8 smoothing what is a smoother? 16 smoother 18 nonparametric nature rigid form 21 parametric regression 24 running mean moving average 33 smoothing smoothing 42 fitted smooth 50 smooth 56 smoothing single predictor scatterplot smoothing 62 scatterplot smoother 67 scatterplot smoother 69 scatterplot smooth 70 building block 76 scatterplot smoother 78 categorical 83 scatterplot smooth 85 smoothing smoothing 91 local averaging target value 96 neighbourhoods around target value 97 scatterplot smoothing 109 smoothing parameter 112 three interior knots 112 smoothing parameter 113 brand 132 smoothing parameter 137 fundamental tradeoff between bias and variance smoothing parameter 141 smoothing parameter 146 fitted smooth 151 robustified 154 bibliographic notes 158 scatterplot smoothing: definition scatterplot smoothing 183 scatterplot smooth 190 parametric regression 195 rigid form 202 infinitely smooth scatterplot smoother 214 parametric fitting scatterplot smoothing 218 regression splines parametric fitting 220 bin smoother bin smoother 226 data points 245 running-mean and running-lines smoothers target value 251 close 256 symmetric nearest neighbourhood 259 running mean 265 symmetric nearest neighbourhood 267 symmetric nearest neighbours target point 272 nearest neighbourhood 280 symmetric nearest neighbourhood 281 target point 281 target point running mean 281 running mean 282 moving average 293 running mean 295 running mean least-squares line 298 data points 310 running-lines smooth 313 running-lines smooth 322 running-lines smooth least-squares line 328 running-lines smoother 332 building block 334 smoothing 335 running-lines smooth weighted least-squares 341 running-lines smoother 350 locally-weighted running-lines loess 353 unweighted running-lines smoother 354 running-lines smoother 356 running-lines smoother 358 unweighted running-lines smoother running-lines smoother 360 kernel smoother kernel smoother kernel target value 364 kernel smoother target point 366 kernel smoother 385 standard gaussian density 393 metric distance metric rank distance 394 kernel smoother 398 kernel smooth 400 fitted smooth 404 computational issues kernel smooth weight function 411 kernel smooth 417 spaced data fine grid 419 running medians and enhancements smoothing running mean 424 running mean 427 hanning twicing 429 spaced data time series smoothing 433 interested reader bibliographic notes 434 spaced data regression smoothers 441 linear 441 degrees of freedom 443 time series 447 equivalent kernel 452 target point 452 equivalent kernel 459 equivalent kernels 461 linear smoothing parameter 463 equivalent kernel 465 running mean 468 equivalent kernel 472 running mean 475 loess smooth 476 smoothing smoothing parameter equivalent kernel 477 equivalent degrees of freedom 479 smoothing 479 equivalent kernel 485 regression splines 490 regression splines piecewise 491 knots 493 piecewise polynomials 494 bin smoother piecewise cubic polynomials 509 interior knots 514 multiple regression 514 basis functions piecewise cubic polynomials 520 natural splines piecewise polynomials cubic splines 521 boundary knots 524 interior knots boundary knots 525 natural splines regression splines 527 interior knots 527 computational aspects 532 number interior knots 535 three interior knots 539 basis functions 541 interior knots boundary knots 542 basis functions 545 third derivative 550 basis functions 555 piecewise cubics 556 matrix 561 basis functions 565 interior knots 567 basis functions 572 matrix 574 piecewise cubics 577 regression splines when the knots are given 579 regression splines 583 equivalent kernel three interior knots 586 regression splines smoothing parameter 590 cubic smoothing spline 602 scatterplot smoother 608 natural cubic spline 611 interior knots 612 effective dimension 613 smoothing 616 running-lines smooth 621 running-lines smooth 625 least-squares line 631 computational aspects natural cubic spline interior knots 635 basis functions 637 natural splines 640 matrix matrix 648 sorted values 652 matrix 662 natural-spline basis matrix 667 cubic smoothing spline 671 cubic smoothing spline local averaging 673 equivalent kernel target point 674 equivalent kernel equivalent kernel smoothing splines 677 smoothing kernel 678 smoothing parameter weight function 682 target point 685 smoothing splines 687 locally weighted running-line smoothers loess 692 nearest neighbours nearest neighbours 698 tri-cube weight 703 weighted least-squares fit 704 target point 711 target point 712 target point 720 smoothing 725 span data points smoothing parameter 730 nearest neighbourhood symmetric nearest neighbourhood target point nearest neighbourhood 737 symmetric nearest neighbourhood running-lines smoother 739 nearest neighbourhood data points target point 744 symmetric nearest neighbourhood 747 nearest neighbourhood loess 750 kernel smoother 756 smoothers for multiple predictors single predictor scatterplot smoother 764 multiple regression 766 running mean kernel smoother 769 nearest neighbours 771 nearest euclidean distance 773 symmetric nearest neighbours 775 running mean target point 779 euclidean distance matrix 785 target point 791 smoothing 796 running mean 800 predictor space 812 weight function 813 weight function kernel smoother 817 predictor space 822 matrix 835 smoothing thin-plate spline 843 interested reader bibliographic notes 845 tensor product 846 regression splines 847 basis functions basis functions 848 regression splines 849 smoothing 850 basis functions 852 multi-predictor smoothers 859 parametric regression 860 smoothing 863 multi-predictor smoothers 864 scatterplot smoother running-lines smoother smoothing locally-weighted running-lines kernel smoother 873 per~se building block 876 smoothing parameter locally-weighted running-lines smoothing kernel smoother 890 smoothing splines 891 smoothing splines 926 natural splines 927 matrix containing basis functions interior knots 928 matrix containing basis functions 930 matrix interior knots boundary knots 932 derivation of smoothing splines 950 third derivative 954 least-squares line 964 cubic smoothing spline 967 smoothing 969 semi-parametric regression 972 matrix 975 estimating equations smoothing 979 efficient kernel smoothing 980 kernel smooth 982 fine grid 992 matrix smoothing 997 smoothing splines 1003 equivalent kernel