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1.1     ! root        1: All the smoothers we shall describe can easily be {\em robustified} by replacing the averaging or least-squares operation by a more robust procedure.
        !             2: known as   {\em Hanning}, {em splitting} and {\em twicing}, in various combinations.
        !             3: called {\em regression smoothers}.
        !             4: Many of the smoothers that we discuss are {\em linear} (section~3.4.2), and this facilitates
        !             5: an approximate assessment of their {\em degrees of freedom} (section~3.5).
        !             6: Thus the weight in the fit at $x_0$ which is associated with the point $x_j$ is $S_{0j}$, and this sequence of weights is known as the {\em equivalent kernel} at $x_0$. 
        !             7: The {\em loess} smooth on the other hand has a strictly local neighbourhood yet the weights die down smoothly to zero.
        !             8: itself; hence the name {\sl smoother}.
        !             9: An important property of a smoother is its {\sl nonparametric} nature; that is,
        !            10: We call the estimate produced by a smoother a {\sl smooth}.
        !            11: setting, usually referred to as {\sl scatterplot
        !            12: The  simplest smoother occurs in the case of a {\sl categorical}
        !            13: While the reader might not normally think of this as {\sl smoothing}, this simple
        !            14:  {\sl local averaging}, that is,
        !            15: The averaging is done in {\sl neighbourhoods} around the target value.
        !            16: question of which {\sl brand} of smoother to use, because  smoothers
        !            17: adjustable {\sl smoothing parameter}.
        !            18: Thus there is a {\sl fundamental tradeoff between bias and variance},
        !            19: In a sense the regression line is an {\sl infinitely smooth} function, and
        !            20: are {\sl close} to $x_i$?
        !            21: This is called a {\sl symmetric nearest neighbourhood} and
        !            22: the {\sl running mean}
        !            23: of which side they are on; this is called a {\sl nearest neighbourhood}.
        !            24: also called a {\sl moving average}, and is popular for evenly-spaced time-series data.
        !            25: in practice it does not work very well.  It tends to be  so wiggly that it hardly deserves the name {\sl smoother.} 
        !            26: The {\sl running
        !            27: the use of a {\sl weighted} least-squares fit in each
        !            28: ^{Cleveland's (1979)} implementation of a locally-weighted running-lines smoother, {\sl loess},   
        !            29: A kernel smoother uses an explicitly defined set of local weights, defined by the {\sl kernel}, to
        !            30: function only of its {\sl metric} distance from $x_0$, while the weights used by the nearest-neighbour    smoothers are typically a function of both {\sl metric} and {\sl rank} distance. 
        !            31: Their {\sl equivalent kernels} are one way to compare
        !            32: All the smoothers studied in this chapter  are {\sl linear} in $Y$, which means that the fit at a point $x_0$ can be written as $\gsmooth(x_0)=\sum_{j=1}^n S_{0j}y_j$, and the $S_{0j}$ depend on all the $x_i$ and on the smoothing parameter $\lambda$.
        !            33: We do this using the {\sl equivalent degrees of freedom}, which we describe in the next chapter. 
        !            34: Regression splines offer a compromise by representing the fit as a {\sl piecewise}
        !            35: of {\sl knots} or breakpoints, $\xi_1,\ldots,\xi_K$. 
        !            36: A variant of polynomial splines are the natural splines; although they are defined for  all piecewise polynomials of odd degree, we  discuss the natural {\sl cubic} splines. 
        !            37: A very simple approach (referred to as cardinal splines) requires a single parameter, the {\sl number} of interior knots. 
        !            38: In summary, regression splines are attractive because of their computational neatness, {\sl when the knots are given}.
        !            39: that minimizer is  a {\sl natural cubic spline} with knots at the unique values of $x_i$
        !            40: This would result in $n+2$ parameters, although the  constraints on each end bring it down to $n$.  We'll see however that the coefficients are estimated in a constrained way as well, and this can bring the {\sl effective} dimension down dramatically.
        !            41: Since the columns of $\bB$ are the evaluated  $B$-splines, in order from left to right and evaluated at the {\sl sorted} values of $X$, and the cubic $B$-splines have local support, $\bB$ is lower 4-banded. 
        !            42: Let $\bN$ be an $n\times n$ nonsingular {\sl natural-spline} basis matrix for representing the solution (Exercise~2.5). 
        !            43: to use {\sl local averaging}.
        !            44: One might say, then,  that a cubic  smoothing spline is  approximately a {\sl kernel} 
        !            45: Here we describe in more detail  the locally-weighted smoother of ^{Cleveland (1979)}, currently called {\sl loess} in the S statistical-computing language. 
        !            46: \item{(iii)}Weights $w_i$ are assigned to each point in $\NN(x_0)$, using the  {\sl tri-cube} weight function:
        !            47: as a percentage or {\sl span} of the data points, is the smoothing parameter. 
        !            48: Nearest neighbourhoods work satisfactorily with {\sl loess} at the endpoints, however,
        !            49: The first two smoothers require a definition of {\sl nearest neighbours}
        !            50: {\sl Nearest} is determined by a distance measure and for this 
        !            51: dimensions: the so-called {\sl thin-plate spline} is one such
        !            52: Another generalization is known as multivariate {\sl tensor product} splines. 
        !            53: In addition, the emphasis in the book is not on smoothers {\sl per~se}, but 
        !            54: \exercise {\sl Updating formula for running-line smooth.}
        !            55: \exercise {\sl Basis for natural splines.} Suppose $B$ is an $n\times (K+4)$ matrix containing the evaluations of the cubic  $B$-spline basis functions with $K$ interior knots evaluated at the $n$ values
        !            56: \exercise {\sl Derivation of smoothing splines; ^{Reinsch (1967)}.} Consider the following optimization problem: minimize 
        !            57: \exercise {\sl Semi-parametric regression; ^^{Green, P.J.}^^{Jennison, C. }^^{Seheult, A.} Green, Jennison, and Seheult (1985).} Suppose we have a set of $n$ observations of $p$ predictors arranged in
        !            58: Construct an  appropriate penalized residual sum of squares, and show that the minimizers must satisfy the following pair of {\sl estimating} equations:
        !            59: \exercise {\sl Efficient kernel smoothing; ^{Silverman (1982)}, ^{H\"ardle (1986)}}

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