Random processes. Matlab What is a random process?
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Transcript of 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 finite-
dimensional 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 10am sharp. The value is -75.3 mV.
• 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 10am sharp. The value is -80.9 mV.
• 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 10am 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)
Go to Matlab
Theoretically,
• Repeated measurement 1:– I arrive this morning to the lab 600 times,
prepare a neuron for recording and measure its membrane potential at 10am 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 10am 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)
What to do?
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
Go to Matlab
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)
Go to Matlab
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 finite-dimensional 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)