Selected Mathematical Theories Underpinning Decision Models

Selected Mathematical Theories Underpinning Decision Models

DOI: 10.4018/978-1-4666-9873-4.ch005
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Abstract

In Chapter 4, various decision support systems have been examined. The rational for Chapter 4 was to appraise the diiferent decision-support systems that have been used in construction without necessarily detailing the complexities and mathematical underpinnings. This chapter will provide the theory that underpins some selected decision support systems. These are regression models (RLM), artificial neural networks (ANN), Matrices, Markov decision processes (MDP) and the ontology rule-based decision support systems.
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Matrices Theory

Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations.

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns that is treated in certain prescribed ways. The individual items in a matrix are called its elements (or entries. Each element of a matrix is often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A in the Figure 1 below.

Figure 1.

General matrix form

A is an m-by-n matrix. The (i,j) entry of the above matrix A is denoted aij, ai,j, Aij or Ai,j.

The matrix A above is an m × n matrix, and can be defined as A = [Aij] (i = 1, 2, m; j = 1, ..., n), or A = [Ai,j]m×n.

The identity matrix In of size n is the n-by-n matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0.

A row vector is a 1-by-n matrix. It is a matrix with one row and n culums which is sometimes used to represent a vector.

A Column vector is an m-by-1 matrix. It is a matrix with m row and 1 culum which is sometimes used to represent a vector.

The sum A+B of two m-by-n matrices A and B is calculated entrywise:

(A + B)i,j = Ai,j + Bi,j, where 1 ≤ i ≤ m and 1 ≤ j ≤ n.

The transpose of an m-by-n matrix A is the n-by-m matrix AT (also denoted Atr or tA) formed by turning rows into columns and vice versa: (AT)i,j = Aj,i.

Multiplication of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B in Figure 2.

Figure 2.

Product of 2 matrices

(1) where 1 ≤ i ≤ m and 1 ≤ j ≤ pfor example, could indicates an l-by-c matrix, and its elements.

The transpose of the matrix of will be noted as.

The product of two matrices is well-defined by their units in the real numbers field.

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