Introducing linear functions: an alternative statistical approach

The introduction of linear functions is the turning point where many students decide if mathematics is useful or not. This means the role of parameters and variables in linear functions could be considered to be ‘threshold concepts’. There is recognition that linear functions can be taught in context through the exploration of linear modelling examples, but this has its limitations. Currently, statistical data is easily attainable, and graphics or computer algebra system (CAS) calculators are common in many classrooms. The use of this technology provides ease of access to different representations of linear functions as well as the ability to fit a least-squares line for real-life data. This means these calculators could support a possible alternative approach to the introduction of linear functions. This study compares the results of an end-oftopic test for two classes of Australian middle secondary students at a regional school to determine if such an alternative approach is feasible. In this study, test questions were grouped by concept and subjected to concept by concept analysis of the means of test results of the two classes. This analysis revealed that the students following the alternative approach demonstrated greater competence with non-standard questions.

Design of linear functional observers for time-delay systems of the neutral-type

This paper presents the design of reduced-order linear functional observers for a class of linear time-delay systems of the neutral-type. The type of the observer proposed in this paper is without internal delay and its order is the same as the number of linear functions to be estimated. First, conditions for the existence of the reduced-order functional observers that are capable of asymptotically estimating any given function of the state vector are derived. Then, based on the newly derived existence conditions, a procedure is given for the determination of the observer parameters. The results derived in this paper include a range of linear systems and extend some existing results of linear functional observers to linear neutral delay systems. A numerical example is given to illustrate the design procedure.

A model for compressing probabilities in belief networks

Probabilistic reasoning with belief (Bayesian) networks is based on conditional probability matrices. Thus it suffers from NP-hard implementations. In particular, the amount of probabilistic information necessary for the computations is often overwhelming. So, compressing the conditional probability table is one of the most important issues faced by the probabilistic reasoning community. Santos suggested an approach (called linear potential functions) for compressing the information from a combinatorial amount to roughly linear in the number of random variable assignments. However, much of the information in Bayesian networks, in which there are no linear potential functions, would be fitted by polynomial approximating functions rather than by reluctantly linear functions. For this reason, we construct a polynomial method to compress the conditional probability table in this paper. We evaluated the proposed technique, and our experimental results demonstrate that the approach is efficient and promising.

Monotone approximation of aggregation operators using least squares splines

The need for monotone approximation of scattered data often arises in many problems of regression, when the monotonicity is semantically important. One such domain is fuzzy set theory, where membership functions and aggregation operators are order preserving. Least squares polynomial splines provide great flexbility when modeling non-linear functions, but may fail to be monotone. Linear restrictions on spline coefficients provide necessary and sufficient conditions for spline monotonicity. The basis for splines is selected in such a way that these restrictions take an especially simple form. The resulting non-negative least squares problem can be solved by a variety of standard proven techniques. Additional interpolation requirements can also be imposed in the same framework. The method is applied to fuzzy systems, where membership functions and aggregation operators are constructed from empirical data.

Knowledge extraction from a mixed transfer function artificial neural network

One of the main problems with Artificial Neural Networks (ANNs) is that their results are not intuitively clear. For example, commonly used hidden neurons with sigmoid activation function can approximate any continuous function, including linear functions, but the coefficients (weights) of this approximation are rather meaningless. To address this problem, current paper presents a novel kind of a neural network that uses transfer functions of various complexities in contrast to mono-transfer functions used in sigmoid and hyperbolic tangent networks. The presence of transfer functions of various complexities in a Mixed Transfer Functions Artificial Neural Network (MTFANN) allow easy conversion of the full model into user-friendly equation format (similar to that of linear regression) without any pruning or simplification of the model. At the same time, MTFANN maintains similar generalization ability to mono-transfer function networks in a global optimization context. The performance and knowledge extraction of MTFANN were evaluated on a realistic simulation of the Puma 560 robot arm and compared to sigmoid, hyperbolic tangent, linear and sinusoidal networks.

Knowledge extraction from a mixed transfer function artificial neural network

One of the big problems with Artificial Neural Networks (ANN) is that their results are not intuitively clear. For example, if we use the traditional neurons, with a sigmoid activation function, we can approximate any function, including linear functions, but the coefficients (weights) in this approximation will be rather meaningless. To resolve this problem, this paper presents a novel kind of ANN with different transfer functions mixed together. The aim of such a network is to i) obtain a better generalization than current networks ii) to obtain knowledge from the networks without a sophisticated knowledge extraction algorithm iii) to increase the understanding and acceptance of ANNs. Transfer Complexity Ratio is defined to make a sense of the weights associated with the network. The paper begins with a review of the knowledge extraction from ANNs and then presents a Mixed Transfer Function Artificial Neural Network (MTFANN). A MTFANN contains different transfer functions mixed together rather than mono-transfer functions. This mixed presence has helped to obtain high level knowledge and similar generalization comparatively to monotransfer function nets in a global optimization context.

Numerical implementation of a functional observability algorithm : a singular value decomposition approach

The paper outlines a numerical algorithm to implement the concept of Functional Observability introduced in [6] based on a Singular Value Decomposition approach. The key feature of this algorithm is in outputting a minimum number of additional linear functions of the state vector when the system is Functional Observable, these additional functions are required to design the smallest possible order functional observer as stated in [6].

Functional observers for dynamical systems

In control theory, a state observer is an auxiliary dynamical system that mirrors the behaviour of a physical system, and it is driven by input and output measurements of the physical system in order to provide an estimate of internal states of the physical system. The primary consideration in the design of an observer is that the estimate of the states should be close to the actual value of the system states. On the other hand, the functional observation problem centers on the construction of an auxiliary dynamical system, known as the functional observer or functional reconstructor, driven by the available system inputs and outputs in order to estimate a linear function or functions of the system states. Obviously, a functional observer is a general form of the state observer because when the linear functions are chosen as the individual states of the system then the problem of functional observation reduces to the problem of state observation.

A system decomposition approach to the design of functional observers

This paper reports a system decomposition that allows the construction of a minimum-order functional observer using a state observer design approach. The system decomposition translates the functional observer design problem to that of a state observer for a smaller decomposed subsystem. Functional observability indices are introduced, and a closed-form expression for the minimum order required for a functional observer is derived in terms of those functional observability indices. © 2014 Taylor & Francis.

Construction of aggregation functions from data using linear programming

This article examines the construction of aggregation functions from data by minimizing the least absolute deviation criterion. We formulate various instances of such problems as linear programming problems. We consider the cases in which the data are provided as intervals, and the outputs ordering needs to be preserved, and show that linear programming formulation is valid for such cases. This feature is very valuable in practice, since the standard simplex method can be used.

Linear optimization for ecological indices based on aggregation functions

We consider an optimization problem in ecology where our objective is to maximize biodiversity with respect to different land-use allocations. As it turns out, the main problem can be framed as learning the weights of a weighted arithmetic mean where the objective is the geometric mean of its outputs. We propose methods for approximating solutions to this and similar problems, which are non-linear by nature, using linear and bilevel techniques.

Smoothing Lipschitz functions

This paper describes a new approach to multivariate scattered data smoothing. It is assumed that the data are generated by a Lipschitz continuous function f, and include random noise to be filtered out. The proposed approach uses known, or estimated value of the Lipschitz constant of f, and forces the data to be consistent with the Lipschitz properties of f. Depending on the assumptions about the distribution of the random noise, smoothing is reduced to a standard quadratic or a linear programming problem. We discuss an efficient algorithm which eliminates the redundant inequality constraints. Numerical experiments illustrate applicability and efficiency of the method. This approach provides an efficient new tool of multivariate scattered data approximation.

A family of simple and robust finite elements for linear and geometrically nonlinear analysis of laminated composite plates

A family of simple, displacement-based and shear-flexible triangular and quadrilateral flat plate/shell elements for linear and geometrically nonlinear analysis of thin to moderately thick laminate composite plates are introduced and summarized in this paper.

The developed elements are based on the first-order shear deformation theory (FSDT) and von-Karman’s large deflection theory, and total Lagrangian approach is employed to formulate the element for geometrically nonlinear analysis. The deflection and rotation functions of the element boundary are obtained from Timoshenko’s laminated composite beam functions, thus convergence can be ensured theoretically for very thin laminates and shear-locking problem is avoided naturally.

The flat triangular plate/shell element is of 3-node, 18-degree-of-freedom, and the plane displacement interpolation functions of the Allman’s triangular membrane element with drilling degrees of freedom are taken as the in-plane displacements of the element. The flat quadrilateral plate/shell element is of 4-node, 24-degree-of-freedom, and the linear displacement interpolation functions of a quadrilateral plane element with drilling degrees of freedom are taken as the in-plane displacements.

The developed elements are simple in formulation, free from shear-locking, and include conventional engineering degrees of freedom. Numerical examples demonstrate that the elements are convergent, not sensitive to mesh distortion, accurate and efficient for linear and geometric nonlinear analysis of thin to moderately thick laminates.