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Summary

under local gauge (or phase) transformations (15.2) Indeed, by insisting on local gauge invariance, we were inevitably led to the Lagrangian of QED, (14.28):

NOTES 6.1a. The terms "parametric" and "nonparametric" are not precise. When Legendre originally proposed the least-squares method, the justification he gave was simply that it minimized the sum of squares of the deviations, with no reference to the distribution of the data. From this viewpoint, least-squares estimation is nonparametric. But least-squares estimates cannot be guaranteed to perform well unless the distribution of the error population is normal. 6.1h. We are using the term "nonparametric regression" to refer to regression procedures based on ranks. The term is also used to refer to

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In this chapter we discussed a simple data structure for maintaining disjoint sets When the union operation is performed, it does not matter, as far as correctness is concerned, which set retains its name A valuable lesson that should be learned here is that considering the alternatives when a particular step is not totally specified can be very important The union step is flexible By taking advantage of this flexibility, we can get a much more efficient algorithm Path compression is one of the earliest forms of self-adjustment, which we have used elsewhere (splay trees and skew heaps) Its use here is extremely interesting from a theoretical point of view because it was one of the first examples of a simple algorithm with a not-so-simple worst-case analysis

Ackermann's function A function that grows very quickly Its inverse is essentially at most 4 (p 865) disjoint set class operations The two basic operations needed for disjoint set manipulation: They are union and f i n d (p 846) disjoint sets Sets having the property that each element appears in only one set (p 846) equivalence class The equivalence class of an element x in set S is the subset of S that contains all the elements related to x (p 846)

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procedures for constructing smooth estimates of nonlinear regression functions, such as in Hardie (1990). 6.1c. The plant heights that Galton analyzed were collected by Charles Darwin in an experiment to compare self-fertilization and cross-fertilization in plants. To see which mode of fertilization tends to produce taller plants, he measured the heights of 15 self-fertilized plants and 15 cross-fertilized plants of the same variety grown under the same conditions. Galton first arranged the heights of the selfed plants in increasing order and did the same for the crossed plants. He then compared the ith tallest selfed plant with the ith tallest crossed plant. In 13 out of the 15 comparisons, the crossed plant was taller. 6.3a. A weighted average of a list of numbers x I' ... , X n is a linear combination WIX I + ... +wnx n in which the coefficients Wi are nonnegative and add to 1. The coefficient Wi' called the weight of Xi' is the proportion of influence that Xi has on the value of the weighted average. The ordinary average i is the special case in which the weights are all equal, that is,

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