Let the labeled samples [z.sub.i] = ([x.sub.i], [y.sub.i]) [member of] X x Y, i [member of] [N.sub.m], be independent random variables
drawn from the probability space (X x Y, P), and the unlabeled samples [x.sub.m+i] [member of] X, i [member of] [N.sub.u], be independent random variables
drawn from the probability space (X, [P.sub.X]).
For instance, the hepatobiliary training data is represented as [D.sub.m] = ([Y.sub.1], [Z.sub.1]), ..., ([Y.sub.m], [Z.sub.m]) where Y and Z are independent random variables
which are the same as the autonomous sample pair (Y, Z).
Let ([OMEGA], S, P) be a probability space and [([X.sub.n]).sub.n[member of]N]] be a sequence of independent random variables
such that [EX.sub.n] = 0 for all n [member of] N and [[summation].sup.[infinity].sub.n=1] var [X.sub.n] < [infinity].
where f is the joint probability density function (PDF) of random vector [xi] which for independent random variables
[[xi].sub.i] can be written as the multiplication of the individual PDF for each variable [f.sub.i]([[xi].sub.i]); i.e.,
Furthermore, he studies some properties of this distribution and presents its stochastic representation as the product of two independent random variables
[square root of [bar.T]] and V, where T ~ [[chi square].sub.(3)] and V is a discrete random variable such that P(V = [+ or -]1) = 1/2; that is, X = [square root of [bar.T]]V has the distribution BN.
Firstly, this procedure is established in the next paragraphs for independent random variables
, based on the relationships between the random variables and random vector to.
[20.] Lugannani, R., and Rice, S., "Saddlepoint Approximation for the Distribution of the Sum of Independent Random Variables
," Advances in Applied Probability, 475-490, 1980.
Kruglov, "On one identity for distribution of sums of independent random variables
," Theory of Probability and its Applications, vol.
The principle of ICA estimation is based on the central limit theorem (CLT), which states that the sum of independent random variables
tends to be distributed towards Gaussian, regardless of the underlying distribution.
where [u.sub.1] and [u.sub.2] are two independent random variables
of uniform distribution over [0, 1].
For cases where [x.sub.n] is a discrete structure, such as a permutation or a graph, and where the input values are realizations of independent random variables
with the same distribution, the output sequence is a Markov chain X = ([X.sub.n])n[member of]N that is adapted to a combinatorial family F in the sense that [X.sub.n] takes its values in the subset [F.sub.n] [subset] F of objects with base parameter n.
where [[epsilon].sub.t] ~ WN(0, [[sigma].sup.2]) represents a white noise, a series of independent random variables
, identically distributed: E[[epsilon].sub.t]=0, [for all]t Var[[epsilon].sub.t] = [[sigma].sup.2], [for all]t [[epsilon].sub.t] si [[epsilon].sub.t + k] independent variables, [for all]k [not equal to] 0.