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Publications about 'statistics'
Articles in journal or book chapters
Dynamic compensation, parameter identifiability, and equivariances.
PLoS Computational Biology,
Note: Preprint was in bioRxiv https://doi.org/0.1101/095828, 2016.[WWW]
A recent paper by Karin et al. introduced a mathematical notion called dynamical compensation (DC) of biological circuits. DC was shown to play an important role in glucose homeostasis as well as other key physiological regulatory mechanisms. Karin et al.\ went on to provide a sufficient condition to test whether a given system has the DC property. Here, we show how DC is a reformulation of a well-known concept in systems biology, statistics, and control theory -- that of parameter structural non-identifiability. Viewing DC as a parameter identification problem enables one to take advantage of powerful theoretical and computational tools to test a system for DC. We obtain as a special case the sufficient criterion discussed by Karin et al. We also draw connections to system equivalence and to the fold-change detection property.
E.D. Sontag and D. Zeilberger.
A symbolic computation approach to a problem involving multivariate Poisson distributions.
Advances in Applied Mathematics,
Note: There are a few typos in the published version. Please see this file for corrections: https://drive.google.com/file/d/0BzWFHczJF2INUlEtVkFJOUJiUFU/view.
Keyword(s): probability theory,
Chemical Master Equation.
Multivariate Poisson random variables subject to linear integer constraints arise in several application areas, such as queuing and biomolecular networks. This note shows how to compute conditional statistics in this context, by employing WZ Theory and associated algorithms. A symbolic computation package has been developed and is made freely available. A discussion of motivating biomolecular problems is also provided.
B.W. Dickinson and E.D. Sontag.
Dynamic realizations of sufficient sequences.
IEEE Trans. Inform. Theory,
Keyword(s): realization theory,
Let Ul, U2, ... be a sequence of observed random variables and (T1(U1),T2(Ul,U2),...) be a corresponding sequence of sufficient statistics (a sufficient sequence). Under certain regularity conditions, the sufficient sequence defines the input/output map of a time-varying, discrete-time nonlinear system. This system provides a recursive way of updating the sufficient statistic as new observations are made. Conditions are provided assuring that such a system evolves in a state space of minimal dimension. Several examples are provided to illustrate how this notion of dimensional minimality is related to other properties of sufficient sequences. The results can be used to verify the form of the minimum dimension (discrete-time) nonlinear filter associated with the autoregres- sive parameter estimation problem.
and E.D. Sontag.
Characterizing innovations realizations for random processes.
In this paper we are concerned with the theory of second order (linear) innovations for discrete random processes. We show that of existence of a finite dimensional linear filter realizing the mapping from a discrete random process to its innovations is equivalent to a certain semiiseparable structure of the covariance sequence of the process. We also show that existence of a finite dimensional realization (linear or nonlinear) of the mapping from a process to its innovations implies that the process have this serniseparable covariance sequence property. In particular, for a stationary random process, the spectral density function must be rational.
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