128 research outputs found

### On hidden Markov chains and finite stochastic systems

In this paper we study various properties of finite stochastic systems or
hidden Markov chains as they are alternatively called. We discuss their
construction following different approaches and we also derive recursive
filtering formulas for the different systems that we consider. The key tool is
a simple lemma on conditional expectations

### Approximate Nonnegative Matrix Factorization via Alternating Minimization

In this paper we consider the Nonnegative Matrix Factorization (NMF) problem:
given an (elementwise) nonnegative matrix $V \in \R_+^{m\times n}$ find, for
assigned $k$, nonnegative matrices $W\in\R_+^{m\times k}$ and
$H\in\R_+^{k\times n}$ such that $V=WH$. Exact, non trivial, nonnegative
factorizations do not always exist, hence it is interesting to pose the
approximate NMF problem. The criterion which is commonly employed is
I-divergence between nonnegative matrices. The problem becomes that of finding,
for assigned $k$, the factorization $WH$ closest to $V$ in I-divergence. An
iterative algorithm, EM like, for the construction of the best pair $(W, H)$
has been proposed in the literature. In this paper we interpret the algorithm
as an alternating minimization procedure \`a la Csisz\'ar-Tusn\'ady and
investigate some of its stability properties. NMF is widespreading as a data
analysis method in applications for which the positivity constraint is
relevant. There are other data analysis methods which impose some form of
nonnegativity: we discuss here the connections between NMF and Archetypal
Analysis. An interesting system theoretic application of NMF is to the problem
of approximate realization of Hidden Markov Models

### A block Hankel generalized confluent Vandermonde matrix

Vandermonde matrices are well known. They have a number of interesting
properties and play a role in (Lagrange) interpolation problems, partial
fraction expansions, and finding solutions to linear ordinary differential
equations, to mention just a few applications. Usually, one takes these
matrices square, $q\times q$ say, in which case the $i$-th column is given by
$u(z_i)$, where we write $u(z)=(1,z,...,z^{q-1})^\top$. If all the $z_i$
($i=1,...,q$) are different, the Vandermonde matrix is non-singular, otherwise
not. The latter case obviously takes place when all $z_i$ are the same, $z$
say, in which case one could speak of a confluent Vandermonde matrix.
Non-singularity is obtained if one considers the matrix $V(z)$ whose $i$-th
column ($i=1,...,q$) is given by the $(i-1)$-th derivative $u^{(i-1)}(z)^\top$.
We will consider generalizations of the confluent Vandermonde matrix $V(z)$
by considering matrices obtained by using as building blocks the matrices
$M(z)=u(z)w(z)$, with $u(z)$ as above and $w(z)=(1,z,...,z^{r-1})$, together
with its derivatives $M^{(k)}(z)$. Specifically, we will look at matrices whose
$ij$-th block is given by $M^{(i+j)}(z)$, where the indices $i,j$ by convention
have initial value zero. These in general non-square matrices exhibit a
block-Hankel structure. We will answer a number of elementary questions for
this matrix. What is the rank? What is the null-space? Can the latter be
parametrized in a simple way? Does it depend on $z$? What are left or right
inverses? It turns out that answers can be obtained by factorizing the matrix
into a product of other matrix polynomials having a simple structure. The
answers depend on the size of the matrix $M(z)$ and the number of derivatives
$M^{(k)}(z)$ that is involved. The results are obtained by mostly elementary
methods, no specific knowledge of the theory of matrix polynomials is needed

### Factor Analysis and Alternating Minimization

In this paper we make a first attempt at understanding how to build an
optimal approximate normal factor analysis model. The criterion we have chosen
to evaluate the distance between different models is the I-divergence between
the corresponding normal laws. The algorithm that we propose for the
construction of the best approximation is of an the alternating minimization
kind

### Explicit computations for some Markov modulated counting processes

In this paper we present elementary computations for some Markov modulated
counting processes, also called counting processes with regime switching.
Regime switching has become an increasingly popular concept in many branches of
science. In finance, for instance, one could identify the background process
with the `state of the economy', to which asset prices react, or as an
identification of the varying default rate of an obligor. The key feature of
the counting processes in this paper is that their intensity processes are
functions of a finite state Markov chain. This kind of processes can be used to
model default events of some companies.
Many quantities of interest in this paper, like conditional characteristic
functions, can all be derived from conditional probabilities, which can, in
principle, be analytically computed. We will also study limit results for
models with rapid switching, which occur when inflating the intensity matrix of
the Markov chain by a factor tending to infinity. The paper is largely
expository in nature, with a didactic flavor

### Large deviations for Markov-modulated diffusion processes with rapid switching

In this paper, we study small noise asymptotics of Markov-modulated diffusion
processes in the regime that the modulating Markov chain is rapidly switching.
We prove the joint sample-path large deviations principle for the
Markov-modulated diffusion process and the occupation measure of the Markov
chain (which evidently also yields the large deviations principle for each of
them separately by applying the contraction principle). The structure of the
proof is such that we first prove exponential tightness, and then establish a
local large deviations principle (where the latter part is split into proving
the corresponding upper bound and lower bound)

### Sample-path Large Deviations in Credit Risk

The event of large losses plays an important role in credit risk. As these
large losses are typically rare, and portfolios usually consist of a large
number of positions, large deviation theory is the natural tool to analyze the
tail asymptotics of the probabilities involved. We first derive a sample-path
large deviation principle (LDP) for the portfolio's loss process, which enables
the computation of the logarithmic decay rate of the probabilities of interest.
In addition, we derive exact asymptotic results for a number of specific
rare-event probabilities, such as the probability of the loss process exceeding
some given function

- â€¦