Cantelli's Inequality Research Articles

Improved Upper Bounds of Cantelli's Inequality Type and Their Practical Applications

Improved Upper Bounds of Cantelli's Inequality Type and Their Practical Applications

A self-triggered stochastic model predictive control for uncertain networked control system

This paper is concerned with the stochastic model predictive control problem (SMPC) for a class of networked control systems with exogenous disturbances and restrictions on communication frequency. A self-triggered SMPC algorithm with an improved triggering condition is designed which integrates the co-design of both the current control inputs and the maximum sampling interval, with the aim of reducing the amounts of communication transmission while guaranteeing the specific performance. To give consideration to both probabilistic guarantee and computability, chance constraints on both states and inputs are transformed into deterministic ones by leveraging Cantelli's inequality and linearisation techniques, so as to be solvable for the optimisation problem. Meanwhile, the nonconvex terms in dynamics of covariance propagation are averted through introducing the covariance upper bound control approach. The formulated online optimisation problem is convex to all decision variables. The theoretical analysis on recursive feasibility and closed-loop stability is addressed for the proposed self-triggered SMPC scheme. Finally, the efficacy and achievable performance of the proposed algorithm are showcased through numerical simulations.

Stochastic model predictive braking control for heavy-duty commercial vehicles during uncertain brake pressure and road profile conditions

When heavy-duty commercial vehicles (HDCVs) must engage in emergency braking, uncertain conditions such as the brake pressure and road profile variations will inevitably affect braking control. To minimize these uncertainties, we propose a combined longitudinal and lateral controller method based on stochastic model predictive control (SMPC) that is achieved via Chebyshev–Cantelli inequality. In our method, SMPC calculates braking control inputs based on a finite time prediction that is achieved by solving stochastic programming elements, including chance constraints. To accomplish this, SMPC explicitly describes the probabilistic uncertainties to be used when designing a robust control strategy. The main contribution of this paper is the proposal of a braking control formulation that is robust against probabilistic friction circle uncertainty effects. More specifically, the use of Chebyshev–Cantelli inequality suppresses road profile influences, which have characteristics that are different from the Gaussian distribution, thereby improving both braking robustness and control performance against statistical disturbances. Additionally, since the Kalman filtering (KF) algorithm is used to obtain the expectation and covariance used for calculating deterministic transformed chance constraints, the SMPC is reformulated as a KF embedded deterministic MPC. Herein, the effectiveness of our proposed method is verified via a MATLAB/Simulink and TruckSim co-simulation.

Some Improvements on Markov's Theorem with Extensions

ABSTRACTMarkov's theorem for an upper bound of the probability related to a nonnegative random variable has been improved using additional information in almost the nontrivial entire range of the variable. In the improvement, Cantelli's inequality is applied to the square root of the original variable, whose expectation is finite when that of the original variable is finite. The improvement has been extended to lower bounds and monotonic transformations of the original variable. The improvements are used in Chebyshev's inequality and its multivariate version.

A minimax probability extreme machine framework and its application in pattern recognition

In this work we propose a minimax probability extreme learning machine framework (MPME), which combines the benefits of minimax probability machine (MPM) with extreme learning machine (ELM). For binary classification problems, we illustrate that the proposed MPME can be interpreted geometrically by minimizing the maximum of Mahalanobis distances to the two classes. Then two variants of the MPME are presented based on the l2-norm loss and l1-norm loss functions (called LSEMPME and LADMPME) respectively. Without making specific assumption on the data distribution, the proposed methods can provide explicit upper-bounds for the generalization error, moreover the LSEMPME and LADMPME minimize empirical risk simultaneously. The decision hyperplanes of the proposed methods pass through the origin in ELM feature space with few decision variables. By using the multivariate Chebyshev–Cantelli inequality, all the proposed problems can be reformulated as second-order cone programming (SOCP) with global solutions. Furthermore, numerical experiments have been carried out on two databases that are drawn from UCI benchmark database and a practical application database. First, the proposed methods are evaluated for a practical application consisting on the analysis of licorice seeds using near-infrared spectral (NIR) data. Experiments in six different spectral regions illustrate that the proposed methods can improve generalization in most cases. Then the proposed methods are evaluated on benchmark datasets. In comparison with traditional methods including MPM, ELM and support vector machine (SVM), experiments show that the proposed methods achieve comparable results in generalization. With few decision variables, the proposed methods are easy to implement for nonlinear classification and to estimate a lower-bound on the prediction accuracy.

The multiple Cantelli inequalities

The Cantelli inequality or the one-sided Chebyshev inequality is extended to the problem of the probability of multiple inequalities for events with more than one variable. The corresponding two-sided Cantelli inequality is extended in a similar manner. The results for the linear combination of several variables are also given as their special cases.

Can Warren Buffett forecast equity market corrections?

ABSTRACTWarren Buffett suggested that the ratio of the market value of all publicly traded stocks to the Gross National Product could identify potential overvaluations and undervaluations in the US equity market when this ratio deviates above 120% or below 80%. We investigate whether this ratio is a statistically significant predictor of equity market corrections and rallies. We find that Buffett's decision rule does not deliver satisfactory forecasts. However, when we adopt a time-varying decision rule, the ratio becomes a statistically significant predictor of equity market corrections. The two time-varying decision rules are: (i) predict an equity market correction when the ratio exceeds a 95% one-tail confidence interval based on a normal distribution, and (ii) predict an equity market correction when the ratio exceeds a threshold computed using Cantelli's inequality. These new decision rules are robust to changes in the two key parameters: the confidence level and the forecasting horizon. This paper also shows that the MV/GNP ratio performs relatively well against the four most popular equity market correction models, but the ratio is not a particularly useful predictor of equity market rallies.

A robust extreme learning machine framework for uncertain data classification

Uncertain or missing data may occur in many practical applications. A principled strategy for handling this problem would therefore be very useful. We consider two-class and multi-class classification problems where the mean and covariance of each class are assumed to be known. With simple structure, fast speed and good performance, extreme learning machine (ELM) has been an important technology in machine learning. In this work, from the viewpoint of probability, we present a robust ELM framework (RELM) for missing data classification. Applying the Chebyshev–Cantelli inequality, the proposed RELM is reformulated as a second-order cone programming with global optimal solution. The proposed RELM only relates to the second moments of input samples and makes no assumption about the data probability distribution. Expectation maximization algorithm is used to fill in missing values and then obtain complete data. Numerical experiments are simulated in various datasets from UCI database and a practical application database. Experimental results show that the proposed method can achieve better performance than traditional methods. These results illustrate the feasibility and effectiveness of the proposed method for missing data classification.

Model predictive control of linear systems with multiplicative unbounded uncertainty and chance constraints

This paper presents a novel stochastic Model Predictive Control algorithm for linear systems characterized by multiplicative and possibly unbounded model uncertainty. Probabilistic constraints on the states and inputs are considered, and a quadratic cost function is minimized. The stochastic control problem, and in particular the probabilistic constraints, are reformulated in deterministic terms by means of the Cantelli inequality, so that the on-line computational burden of the algorithm is similar to the one of a standard MPC method. The properties of the algorithm, namely the recursive feasibility and the pointwise convergence of the state, are proven by suitably selecting the terminal cost and the constraints on the mean and the variance of the state at the end of the prediction horizon, and by considering as additional optimization variables also the mean and the covariance of the state at the beginning of the prediction horizon. An extension to deal with the case of expectation, rather than probabilistic, constraints is reported. The numerical issues related to the off-line selection of the algorithm’s parameters and its on-line implementation are discussed. Simulation results referred to a system with unbounded uncertainty are shown to compare the performances achievable with probabilistic and expectation constraints.

An approach to output-feedback MPC of stochastic linear discrete-time systems

In this paper we propose an output-feedback Model Predictive Control (MPC) algorithm for linear discrete-time systems affected by a possibly unbounded additive noise and subject to probabilistic constraints. In case the noise distribution is unknown, the probabilistic constraints on the input and state variables are reformulated by means of the Chebyshev–Cantelli inequality. The recursive feasibility is guaranteed, the convergence of the state to a suitable neighbor of the origin is proved under mild assumptions, and the implementation issues are thoroughly addressed. Two examples are discussed in detail, with the aim of providing an insight into the performance achievable by the proposed control scheme.

Francesco Paolo Cantelli

Summary Francesco Paolo Cantelli made fundamental contributions to the foundations of probability theory and to the clarification of different types of probabilistic convergence. He is remembered through the Borel–Cantelli Lemma, the Glivenko–Cantelli Theorem, and Cantelli's Inequality. He also made seminal contributions to actuarial science.

Alternative Measures of Agreement for Ranked Data

Combinatorial matching paradigms are used to derive kappa-type statistics for measuring agreement with ranked data. Significance tests for these statistics are presented, based on the Poisson distribution and Cantelli's inequality.