Random processes Matlab What is a random process

Random processes

Matlab ? What is a random process

A random process • Is defined by its finite-dimensional distributions – The probability of events at a finite number of time points • The finite dimensional distributions have to be ‘consistent’ – Integrating over one time point gives the finitedimensional distribution for the other time points • Given a consistent family of finite-dimensional distributions on ‘good enough’ spaces, there is a unique process with those distributions (Kolmogorov) – ‘Good enough’ means Borel

Stationarity and ergodicity How to measure the resting membrane potential of a neuron?

Stationarity and ergodicity • I arrive this morning to the lab, prepare a neuron for recording and measure its membrane potential at 10 am sharp. The value is -75. 3 m. V. • Is this the resting potential of the neuron?

Stationarity and ergodicity • The measurement is noisy • We want to have a number of repeats of the same measurement • How to get repeated measurements?

Stationarity and ergodicity • Repeated measurement: – I arrive this morning a second time to the lab, prepare a neuron for recording and measure its membrane potential at 10 am sharp. The value is -80. 9 m. V. • What is the problem?

Stationarity and ergodicity • Repeated measurement 1: – I arrive this morning to the lab 600 times, prepare a neuron for recording and measure its membrane potential at 10 am sharp. • Repeated measurement 2: – I measure the membrane potential of the same neuron as before once a second from 10: 00 to 10: 10 (I get 600 measurements)

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, Theoretically • Repeated measurement 1: – I arrive this morning to the lab 600 times, prepare a neuron for recording and measure its membrane potential at 10 am sharp. • Repeated measurement 2: – I measure the membrane potential of the same neuron as before once a second from 10: 00 to 10: 10 (I get 600 measurements)

, Practically • Repeated measurement 1: – I arrive this morning to the lab 600 times, prepare a neuron for recording and measure its membrane potential at 10 am sharp. • Repeated measurement 2: – I measure the membrane potential of the same neuron as before once a second from 10: 00 to 10: 10 (I get 600 measurements)

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Ergodicity • For an ergodic process, – Averaging across many repeated trials (repeated measurements 1) – Averaging across time for a single trial (repeated measurements 2) – Are equal • An ergodic process is always stationary, the reverse may not be true

What makes a stationary process ergodic? • Asymptotic independence • Samples that are far enough in time are independent

Correlation, independence, gaussian and non-gaussian processes

Independence vs. lack of correlation • Two variables are independent if knowing anything about one of them doesn’t allow you to make any deductions that you couldn’t already make about the other one • Two variables are uncorrelated if their covariance is 0 • Independence implies lack of correlation • Lack of correlation in general does not imply independence

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Independence vs. lack of correlation • For variables that are jointly Gaussian, lack of correlation implies independence • What are jointly Gaussian variables?

Jointly Gaussian variables • The distribution of each by itself is gaussian • The joint distribution of each pair is gaussian • The joint distribution of each triplet is gaussian • … • (allowing for degeneracy)

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Jointly gaussian variables • Because of the issue of degeneracy, the formal definition is indirect • For example: random variables are jointly gaussian if all linear combinations are gaussian (allowing the degenerate case of identically 0 variables) • Or using characteristic functions

Characterizing jointly gaussian variables • A 1 -d Gaussian variable is fully characterized by its mean and variance • These determine its probability density function and therefore all other quantifiers • An n-d Gaussian variable is fully characterized by the mean of each component and their covariances • These determine the joint probability density and therefore all other quantifiers

Gaussian process • A random process is gaussian if all finitedimensional distributions are jointly gaussian • A Gaussian process is determined by specifying the mean at each moment in time and a matrix of covariances between the values at different moments in time • All finite-dimensional distributions are Gaussian, and are therefore determined by the above data

Stationary Gaussian processes • If the process is in addition stationary – The mean and variances are constant as a function of time – the 2 -d distributions do not depend on the absolute time • In that case, the covariance matrix is constant along the diagonals – ‘Toeplitz matrices’ • The covariance is specified by a function of the delay between samples

Stationary gaussian processes • The autocovariance function is also called – Autocorrelation function – Covariance function – Correlation function –… • Make sure you know the normalization (what is the value of the function at 0)