semipar_article - 1 Nonparametric Regression LIDAR example...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
1 Nonparametric Regression LIDAR example * * * * * * * * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 400 450 500 550 600 650 700 0.0 0.2 0.4 0.6 0.8 LIDAR for measurement of mercury in atmosphere range -logratio Example comes from environmental monitoring Goal is to fit a smooth curve through the data Derivative of the curve is the object of scientific importance It is proportional to the concentration of mercury 1.1 Why nonparametric regression? Parametric models are not always applicable * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 400 450 500 550 600 650 700 LIDAR for measurement of mercury in atmosphere range * data quintic fit loess
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
No polynomial fits well Quintic fit is shown to illustrate the problems with poly- nomials No nonlinear parametric models immediately suggest them- selves Loess is a nonparametric fit Parametric, nonparametric, and semiparametric models Parametric: shape of curve is determined by model Examples: polynomials, exponentials, cosines, sines, and combinations of these Nonparametric: shape of curve is determined by the data Example: Y = f ( X ) + ± where f is a “smooth” function Semiparametric: combines parametric and nonparametric Example: Y = β 1 X 1 + f ( X 2 ) + ± where f is a “smooth” function 1.2 Polynomial approximation Local polynomial approximation 2
Background image of page 2
-1.5 -1.0 -0.5 0.0 0.5 1.0 -3 -2 -1 0 1 x f(x) * function linear quadratic cubic quartic Black curve is the function f ( x ) = 1 - exp( - x ) Polynomials are Taylor approximations to f ( x ) at x = 0 . 5 (in- dicated by an asterick) Non-local polynomial approximation -2 0 2 4 6 -6 -4 2 4 x * function linear quadratic cubic quartic None of the Taylor approximations do well over a wide range Increasing the degree does not help 1.3 Least-squares approximation Least-squares approximation 3
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Assume: want to approximation a function f ( x ) by a polyno- mial of degree p on an interval [ a, b ] Select a grid of points, x 1 , . . . , x N on [ a, b ] x 1 = a and x N = b The x i are equally-spaced Find β 0 , . . . , β p to minimize N X i =1 { f ( x i ) - ( β 0 + β 1 x i + · · · + β p x p i ) } 2 β 0 + β 1 x + · · · + β p x p is the (unweighted) least-squares approx- imation to f on the interval [ a, b ] Taylor approximation versus least-squares approximation -2 0 2 4 6 -6 -4 -2 0 2 x f(x) * function Taylor-cubic Least-squares-cubic On average, the least-squares approximation is best on the en- tire interval The Taylor approximation is best locally, at the point x = 1 / 2 Tweaking the polynomial in one place changes it everywhere (bad) 4
Background image of page 4
Function approximation – what works best? The best strategy is to approximate a function locally by low degree (constant, linear, quadratic, or cubic) polynomials In regression there are two major ways to do this: local fitting splines Both work well and which is used is largely a matter of conve- nience 2 Local fitting Local polynomial regression regression function 0.0 0.2 0.4 0.6 0.8 1.0 2.6 2.8 3.0 3.2 3.4 uv Do a separate weighted linear least-squares fit centered at each of 100 (say) grid points Use each fit to estimate the curve at its grid point (and nowhere else) Interpolate between the grid points 5
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Loess Loess
Background image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 22

semipar_article - 1 Nonparametric Regression LIDAR example...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online