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Presented by Wenli Li, Shuhong Li, and Vivian Tam Venables and Ripley Section 8.7 Novemeber 22, 2004 PowerPoint Presentation
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Presented by Wenli Li, Shuhong Li, and Vivian Tam Venables and Ripley Section 8.7 Novemeber 22, 2004 - PowerPoint PPT Presentation


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One-Dimensional Curve-Fitting. Presented by Wenli Li, Shuhong Li, and Vivian Tam Venables and Ripley Section 8.7 Novemeber 22, 2004. INTRODUCTION. Curve-fitting : Sample data:{(x 0 ,y 0 ), (x 1 ,y 1 ), ... (x n , y n )} interpolation & extrapolation

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Presented by Wenli Li, Shuhong Li, and Vivian Tam Venables and Ripley Section 8.7 Novemeber 22, 2004


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slide1

One-Dimensional Curve-Fitting

Presented by Wenli Li, Shuhong Li,

and Vivian Tam

Venables and Ripley Section 8.7

Novemeber 22, 2004

slide2

INTRODUCTION

  • Curve-fitting:
    • Sample data:{(x0,y0), (x1,y1), ... (xn, yn)}
    • interpolation & extrapolation
  • One-dimensional curve-fitting (section 8.7):
    • The functional form is not pre-specified
    • SPLINES (ns, smooth.spline)
    • Local Regression (LOESS, SUPSMU, KERNEL SMOOTHER and LOCPOLY)
  • Data set:
    • One independent & one dependent

Examples: GAGurine & Mercury level

gagurine mass
Dataset:

Variables:

Age: independent

GAG: dependent

Sample size: 314

Classical way:

library(MASS)

attach(GAGurine)

plot(Age, GAG, main=”Degree 6 polynomial”)

GAG.lm<-lm(GAG~Age+I(Age^2) +I(Age^3) +I(Age^4) +I(Age^5) +I(Age^6) +I(Age^7) +I(Age^8))

anova(GAG.lm)

GAG.lm2<-lm(GAG~Age+I(Age^2) +I(Age^3) +I(Age^4) +I(Age^5) +I(Age^6))

xx<-seq(0, 17, len=200)

lines(xx, predict(GAG.lm2, data.frame(Age=xx), col=“red”)

Age: 0.00 0.00……0.46 0.47.….17.30 7.67

GAG 23.0 23.8……18.6 26.4.…..1.9 9.3

=======================================

Terms added sequentially (first to last)

Df Sum of Sq Mean Sq F-value Pr(F)

Age 1 12590 12590 593.58 0.0000

I(Age^2) 1 3751 3751 176.84 0.0000

I(Age^3) 1 1492 1492 70.32 0.0000

I(Age^4) 1 449 449 21.18 0.00001

I(Age^5) 1 174 174 8.22 0.00444

I(Age^6) 1 286 286 13.48 0.00028

I(Age^7) 1 57 57 2.70 0.10151

I(Age^8) 1 45 45 2.12 0.14667

GAGurine (MASS)
splines
SPLINES
  • Algorithm:
  • Function: ns( )
    • Generate a Basis Matrix for Natural Cubic Splines
    • Usage: ns(x, df, knots, intercept=F, Boundary.knots,derivs)
    • Arguments:
      • Required: x the predictor variable.
      • Optional:
        • Df: degrees of freedom. One can supply df rather than knots; ns then chooses df-1-intercept knots at suitably chosen quantiles of x. This argument is ignored if knots is supplied.
        • Knots: breakpoints that define the spline.
splines5
SPLINES

Function: smooth.spline( )

  • Fits a cubic B-spline smooth to the input data.
  • Usage: smooth.spline(x, y, w = <<see below>>, df = <<see below>>, spar = 0, cv = F, all.knots = F, df.offset = 0, penalty = 1)
  • Arguments:
    • Required: X, values of the predictor variable. There should be at least ten distinct x values.
    • Optional:
      • Y: response variable, of the same length as x.
      • Df:a number which supplies the degrees of freedom = trace(S)rather than a smoothing parameter.
splines6
SPLINES

library(splines)

plot(Age, GAG, type=”n”, main=”Spline”)#splines

lines(Age, fitted(lm(GAG~ns(Age, df=5))), col=”red”)

lines(Age, fitted(lm(GAG~ns(Age, df=10))), lty=3, col=”green”)

lines(Age, fitted(lm(GAG~ns(Age, df=20))), lty=4, col=”blue”) 

lines(smooth.spline(Age, GAG), lwd=3, col=”black”)# Smoothing splines

legend(12, 50, c(“red: df=5”, “green:df=10”, “blue:df=20”, “Smoothing”), lty=c(1,3, 4,1), lwd=c(1, 1,1, 3), bty=”n”)

kernel smooth
KERNEL SMOOTH

Function: ksmooth( )

  • Estimates a probability density or performs scatterplot smoothing using kernel estimates.
  • Usage: ksmooth(x, y=NULL, kernel="box", bandwidth=0.5, range.x=range(x), n.points=length(x), x.points=<<see below>>)
  • Arguments:
    • Required: X, vector of x data
    • Optional:
      • Y: vector of y data. This must be same length as x, and missing values are not accepted.
      • Kernel: "box“,"triangle“,"parzen“,"normal”
    • Bandwidth:Larger values of bandwidth make smoother estimates, smaller values of bandwidth make less smooth estimates.
kernel smoother
Kernel Smoother

#kernel smoother:

plot(Age, GAG, type=”n”, main=”ksmooth”)

lines(ksmooth(Age, GAG, “normal”, bandwidth=1), col=”red”)

lines(ksmooth(Age, GAG, “normal”, bandwidth=5))

legend(12, 50, c(“red: bandwidth=1”, “black: bandwidth=5”),bty=”n”)

loess
LOESS
  • Using Local Polynomial Regression fit a curve determined by one or more numerical predictors
  • gets a predicted value at each point by fitting a weighted linear regression, where the weights decrease with distance from the point of interest
loess parameters
LOESS Parameters
  • f:controls the window size
  • weights: distance from some point x
  • span: the parameter alpha which controls the degree of smoothing
  • degree: the degree of the polynomials to be used, up to 2
slide11

LOESSCode: library(MASS)attach(GAGurine)plot(Age,GAG,type="n",main="loess")lines(loess.smooth(Age,GAG,span=2/3,degree=1),col="red",lwd=1)lines(loess.smooth(Age,GAG,span=2/3,degree=4),col="blue",lwd=2)lines(loess.smooth(Age,GAG,span=1/3,degree=4),col="green",lwd=1)legend(10,45, c("Red: span=2/3,deg=1","Blue: span=2/3,deg=4",”green: span=1/3,deg=4"),bty="n")

supsmu
SUPSMU
  • Serves a purpose similar to that of the function loess
  • The best of the three smoothers is chosen by cross-validation
  • If there are substantial correlations in x-value, then a pre-specified fixed span smoother should be used. Reasonable span values are 0.2 to 0.4
supsmu parameters
SUPSMU Parameters:
  • span: the fraction of the observations in the span of the running(lines smoother, or ‘“cv”’ to choose this by leave-one-out cross-validation)
  • bass: controls the smoothness of the fitted curve. Values of up to 10 indicate increasing smoothness
  • periodic: if TRUE, the smoother assumes x is a periodic variable with values in the range [0.0, 1.0] and period 1.0. An error occurs if x has values outside this range

References:

Friedman, J. H. (1984) A variable span scatter-plot smoother. Laboratory for Computational Statistics, Stanford University Technical Report No. 5

slide15

Code:plot(Age,GAG,type="n",main="supsmu")lines(supsmu(Age,GAG))lines(supsmu(Age,GAG,bass=3),lty=3)lines(supsmu(Age,GAG,bass=10),lty=4)legend(12,50,c("default","bass=3","bass=10"),lty=c(1,3,4),bty="n")Code:plot(Age,GAG,type="n",main="supsmu")lines(supsmu(Age,GAG))lines(supsmu(Age,GAG,bass=3),lty=3)lines(supsmu(Age,GAG,bass=10),lty=4)legend(12,50,c("default","bass=3","bass=10"),lty=c(1,3,4),bty="n")

locpoly
LOCPOLY
  • Estimates a probability density function using local polynomials
  • A fast binned implementation over an equally-spaced grid is used
  • Use approximations over an equally-spaced grid for fast computation
  • In a simple form : locpoly(x, y, degree=#, bandwidth=# )

Parameters:

  • locpoly(x, y, drv=0, degree=<<see below>>, kernel="normal“

bandwidth,gridsize=401, bwdisc=25, range.x=<<see below>>,

binned=FALSE, truncate=TRUE )

  • drv: order of derivative to be estimated
  • degree: degree of local polynomial used
  • bandwidth: the kernel bandwidth smoothing parameter
  • range.x: vector containing the minimum and maximum values of 'x' at which to compute the estimate
locpoly17
LOCPOLY

Code:

library(MASS)

attach(GAGurine)

library(KernSmooth)

plot(Age, GAG, type="n", main="(Age, GAG) Locpoly")

(h<- dpill(Age, GAG))

lines(locpoly(Age, GAG, degree=0, bandwidth=h), col="red",lty=1,lwd=2)

lines(locpoly(Age, GAG, degree=1, bandwidth=h), col="blue",lty=3,lwd=3)

lines(locpoly(Age, GAG, degree=2, bandwidth=h), col="green",lty=4,lwd=3)

legend(10,40,c("const=0 red","linear=1 blue","quad=2 green"),lty=c(1,3,4),bty="n")

detach()

example mercury level
Example: Mercury Level
  • Model : Mercury and Alkalinity
  • In 1990 to 1991, largemouth bass fish were studied in 53 different Florida lakes to examine the Mercury contamination level and the factors that influenced the level of mercury absorpsion in the fish
  • One factor studied was the Alkaliniity level of the water
  • The graph of Mercury level and Alkalinity level is plotted to study the relationship
mercury level graphs coding
Mercury Level Graphs Coding:
  • #1 loess
  • plot( Alkalinity, Mercury, main="Alkalinity and Mercury, Loess")
  • lines(loess.smooth(Alkalinity,Mercury,span = 2/3, degree = 1), col="red",lwd=2)
  • lines(loess.smooth(Alkalinity,Mercury,span = 2/3, degree = 2), col="blue",lwd=2)
  • legend(65,1.0, c("deg=1 Red","deg=2 Blue"),bty="n")
  • #2 supsmu
  • plot( Alkalinity, Mercury, main="Alkalinity and Mercury, Supsmu")
  • lines(supsmu(Alkalinity,Mercury, bass=1), lty=1,col="red",lwd=2)
  • lines(supsmu(Alkalinity,Mercury, bass=10), lty=3,col="blue",lwd=3)
  • legend(58,1.0, c("base=1red","base=10blue"),lty=c(1,3),bty="n",lwd=2)
  • #3 ksmooth
  • plot(Alkalinity, Mercury, type="n", main="Alkalinity and Mercury, Ksmooth")
  • lines(ksmooth(Alkalinity, Mercury, "normal", bandwidth=1),col="green",lwd=2)
  • lines(ksmooth(Alkalinity, Mercury, "normal", bandwidth=5),col="red", lty=2,lwd=2)
  • legend(75,1.0, c("bw=1","bw=5"),lty=c(1,2),bty="n")
  • #4 locpoly
  • library(KernSmooth)
  • plot( Alkalinity, Mercury, type="n",main="Alkalinity and Mercury, Locpoly")
  • #select bandwidth
  • (h <- dpill(Alkalinity,Mercury))
  • lines(locpoly(Alkalinity,Mercury,degree=0, bandwidth=h),lty=1,col="green",lwd=2)
  • lines(locpoly(Alkalinity,Mercury,degree=1, bandwidth=h),lty=2,col="red",lwd=2)
  • lines(locpoly(Alkalinity,Mercury,degree=2, bandwidth=h),lty=3,col="purple",lwd=3)
  • legend(75,1.0, c("const","linear","quad"),lty=c(1,2,3),bty="n")
summary
SUMMARY
  • Use One-Dimensional Curve-Fitting when:

Scatter Plot does not result in a Linear Model

Data Transformation does not give satisfactory

Linear Model result

Accommodate future data

Include previous outliers

Business applications

  • Several methods discussed including:

1. SPLINES

2. LOESS

3. SUPSMU

4. KSMOOTH

5. LOCPOLY

  • Parameters: such as bandwidth, df, derivative, smoothness, degree etc can help the curve fitting.