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1.1 root 1: \ifnum\pageno=1 \input macros.tex \immediate\openout\inx=chap5.index\makecontents\pageno=105 \fi
2: \proofmodefalse
3: \draft{11/24/89 by Trevor}
4: \def \sH {\script H}
5: \def\SQ{\tilde{\bS}}
6: \def\scM {\script M}
7: \chapter 5 {Some theory for additive models}
8: \Section{Introduction}
9: In the previous chapter we introduce the nonparametric additive model
10: and the backfitting procedure, a heuristic method for estimation.
11: In this chapter we provide some theoretical underpinning for these ideas.
12: The technical content and level of this chapter is somewhat
13: higher than the others, so the reader
14: interested only in applications
15: may well decide not to tackle it.
16:
17: The chapter has two main parts.
18: In the first half the backfitting algorithm is justified as a method
19: for estimating the additive model.
20: Several justifications are provided.
21: We first introduce an $L_2$ version of backfitting, that is,
22: a backfitting algorithm for square integrable random variables, and
23: show how the intuitive procedure introduced in the last chapter
24: can be viewed as a data analogue of this.
25: A second, different, justification for backfitting,
26: comes from a penalized least-squares framework.
27: Unlike the $L_2$ argument, it makes no appeal to random
28: variables:
29: instead, it applies to finite-sample backfitting procedures that use linear smoothers
30: (recall the discussion of linear smoothers in sections~2.8 and 3.4.2).
31:
32: An important by-product of these justifications
33: is the set of {\em estimating equations} that are solved by backfitting.
34: This linear system can be solved noniteratively and in some
35: special cases such a direct solution is more appropriate
36: than backfitting.
37:
38: We also describe very briefly a more technical abstract derivation using the theory of reproducing-kernel Hilbert-spaces.
39:
40: In the second half of the chapter
41: we study the existence and uniqueness of the solutions to the additive-model
42: estimating equations and
43: the convergence of the backfitting algorithm.
44: The results pertain to linear smoothers only.
45: One of the interesting notions that arises is {\em concurvity}, the
46: analogue of collinearity.
47: We also discuss some theory for standard-error bands and degrees of freedom of the estimated
48: smooths,
49: and the relationship of backfitting to the Gram-Schmidt and Gauss-Seidel
50: techniques.
51:
52: Related theory for generalized additive models
53: is not covered here but
54: is discussed
55: in the
56: next chapter.
57: We don't devote much space to asymptotic issues such as consistency or rates of convergence for
58: additive models; these are briefly mentioned in the
59: bibliographic notes.
60:
61: \Sectionskip\Section{Estimating equations for additive models}
62: \Mark{ESTIMATING EQUATIONS}
63: The additive model
64: $$\ev(Y\given \rvX)=\sum_{j=1}^p f_j(X_j)\eqn{\addm}$$
65: can be estimated by the backfitting algorithm, which we
66: give again below:
67:
68: \setbox1=\vbox{\hsize \algwidth {\setnine\parindent 20pt
69: \item{(i)} {\em Initialize:}$ \quad f_j=f_j^{\,(0)},
70: j=1,\ldots, p$
71: \item{(ii)} {\em Cycle:} $\quad j=1,\ldots, p,1,\ldots
72: p,\ldots$
73: $$f_j=\smooth_j(\vec y-\sum_{k\neq j}\vec f_k\given \vec x_j) $$
74: \item{(iii)} Continue (ii) until the individual functions don't
75: change.
76: }%end algorithm
77: \smallskip
78: } %end box1
79: \midinsert \algorithm{{\ninerm\noindent Algorithm \chapnodot 1}
80: The backfitting algorithm}{\box1} \endinsert
81:
82: In the above, the $\vec f_j$ are the $n$-vectors
83: $\{f_j(x_{1j}),\ldots, f_j(x_{nj})\}^T$, with $x_{ij}$ in the order of $y_i$.
84: We have omitted the constant term $\alpha$ in \addm; we see later that
85: this does not change the resulting estimates.
86:
87: In order to justify this procedure, we need some way of introducing
88: the smoothness that is provided by the scatterplot smoothers
89: in the algorithm.
90: To put it another way,
91: if we naively
92: tried to minimize
93: $$\sum_{i=1}^n \Bigl\{y_i-\tsum_{j=1}^p f_j(x_{ij})\Bigr\}^2\eqn{\naive}$$
94: then the solution would be any set of functions $(\,f_j:j=1,\ldots,p\,)$ that interpolated the data (assuming for the moment that the $X$-values are distinct).
95: For example, $f_1(x_{i1})=y_i$ $\forall i$ and $f_j\equiv 0$
96: for $j >1$.
97: We discuss several ways of introducing smoothness.
98: One approach is explicitly to add a term to $\naive$ that penalizes for lack of
99: smoothness, the
100: {\em penalized least-squares} approach.
101: Another approach, described next, is to step back and
102: consider random variables instead of data.
103: A Hilbert-space version of the additive model and backfitting can be
104: formulated,
105: with conditional expectation operators playing the role of smoothers.
106: Besides its use here,
107: this formulation is
108: mathematically
109: interesting
110: in its own right.
111: The data version of backfitting is then derived as an empirical version
112: of this Hilbert-space procedure, the smoothness
113: entering when one considers how best to estimate the conditional
114: expectations.
115: A third approach, based on reproducing-kernel Hilbert-spaces, is a more abstract version of the penalized least-squares approach. Later on in the chapter
116: we give yet another derivation, based on a Bayesian stochastic model.
117:
118: \sectionskip\section{$L_2$ function spaces}
119: Let $\sH_j$ for $j=1,\ldots,p$ denote the Hilbert spaces of measurable
120: functions $\phi_j (X_j)$ with $\ev\phi_j(X_j)=0$, $\ev\phi^2_j(X_j) <
121: \infty$, and inner product
122: $\inner{\phi_j(X_j),\phi_j'(X_j)}=\ev \phi_j(X_j)\phi_j'(X_j) $. In
123: addition, denote by $\sH$ the space of arbitrary centered, square
124: integrable functions of $X_1,\ldots,X_p$. We consider the $\sH_j$
125: as subspaces of $\sH$ in a canonical way. Furthermore, denote by
126: $\sH^{add} \subset \sH$ the linear subspace of additive
127: functions: $\sH^{add}=\sH_1+\cdots +\sH_p$, which is closed under some technical assumptions.
128: These are all subspaces of $\sH_{YX}$,
129: the space of centered square integrable functions of $Y$ and $X_1,\ldots,X_p$.
130:
131: The optimization problem in this population setting is
132: to minimize
133: $$\ev\{Y-g(\rvX)\}^2\eqn{\backc}$$
134: over $g(\rvX)=\sum_j f_j(X_j)\in\sH^{add}$.
135: Of course, without the additivity
136: restriction, the solution is simply $\ev(Y\given \rvX)$; we seek the
137: closest additive approximation to this function. Since by assumption $\sH^{add}$ is a
138: closed subspace of $\sH$ this minimum exists and is unique; the
139: individual functions $f_j(X_j)$, however, may not be uniquely
140: determined. Denote by $P_j$ the conditional expectation operator $\ev(\cdot\given
141: X_j)$; as such $P_j$ is an orthogonal projection onto $H_j$.
142:
143: The minimizer $g(\rvX)$ of \backc\ can be characterized by residuals $Y-g(\rvX)$ which are orthogonal to the space of fits:
144: $Y-g(\rvX) \perp \sH^{add}$.
145: Since $\sH^{add}$ is generated by $\sH_j\ (\subset \sH^{add})$,
146: we have equivalently:
147: $Y-g(\rvX)\perp \sH_j,\quad\forall j$
148: or:\
149: $\;P_j\{Y-g(\rvX)\}=0\quad\forall j$.
150: Component-wise this can be written as
151: $$\eqalign{f_j(X_j) &=P_j\Bigl\{Y-\sum_{k\neq j} f_k(X_k)\Bigr\}\cr &=
152: \ev\Bigl\{Y-\sum_{k\neq j}f_k(X_k)\given X_j\Bigr\}.\cr}\eqn{\backe}$$
153: Equivalently, the following system of {\em estimating equations}
154: is necessary and sufficient for $\vec f=
155: (f_1,\ldots,f_p)$ to minimize \backc:
156: $$\pmatrix{I&P_1&P_1&\cdots&P_1\cr
157: P_2&I&P_2&\cdots&P_2\cr\vdots&\vdots&\vdots&\ddots&\vdots\cr
158: P_p&P_p&P_p&\cdots&I\cr}\pmatrix{f_1(X_1)\cr f_2(X_2)\cr\vdots\cr
159: f_p(X_p)}= \pmatrix{P_1 Y\cr P_2Y\cr \vdots\cr P_pY}\eqn{\backd}$$
160: or
161: $$\bP\vec f=\vec Q Y,$$
162: where $\vec P $ and $ \vec Q$ represent a matrix and vector of operators, respectively,
163: and operator matrix multiplication is defined in the obvious way.
164:
165: The reader familiar with the Gauss-Seidel method for solving
166: linear systems of equations will recognize backfitting as a formal
167: Gauss-Seidel algorithm for solving the system $\backd$.
168: (We say {\em formal} because the elements in the left matrix
169: of $\backd$ are not real numbers or real-valued matrices but
170: conditional expectation operators.)
171: Suppose we wish to solve a linear system of equations $\bA\vec z=\vec b$, with
172: $\bA=\{a_{ij}\}$ an $m\times m$ matrix, $\vec b$ an $m$ vector and $\vec z$
173: the vector of $m$ unknown coefficients.
174: The Gauss-Seidel iterative method
175: solves for each $z_i$ in turn from the
176: relation in the $i$th row
177: $\sum_{j=1}^m a_{ij} z_j=b_i$.
178: This process is repeated for $i=1,\ldots, m,1,\ldots,m,\ldots$,
179: using the latest values of each $z_j$ at each step, until convergence.
180:
181: The connection between backfitting and the Gauss-Seidel method becomes more precise
182: when we consider the corresponding data version of $\backd$ using linear smoothers.
183: Recall from section~2.8 that a linear smoother can be written as
184: a {\em smoother matrix} times the response vector $\vec y$,
185: that is $\hatvec f=\bS\vec y$.
186: Examples of linear smoothers include the running-mean, locally-weighted running-line,
187: smoothing splines and kernel smoothers.
188: Consider then a backfitting algorithm that estimates the conditional
189: expectation operator $P_j$ by a linear scatterplot smoother
190: with smoother matrix $\bS_j$.
191: Then the data version of the estimating equations $\backd$ is
192: the $np\times np$ system
193: $$\pmatrix{\bI&\bS_1&\bS_1&\cdots&\bS_1\cr
194: \bS_2&\bI&\bS_2&\cdots&\bS_2\cr\vdots&\vdots&\vdots&\ddots&\vdots\cr
195: \bS_p&\bS_p&\bS_p&\cdots&\bI\cr}\pmatrix{\vec f_1\cr \vec f_2\cr\vdots\cr \vec
196: f_p}= \pmatrix{\bS_1 \vec y\cr \bS_2\vec y\cr \vdots\cr \bS_p\vec
197: y}.\eqn{\backdd}$$
198:
199: In short form we write
200: $$\hat {\vec P}\vec f=\hat {\bQ}\vec y.$$
201: Backfitting is a Gauss-Seidel procedure for solving the above
202: system.
203: The only nonstandard aspect is that we solve for $n$ elements at each step instead of one (block Gauss-Seidel), although for linear smoothers this distinction can be dropped.
204:
205: Now
206: suppose we start with $\backdd$.
207: Why use an iterative procedure like backfitting to find its solution?
208: Why not use a standard, noniterative method like a QR decomposition?
209: The difficulty is that in general, $\backdd$ is an $np\times np$ system
210: and since methods like QR require $O(m^3)$ operations to solve an $m\times m$
211: system, our problem would cost $O\{(np)^3\}$ operations.
212: On the other hand, backfitting exploits the special structure in $\backdd$,
213: and if the smoothers can be applied in $O(n)$ computations (as is the
214: case for running-lines and smoothing splines), then backfitting
215: requires only $O(np)$ computations.
216: (This assumes that a fixed number of iterations is sufficient for convergence).
217: If, however, the effective dimension of the system $\backdd$ is really less than $np$,
218: there may be better methods than backfitting for solving the problem.
219: In particular, if each $\bS_j$ is an orthogonal projection and the
220: union of the projection spaces has rank $m$, then $\backdd$ is equivalent
221: to an $m\times m$ least-squares problem (Exercise~5.3) and least-squares methods
222: are likely to be preferable if $m << n$.
223: This is the case if the $\bS_j$s produce linear or polynomial fits,
224: or if we use regression splines with a small number of knots.
225:
226: Later in this chapter the properties of the estimating equations \backdd\ are
227: studied.
228: We find that there is an intimate connection between the
229: existence of solutions of this system and the convergence of the
230: backfitting procedure for finding these solutions.
231:
232: \sectionskip\section{Penalized least-squares}
233: In this section we provide a different justification for backfitting
234: from that given in
235: the Hilbert-space framework in the previous section.
236: In section~2.10 we derive the cubic smoothing spline
237: as the minimizer over all twice continuously differentiable functions of the
238: penalized least-squares criterion
239: $$\sum_{i=1}^n\{y_i-f(x_i)\}^2+\lambda\int \{f''(x)\}^2 \, dx.\eqn{\qb}$$
240: We establish this by using the fact that the solution to $\qb$ is a cubic
241: spline,
242: and hence we simplified $\qb$ by writing it as a function of $f_i=f(x_i)$, the $n$ evaluations of the minimizing function~$f$.
243: This gave the equivalent form
244: $$(\vec y-\vec f)^T(\vec y-\vec f) +\lambda\vec f^T\bK\vec f\eqn{\nqd}$$
245: where $\bK$ is a certain quadratic penalty matrix.
246: The quantity
247: $\nqd$ is easily shown to have a minimum given by
248: $$\hat\vec f=(\bI+\lambda \bK)^{-1}\vec y.
249: \eqn{\css}$$
250: We also argued in the opposite direction for
251: other symmetric linear smoothers.
252: That is, given a symmetric linear smoother based on the smoother matrix
253: $\bS$, the smooth $\hat\vec f=\bS\vec y$ minimizes
254: $$(\vec y-\vec f)^T(\vec y-\vec f) +\vec f^T(\bS^- -\bI)\vec f\eqn{\newm}$$
255: over all $\vec f\in\script{R}(\bS)$ (the range of $\bS$), where $\bS^-$ is any generalized inverse of $\bS$.
256:
257: In order to extend this idea to the estimation of the additive model,
258: we generalize the criterion $\qb$ in an obvious way.
259: We seek to minimize
260: $$\sum_{i=1}^n\Bigl\{y_i-\tsum_{j=1}^pf_j(x_{ij})\Bigr\}^2 +\sum_{j=1}^p
261: \lambda_j\int \{f_j''(t)\}^2 \, dt\eqn{\splinpen}$$
262: over all twice
263: continuously differentiable functions $f_j$.
264: Before deriving the solution to $\splinpen$, let's take note of some
265: of its features.
266: Notice that each function in
267: $\splinpen$ is penalized by a separate constant $\lambda_j$.
268: This in turn determines the smoothness of that function in the solution.
269: Note also that if the $\lambda_j$s are all zero (no smoothness penalty)
270: the solution to \splinpen\ is any interpolating set of functions whose evaluations satisfy $\sum_{j=1}^p f_j(x_{ij})=y_i$ for $i=1,\ldots,n$.
271: On the other hand, if each $\lambda_j$ goes to infinity, the
272: penalty term goes to infinity unless $f_j''(t)=0$ for all $j$, that is,
273: unless
274: each $f_j$ is linear.
275: Hence the problem reduces to standard linear least-squares.
276:
277: Using a straightforward
278: extension of the arguments used in the single-predictor
279: case,
280: the solution to \splinpen\ is shown to be a cubic spline in each of the predictors.
281: As before we parametrize by evaluations at the $n$ observations.
282: Thus we may rewrite $\splinpen$ as
283: $$\Bigl(\vec y-\sum_{j=1}^p \vec f_j\Bigr)^T\Bigl(\vec y-\sum_{j=1}^p\vec f_j\Bigr)+\sum_{j=1}^p
284: \lambda_j \vec f_j^T \bK_j \vec f_j \eqn{\nsplinpen}$$
285: where the $\bK_j$s are penalty matrices for each predictor, defined
286: analogously to the $\bK$ for a single predictor given in section~2.10.
287: Now if we differentiate $\nsplinpen$ with respect to the function $\vec f_k$
288: we obtain $-2(\vec y-\sum_j \vec f_j)+2\lambda_k \bK_k\vec f_k=\bf 0$
289: or
290: $$\hat\vec f_k=\Bigl(\bI+\lambda_k \bK_k\Bigr)^{-1}\Bigl(\vec y-\tsum_{j\neq k} \hat\vec f_j\Bigr).\eqn{\solsplin}$$
291: As noted earlier, $(\bI+\lambda_k \bK_k)^{-1}$ is the smoother matrix for
292: a cubic smoothing spline, and hence $\solsplin$, for $k=1,\ldots, p$,
293: are just
294: the estimating equations $\backdd$.
295:
296: Arguing in the opposite direction, the minimizers of the
297: penalized least-squares criterion
298: $$\Bigl(\vec y-\sum_{j=1}^p \vec f_j\Bigr)^T\Bigl(\vec y-\sum_{j=1}^p\vec f_j\Bigr)+\sum_{j=1}^p
299: \vec f_j^T(\bS_j^{-}-\bI)\vec f_j,\eqn{\gsplinpen}$$
300: over all $\vec f_j\in\script{R}(\bS_j)$,
301: are the solutions to the estimating equations $\backdd$ (Exercise~5.1).
302:
303:
304:
305: As we do in the single predictor case (Exercise~3.6), we can interpret
306: each of the penalty terms in $\nsplinpen$ as a down-weighting of each
307: of the components of $\vec f_j$, the down-weighting determined by
308: the corresponding eigenvalue of that component
309: and
310: $\lambda_j$.
311:
312: \def\sH{{\script H}}
313: \sectionskip
314: \section{Reproducing-kernel Hilbert-spaces}
315: This section describes a more abstract framework for defining and estimating general nonparametric regression models which includes additive models as a special case.
316: We present these results to give the reader a taste of this rich
317: area;
318: the level of mathematics is somewhat higher than the rest of the chapter.
319: The description is close to that of ^{Chen, Gu and Wahba (1989)}.
320:
321: A Hilbert space $\sH$ of real-valued functions of $t\in \Omega$ is a {\em reproducing-kernel} Hilbert-space if evaluation is a continuous linear functional.
322: By the Riesz representation theorem, there exist {\em representers of evaluation}
323: $e_t\in\sH$ such that $f(t)=\langle f,e_t\rangle_\sH$ for $f\in \sH$, where $\langle \cdot,\cdot\rangle_{\script H}$ denotes the inner-product on $\sH$.
324: The consequences of these properties will become clearer as we proceed.
325:
326: The reproducing kernel itself,
327: $Q(\cdot,\cdot):\Omega\times\Omega\mapsto\R 1$, is defined by
328: $Q(s,t)=\langle e_s,e_t\rangle_\sH$, and consequently
329: $e_s=Q(s,\cdot)$, considered as a function of the second argument, with
330: the first held fixed at $s\in \Omega$.
331: We will see that the kernel $Q$, evaluated at the realizations of $t$, provides a finite dimensional basis for representing the solution to a class of optimization problems.
332:
333: Now suppose $\Omega$ is a space of vector predictors $\fat{X}=(X_1,\ldots,X_p)$ and that $\sH$ has the decomposition
334: $$
335: \sH=\sH_0+\sum_{k=1}^q\sH_k,
336: $$
337: where $\sH_0$ is spanned by $\phi_1,\ldots,\phi_M$, and $\sH_k$ has the reproducing kernel $Q_k(\cdot,\cdot)$.
338: The space $\sH_0$ is the projection component of $\sH$, that is, the
339: space of functions that are not to be penalized in the optimization.
340: In the previous section $\sH_0$ is the space of functions linear in $t$.
341:
342: We are now set up to pose the optimization problem.
343: For a given set of predictors $\vec x^1,\ldots,\vec x^n$ (with each $\vec x^i\in\Omega$), find $f=\sum_{k=0}^q f_k$ with $f_k\in \sH_k$, \ $k=0,\ldots, q$, to minimize
344: $$
345: \sum_{i=1}^n\Bigl\{y_i-\tsum_{k=0}^qf_k(\vec x^i)\Bigr\}^2+ \sum_{k=1}^q\lambda_k\norm{f_k}^2_{\sH_k}.\eqn{\rkcrit}
346: $$
347:
348: The first part of the criterion is discrete in nature, and is the reason why
349: reproducing-kernel spaces are natural for these kinds of problems.
350: We do not want small changes in the $\vec x^i$ to result in vastly different solutions; this is why it is desirable for evaluation to be continuous.
351:
352:
353: The theory of reproducing kernels guarantees that a minimizer exists, and has the form
354: $$\eqalign{
355: \hat{f}_0(\fat{X})&=\sum_{j=1}^M\beta_{j0}\phi_j(\fat{X})\cr
356: \hat{f}_k(\fat{X})&=\sum_{i=1}^n\beta_{ik}Q_k(\fat{X},\vec x^i).\cr
357: }
358: \eqn{\rksoln}
359: $$
360: Furthermore, if the projection onto $\sH_0$ is unique, then so is the solution
361: to the larger problem.
362: So even though the problem is posed in an infinite-dimensional space, the minimizing $\hat f$ is finite-dimensional, and the $Q_k$ supply bases for representing the solution.
363: The parameters are found by minimizing the finite dimensional quadratic criterion
364: $$
365: \norm{\vec y-\vec T\fat{\beta_0}-\sum_{k=1}^q\vec Q_k\fat{\beta}_k}^2 +
366: \sum_{k=1}^q\lambda_k\fat{\beta}_k^T\vec Q_k\fat{\beta}_k\eqn{\rkdata}
367: $$
368: where $\vec T$ is the $n\times M$ matrix of evaluations of $\phi_j$, with
369: $ij$th entry $T_{ij}=\phi_j(\vec x^i)$, and $\vec Q_k$ is the $n\times n$
370: { evaluated} kernel with $ij$th entry $Q_k(\vec x^i,\vec x^j)$.
371: This problem is of dimension at most $qn+M$.
372:
373: At this point a number of specializations are possible:
374: \smallskip
375: {\parindent 20pt
376: \item{(i)} If $q=p$ and each of the $\sH_k$ are the canonical subspaces of $\sH$,
377: then the additive model consists of a sum of univariate functions. Furthermore, by choosing an inner product appropriate for cubic smoothing splines, the problem reduces exactly to \splinpen, although the solution is typically represented by a different basis.
378: \item{(ii)} The current specification has $nq+M$ parameters. If attention is restricted to $f_0$ and $f_+=\sum_{k=1}^qf_k$, with $Q_+=\sum_{k=1}^q Q_k/\lambda_k$, then the dimension of the solution is reduced to $M+n$.
379: \item{(iii)} The general problem as specified by \rksoln\ can potentially be solved more cheaply by backfitting; the system of estimating equations that characterize the minimum of \rksoln\ can be written in a form similar to \backdd:
380: $$\pmatrix{\bI&\bS_0&\bS_0&\cdots&\bS_0\cr
381: \bS_1&\bI&\bS_1&\cdots&\bS_1\cr\vdots&\vdots&\vdots&\ddots&\vdots\cr
382: \bS_q&\bS_q&\bS_q&\cdots&\bI\cr}\pmatrix{\vec f_0\cr \vec f_1\cr\vdots\cr \vec
383: f_q}= \pmatrix{\bS_0 \vec y\cr \bS_1\vec y\cr \vdots\cr \bS_q\vec
384: y}\eqn{\backrk}$$
385: where $\bS_0=\bT(\bT^T\bT)^{-1}\bT$ and $\bS_k=\bQ_k(\bQ_k+\lambda_i\bI)^{-1}$.
386: If the computational complexity of the individual operators $\bS_k$ is significantly lower than that of the full problem, savings can be made using backfitting-type algorithms.
387:
388: }\smallskip
389: This is a very brief summary of some powerful machinery; we cite a number of relevant references in the bibliographic section for more details of this approach and for pointers to the large application area.
390:
391: \Sectionskip\Section{Solutions to the estimating equations}
392: %\Mark{THE BACKFITTING ALGORITHM}
393: \section{Introduction}
394: The remainder of the chapter
395: focuses attention on additive models with linear smoothers $\smooth_1,\ldots,\smooth_p$, and the algorithms for estimating them.
396:
397: Most of the smoothers that we have discussed produce function estimates, and so we could discuss issues such as convergence in terms of these functions as well.
398: Instead we restrict attention to the evaluation of these functions at the $n$ realizations of the predictors.
399: We do this mainly for simplicity and clarity, but point out that most of the results cited here, in particular those pertaining to convergence, can be modified to include this more general case.
400: As a consequence, we usually refer to a smoother by its matrix representation $\bS$ rather than in the operator form $\smooth$.
401:
402: The centerpiece of the discussion is the set of estimating equations \backdd.
403: Before one delves into methods for solving such a system, questions
404: of consistency and degeneracy have to be answered.
405: In other words, we
406: must confirm that the system has at least one solution, and find out
407: whether this solution is unique.
408: We first look at a few special cases which are are easy to
409: work out and illustrate the
410: main issues.
411: Later in the chapter we answer these questions in some generality.
412:
413:
414: In our discussion we sometimes assume implicitly that the same
415: smoother is used for each of the variables, but this is only for ease
416: of presentation. The results are general in nature and apply to any
417: backfitting procedure in which some linear smoother is used for each
418: of the variables.
419: In fact, there is no need even to assume that each smoother is based
420: on a single predictor:
421: for example a two-dimensional smoother or a least-squares fit on some
422: set of predictors could be included.
423: Even more generally
424: one can think of $\bS_1,\ldots,\bS_p$ as a
425: set of linear transformations, without reference to predictor variables at all.
426:
427: Throughout the chapter we assume
428: that the $\bS_j$s all reproduce constant functions.
429: Note that this
430: causes a simple kind of non\-unique\-ness in the backfitting algorithm.
431: Suppose the starting functions are all zero.
432: Then at every stage of the procedure $\hat \vec f_1$ has
433: the same mean as $\vec y$, but the other $\hat\vec f_j$s have mean $\bf 0$.
434: If the procedure started at $j=2$, however, the mean of $\vec y$ would go into
435: $\hat\vec f_2$ instead.
436: A closer look reveals that
437: nonzero starting functions cause a dependence
438: of the final iterates on the values of the starting functions.
439: It is also clear that unless special constraints are built in, the constant in the additive model is not identifiable.
440: This is a special instance of what we call {\em concurvity}, the analogue of {\em collinearity} in linear models.
441:
442: It turns out that such degeneracies do not affect convergence in any important way.
443: In this case a simple fix is possible:
444: assume that $\vec y$ has been centered to have mean $\bf 0$,
445: and replace $\bS_j$ by
446: the matrix that smooths then subtracts off the average of the smooth.
447: It is easy to see that the resultant smoother matrix is
448: $(\bI-{\bf 1}{\bf 1}^T/n)\bS_j$,
449: what we call a {\em centered} smoother.
450: This ensures that at every stage of the procedure the $\hatvec f_j$s have
451: mean $\bf 0$.
452:
453:
454: \sectionskip\section{Projection smoothers}
455: The additive model is introduced as a generalization of the linear regression model.
456: What if we use a linear least-squares fit,
457: or any other orthogonal projection,
458: for each predictor?
459: As mentioned in section~5.2.1, the set of estimating equations \backdd\ is
460: equivalent to the usual normal equations for linear regression
461: (Exercise~5.3).
462: Thus if
463: $\bS_j$ is an orthogonal projection
464: and $\script L_{col}(\bS_j)$ denotes the subspace spanned by the columns
465: of $\bS_j$,
466: we expect
467: the backfitting solution $\hatvec y=\sum_1^p \hatvec f_j$ to converge to
468: the projection of $\vec y$ onto $V=\script L_{col}(\bS_1)\oplus
469: \script L_{col}(\bS_2)\oplus \ldots\oplus \script L_{col}(\bS_p)$.
470: Indeed if this wasn't the case we might well question the entire
471: backfitting paradigm.
472: Fortunately, it is fairly easy to show that backfitting does the
473: expected in this special case.
474: We sketch the proof here, leaving the details to the
475: reader (Exercise~5.2).
476: The idea is to show that the residual vector from backfitting
477: converges to the least-squares residual vector,
478: that is, the projection of $\vec y$ onto the orthogonal
479: complement of $V$.
480: After one cycle of backfitting, the residual vector
481: from backfitting, say
482: $\vec r$, is $\bC\vec y$ where
483: $$\bC=(\bI-\bS_p)(\bI-\bS_{p-1})\cdots (\bI-\bS_1).\eqn{\resv}$$
484: Thus after $m$ cycles the residual is $\vec r^{\,(m)}=\bC^m\vec y$.
485: We can split $\vec y$ into its components in the
486: projection space $V$ and its orthogonal complement,
487: that is $\vec y=\hat\vec y+\vec y^{\perp}$.
488: Now the operator $\bC$ {\em takes residuals} along each predictor in turn, and
489: hence leaves $\vec y^{\perp}$ unchanged.
490: Thus
491: $$\vec r^{\,(m)}=\bC^m\hat\vec y+\vec y^{\perp}.\eqn{\splitup}$$
492: The proof is then completed by showing that $\norm{\bC^m\hat\vec y}
493: \rightarrow 0$ and hence $\vec r^{\,(m)}\rightarrow \vec y^{\perp}$ (Exercise~5.2).
494:
495: Hence we see that the backfitting procedure provides an alternative
496: method for computing least-squares fits.
497: Examination of $\splitup$ reveals that it works by successively projecting
498: the current residual into the space orthogonal to each $\script L_{col}(
499: \bS_j)$.
500: \input zigfig.tex
501: \par
502: Figure~\zigfig\ gives a picture of this
503: in the two-predictor case.
504: Each ${\vec x}_j^\perp$ denotes the vector orthogonal to $\vec x_j$ in the span of $\script L_{col}(\vec x_1,\vec x_2)$.
505: It shows the backfitting residual $\vec r^{\,(m)}$ converging to the least-squares residual $\vec y^{\perp}$ in a zig-zag fashion.
506: At convergence the backfitting residual vector is orthogonal to
507: each $\script L_{col}(\bS_j)$ and equals the least-squares residual
508: $\vec y^{\perp}$.
509: This is a novel but not very practical way of finding the least-squares fit, for it can be very slow if the predictors are correlated.
510: In particular, one can show that
511: for two predictors
512: the difference between the $i$th iterate and the solution converges to zero
513: geometrically at rate
514: $\cos(\theta)$, where $\theta$ is the angle between
515: the two predictor vectors
516: (Exercise~5.4).
517: This is intuitively plausible from Fig.~\zigfig.
518: There is a close connection between backfitting and the iterative proportional scaling algorithm for fitting log-linear models to contingency tables (^{Bishop \etal,~1975}).
519: They both cycle through a system of estimating equations and update one component at a time; convergence is geometric in both cases.
520:
521: It is interesting to look at two extreme situations that can occur.
522: First, suppose that the predictors are perfectly
523: collinear
524: ($\theta=0$ in the two predictor case).
525: Then one can easily check that backfitting converges after a single
526: cycle, with $\hat\vec f_1=\hat\vec y$ and $\hat\vec f_j =\bf 0$ for $j>1$.
527: More interestingly, suppose that the predictors are mutually uncorrelated
528: ($\theta=90^\circ$ in the two-predictor case).
529: Then again we have convergence after a single cycle.
530: This brings up the connection of backfitting with the Gram-Schmidt method
531: for solving the least-squares normal equations.
532: Like backfitting, the Gram-Schmidt method works by regressing the
533: current residual onto each predictor in turn.
534: However there is one important difference.
535: After regressing on the $j$th predictor, the $(j+1)$th through $p$th
536: predictors are also regressed
537: on the $j$th predictor and the residual vector from each regression
538: is used in place of the predictor in the remaining steps.
539: This process orthogonalizes the predictors and because of this,
540: the Gram-Schmidt procedure converges after a single
541: cycle.
542: In the orthogonal-projection setting, clearly the Gram-Schmidt method is
543: superior to backfitting and this suggests that a sweeping-out operation be used to improve the backfitting
544: algorithm with general smoothers.
545: This is not useful, however, because the
546: resultant model would no longer be additive in the predictors.
547:
548: We note also that the above proof of the convergence of the residual
549: vector does not establish that the estimated functions converge to
550: the correct ones.
551: Indeed, if there exists strict collinearity among the predictors
552: it wouldn't be clear to which solution the functions $\hat \vec{f}_j$ produced by backfitting
553: would converge, if indeed they converge at all.
554: It turns out that backfitting always does converge to a solution representing
555: the projection of $\vec y$ onto $V$, but
556: the fixed point depends on the starting functions.
557:
558: \sectionskip\section{Semi-parametric models}
559: Consider an additive model in which all but one term is assumed
560: to be linear --- the so called {\em semi-parametric} model.
561: The backfitting algorithm for estimating such
562: a model can be thought
563: of as having two smoothers: one a projection $\bS_1=\bX(\bX^T\bX)^{-1}\bX^T$ producing a least-squares fit
564: $\bX\hatfat \beta$ on one or more covariates (represented by the full-rank design
565: matrix $\bX$), and the other a smoother $\bS_2$ producing an estimate $\hat\vec
566: f_2$. The backfitting steps are
567: $\vec f_1=\bS_1(\vec y-\vec f_2)=\bX(\bX^T\bX)^{-1}\bX^T(\vec y-\vec f_2)\equiv
568: \bX {\fat\beta}$, and $\vec f_2=\bS_2(\vec y-\bX{\fat\beta})$.
569: It turns out that we can solve for $\hatfat\beta$ and
570: $\hat\vec f_2$ explicitly (Exercise~2.8):
571: $$\eqalign{{\hatfat\beta}
572: &=\{\bX^T(\bI-\bS_2)\bX\}^{-1}\bX^T(\bI-\bS_2)\vec y\cr
573: { \hat\vec f}_2 &= \bS_2(\vec
574: y-\bX\hatfat\beta)\cr}\eqn{\simmm}$$
575: so that iteration is unnecessary.
576: Although $\bS_2$ is an $n\times n$ matrix, all we have to do is smooth each of
577: the $p$ columns of $\bX$, an operation that can usually be performed in $O(np)$ operations.
578: This provides a computationally simple method for nonparametric
579: analysis of covariance; it is interesting that the smoother matrix
580: $\bS_2$ enters as the weight matrix for the regression on $\bX$.
581: This manipulation also shows that in this special case, the estimating equations
582: $\backdd$ are consistent and have a unique solution as long a
583: $\bX^T(\bI-\bS_2)\bX$ is invertible.
584: In section~6.7
585: we discuss this model in more detail.
586:
587: \sectionskip\section{Backfitting with two smoothers}
588: The third special case that we consider is an additive model
589: that involves two linear smoothers.
590: It turns out that one can analyse the properties of the estimating
591: equation solutions and the convergence of backfitting with some
592: fairly elementary calculations, and this exercise sheds light on the main
593: issues that arise in the more difficult $p$-predictor case.
594:
595:
596:
597: It is possible to determine general conditions under which the system \backdd\ is consistent by
598: checking whether the rank of $\hatvec P$ is the same as that of the
599: augmented matrix $[\hatvec P\colon\hatvec Q\vec y]$.
600: However, it is easier to proceed by constructing the solutions to \backdd\
601: through the backfitting procedure.
602:
603: Recall that the components of each $\vec x_j$ are in the same order as
604: the components of $\vec y$.
605: As a technical point, this means that
606: $\bS_j$ really means
607: $\bE_j^{-1}\bS_j\bE_j$ where $\bE_j$ is the permutation matrix that sorts
608: in the order of $\vec x_j$ (Exercise~5.19).
609:
610: Let
611: $\norm{\bC}=\sup_{\vec a\neq \bf 0}\norm{\bC\vec a}/\norm{\vec a}$, the
612: 2-norm of the matrix $\bC$
613: (this choice is made for convenience; any matrix norm would do).
614: The system \backdd\ can be written $$\eqalign{\vec f_1&=\bS_1(\vec y-\vec
615: f_2)\cr \vec f_2&=\bS_2(\vec y-\vec f_1).\cr}\eqn{\back}$$ Let $\vec
616: f_1^{\,(m)}$ and $\vec f_2^{\,(m)}$ denote the estimates at the $m$th stage of the
617: backfitting algorithm, with $m=0$ denoting the starting functions.
618: Backfitting consists of alternating the steps
619: $$\eqalign{\vec
620: f_1^{\,(m)}&=\bS_1(\vec y-\vec f_2^{\,(m-1)})\cr \vec f_2^{\,(m)}&=\bS_2(\vec y-\vec
621: f_1^{\,(m)}).\cr}\eqn{\backit}$$ Using induction one shows that for $m\ge 1$
622: $$\eqalign{\vec f_1^{\,(m)}&=\vec y-\sum_{j=0}^{m-1}(\bS_1\bS_2)^j(\bI-\bS_1)\vec
623: y-(\bS_1\bS_2)^{m-1}\bS_1\vec f _2^{\, (0)},\cr \vec
624: f_2^{\,(m)}&=\bS_2\sum_{j=0}^{m-1}(\bS_1\bS_2)^j(\bI-\bS_1)\vec y
625: +\bS_2(\bS_1\bS_2)^{m-1}\bS_1 \vec f_2^{\, (0)}. \cr}\eqn{\qbac}$$
626:
627: Then a sufficient condition for $\vec
628: f_1^{\,(m)}$ and $\vec f_2^{\,(m)}$ to converge is $\norm{\bS_1\bS_2} < 1$. If this is
629: the case, we can solve $\qbac$ to obtain
630: $$\eqalign{ \vec
631: f_1^{\,(\infty)}&=\{\bI-(\bI-\bS_1\bS_2)^{-1}(\bI-\bS_1)\}\vec y\cr \vec
632: f_2^{\,(\infty)}&=\bS_2(\bI-\bS_1\bS_2)^{-1}(\bI-\bS_1) \vec y\cr
633: &=\{\bI-(\bI-\bS_2\bS_1)^{-1}(\bI-\bS_2)\}\vec y.\cr }\eqn{\qconv}$$
634: The fit
635: $\hat{\vec y}$ is given by
636: $$\eqalign{\hat{\vec y}&=\vec
637: f_1^{\,(\infty)}+\vec f_2^{\,(\infty)}\cr
638: &=\{\bI-(\bI-\bS_2)(\bI-\bS_1\bS_2)^{-1}(\bI-\bS_1)\}\vec y\cr}\eqn{\qconvsum}$$ which
639: is symmetric in $\bS_1$ and $\bS_2$, as some simple calculations show.
640:
641: This proves that if
642: $\norm{\bS_1\bS_2}<1$, the estimating equations are consistent, and have
643: a unique solution.
644: In addition, the final iterates from the
645: backfitting procedure are independent of the starting
646: values and starting order.
647:
648: Is $\norm{\bS_1\bS_2}<1$ typically?
649: If the smoothers are not centered, we
650: would have $\bS_1\bS_2\bf 1=\bf 1$ so that $\norm{\bS_1\bS_2}=1$,
651: but the centering makes $\bS_1\bS_2\bf 1=0$.
652: However, smoothers like the cubic spline
653: smoother have a second unit eigenvalue corresponding to the linear
654: functions. Consider a backfitting algorithm with two covariates
655: $\vec x_1$ and $\vec x_2$ using cubic smoothing splines.
656: If the data show strict collinearity through the origin (i.e. $\vec
657: x_2=c\vec x_1$), we still have $\norm{\bS_1\bS_2}=1$.
658: With higher order splines, for example, quintic smoothing splines which also have unit quadratic
659: eigenvectors, similar situations involving linear and quadratic functions are possible.
660: The condition $\norm{\bS_1\bS_2}=1$ is an example of {\em concurvity}, a phenomenon that we study later in
661: the general $p$-covariate case.
662:
663: As it turns out, if $\bS_1$ and $\bS_2$ are symmetric
664: with eigenvalues in $(-1,1]$, one can prove that
665: the estimating equations are consistent.
666: Furthermore, the backfitting algorithm converges despite the presence of concurvity,
667: and
668: the fitted values
669: $\vec
670: f_1^{\,(\infty)}+\vec f_2^{\,(\infty)}$
671: are independent of the starting functions.
672: Concurvity will, however, lead to a dependence of the limits $\vec
673: f_1^{\,(\infty)}$ and $\vec f_2^{\,(\infty)}$ on the starting guess $\vec
674: f_2^{\, (0)}$.
675: That is $\vec f_+^{\,(\infty)}=
676: \vec f_1^{\,(\infty)}+\vec f_2^{\,(\infty)}$ is unique but $\vec
677: f_1^{\,(\infty)}$ and $\vec f_2^{\,(\infty)}$ are not.
678: Cubic smoothing splines satisfy these conditions; however, the conditions are sufficient and not necessary.
679: Empirical evidence suggests that the results may also hold for smoothers such as locally-weighted lines, for which the smoother matrix is asymmetric and has modulus greater than one.
680:
681: We can usefully view these results as an extension of those for linear regression with a singular regression-matrix $\bX$: the fit $\hatfat{\mu}=\bX\hatfat\beta$ is unique, but $\hatfat\beta$ is not.
682:
683: It is not surprising that the condition $\norm{\bS_1\bS_2}=1$ leads to
684: a dependence of the backfitting solutions on the starting functions,
685: for it is immediate in this case that the backfitting solutions themselves
686: are not unique.
687: Since each $\bS_j$ is assumed to have eigenvalues in $(-1,1]$,
688: $\norm{\bS_1\bS_2}$ can only equal 1
689: if there is some vector $\vec a$ that is
690: reproduced by both $\bS_1$ and $\bS_2$.
691: But this will lead to nonuniqueness of the solution to
692: backfitting, for if $\hat\vec f_1$ and $\hat\vec f_2$ are solutions, then
693: so are $\hat\vec f_1+\vec a$ and $\hat\vec f_2-\vec a$,
694: evident upon examination of $\back$.
695:
696: \sectionskip\section{Existence and uniqueness: $p$-smoothers}
697: The two-smoother problem has revealed important aspects of the estimating equations
698: and backfitting which can be used to motivate the more general results that
699: we discuss here.
700: In fact, the results in the general case are qualitatively the
701: same as in the simpler setting.
702:
703: In the two-smoother problem, we are forced to restrict attention to
704: symmetric smoother matrices with eigenvalues in $(-1,1]$.
705: The exclusion of $-1$ as an eigenvalue is necessary to avoid oscillatory behaviour in
706: the algorithm.
707: In the $p$-smoother case, it turns out that we need to assume that
708: each smoother is symmetric and has
709: eigenvalues in $[0,1]$.
710: We can give no intuitive reason for this stronger condition being
711: necessary here, but note that this is not a
712: practical limitation of the results, because any
713: reasonable symmetric smoother should satisfy this property.
714:
715: The first issue to be settled is the existence of at least one solution
716: to the estimating equations
717: $\backdd$. It turns out that if $\bS_1, \ldots,\bS_p$ are symmetric
718: with eigenvalues in $[0,1]$,
719: the estimating equations $\backdd$ have at least one solution for every $\vec y$.
720: Given that at least one solution exists, is it unique?
721: Nonuniqueness occurs in
722: the
723: two smoother case when $\norm{\bS_1 \bS_2}=1$.
724: This leads us to ask: what interrelation among the $\bS_j$s in the $p$-smoother case will lead to nonuniqueness of the solution
725: to backfitting?
726: The answer lies
727: in the estimating equations $\hat\vec P \vec f=\hat\vec Q\vec y$.
728: Suppose that there is a $\vec g$ such that $\hat\vec P\vec g=\bf 0
729: $.
730: Then the system $\hat\vec P\vec f=\hat\vec Q\vec y$ has an infinite number of solutions because if
731: $\{\,\vec f_j^{\,(\infinity)}:j=1,\ldots,p\,\}$ is a solution, then
732: so is
733: $\{\,\vec f_j^{\,(\infinity)} +c\vec g_j:j=1,\ldots,p\,\}$ for any $c$.
734:
735: We think of this phenomenon as the analogue of collinearity, and define
736: the {\em concurvity space} of the system \backdd\
737: to be the space of functions $\vec g$
738: satisfying
739: $\hat{\vec P}\vec g = \bf 0$. Concurvity in function space is similarly
740: defined, with regard to the system \backd.
741:
742: A quick word on notation. An unsubscripted vector $\vec g$ denotes the $np$ vector of evaluations $\vec g^T=(\vec g_1^T, \ldots,\vec g_p^T)$, while an additive fit is denoted by $\vec f_+=\sum_{j=1}^p\vec f_j$.
743:
744:
745: In the two variable case it is easy to show that
746: $\norm{\bS_1\bS_2}<1 $ and $\norm{\bS_2\bS_1}<1$ if and only the concurvity
747: space is empty.
748: Thus we see that concurvity plays an important role in the behaviour
749: of backfitting.
750: It is natural, then, to try to pin down exactly how concurvity can
751: occur.
752: We can do this, after a bit of preparation.
753: Let $\bS_j$, \ $j=1,\ldots,p$ be symmetric smoother
754: matrices with eigenvalues in $[0,1]$. Let $\scM_1(\bS_j)$
755: be the space spanned by the
756: eigenvectors of $\bS_j$ with eigenvalue +1 (that is, they pass through the smoother
757: unchanged), for $j=1,\ldots,p$.
758: Then $\hat \vec P
759: \vec g=\bf 0$ if and only if $\vec g_j\in \scM_1(\bS_j)\;\forall j$ and $\vec
760: g_+=\bf 0$.
761:
762: In other words, we have concurvity if and only if the spaces
763: $\scM_j(\bS_j)$ are linearly dependent; that is, there exist
764: $\vec g_j\in \scM_1(\bS_j)$ not all zero satisfying $\vec
765: g_+=\bf 0$. Given such a linear degeneracy, any solution $\vec
766: f_1,\ldots,\vec f_p$ of $\hat{\vec P}\vec f=\hat{\vec Q}\vec y$ leads
767: to nonuniqueness in the form of additional solutions $\vec f_1+c\vec
768: g_1,\ldots,\vec f_p+c\vec g_p$.
769:
770: The result above says that concurvity involves only functions in the eigenspaces
771: corresponding to eigenvalue +1.
772: In the case of cubic smoothing splines, those eigenspaces correspond to
773: linear functions of each predictor,
774: and thus exact concurvity only exists if the predictors are exactly collinear.
775: However, approximate concurvity is of practical concern, when the predictors are clustered around some lower dimensional manifold.
776: Note that if quintic splines or
777: quadratic regression are used, the eigenspaces $\scM_1(\bS_j)$ consist of
778: the quadratic functions in the $j$th variable; hence concurvity may
779: involve truly nonlinear degeneracies between the variables.
780:
781: \sectionskip\section{Convergence of backfitting: $p$-smoothers}
782: With this definition of concurvity in hand, we can state the main result for the
783: convergence of backfitting.
784: Consider a backfitting algorithm with
785: symmetric smoothers
786: $\bS_j$, \ $j=1,\ldots,p$,
787: having eigenvalues in $[0,1]$.
788: Then if the $\bS_j$ do not exhibit concurvity, it can be shown that backfitting converges to
789: the unique solution of $\backdd$, independent of the starting functions.
790: If there is concurvity, backfitting converges to one of the
791: solutions of $\backdd$, the starting functions determining the final
792: solutions.
793:
794: Note that these results apply to,
795: amongst others, smoothing splines,
796: regression splines,
797: and simple linear and polynomial
798: regression.
799: They also can be applied to a backfitting algorithm
800: that uses a mixture of these smoothers, for example a cubic smoothing spline
801: for one variable, a simple linear fit for another variable,
802: etc.
803: The smoothers need not even be univariate; the results apply to
804: two or higher-dimensional surface smoothers as well.
805:
806: \sectionskip\section{Summary of the main results of the section}
807: For two smoothers $\bS_1$ and $\bS_2$:
808: \smallskip
809: {\parindent 20pt
810: \item{(i)} if $\norm{\bS_1\bS_2}<1$, then
811: the estimating equations \back\ have a unique solution \qconv\ and
812: the backfitting algorithm converges to this unique solution;
813: \item{(ii)} if $\bS_1$ and $\bS_2$ are symmetric
814: with eigenvalues in $(-1,1]$,
815: then
816: the estimating equations \back\ have at least one solution,
817: and the backfitting algorithm converges to one of the solutions.
818: This solution is dependent on the starting function
819: ${\vec f}_2^{\, (0)}$.
820:
821: }\smallskip
822: In general for $p$ symmetric smoothers $\bS_1,\ldots,\bS_p$ with eigenvalues in $[0,1]$:
823: \smallskip
824: {\parindent 20pt
825: \item{(i)} The estimating equations $\backdd$ have at least one solution for every $\vec y$.
826: \item{(ii)} Let $\scM_1(\bS_j)$
827: be the space spanned by the
828: eigenvectors of $\bS_j$ with eigenvalue +1 (that is, they pass through the smoother
829: unchanged), for $j=1,\ldots, p$.
830: Then $\hat \vec P
831: \vec g=\bf 0$ if and only if $\vec g_j\in \scM_1(\bS_j)\;\forall j$ and $\vec
832: g_+=\bf 0$. Either of these conditions characterize the {\em concurvity} space of the
833: estimating equations.
834: \item{(iii)} If the concurvity space is empty, backfitting converges to
835: the unique solution of $\backdd$, independent of the starting functions.
836: \item{(iv)}
837: If the concurvity space is not empty, backfitting converges to one of the
838: solutions of $\backdd$, and the starting functions determine the final
839: solutions.
840:
841: }
842: \Sectionskip\Section{Special topics}
843: \section{Weighted additive models}
844: Consider a weighted penalized least-squares criterion of the form
845: $$\Bigl(\vec y-\sum_j\vec f_j\Bigr)^T\bW\bigl(\vec y-\sum_j\vec f_j\Bigr)+\sum_j
846: \lambda_j \vec f_j^T \bK_j \vec f_j \eqn{\wsplinpen}$$
847: where $\bW$ is a diagonal
848: matrix of weights,
849: and $\lambda_j$ is a smoothing parameter and $\bK_j$ is a smoothing-spline penalty matrix for the $j$th predictor.
850: These weights might represent the relative precision of each observation or
851: might arise as part of another iterative procedure,
852: for example the local-scoring procedure
853: described in Chapters~4 and 6.
854: The estimating equations for this problem have the same form as for the unweighted case, except that
855: the smoothers are
856: weighted smoothing splines given by
857: $\bS_j=(\bW+\lambda_j\bK_j)^{-1}\bW$.
858: We could generalize all the results presented so far to deal with the weighted case by simply computing norms and inner products in the metric of $\bW$.
859: However, it is simpler
860: to map the problem back to the unweighted case,
861: using the transformations
862: $\by'=\bW^{1/2}\by$,
863: $\vec f_j'=\bW^{1/2}\vec f_j$,
864: $\bK_j'=\bW^{-1/2}\bK_j\bW^{-1/2}$.
865: Note that $\bS_j$ is not symmetric, but
866: $\bW^{1/2}\bS\bW^{-1/2}$ is symmetric with eigenvalues in
867: $[0,1]$, and unit eigenvalues
868: corresponding to linear functions of the $j$th variable.
869: Thus the convergence results for the unweighted case can be directly applied.
870:
871:
872: \sectionskip\section{A modified backfitting algorithm}
873: In the previous chapter we mention the possibility of modifying the backfitting
874: algorithm to improve its efficiency.
875: The basic idea is as follows.
876: Many smoothers have a {\em projection} part and a {\em shrinking} part.
877: For example, a cubic smoothing spline has unit eigenvalues that are constant
878: and linear functions of the predictor (its projection part), and
879: eigenvalues less than one for other eigenvectors.
880: The idea is to combine all of the projection operations for
881: all of the predictors into one large projection, and use only
882: the nonprojection parts of each smoother in an iterative backfitting-type
883: operation.
884:
885: This modification has several advantages.
886: When a smoothing-spline or running-line smoother is used for
887: several predictors,
888: practical experience
889: has shown that
890: if the predictors are correlated,
891: many iterations may be required to get the correct
892: average slope of the functions.
893: By performing all of the projections in one operation,
894: all of the function slopes are simultaneously estimated.
895: A second advantage is in collinearity/concurvity situations.
896: In a backfitting problem with symmetric smoothers having
897: eigenvalues in $[0,1]$, we have seen in the previous section that the
898: only nonuniqueness ({\em concurvity}) occurs in the eigenspaces
899: with eigenvalue one.
900: By separating out the estimation of these components of the functions,
901: the nonunique part of the solutions is
902: conveniently allocated to the projection step.
903: This makes it easy to characterize the solutions and eliminates the
904: dependence of the final solutions on the starting functions.
905:
906: Let us now be more specific.
907: Let $\bG_j$ be the matrix that projects onto $\scM_1(\bS_j)$, the space of eigenvalue
908: one for the $j$th smoother.
909: Using $\bG_j$, we define the modified smoother matrices
910: $$\tilde \bS_j=(\bI-\bG_j)\bS_j.$$
911: Note that $\tilde \bS_j$ has the effect of subtracting out the component
912: of the smoothed value that lies in $\scM_1(\bS_j)$.
913: The general form of the modified backfitting algorithm is given below.
914: \setbox2=\vbox{\hsize \algwidth {\setnine\parindent 20pt
915:
916: \item{(i)} Initialize $\tilde{\vec f}_1, \ldots,\tilde{\vec f}_p$ and set $\tilde{\vec
917: f}_+=\tilde{\vec f}_1+\cdots +\tilde{\vec f}_p$.
918: \item{(ii)} Regress $\vec y-\tilde{\vec f}_+$ onto the space $\scM_1(\bS_1)+\cdots + \scM_1(\bS_p)$,
919: that is,
920: set $\vec g = \bG(\vec y-\tilde{\vec f}_+)$, where $\bG$ is the orthogonal projection
921: onto $\scM_1(\bS_1)+\cdots +\scM_1(\bS_p)$ in $\R n$.
922: \item{(iii)} Apply one cycle of backfitting to $\vec y-\vec g$
923: using smoothers $\tilde{\bS}_i$; this step yields an updated
924: additive fit $\tilde{\vec f}_+ = \tilde{\vec f} _1 + \cdots
925: +\tilde{\vec f} _p$.
926: \item {(iv)} Repeat steps (ii) and (iii) until
927: convergence. The final estimate for the overall fit is $\vec f_+ =\vec g + \tilde{\vec f}_+ $.
928:
929: }%end algorithm
930: \smallskip
931: } %end box 2
932:
933: \midinsert
934: \algorithm{{\ninerm\noindent Algorithm \chapnodot 2} The modified backfitting algorithm}{\box2}
935: \endinsert
936:
937: Note that it is not sufficient
938: to perform the projection step only once.
939: It must be iterated with the
940: other steps because
941: when $\tilde{\vec f}_+$ is changed in step (ii), the projection component
942: $\vec g$ no longer equals $\bG(\vec y-\tilde{\vec f}_+)$.
943: An alternative to step (iii) is to iterate it to convergence rather than cycling through once; we find that this tends to slow down convergence in terms of the number of smooths performed.
944:
945: In order justify this procedure, we must not
946: only show that it does converge, but that it converges to the same solution as the
947: original backfitting procedure.
948: It turns out (to follow) that in the case of symmetric
949: smoothers with eigenvalues in $[0,1]$, the modified backfitting
950: procedure does solve the original problem.
951: For other linear smoothers, the modified backfitting procedure might
952: still make sense, but it solves a slightly different problem.
953:
954: We now state some convergence results about modified backfitting algorithms:
955: \smallskip
956: {\parindent 20pt
957: \item{(i)}If $\bS_j$, \ $j=1,\ldots,p,$ are symmetric and have
958: eigenvalues in $[0,1]$,
959: then the modified
960: backfitting algorithm converges in the sense that $\vec g$ and $ \tilde{\vec f} _1,
961: \ldots, \tilde{\vec f} _p$ converge.
962: \item{(ii)}Suppose the modified
963: backfitting algorithm has converged with smoothers $\tilde{\bS}_j$,
964: yielding functions $\tilde{\vec f}_j$ and $\vec g_j\in \scM_1(\bS_j)$. Then the
965: components $\vec f_j=\vec g_j + \tilde{\vec f}_j$ are solutions to
966: the estimating equations with smoothers
967: $\bS^*_j=\bG_j+(\bI-\bG_j)\bS_j$.
968:
969: }\smallskip
970:
971: Notice that if $\bS_j$ is symmetric, we have
972: $\bS_j^*=\bS_j$ and thus the solutions to the modified algorithm solve the
973: estimating equations with smoothers $\bS_j$.
974:
975: If the $\bS_j$ are symmetric and have eigenvalues in $[0,1]$ then $\tilde{\bS}_j=\bS_j-\bG_j$,
976: and $\norm{\tilde{\bS}_j }<1$. Smoothing splines belong to this
977: class, and hence the algorithm always converges for them.
978: If cubic smoothing splines are used for all predictors, $\bG$ is the {\em hat} matrix
979: corresponding to the least-squares regression on $({\bf 1},{\vec x}_1,\ldots,{\vec x}_p)$.
980: The nonlinear functions $\tilde{\vec f_j}$ are uniquely determined.
981: Concurvity (collinearity) can show up only in the $\bG$ step, where it is dealt with in the standard linear least-squares fashion.
982: At convergence, one may then decompose $\vec g=\sum \vec g_j$ and reconstruct final components $\vec f_j=\vec
983: g_j+\tilde{\vec f_j}$.
984: If $\bS_j$ is a cubic smoothing spline and if $\by$
985: is centered initially, then $\vec g_j=\hat{\beta}_j\cdot\vec x_j$,
986: where $\hat\beta_1,\ldots,\hat\beta_p$ are the coefficients from the
987: multiple linear regression of $\vec y-\tilde{\vec f}_+$ on $\vec
988: x_1,\ldots,\vec x_p$.
989:
990: \sectionskip\section{Explicit solutions to the estimating equations}
991: By manipulating the fixed points of the modified backfitting procedures,
992: an expression for the solutions to the estimating equations \backdd\ can be derived
993: (Exercise~5.6).
994: Let $\tilde \bA_j=(\bI-\tilde \bS_j)^{-1}\tilde \bS_j$, $\tilde \bA=\sum_1^p \tilde \bA_j$,
995: and $\bB=(\bI+\tilde \bA)^{-1}\tilde \bA$.
996: Then the solutions are $\tilde{\vec f}_+=(\bI-\bB\bG)^{-1}\bB(\bI-\bG)\vec y$ and
997: $\vec g=\bG(\vec y-\tilde{\vec f}_+)$.
998: These can be combined to obtain
999: $$\eqalign{
1000: \vec f_+=&\{\bG+(\bI-\bG)(\bI-\bB\bG)^{-1}\bB(\bI-\bG)\}\vec y\cr
1001: \tilde{\vec f}_j=&(\bI-\tilde \bS_j)^{-1}(\vec y-\vec g-\tilde{\vec f}_+)\cr
1002: \vec f_j=&\vec g_j+\tilde{\vec f}_j\cr}\eqn{\mbsol}$$
1003: and the individual $\vec g_j$s are any
1004: vectors $\vec g_j\in \scM_1(\bS_j)$ such that $\sum_1^p \vec g_j=\vec g$.
1005: Interestingly, $\mbsol$ reveals that for symmetric
1006: smoother matrices with eigenvalues in $[0,1]$, a direct solution can be obtained in $O(n^3 p)$ operations,
1007: the number required for computing
1008: $(\bI-\tilde \bS_j)^{-1}$ for $j=1,\ldots, p$.
1009: This is less than the $O\{(np)^3\}$ operations that are needed to solve the
1010: estimating equations $\backdd$ in general.
1011:
1012: \sectionskip\section{Standard errors}
1013: From the previous sections we note that each estimated function
1014: in the additive fit is the result of a
1015: linear mapping or smoother applied to $\vec y$. This
1016: means
1017: that the variance formula developed in Chapter~3
1018: can be applied to the additive model. At convergence, we can
1019: express $\hatvec f_j$ as
1020: $\bR_j\vec y$ for some $n\times n$ matrix $\bR_j$.
1021: If the observations have independent and identically distributed errors,
1022: then $\cov(\hatvec{f}_j)=\bR_j\bR_j^T\sigma^2$ where
1023: $\sigma^2=\var(Y_i)$. As in the least-squares case, if $\hat{\vec P}$ in equation \backdd\
1024: has singular values close to $\bf 0$, this will be reflected in $\cov(\hatvec
1025: f\,)$ as large variances and covariances.
1026:
1027: Direct computation of $\bR_j$ is
1028: formidable, except in very special cases such as the semi-parametric model.
1029: Our best general approach to date is to apply the backfitting procedure to the each of the $n$ unit
1030: $n$-vectors that are the columns of $\bI_n$, the $n\times n$ identity matrix.
1031: The result of backfitting applied to the $i$th unit vector produces fitted vectors $\hatvec{f}_j^{\,i}$, $j=1,\ldots,p$, where $\hatvec{f}_j^{\,i}$ is the $i$th column
1032: of $\bR_j$.
1033: Similarly, $\hatvec{f}_+^{\,i}$ is the $i$th column of $\bR$.
1034: The standard-error bands in Fig.~\fone\ of Chapter~4 are constructed
1035: using $\pm$ twice the square root of the diagonal elements of $\hat{\sigma}^2\bR_j\bR_j^T$.
1036: Since the backfitting algorithm is $O(kn)$ for $O(n)$ smoothers, this procedure is $O(kn^2)$.
1037: Now $k=pmC$, where $p$ is the number of predictors, $m$ is the number of backfitting iterations, and $C$ is the constant for the particular smoother.
1038: For smoothing splines, typical numbers might be $k=5\times 5\times 35= 875$, which is likely to be larger than $n$, so this task can be tedious in practice.
1039: We have nevertheless used it in many examples, although usually not often
1040: within any single analysis.
1041:
1042: The global confidence set techniques described in Chapter~3 can be extended to apply to additive models;
1043: the procedure is similar to the univariate case.
1044: Suppose the additive model is correct, i.e., $Y_i=f_+(\vec X_i)+\varepsilon_i$,
1045: and our estimate of $\bR\vec f_+=\vec g_+ $ is $\hatvec{ f}_+=\bR\vec y$.
1046: Then an approximate pivotal for $\vec g_+$ is
1047: $$\nu(\vec g_+)=(\hatvec f_+-\bg_+)^T(\bR\bR^T\hat\sigma^2)^{-1}
1048: (\hatvec f_+-\bg_+).\eqn{\nuu}
1049: $$
1050: Assuming that we have an estimate of the dispersion parameter $\hat\sigma^2$ and the distribution $G$ of $\nu$, then we can construct a simultaneous $1-\alpha$ confidence set of all the component functions:
1051: $$C(\bg_1,\ldots,\bg_p)=\{\bg_1,\ldots,\bg_p;\nu(\bg_+) \leq G_{1-\alpha}\}.\eqn{\bogplus}$$
1052: We will not pursue this topic further here; it is an area of current research and we need to gain more experience with it.
1053:
1054: \sectionskip\section{Degrees of freedom}
1055: Each of the definitions for degrees of freedom given in Chapter~3 has a natural analogue here.
1056: The overall degrees of freedom $\df$ is simply $\tr(\bR)$,
1057: where $\bR$ is the (smoother) matrix that produces $\hatvec f_+=\bR \vec y$.
1058: In addition, the posterior covariance of $\vec f_+$,
1059: in the Bayesian treatment of the additive model given in section~5.4.6,
1060: is proportional to $\bR$, for appropriate choice of
1061: the prior covariances.
1062:
1063: Similarly, the degrees of freedom for error is
1064: $\dferr=n-\tr(2\bR-\bR\bR^T)$.
1065: More usefully,
1066: for model comparison, we need a notion of the change in the error degrees of freedom $\Delta\dferr$ due to an individual term.
1067: Let $\bR_{(j)}$ denote that operator that produces the additive fit with
1068: the $j$th term removed.
1069: Then we define $\dferr_j$, the degrees of freedom for error due to the $j$th term:
1070: $$\dferr_j=\trace(2\bR-\bR\bR^T)-\trace(2\bR_{(j)}-\bR_{(j)}\bR_{(j)}^T).$$
1071: This is the expected increase in the residual sum of squares (up to a scale factor) if the $j$th predictor is excluded from the model, assuming its exclusion does not increase the bias.
1072: Approximate $F$ and $\chi^2$ tests that make use of $\dferr_j$ are discussed
1073: in section~6.8.
1074:
1075: The sum of the variances of the fitted values is a meaningful concept
1076: for an additive fit and thus
1077: $\dfvar=\trace(\bR\bR^T)$.
1078: Further, the sum of the variances of the fitted component function $\hat\vec f_j$ is
1079: $\sigma^2\trace(\bR_j\bR_j^T)$;
1080: the effect of predictor correlation on this quantity is
1081: explored in Exercise~5.17.
1082:
1083:
1084:
1085:
1086:
1087: None of these definitions are attractive from a computational point of view.
1088: In particular, it would be convenient to use $\tr(\bS_j)-1$ or
1089: even $\tr(2\bS_j-\bS_j\bS_j^T)-1$ to select the amount of smoothing
1090: prior to including the $j$th predictor in a model,
1091: and as an approximation to $\dferr_j$ for model comparison.
1092: We subtract one since there is a redundant constant in $p-1$ of the $p$ terms
1093: in the model; in general we subtract the dimension of $\bigcap_j\scM_1(\bS_j)$.
1094: In the extreme case of exact concurvity, it is possible to show that $\tr(2\bS_j-\bS_j\bS_j^T)-1$
1095: is an upper bound for $\dferr_j$ (Exercise~5.7);
1096: for a balanced additive model (the other extreme; Exercise~5.18) it is equal to $\dferr_j$.
1097: ^{Buja, Hastie and Tibshirani (1989)} carried out some small simulation experiments
1098: and found that adding up the
1099: individual degrees of freedom gave a good approximation to the
1100: true degrees of freedom.
1101: The only exceptions occurred when the predictors had extremely high
1102: correlation or when a very small smoothing parameter was used.
1103:
1104: In the examples in this book, we use the convenient
1105: approximation $\tr(\bS_j)-1$ to select
1106: the amount of smoothing, while we use the exact quantities $\dferr_j$
1107: for model comparisons via approximate $F$ or $\chi^2$ tests.
1108: Further details are given in section~6.8.
1109:
1110: \sectionskip\section{A Bayesian version of additive models}
1111: Just as in the case of a single smoother, there is a rather simple Bayesian
1112: approach to additive models.
1113: As in section~3.6, there is a functional (stochastic process) version,
1114: and a finite dimensional (sampled) version; we focus on the latter.
1115:
1116: The model is
1117: $$
1118: \vec y = \vec f_1+\cdots +\vec f_p+\fat{\varepsilon}\eqn{\bayesadd}$$
1119: with $\vec f_j \sim N({\bf 0},\sigma^2 \bQ_j)$ independently for all $j$ and independent of ${\fat\varepsilon}\sim N({\bf 0},\sigma^2 \bI)$.
1120: Note from section~3.6 that each $\bQ_j$ corresponds to the inverse of some penalty matrix $\bK_j$; also, from section~5.2.3, $\bQ_j$ can be identified with the realization of a reproducing kernel. For the moment we assume that the priors are proper and nonsingular.
1121:
1122: Straightforward derivations (Exercise~5.9) show the following:
1123: \smallskip
1124: {\parindent 20pt
1125: \item{(i)} The prior for $\vec f_+=\vec f_1+\cdots +\vec f_p$
1126: is $N({\bf 0},\bQ_+)$ with $\bQ_+=\sum_j\bQ_j$.
1127: \item{(ii)} The posterior mean for $\vec f_+$ is $\ev(\vec f_+\given \vec y)= \bQ_+(\bI+\bQ_+)^{-1}\vec y$.
1128: Similarly the posterior means for the individual functions are $\ev(\vec f_j\given \vec y)= \bQ_j(\bI+\bQ_+)^{-1}\vec y$.
1129: \item{(iii)} The posterior covariance of $\vec f_+$ is $\sigma^2\bQ_+(\bI+\bQ_+)^{-1}$, and for $\vec f_j$ they are $\sigma^2(\bQ_j-\bQ_j(\bI+\bQ_+)^{-1}\bQ_j)$.
1130:
1131: }\smallskip
1132:
1133: Notice that the solution involves the inversion of an $n\times n$ {\em unstructured} matrix, which takes $O(n^3)$ operations, unless of course backfitting is used.
1134:
1135: Similar equations can be derived for partially improper priors.
1136: These give infinite variance to certain components in $\R n$, which correspond
1137: to components in $\scM_1(\bS_j)$.
1138: The simplest way to formulate the problem is along the lines taken in section~5.2.3, where the projection space is explicitly isolated.
1139: In addition, smoothing parameters are thought of as prior variances
1140: for the $\vec f_j$s; we have absorbed these into the $\bQ_j$s in the
1141: above formulation.
1142:
1143: \Sectionskip\Section{Bibliographic notes}
1144: The theory of nonparametric additive modelling is relatively recent.
1145: ^{Breiman and Friedman (1985)} proposed the ACE algorithm, a procedure
1146: more general than the additive model (allowing response transformations, and discussed in
1147: Chapter~7) and proved many results on convergence and consistency
1148: of backfitting in both Hilbert-space and data settings.
1149: Most of this chapter is based on the paper by Buja, Hastie, and Tibshirani (1989)
1150: in which some of Breiman and Friedman's convergence results were
1151: extended and the concurvity and degrees of freedom results are
1152: presented.
1153: In some cases, results stronger than those given in this chapter
1154: can be derived; see ^{Buja \etal~(1989)} and the discussions.
1155: We have traded generality for simplicity and interpretability
1156: in this chapter.
1157:
1158: Backfitting goes back a long way.
1159: ^{Friedman and Stuetzle (1981)} defined the term in the context of projection-pursuit regression,
1160: while ^{Wecker and Ansley (1983)} suggested its use in the context of economic models.
1161: Some of the earlier references include ^{Papadakis (1937) } for the separation of fertility trends in the analysis of field trials, and ^{Shiskin, Young and Musgrave (1967)} in the X-11 system for decomposing time series (Chapter~8).
1162: ^{Kohn and Ansley (1989)} independently studied the properties of additive models in the stochastic setting, including convergence.
1163:
1164: Much of the statistical theory and practice of one and higher-dimensional spline models is due to Grace Wahba and her co-workers.
1165: Some relevant references are ^{ Kimeldorf and Wahba (1971)}, ^{Wahba (1978, 1980, 1986)}, ^{O'Sullivan (1983)}, ^{Gu and Wahba (1988)}, and ^{Chen, Gu and Wahba (1989)}.
1166: Section~5.2.3 is based almost entirely on this last reference.
1167: Cox (1989) ^^{Cox, D.D., 1989} described the Bayesian formulation of additive models,
1168: summarized in section~5.4.6.
1169:
1170: The notion of concurvity was introduced by ^{Buja, Donnell and Stuetzle}
1171: (1986)
1172: and was also discussed in ^{Buja \etal~(1989)}.
1173: ^{Bickel, Klaassen, Ritov, and Wellner (1990) } studied the theory
1174: for semi-parametric models,
1175: and in the two predictor projection case, showed convergence of the
1176: backfitting functions and computed the rates of convergence.
1177:
1178: The semi-parametric approach discussed in section~5.3.3 was considered
1179: by ^{Engle, Granger, Rice and Weiss (1986)}, ^{Denby (1986)}, ^{Wahba (1986)}, ^{Green and
1180: Yandell (1985)}, ^{Green (1985)},
1181: ^{Eubank (1985)},
1182: ^{Heckman (1986, 1988)},
1183: ^{Speckman (1988)}, and ^{Shiau and Wahba (1988)}.
1184: Green (1985), and Green and Yandell (1985) looked at
1185: regression and more general models
1186: with a single nonparametric term
1187: (we discuss their work in Chapter~6).
1188: ^{Heckman (1986)} proved consistency of the regression estimate in a regression
1189: model with a single cubic spline term and showed that the estimates
1190: of the regression coefficients and nonparametric function
1191: are Bayes estimates under an appropriate diffuse prior,
1192: generalizing the work of ^{Wahba (1978)}.
1193: ^{Rice (1986)} studied the convergence rates for these partially-splined models.
1194: Speckman (1988) compared the bias and variance of estimators for this model,
1195: and proposed a new estimate (Exercise~5.13) with asymptotically lower-order bias.
1196: This estimate was also suggested by ^{Denby (1986)}.
1197: ^{Shiau and Wahba (1988)} did a thorough study of bias and variance for these
1198: models, and
1199: ^{Heckman (1988) }studied minimax estimators.
1200:
1201: Stone (1982) ^^{Stone (C.J., 1982)} studied rates of convergence for additive models, with the
1202: functions estimated by polynomials or regression splines.
1203: He proved the interesting result
1204: that the optimal rate of convergence for an estimate
1205: of the additive model
1206: is the same as
1207: that for a single function, discussed in section~3.10.
1208: Thus an increase in the dimension $p$ does not decrease the rate of convergence,
1209: as it does if one is estimating a general (nonadditive) $p$-dimensional
1210: function.
1211:
1212: The Gauss-Seidel algorithm is discussed in most textbooks on
1213: numerical analysis; see for example ^{Golub and Van Loan (1983)}.
1214: For a description of the QR algorithm, see ^{Thisted (1988)}.
1215:
1216: \Sectionskip\Section{Further results and exercises 5}
1217: \Mark{EXERCISES \ 5}%
1218: \beginexercises
1219: \exercise
1220: Consider an additive model with
1221: symmetric but not necessarily invertible smoother matrices.
1222: Show that the minimizers of the penalized least-squares criterion $\gsplinpen$
1223: are the solutions to the estimating equations $\backdd$.
1224:
1225: [^{Buja, Hastie and Tibshirani, 1989]}
1226: \exercise
1227: Complete the proof of convergence of backfitting in the case of
1228: orthogonal projections (section~5.3.2) by showing that
1229: $\norm{\bC^m\hat\vec y}\rightarrow 0$.
1230: (Hint: show that $\norm{\bC\vec a}\leq \norm{\vec a}$ for all $\vec a$,
1231: with equality if and only if $\vec a$ is in the orthogonal complement
1232: of $V$).
1233: \exercise
1234: Consider a backfitting procedure with orthogonal projections,
1235: and
1236: let $ \bD$ be the overall design matrix whose columns span
1237: $V=\script L_{col}(\bS_1)\oplus
1238: \script L_{col}(\bS_2)\oplus \ldots \oplus\script L_{col}(\bS_p)$.
1239: Show that the estimating equations $\hat {\vec P}\vec f=\hat {\bQ}\vec y$
1240: are equivalent to the least-squares normal equations
1241: $\bD^T\bD\fat \beta=\bD^T\vec y$ where $\fat \beta$ is the vector
1242: of coefficients.
1243: \exercise
1244: In a backfitting procedure with two least-squares projections $\bS_1$ and
1245: $\bS_2$ based on predictors $\vec x_1$ and $\vec x_2$, show that
1246: if $\theta\neq 0$, then
1247: the difference between the $i$th iterate and the solution converges to zero
1248: geometrically at rate
1249: $\cos{\theta}$,
1250: where
1251: $\theta$ is the angle between $\vec x_1$ and $\vec x_2$.
1252:
1253: [^{Deutsch, 1983}]
1254: \exercise Prove that in the case of the semi-parametric model, backfitting converges to the solution $\simmm$.
1255: Give conditions that guarantee the solution is unique.
1256:
1257: [^{Green, 1985}]
1258: \exercise Prove item (ii) in section~5.4.2, stating that the modified backfitting
1259: procedure provides a solution to the estimating equations $\backdd$.
1260: Derive the explicit solutions $\mbsol$ to the estimating equations from the fixed
1261: points of the modified backfitting algorithm.
1262:
1263: [^{Buja, Hastie and Tibshirani, 1989}]
1264: \exercise
1265: As an extreme case of concurvity, consider an additive model with $p$
1266: identical smoothers $\bS$.
1267: Assume that $\bS$ is centered and symmetric,
1268: with eigenvalues in $[0,1]$, and that the dimension of $\scM_1(\bS)$ is $q$.
1269: Show that
1270: {\parindent 20pt
1271: \item{(i)} $\trace(2\bS-\bS^2)\leq \trace(2\bR-\bR^2)\leq p\;\trace(2\bS-\bS^2)-(p-1)q$.
1272: \item{(ii)} $\trace(2\bR-\bR^2)-\trace(2\bR_{(j)}-\bR_{(j)}^2)\leq \trace(2\bS-\bS^2)-q.$
1273:
1274: }
1275: Hence conclude that
1276: {\parindent 20pt
1277: \item{(a)} including the same predictor twice or more increases the degrees of
1278: freedom (in contrast to projections such as linear regression);
1279: \item{(b)} the sum of the degrees of freedom of the individual function estimates
1280: provides an upper bound on the degrees of freedom of the fitted model;
1281: \item{(c)} $\trace(2\bS-\bS^2)-q$ provides an upper bound on the degrees of freedom contribution $\dferr_j$.
1282:
1283: }\smallskip
1284:
1285: [^{Buja, Hastie and Tibshirani, 1989]}
1286:
1287: \exercise Compute the smoother matrix for a running-mean and kernel smoother
1288: for a small dataset, and hence find empirically that these smoothers are
1289: not symmetric
1290: and that their eigenvalues can have moduli larger than one.
1291: \exercise Derive the expressions (i)--(iii) in section~5.4.6 for the Bayes posterior means and variances for a stochastic additive model.
1292: \exercise Starting with equations \backdd\ with each of the smoothers a smoothing spline, derive an equivalent form for $\hatvec{f}_+$ as in the previous exercise.
1293: \exercise Suppose that the $B$-spline basis functions are to be used to represent the smoothing splines in an additive model fit.
1294: \smallskip
1295: {\parindent 20pt
1296: \item{(i)}Derive closed form expressions for the $B$-spline coefficients of the additive model solution.
1297: \item{(ii)} Derive expressions for $\hat{f}_+(\vec x^0)$ and $\hat{f}_j(x_{0j})$, the fitted functions evaluated at an arbitrary point $\vec x^0$.
1298:
1299: }
1300: \exercise What is the rank of $\hat {\vec P}$ in \backdd, if the smoothers are all cubic smoothing splines, and the $\vec x_j$ span a $p$ dimensional space?
1301: Identify the null space.
1302: \exercise Derive the value of $\fat\beta$ that minimizes
1303: $$\norm{\vec y-\vec X\fat{\beta}-\bS(\vec y- \vec X\fat{\beta})}\eqn{\twicing}$$
1304: and compare it with the semi-parametric estimator \simmm\ in section~5.3.3, using the same smoother $\bS$ in both cases.
1305: \exercise Suppose the same smoother $\vec S$ is used to estimate both the terms
1306: in a two-term additive model (that is, both variables are identical).
1307: Show that the backfitting residual converges to $(\bI+\bS)^{-1}(\bI-\bS)\vec y$, and that the residual sum of squares converges upwards. Can the residual sum of squares converge upwards in less structured situations?
1308: \exercise Consider a semi-parametric model with $p$ predictors (including the constant) and an additional smoother $\bS$.
1309: Derive explicit expressions for $\bR$ and $\bR_j$, and show that $\trace(\bR)\leq p+\trace(\bS)-1$.
1310: \exercise Suppose $\bS_1$ and $\bS_2$ both have the same eigenspaces; this occurs frequently in time-series applications, as in Chapter~8, where smoothing takes place in the Fourier domain. Let $\lambda_{1k}$ and $\lambda_{2k}$, $k=1,\ldots,n$,
1311: be the eigenvalue pair for the $k$th eigenvector, with $\lambda_{jk}\in [0,1]\;\forall k,j$. Assume for convenience that the constant term has been removed, and
1312: that $\lambda_{1k}\lambda_{2k}<1$.
1313: \smallskip
1314: {\parindent 20pt
1315: \item{(i)}Show that the additive-fit operators $\bR=\bR_1+\bR_2$ have eigenvalues
1316: $$\eqalign{\lambda_{k}(\bR)&=1-{(1-\lambda_{1k})(1-\lambda_{2k})\over 1-\lambda_{1k}\lambda_{2k}}\cr
1317: \lambda_{k}(\bR_1)&={\lambda_{1k}(1-\lambda_{2k})\over 1-\lambda_{1k}\lambda_{2k}}\cr
1318: \lambda_{k}(\bR_2)&={\lambda_{2k}(1-\lambda_{1k})\over 1-\lambda_{1k}\lambda_{2k}}\cr}
1319: $$
1320: \item{(ii)}Conclude from (i) that components with eigenvalue one for either smoother
1321: get totally absorbed into the corresponding term.
1322: \item{(iii)}Show that $\trace(\bR)-\trace(\bS_1)\leq\trace(\bS_2)$
1323: \item{(iv)}Show that $\trace(\bR_j\bR_j^T)\leq\trace(\bS_j\bS_j^T)$, and interpret this result in terms of pointwise variances.
1324:
1325: }
1326: \exercise Consider two extremely simple smoothers: $\bS_1=\vec u\lambda\vec u^T$ and $\bS_2=\vec v\lambda\vec v^T$, with $\lambda\in[0,1]$, and $\norm{\vec u}=\norm{\vec v}=1$.
1327: Let $\bR_j$ be the additive operators as above.
1328: Show that
1329: $$\trace(\bR_j\bR_j^T)={\lambda^2\{1-c^2\lambda(2-\lambda)\}\over(1-c^2\lambda^2)^2},$$ where $c=\langle \vec u,\vec v\rangle$.
1330: For $c=0$, $\trace(\bR_j\bR_j^T)=\lambda^2=\trace(\bS_j\bS_j^T)$, while for $c=1$, $\trace(\bR_j\bR_j^T)= \{\lambda/(1+\lambda)\}^2$.
1331: Investigate for values of $c\in (0,1)$ and a range of values of $\lambda$.
1332:
1333: \exercise
1334: Consider a balanced additive model as defined in Exercise~4.8.
1335: Let $\bS_0={\bf 11}^T/n$ denote the {\em mean smoother}.
1336: Using the notation of section~5.4.5,
1337: show that
1338:
1339: {\parindent 20pt
1340: \item{(i)}$\bR_j$=$\bS_j-\bS_0$, $\bR=\bS_0+\sum_{j=1}^p\bR_j$, and $\bR_j\bR_k =\bf 0$ for $j\neq k$.
1341: \item{(ii)} The full and marginal definitions of $\dferr$ and $\dfvar$ coincide, that is
1342: $\tr(2\bR-\bR\bR^T)-\tr(2\bR_{(j)}-\bR_{(j)}\bR_{(j)}^T)=\tr(2\bS_j-\bS_j\bS_j^T)-1$ and $\tr(\bR_j\bR_j^T)=\tr(\bS_j\bS_j^T)-1$.
1343:
1344: }
1345: \exercise
1346: Suppose each of the predictors in an additive model have ties, and smoothing splines are to be used in the fit. Describe the smoother matrices $\bS_j$, as well as their eigenstructure. Show how the estimating equations \backdd\ can be reduced from the
1347: default dimension $np$ to $\sum_{j=1}^pm_j$, where $m_j$ is the number of unique values in $\vec x_j$.
1348: \endexercises
1349: \vfill\supereject
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