By Banesh Hoffmann

ISBN-10: 0486604896

ISBN-13: 9780486604893

No calculus wanted, yet this isn't an straight forward publication. Introduces vectors, algebraic notation and uncomplicated principles, vector algebra and scalars. Covers components of parallelograms, triple items, moments, angular speed, components and vectorial addition, extra concludes with dialogue of tensors. 386 workouts.

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**Sample text**

The main summary measures can be expressed directly in terms of matrix operations on X. For example, the arithmetic mean of the variables, described by a p-dimensional vector X, can be obtained directly from the data matrix as X= 1 1 X, n where 1 indicates a (row) vector of length n with all elements equal to 1. As previously mentioned, it is often better to standardise the variables in X. To achieve this aim, we first need to subtract the mean from each variable. The matrix containing the deviations from each variable’s mean is given by ˜ = X − 1 J X, X n where J is a n × n matrix with all the elements equal to 1.

Din ⎟ ⎟, ⎜ .. . .. ⎟ ⎝ . . ⎠ dn1 . . dni . . 0 where the generic element dij is a measure of distance between the row vectors xi and xj . The Euclidean distance is the most commonly used distance measure. It is defined, for any two units indexed by i and j , as the square root of the difference between the corresponding vectors, in the p-dimensional Euclidean space: 1/2 p 2 dij = d(xi , xj ) = xis − xj s 2 . s=1 The Euclidean distance can be strongly influenced by a single large difference in one dimension of the values, because the square will greatly magnify that difference.

N1+ nI + n j = 1, 2, . . , J. If this occurs it means that, with reference to the first equation, the (bivariate) joint analysis of the two variables X and Y does not given any additional knowledge about X than can be gained from the univariate analysis of the variable X; the same is true for the variable Y in the second equation. When this happens Y and X are said to be statistically independent. Note that the concept of statistical independence is symmetric: if X is independent of Y then Y is independent of X.