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Stochastic processes and weak description

A signal is a function of time, usually symbolized v(t,s)v(t,s). In a noisy signal, the exact value of the signal is random. Therefore, we will model noisy signals as a random function v(t,s)v(t,s), where at each time tt, v(t,s)v(t,s) 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 y(1),y(2),...,y(w)y(1),y(2),...,y(w) will be interpreted as realization of a stochastic process which is sequence of random variables defined on the same probabilistic space.

We denote v(t,s)v(t, s) with t=0,±1,±2,...t=0,\pm 1,\pm 2,... where ss 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 ss.

We will denote v(t)v(t) for simplicity but we mean the stochastic process v(t,s)v(t,s).

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 mv(t)m_v(t) of a random process v(t,s)v(t,s) is a function that specifies the expected value at each time tt. It is the sequence of the mean of v(t,s)v(t,s) which coincide with the expected value computed over the variability of ss.

mv(t)=E[v(t,s)]=sv(t,s)Pdsm_v(t)=\mathbb{E}[v(t,s)]=\int_{s}{v(t,s)\mathbb{P}ds}

The mean function is interpreted as the central (baricentric) value around which the realization of v(t,s)v(t,s) take value.

Variance function

The variance function Vv(t)V_v(t) of a random process v(t,s)v(t,s) 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.

Vv(t)=Var[v(t,s)]=E[(v(t,s)mv(t))2]V_v(t)=Var[v(t,s)]=\mathbb{E}[(v(t,s)-m_v(t))^2]

Covariance function

The covariance function λv(t,τ)\lambda_v(t,\tau) specifies the covariance between the value of the process at time tt and the value at time tτt-\tau:

$$\lambda_v(gamma_v(t,\tau)=\mathbb{E}[(v(t,s)-m_v(t))(v(t-\tau ,s)-m_v(t-\tau ))]$$

It provides an indication of correlation of variables defining the stochastic process at different time instants:

  • $\lambda_v(gamma_v(t,\tau)>0indicatesatendencytopreservethesignfrom indicates a tendency to preserve the sign from t-\tauto to t$
  • $\lambda_v(gamma_v(t,\tau)<0indicatesatendencytochangethesignfrom indicates a tendency to change the sign from t-\tauto to t$

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.