random function , where at each time , is a random variable. These “noisy signals” are formally called random processes or stochastic processes.
A stochastic or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner.
Observations will be interpreted as realization of a stochastic process which is sequence of random variables defined on the same probabilistic space.
We denote with where is the outcome in a probabilistic space. "Outcome" implies that the quantities of interest depends on some variables. For a stochastic process the are multiple realizations, one for each value of .
We will denote for simplicity but we mean the stochastic process .
Weak description of stochastic processes
A complete, strong, description of stochastic processes is given by the probability distribution but in practice it is hard to obtain this description. Therefore we use some partial idicators to conceive a weak description.
In particular the weak (wide sense) characterization of a stochastic process is given by:
- The mean function
- The covariance function This means that two processes with the same mean and covariance are considered equal.
Mean function
The mean function of a random process is a function that specifies the expected value at each time . It is the sequence of the mean of which coincide with the expected value computed over the variability of .
The mean function is interpreted as the central (baricentric) value around which the realization of take value.
Variance function
The variance function of a random process is a function that specifies the variance of the process at each time t. The variance functions provides a quantification of the dispersion around the mean.
Covariance function
The covariance function specifies the covariance between the value of the process at time and the value at time :
$$\lambda_v(gamma_v(t,\tau)=\mathbb{E}[(v(t,s)-m_v(t))(v(t-\tau ,s)-m_v(t-\tau ))]$$
It provides an indication of correlation of variables defining the stochastic process at different time instants:
- $\
lambda_v(gamma_v(t,\tau)>0t-\taut$ - $\
lambda_v(gamma_v(t,\tau)<0t-\taut$
Observation: the variance function can be obtained from the covariance function:
$$V_v(t)=\lambda_v(gamma_v(t,0)$$
Questions from past exams
Give the definition of the mean function and the covariance function for a generic stochastic process and explain what is the interpretation of these two indicators.
See above.