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Clustering validation is applied to evaluate the quality of classifications. This step is crucial for unsupervised machine learning. A plethora of methods exist for this purpose; however, a common drawback is that statistical inference is not possible. In this study, we construct a density function for the cluster number. For this purpose, we use smooth techniques. Then, we apply non-negative matrix factorization using the Kullback–Leibler divergence. Employing a unique linearly independent uncorrelated observational variable hypothesis, we construct a sequence by varying the dimension of the span space of the factorization only using analytical techniques. The expectation of the limit of this sequence follows a gamma probability density function. Then, identifying the dimension of the factorization of the space span with clusters, we transform the estimation of the suitable dimension of the factorization into a probabilistic estimate of the number of clusters. This approach is an internal validation method that is suitable for numerical and categorical multivariate data and independent of the clustering technique. Our main achievement is a predictive clustering validation model with graphical abilities. It provides results in terms of credibility, thus making it possible to compare results such as expert judgment on a quantitative basis.

This study introduces a novel method for identifying dynamic systems aimed at deriving reduced-fractional-order models. Applicable to processes exhibiting an S-shaped step response, the method effectively characterizes fractional behavior within the range of fractional orders (α∈[0.5,1.0]). The uniqueness of this approach lies in its hybrid nature, combining one-variable optimization techniques for estimating the model fractional order α with analytical expressions to estimate parameters T and L. This hybrid approach leverages information from the reaction curve obtained through an open-loop step-test experiment. The proposed method demonstrates its efficacy and simplicity through several illustrative examples, showcasing its advantages over established analytical and optimization-based techniques. Notably, the hybrid approach proves particularly advantageous compared to methods relying on the process reaction curve. To highlight its practical applicability, the identification algorithm based on this hybrid approach is implemented on hardware using a microprocessor. The experimental prototype successfully identifies the First-Order Plus Dead Time (FFOPDT) model of a thermal-based process, validating the proposed method's real-world utility.

This manuscript provides a comprehensive exploration of Probabilistic latent semantic analysis (PLSA), highlighting its strengths, drawbacks, and challenges. The PLSA, originally a tool for information retrieval, provides a probabilistic sense for a table of co-occurrences as a mixture of multinomial distributions spanned over a latent class variable and adjusted with the expectation–maximization algorithm. The distributional assumptions and the iterative nature lead to a rigid model, dividing enthusiasts and detractors. Those drawbacks have led to several reformulations: the extension of the method to normal data distributions and a non-parametric formulation obtained with the help of Non-negative matrix factorization (NMF) techniques. Furthermore, the combination of theoretical studies and programming techniques alleviates the computational problem, thus making the potential of the method explicit: its relation with the Singular value decomposition (SVD), which means that PLSA can be used to satisfactorily support other techniques, such as the construction of Fisher kernels, the probabilistic interpretation of Principal component analysis (PCA), Transfer learning (TL), and the training of neural networks, among others. We also present open questions as a practical and theoretical research window.

In this paper, a general procedure for identifying a fractional first-order plus dead-time (FFOPDT) model is presented. This procedure is based on fitting three arbitrary points on the process reaction curve, where process information is obtained from a simple open-loop test. A simplification of the general identification procedure is also considered, where only points symmetrically located on the reaction curve are selected. The proposed symmetrical procedure has been applied to the following sets of representative points: (5–50–95%), (10–50–90%), (15–50–85%), (20–50–80%), (25–50–75%), and (30–50–70%). Analytical expressions of the corresponding FFOPDT model parameters for these sets of symmetrical points have been obtained. In order to show the effectiveness and applicability of this procedure for the identification of fractional-order models and to get insight into the influence of selection of the set of symmetrical points on the accuracy of the identified model, some numerical examples are proposed. This identification procedure gives good results in comparison with other integer- and fractional-order identification methods. Finally, some conclusions and final remarks are offered in this context.

In this paper, the conceptualization of a control hardware architecture aimed to the implementation of integer- and fractional-order identification and control algorithms is presented. The proposed hardware architecture combines the capability of implementing PC-based control applications with embedded applications on microprocessor- and FPGA-based real-time targets. In this work, the potential advantages of this hardware architecture over other available alternatives are discussed from different perspectives. The experimental prototype that has been designed and built to evaluate the control hardware architecture proposed in this work is also described in detail. The thermal-based process taking place in the prototype is characterized for being reconfigurable and exhibiting fractional behaviour, which results in a suitable equipment for the purpose of fractional-order identification and control. In order to demonstrate the applicability and effectiveness of the proposed control hardware architecture, integer- and fractional-order identification and control algorithms implemented in various control technologies have been applied to the temperature-based experimental prototype described before. Detailed discussion about results and identification and control issues are provided. The main contribution of this work is to provide an efficient and practical hardware architecture for implementing fractional-order identification and control algorithms in different control technologies, helping to bridge the gap between real-time hardware solutions and software-based simulations of fractional-order systems and controllers. Finally, some conclusions and concluding remarks are offered in the industrial context.

This paper aims to present a general identification procedure for fractional first-order plus dead-time (FFOPDT) models. This identification method is general for processes having S-shaped step responses, where process information is collected from an open-loop step-test experiment, and has been conducted by fitting three arbitrary points on the process reaction curve. In order to validate this procedure and check its effectiveness for the identification of fractional-order models from the process reaction curve, analytical expressions of the FFOPDT model parameters have been obtained for both situations: as a function of any three points and three points symmetrically located on the reaction curve, respectively. Some numerical examples are provided to show the simplicity and effectiveness of the proposed procedure. Good results have been obtained in comparison with other well-recognized identification methods, especially when simplicity is emphasized. This identification procedure has also been applied to a thermal-based experimental setup in order to test its applicability and to obtain insight into the practical issues related to its implementation in a microprocessor-based control hardware. Finally, some comments and reflections about practical issues relating to industrial practice are offered in this context.

The Probabilistic Latent Semantic Analysis has been related with the Singular Value Decomposition. Several problems occur when this comparative is done. Data class restrictions and the existence of several local optima mask the relation, being a formal analogy without any real significance. Moreover, the computational difficulty in terms of time and memory limits the technique applicability. In this work, we use the Nonnegative Matrix Factorization with the Kullback–Leibler divergence to prove, when the number of model components is enough and a limit condition is reached, that the Singular Value Decomposition and the Probabilistic Latent Semantic Analysis empirical distributions are arbitrary close. Under such conditions, the Nonnegative Matrix Factorization and the Probabilistic Latent Semantic Analysis equality is obtained. With this result, the Singular Value Decomposition of every nonnegative entries matrix converges to the general case Probabilistic Latent Semantic Analysis results and constitutes the unique probabilistic image. Moreover, a faster algorithm for the Probabilistic Latent Semantic Analysis is provided.