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From these deﬁnitions it follows that {g(t) − Y¯ (t)}2 dt + λ g (t)2 dt = Y¯ (t)2 dt + γ Jγ + λγ Kγ − 2γ L¯ a. By standard linear algebra, this expression is minimized when γ is the vector of coeﬃcients γ (λ) given by (J + λK)γ (λ) = L¯ a. 12) (λ) γj βj (t). 3. Suppose that ξ(t) is a possible principal component weight function, and that the vector f gives the coeﬃcients of the basis expansion of ξ(t) in terms of the βj (t), so that m fj βj (t). ξ(t) = j=1 38 2. Life Course Data in Criminology The vector of principal component scores of the data is then ξ(t)Yi (t)dt = AL f.

37 of a basis δ1 (t), . . , δm (t). Thus there is an n × m matrix A = (aij ) of coeﬃcients such that m Yi (t) = aij δj (t). j=1 If we let a ¯j = n−1 i aij , then we have m Y¯ (t) = a ¯j δj (t). j=1 Because the basis functions δj (t) may not be suﬃciently smooth to allow the appropriate roughness penalty to be deﬁned, we may wish to use a diﬀerent basis βk (t) of size M when expanding the estimated mean curve. Given an M -vector γ of coeﬃcients, consider the function g with these basis function coeﬃcients in the new basis: m g(t) = γj βj (t).

2. 2. First steps in a functional approach 21 numbers of arrests in each year, but because of the skewness of the annual counts this would give inordinate weight to high values in the original data. In order to stabilize the variability somewhat, we start by taking the square root of the number of arrests each year. 3 that the skewness is somewhat reduced. In addition, if the numbers of arrests are Poisson counts, then the square root is the standard variance-stabilizing transformation. 2. 4. We now throw away the original points and regard this function as a whole as being the datum for this individual.