Krus,
D.J. (2002) Imaging higher transcendental functions in 3-dimensions. Journal of Visual Statistics, 1, 6-9.
IMAGING Higher Transcendental Functions in 3-DIMENSIONS
Novel conceptualization of matrix addition can be used for computation of all possible outcomes of discrete events. Transformation of the matrix of summands to a frequency matrix allows for three-dimensional visualization of binomial distributions and for approximations of the normal distribution by factorials of its underlying binary components.
The necessity to conceptualize algorithms for computation of higher transcendental functions (of the length n) within the matrix algebra framework became obvious during my work with the OpenGL rendering routines for 3-D representations of these functions, as the 3-D renderings of these functions generally require n2 data points. As many problems within this area are based on combinations and permutations, a matrix algebra operation of addition of matrices appeared as useful, as this operation can be used to obtain a matrix of all possible outcomes of finite discrete events under scrutiny.
Let us begin our discussion by considering the matrix operation of addition of a variable x and its transpose,
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The matrix A is formed by substituting sums for the products of its vector multiplication xx’ analog and should not to be confused with addition of matrix elements (cf., Krus and Wilkinson (1986), who described a similar operation for subtraction of matrices). The convention adapted here is to use operands in parentheses for operations on matrix elements, reserving the symbols such as + or - for the operations on matrices. Thus the operand of the above equation symbolizes matrix algebra operation of addition of vectors, not elements of vectors, and in the further discussion will represent addition of matrices, not elements of matrices.
As this is a novel definition of vector addition, let us introduce an example where x [0 1] represents faces of a coin and matrix A all possible outcomes of throwing two coins, thus, for the example,
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Next, decompose matrix A into vector and define matrix of all possible outcomes of a set of k discrete events pX, adjacent to vector , as
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By adding the pX and its transpose,
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for the example
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and by computing frequencies of the matrix P, we can obtain values identical to those of the Pascal’s triangle. For the example, frequencies of the above matrix [1 4 6 4 1] equal the 4th row of the Pascal’s triangle [4! / 0!4! 4! / 1!3! 4! / 2!2! 4! / 3!1! 4! / 4!0!]
The binomial distribution defined by the vector u can be plotted in two dimensions, as shown in Fig. 1,
Fig. 1. Theoretical outcomes of throws of four coins, plotted in two dimensions.
or in its three dimensional representation, by denoting frequencies of the matrix P
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as
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The 3-D representations of frequencies of the matrix P, for repeated throws of four coins, are shown in Fig.2.
Fig. 2. Theoretical outcomes of throws of four coins, plotted in three dimensions.
You may repeat the algorithm described above as many times as deemed necessary. By repeating the algorithm two more times, we can get a good approximation of the normal distribution, shown in Fig.3.
Fig 3. Three-dimensional plot of normal distribution, approximated by binomials
Discussion
Matrix algebra assisted decomposition of binomial functions into their binary components provides for better understanding of translations of standard renderings of the probability distributions into their polynomial equivalents, necessary for plotting these functions either by modern spreadsheets (such as Microsoft Excel), or by a new-generation statistical programs (such as Cruise Scientific Visual Statistics Studio used for renderings of the 3-D plots shown here). Unlike the typical two-dimensional routines, typically requiring an abscissa-ordinate series, the 3-dimensional objects require for it’s rendering matrices. The discussed matrix operation of addition can be useful for visualization of structures, reflecting all possible outcomes of discrete finite phenomena.
Reference
Krus, D.J., & Wilkinson, S.M. (1986) Matrix differencing as a concise expression of test variance. Educational and Psychological Measurement, 46, 179-183.