2 CHAPTER 1.NONPARAMETRIC REGRESSION Properties of splines For simplicity,let us consider a general pth degree spline with a single knot at t.Let P(r) denote an arbitrary pth degree polynomial P(x)=+x+32x2+…+3nxP. Then S(x)=P(x)+Bp+1(x-t)4 (1.1) takes on the value P(z)for any <t,and it takes on the value P(x)+p+i(x-t)p for any r>t.Thus,restricted to each region,the function is a pth degree polynomial.As a whole,this function is a pth degree piecewise polynomial;there are two pieces. Note that we require p+2 coefficients to specify this piecewise polynomial.This is one more coefficient than the number needed to specify a pth degree polynomial,and this is a result of the addition of the truncated power function specified by the knot at t.In general,we may add k truncated power functions specified by knots at t1,t2,...,ts,each multiplied by different coefficients.This would result in p+k+1 degrees of freedom. An important property of splines is their smoothness.Polynomials are very smooth, possessing all derivatives everywhere.Splines possess all derivatives only at points which are not knots.The number of derivatives at a knot depends on the degree of the spline. Consider the spline defined by (1.1).We can show that S(x)is continuous at t,when p>0,by noting that S(t)=P(t) and lim9p+1(x-t)4=0 clt so that lim S(x)=P(t). xlt In fact,we can argue similarly for the first p-1 derivatives: S0)(t)=P6(t),j=1,2,.,p-1 and lim6p+1p(p-1).(p-j+1)(x-t)P-j=0 1 so that lim S()()=p)(t) xlt The pth derivative behaves differently: S(P)(t)=p!Bp and lim S(P)(x）=plpp+p9p+1 xlt2 CHAPTER 1. NONPARAMETRIC REGRESSION Properties of splines For simplicity, let us consider a general pth degree spline with a single knot at t. Let P(x) denote an arbitrary pth degree polynomial P(x) = β0 + β1x + β2x 2 + · · · + βpx p . Then S(x) = P(x) + βp+1(x − t) p + (1.1) takes on the value P(x) for any x ≤ t, and it takes on the value P(x) + βp+1(x − t) p for any x > t. Thus, restricted to each region, the function is a pth degree polynomial. As a whole, this function is a pth degree piecewise polynomial; there are two pieces. Note that we require p + 2 coefficients to specify this piecewise polynomial. This is one more coefficient than the number needed to specify a pth degree polynomial, and this is a result of the addition of the truncated power function specified by the knot at t. In general, we may add k truncated power functions specified by knots at t1, t2, . . . , tk, each multiplied by different coefficients. This would result in p + k + 1 degrees of freedom. An important property of splines is their smoothness. Polynomials are very smooth, possessing all derivatives everywhere. Splines possess all derivatives only at points which are not knots. The number of derivatives at a knot depends on the degree of the spline. Consider the spline defined by (1.1). We can show that S(x) is continuous at t, when p > 0, by noting that S(t) = P(t) and lim x↓t βp+1(x − t) p + = 0 so that lim x↓t S(x) = P(t). In fact, we can argue similarly for the first p − 1 derivatives: S (j) (t) = P (j) (t), j = 1, 2, . . . , p − 1 and lim x↓t βp+1p(p − 1)· · ·(p − j + 1)(x − t) p−j = 0 so that lim x↓t S (j) (x) = P (j) (t) The pth derivative behaves differently: S (p) (t) = p!βp and lim x↓t S (p) (x) = p!βp + p!βp+1