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presentation in this thesis eliminates the discussion of polynomial chaos be- yond this paragraph. Finally, Constantine [2009] explores the general setting of parameterized matrix equations with one or many parameters. In what follows, we explain and analyze the RAPr model. This model has strong connections with other path damping approaches to PageRank computation, which we show in sections 4.3.3 and 4.4.3. For the interested reader, we present our algorithms with actual Matlab code from our RAPr suite of codes. 4.1 notation For this chapter, we’ll need to introduce additional notation to handle the concepts from probability. See table 4.1 for a summary. Random variables are denoted by capital, non-subscribed, roman letters. Only A, the random α, is used frequently. Another small change from the standard notation is that lower case roman letters a, b, l , r denote scalar parameters or bounded intervals.1 These will be clear from context. For instance, they are used in the example below. There are two other novelties: the expectation operator E [⋅] and the stan- dard deviation operator Std [⋅]. Given a continuous random variable2 A with a density function ρ(x) on the interval [l , r], then we define 1 In the notation defined in sec- tion 2.1, scalars were Greek symbols. Switching to roman for a few scalars helps interpretability for this chap- ter. 2 Such random variables are quite special, but they suffice for this chapter. E[A] = Evaluating the expectation of a function corresponds to (4.1) tor r l r l √ E[f(A)] = The standard deviation operator is defined in terms of the expectation opera- ρ(x)dx. f(x)ρ(x)dx. (4.2) (4.3) 4.1 ⋅ notation 65 E [(A − E [A])2]. A good background on probability is Grinstead and Snell [1997]. Std [A] = Symbol A Beta(a,b,[l,r]) Beta(⋅, ⋅) δij E [⋅] Std [⋅] Meaning a random variable for α the Beta distribution, see section 4.4.1 the Beta function the Kronecker delta the expectation operator the standard deviation operator Table 4.1 – Additional notation for the random alpha PageRank model. Capital roman letters, such as A, denote random variables in this chapter.PDF Image | Instagram Cheat Sheet
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