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Introduction
Published in Bo Shen, Zidong Wang, Qi Li, Control and State Estimation for Dynamical Network Systems with Complex Samplings, 2023
Another way to describe the sampling intervals is treating them as random variables obeying certain probability distributions. For example, in [2], the sampling interval has been modeled by a stochastic variables following Erlang distribution, and a stochastic dynamic programming framework has been used for the solution of the optimal control problems. Following a similar line, in [68], the H2 performance constraint has been introduced to the optimal control context, and an H2 state feedback controller has been designed for the sampled-data system with randomly sampled measurement. Recently, in [134], a new quantized/saturated control problem has been addressed for a class of sampled-data systems with noisy sampling intervals obeying Erlang distribution, where the confluent Vandermonde matrix approach has been exploited to deal with multiple control inputs. Later, the random variable obeying categorical distribution has been introduced in [61] to describe the stochastic sampling phenomenon, and a probability-distribution-dependent control scheme has been proposed to guarantee the mean-square exponential synchronization of the error dynamical network. On the other hand, in [136], the sampling intervals have been represented by an arbitrary stochastic variable with known probability density function, and a fundamental stabilization problem has been investigated for such stochastic sampled-data systems.
Applications in Industry
Published in Sylvia Frühwirth-Schnatter, Gilles Celeux, Christian P. Robert, Handbook of Mixture Analysis, 2019
Kerrie Mengersen, Earl Duncan, Julyan Arbel, Clair Alston-Knox, Nicole White
The multinomial distribution provides a natural framework when the sampling process consists of independent and identically distributed (i.i.d.) observations of a fixed number of species, say G (see applications in ecology such as Fordyce et al., 2011; De’ath, 2012; Holmes et al., 2012). Within this framework, individual observations follow a categorical distribution, which is a generalization of the Bernoulli distribution when the sample space is a set of G > 2 items. Namely, the probability mass function of an individual observation yn, which can take on a value of a species in {1,…, G}, is P(yn = g) = ηg. A tantamount notation, also more reminiscent of mixture models, is () yn~ind∑g=1Gηgδg,
Overview
Published in Kim H. Pries, Jon M. Quigley, Testing Complex and Embedded Systems, 2018
We can use nonparametric tests such as Tukey’s quick test to analyze the results of semi-quantitative testing with high confidence. The t-test is not as useful—because the values for the t-test should be variables, not counts. Note that many of the familiar statistical tests (e.g., t-tests and z-tests) are intended for use with measured variables; we must use the appropriate categorical statistics when dealing with count data. For example, we might consider the following: ■ Categorical distribution, general model■ Stratified analysis■ Chi-square test■ Relative risk■ Binomial regression■ McNemar’s test■ Kappa statistics■ Generalized linear models■ Wald test■ Correspondence analysis
Jaccard matrix for nonlinear filter statistics
Published in SICE Journal of Control, Measurement, and System Integration, 2023
Secondly, we define Jaccard cell as a generalization of the index ψ in (3) to that of multi-dimensional Bernoulli random vectors. Let and be one-hot vectors following the categorical distributions, where the one-hot encoding or 1-of-K encoding, like , is well known in machine learning. A categorical distribution , also called a generalized Bernoulli distribution, is a particular case of the multi-nomial distribution [17]. Given the above and , we define Jaccard cell as the following: The given and equal following the Bernoulli random variables under the condition, Consequently, the Jaccard cell (5) is a natural extension of the index (3) that is equivalent to the Jaccard index and pARIs. Moreover, we define the Jaccard matrix as the following matrix: given by aggregating the Jaccard cell (5).
Bayesian Convolutional Neural Network-based Models for Diagnosis of Blood Cancer
Published in Applied Artificial Intelligence, 2022
Mohammad Ehtasham Billah, Farrukh Javed
Given the dataset and its corresponding label set , in a classical setting, the CNN maps the input to the output using the set of weights . The resulting model can therefore be seen as a probabilistic model, where the output is a categorical distribution. By placing a prior over the kernels, the CNN can be converted to a Bayesian CNN. After setting the prior distribution over kernels, the posterior distribution becomes as follows:
State-space modeling for degrading systems with stochastic neural networks and dynamic Bayesian layers
Published in IISE Transactions, 2023
Md Tanzin Farhat, Ramin Moghaddass
Now for a system with variables in the observation process, where nc, nb, and na are respectively the number of continuous, binary, and categorical variables, the stochastic layer would need to include a nc-vector for the mean, a matrix for the covariance matrix, nb neurons to cover binary variables, and neurons to cover categorical variables. In other words, the layer of should include all neurons that characterize the stochastic behavior of sensor outputs. We should point out that the framework discussed above does not assume that output variables are independent. In fact, since all output variables are at the same level and dependent on previous layers, they are marginally dependent (or only conditionally independent given neurons in layer ). If prior knowledge is available with regards to the causal dependencies between sensor outputs, one can incorporate such dependencies by changing the inputs and outputs in (6)-(7). The second way to incorporate dependency is from the probability distribution used in the output layer that provides a joint distribution of output variables. For instance, when multivariate Gaussian distribution is used, then the correlation between continuous sensor outputs may be captured with the non-diagonal elements of the covariance matrix. For categorical distribution, the SoftMax function provides a joint probability distribution where the sum of probabilities of all categories is one. Our framework can handle two important special cases with respect to the observation process, namely, missing data and bounded data. We have shown in Appendices 1 and 2 how these concepts can be taken into account by modifying the probability distribution functions. The ability to handle these two concepts helps with numerical stability and extends its application to more complex degrading systems.