Transcript PPT

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 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 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)