spreg                package:funfits                R Documentation

_S_m_o_o_t_h_i_n_g _s_p_l_i_n_e _r_e_g_r_e_s_s_i_o_n

_D_e_s_c_r_i_p_t_i_o_n:

     A smoothing spline is a locally weighted average of the data y's
     based on the relative locations of the x values. Formally the
     estimate is the curve that minimizes the criterion: (1/n)
     sum(k=1,n) ( Y_k - f( X_k))**2  + lambda* R(f) where R(f) is the
     integral of the squared second derivative of f over the range of
     the X values. The solution is a piecewise cubic polynomial with
     the join points at the unique set of X values. The polynomial
     segments are constructed so that the entire curve has continuous
     first and second derivatives and the second and third derivatives
     are zero at the boundaries.  The smoothing parameter has the range
     [0,infinity]. Lambda equal to  zero gives a cubic spline
     interpolation of  the data. As lambda diverges to infinity ( e.g
     lambda =1e20) the  estimate will converge to the straight line
     estimated by least squares.

     The values of the estimated function at the data points can be
     expressed in the matrix form:

     predicted.values= A(lambda)Y where A is an nXn symmetric matrix
     that does NOT depend on Y. The diagonal elements are the leverage
     values for the estimate and the sum of these  (trace(A(lambda))
     can be interpreted as the effective number of parameters that are
     used to define the spline function.

_U_s_a_g_e:

     spreg(x, y, lambda, xgrid, weight=rep(1, length(x)), derivative=0, 
     Adiag=T, cost=1)

_A_r_g_u_m_e_n_t_s:

       x: Vector of x values  

       y: Vector of y values 

  lambda: Smoothing parameter. If omitted this is estimated by GCV. 

   xgrid: Vector of points to evaluate the estimated curve. Default is
          unique sorted x's. 

  weight: A vector that is proportional to the standard deviation of
          the errors. 

derivative: If equal to 1 or 2 returns the estimated first or second
          derivative of the estimate 

   Adiag: If true will compute leverage values for the estimate 

    cost: Cost value to be used in the GCV criterion. 

_V_a_l_u_e:

     A list of class spreg. The values of the GCV function and the
     effective number of parameters are tabulated in the component
     gcv.grid. The component predicted is a two column matrix that
     contains the values from xgrid (or sorted unique x's) and the
     estimated curve at these points.

_R_e_f_e_r_e_n_c_e_s:

     Additive Models by Hastie and Tibishirani

_S_e_e _A_l_s_o:

     predict.spreg, splint, tpsreg

_E_x_a_m_p_l_e_s:

     spreg( auto.paint$thick, auto.paint$DOI)-> out
     plot(out)
     lines(out$predicted)

