Motorcycle accident data

A series of 133 measurements of head acceleration in a simulated motorcycle accident, used to test crash helmets.

times – time in milliseconds after impact

accel – acceleration in g

mcycle.nw=npreg(accel~times,ckertype="epanechnikov",regtype="lc")

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mcycle.lr=npreg(accel~times,ckertype="epanechnikov",regtype="ll")

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mcycle.spline=smooth.spline(accel~times,all.knots=T)
mcycle.loess=loess(accel~times,span=1/4)

## 
## Regression Data: 133 training points, in 1 variable(s)
##                   times
## Bandwidth(s): 0.6804928
## 
## Kernel Regression Estimator: Local-Constant
## Bandwidth Type: Fixed
## 
## Continuous Kernel Type: Second-Order Epanechnikov
## No. Continuous Explanatory Vars.: 1

## 
## Regression Data: 133 training points, in 1 variable(s)
##                 times
## Bandwidth(s): 1.53408
## 
## Kernel Regression Estimator: Local-Linear
## Bandwidth Type: Fixed
## 
## Continuous Kernel Type: Second-Order Epanechnikov
## No. Continuous Explanatory Vars.: 1

## Call:
## smooth.spline(x = accel ~ times, all.knots = T)
## 
## Smoothing Parameter  spar= 0.7670834  lambda= 0.000110663 (12 iterations)
## Equivalent Degrees of Freedom (Df): 12.2553
## Penalized Criterion (RSS): 38606.57
## GCV: 565.4861

Equivalent kernel for cubic spline smoothing (Silverman, 1984)

\[ K(u)=\frac{1}{2}e^{-|u|/\sqrt{2}} \sin(|u|/\sqrt{2}+\pi/4)) \] with variable bandwidth \(\lambda^{1/4}f(x)^{-1/4}\)