Almost Sure Invariance Principles for Partial Sums of Weakly by Walter Philipp

By Walter Philipp

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X6 of the bank notes. Genuine notes are circles, counterfeit notes are stars. 13. It becomes apparent from the location of the point clouds that a better separation is obtained. We have rotated the three dimensional data until this satisfactory 3D view was obtained. Later, we will see that rotation is the same as bundling a high-dimensional observation into one or more linear combinations of the elements of the observation vector. 13 a plane and no longer parallel to one of the axes. The formula for such a separation plane is a linear combination of the elements of the observation vector: a1 x1 + a2 x2 + .

One sees an overlap of the coordinate values for indices 1–3 and an increased separability for the indices 4–6. 44 1 Comparison of Batches Summary → Parallel coordinates plots overcome the visualization problem of the Cartesian coordinate system for dimensions greater than 4. → Outliers are visible as outlying polygon curves. → The order of variables is still important, for example, for detection of subgroups. → Subgroups may be screened by selective coloring in an interactive manner. 8 Boston Housing Aim of the analysis The Boston Housing data set was analyzed by Harrison and Rubinfeld (1978) who wanted to find out whether “clean air” had an influence on house prices.

The bandwidth h determines the degree of smoothness of the estimate f . → Kernel densities are smooth functions and they can graphically represent distributions (up to 3 dimensions). 06σn−1/5 . This bandwidth is to be used only in combination with a Gaussian kernel ϕ. → Kernel density estimates are a good descriptive tool for seeing modes, location, skewness, tails, asymmetry, etc. 4 Scatterplots Scatterplots are bivariate or trivariate plots of variables against each other. They help us understand relationships among the variables of a data set.

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