Nonparametric Regression (ELMR Chapter 14)

Parametric vs nonparametric models
Kernel estimators
Splines
- Smoothing splines
- Regression splines
Local polynomials
Wavelets (not covered in this course)

Display system information and load tidyverse and faraway packages

sessionInfo()

## R version 3.6.2 (2019-12-12)
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library(faraway)

Parametric vs nonparametric models

Given a regressor/predictor \(x_1,\ldots,x_n\) and response \(y_1, \ldots, y_n\), where \[ y_i = f(x_i) + \epsilon_i, \] where \(\epsilon_i\) are iid with mean zero and unknown variance \(\sigma^2\).
Parametric approach assumes that \(f(x)\) belongs to a parametric family \(f(x \mid \boldsymbol{\beta})\). Examples are \[\begin{eqnarray*} f(x \mid \boldsymbol{\beta}) &=& \beta_0 + \beta_1 x \\ f(x \mid \boldsymbol{\beta}) &=& \beta_0 + \beta_1 x + \beta_2 x^2 \\ f(x \mid \boldsymbol{\beta}) &=& \beta_0 + \beta_1 x^{\beta_2}. \end{eqnarray*}\]
Nonparametric approach assumes \(f\) is from some smooth family of functions.
Nonparametric approaches offers flexible modelling of complex data, and often serve as valuable exploratory data analysis tools to guide formulation of meaningful and effective parametric models.

Three datasets.

exa: \(f(x) = \sin^3 (2 \pi x^3)\).

exa <- as_tibble(exa) %>% print()

## # A tibble: 256 x 3
##          x       y     m
##      <dbl>   <dbl> <dbl>
##  1 0.00480 -0.0339     0
##  2 0.0086   0.165      0
##  3 0.0117   0.0245     0
##  4 0.017    0.178      0
##  5 0.0261  -0.347      0
##  6 0.0299  -0.755      0
##  7 0.0307   0.355      0
##  8 0.0315   0.0401     0
##  9 0.0331   0.106      0
## 10 0.034    0.122      0
## # … with 246 more rows

exa %>%
  ggplot(mapping = aes(x = x, y = y)) + 
  geom_point() +
  geom_smooth(span = 0.2) + # small span give more wiggleness
  geom_line(mapping = aes(x = x, y = m)) # true model (black line)

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

exb: \(f(x) = 0\).

exb <- as_tibble(exb) %>% print()

## # A tibble: 50 x 3
##        x       y     m
##    <dbl>   <dbl> <dbl>
##  1  0.02  0.781      0
##  2  0.04  1.64       0
##  3  0.06 -0.363      0
##  4  0.08 -0.0540     0
##  5  0.1  -0.804      0
##  6  0.12  0.907      0
##  7  0.14 -0.843      0
##  8  0.16 -0.229      0
##  9  0.18 -0.499      0
## 10  0.2   1.20       0
## # … with 40 more rows

exb %>%
  ggplot(mapping = aes(x = x, y = y)) + 
  geom_point() +
  geom_smooth() +
  geom_line(mapping = aes(x = x, y = m))

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

faithful: data on Old Faithful geyser in Yellowstone National Park.

faithful <- as_tibble(faithful) %>% print()

## # A tibble: 272 x 2
##    eruptions waiting
##        <dbl>   <dbl>
##  1      3.6       79
##  2      1.8       54
##  3      3.33      74
##  4      2.28      62
##  5      4.53      85
##  6      2.88      55
##  7      4.7       88
##  8      3.6       85
##  9      1.95      51
## 10      4.35      85
## # … with 262 more rows

faithful %>%
  ggplot(mapping = aes(x = eruptions, y = waiting)) + 
  geom_point() +
  geom_smooth() # small span give more wiggleness

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

Kernel estimators

Moving average estimator \[ \hat f_\lambda(x) = \frac{1}{n\lambda} \sum_{j=1}^n K\left( \frac{x-x_j}{\lambda} \right) Y_j = \frac{1}{n} \sum_{j=1}^n w_j Y_j, \] where \[ w_j = \lambda^{-1} K\left( \frac{x-x_j}{\lambda} \right), \] and \(K\) is a kernel such that \(\int K(x) \, dx = 1\). \(\lambda\) is called the bandwidth, window width or smoothing parameter.
When \(x\)s are spaced unevenly, the kernel estimator can give poor results. This is improved by the Nadaraya-Watson estimator \[ f_\lambda(x) = \frac{\sum_{j=1}^n w_j Y_j}{\sum_{j=1}^n w_j}. \]
Asymptotics of kernel estimators \[ \text{MSE}(x) = \mathbb{E} [f(x) - \hat f_\lambda(x)]^2 = O(n^{-4/5}). \] Typical parametric estimator has \(\text{MSE}(x) = O(n^{-1})\) if the parametric model is correct.
Choice of kernel. Ideal kernel is smooth, compact, and amenable to rapid computation. The optimal choice under some standard assumptions is the Epanechnikov kernel \[ K(x) = \begin{cases} \frac 34 (1 - x^2) & |x| < 1 \\ 0 & \text{otherwise} \end{cases}. \]

Choice of smoothing parameter \(\lambda\). Small \(\lambda\) gives more wiggly curves, while large \(\lambda\) yields smoother curves.

for (bw in c(0.1, 0.5, 2)) {
  with(faithful, {
    plot(waiting ~ eruptions, col = gray(0.75))
    # Nadaraya–Watson kernel estimate with normal kernel
    lines(ksmooth(eruptions, waiting, "normal", bw))
  })
}

Cross-valiation (CV) choose the \(\lambda\) that minimizes the criterion \[ \text{CV}(\lambda) = \frac 1n \sum_{j=1}^n [y_j - \hat f_{\lambda(j)}(x_j)]^2, \] where \((j)\) indicates that point \(j\) is left out of the fit.

library(sm)

## Package 'sm', version 2.2-5.6: type help(sm) for summary information

with(faithful,
     sm.regression(eruptions, waiting, h = h.select(eruptions, waiting)))

with(exa, sm.regression(x, y, h = h.select(x, y)))

with(exb, sm.regression(x, y, h = h.select(x, y)))

Splines

Smoothing splines

Smoothing spline approach chooses \(\hat f\) to minize the modified least squares criterion \[ \frac 1n \sum_i [y_i - f(x_i)]^2 + \lambda \int [f''(x)]^2 \, dx, \] where \(\lambda > 0\) is the smoothing paramter and \(\int [f''(x)]^2 \, dx\) is a roughness penalty. For large \(\lambda\), the minimizer \(\hat f\) is smoother; for smaller \(\lambda\), the minizer \(\hat f\) is rougher. This is the smoothing spline fit.
The minimizer takes a special form: \(\hat f\) is a cubic spline (piecewise cubic polynomial in each interval \((x_i, x_{i+1})\)).
The tuning parameter \(\lambda\) is chosen by cross-validation (either leave-one-out (LOO) or generalized (GCV)) in R.

with(faithful, {
  plot(waiting ~ eruptions, col = gray(0.75))
  lines(smooth.spline(eruptions, waiting), lty = 2)
})

with(exa, {
  plot(y ~ x, col = gray(0.75))
  lines(x, m) # true model
  lines(smooth.spline(x, y), lty = 2)
})

with(exb, {
  plot(y ~ x, col = gray(0.75))
  lines(x, m) # true model
  lines(smooth.spline(x, y), lty = 2)
})

The last example exb shows that automatic choice of tuning parameter is not foolproof.

Regression splines

The regresison spline fit differs from the smoothing splines in that the number of knots can be much smaller than the sample size.
Piecewise linear splines:

# right hockey stick function (RHS) with a knot at c
rhs <- function(x, c) ifelse(x > c, x - c, 0)
curve(rhs(x, 0.5), 0, 1)

Define some knots for Example A

(knots <- 0:9 / 10)

##  [1] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

and compute a design matrix of splines with knots at these points for each \(x\):

# each column is a RHS function with a specific knot 
dm <- outer(exa$x, knots, rhs)
dim(dm)

## [1] 256  10

matplot(exa$x, dm, type = "l", col = 1, xlab = "x", ylab="")

Compute and dipslay the regression spline fit.

lmod <- lm(exa$y ~ dm)
plot(y ~ x, exa, col = gray(0.75))
lines(exa$x, predict(lmod))

We can acheive better fit by using more knots in denser regions of greater curvature.

newknots <- c(0, 0.5, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95)
dmn <- outer(exa$x, newknots, rhs)
lmod <- lm(exa$y ~ dmn)
plot(y ~ x, data = exa, col = gray(0.75))
lines(exa$x, predict(lmod))

High-order splines can produce a smoother fit. The bs() function can be used to generate the appropriate spline basis. The default is cubic B-splines.

library(splines)

matplot(bs(seq(0, 1, length = 1000), df = 12), type = "l", ylab="", col = 1)

# generate design matrix using B-splines with df = 12
lmod <- lm(y ~ bs(x, 12), exa)
plot(y ~ x, exa, col = gray(0.75))
lines(m ~ x, exa) # true model
lines(predict(lmod) ~ x, exa, lty = 2) # Cubic B-spline fit

Local polynomials

Both kernel and spline smoothers are vulnerable to outliers.
Local polynomial method fit a polynomial in a window using robust methods, then the predicted response at the middle of the window is the fitted value. This procedure is repeated by sliding the window over the range of the data.
lowess (locally weighted scatterplot smoothing) or loess (locally estimated scatterplot smoothing) functions in R.
We need to choose the order of polynomial and window width. Default window width is 0.75 of data. Default polynomial order is 2 (quadratic).
Examples.

with(faithful, {
  plot(waiting ~ eruptions, col = gray(0.75))
  f <- loess(waiting ~ eruptions)
  i <- order(eruptions)
  lines(f$x[i], f$fitted[i])
})

with(exa, {
  plot(y ~ x, col = gray(0.75))
  lines(m ~ x)
  f <- loess(y ~ x) # default span = 0.75
  lines(f$x, f$fitted, lty = 2)
  # try smaller span (proportion of the range)
  f <- loess(y ~ x, span = 0.22)
  lines(f$x, f$fitted, lty = 5)
})

with(exb, {
  plot(y ~ x, col = gray(0.75))
  lines(m ~ x) 
  f <- loess(y ~ x) # default span = 0.75
  lines(f$x, f$fitted, lty = 2)
  # span = 1 means whole span of data (smoothest)
  f <- loess(y ~ x, span = 1)
  lines(f$x, f$fitted, lty = 5)
})

Pointwise confidence band is obtained by the local parametric fit for smoothing splines or loess.

ggplot(data = exa, mapping = aes(x = x, y = y)) +
  geom_point(alpha = 0.25) + 
  geom_smooth(method = "loess", span = 0.22) + 
  geom_line(mapping = aes(x = x, y = m), linetype = 2)

## `geom_smooth()` using formula 'y ~ x'

Simultaneous confidence band can be constructed by the mgcv package.

library(mgcv)

## Loading required package: nlme

## 
## Attaching package: 'nlme'

## The following object is masked from 'package:dplyr':
## 
##     collapse

## This is mgcv 1.8-31. For overview type 'help("mgcv-package")'.

ggplot(data = exa, mapping = aes(x = x, y = y)) +
  geom_point(alpha = 0.25) + 
  geom_smooth(method = "gam", span = 0.22, formula = y ~ s(x, k = 20)) + 
  geom_line(mapping = aes(x = x, y = m), linetype = 2)

Wavelets (not covered in this course)

In general, we approximate a curve by a family of basis functions \[ \hat f(x) = \sum_i c_i \phi_i(x), \] where the basis functions \(\phi_i\) are given and the coefficients \(c_i\) are estimated.
Ideally we would like the basis functions \(\phi_i\) to be (1) compactly supported (adatped to local data points) and (2) orthogonal (fast computing).
Orthogonal polynomials and Fourier basis are not compactly supported.
Cubic B-splines are compactly supported but not orthogonal.
Wavelets are both compactly supported and orthogonal.
Haar basis. The mother wavelet function on [0, 1] \[ w(x) = \begin{cases} 1 & x \le 1/2 \\ -1 & x > 1/2 \end{cases}. \] Next two members are defined on [0, 1/2) and [1/2, 1) by rescaling the mother wavelet to these two intervals. In general, at level \(j\) \[ h_n(x) = 2^{j/2} w(2^j x - k), \] where \(n = 2^j + k\) and \(0 \le k \le 2^j\).

library(wavethresh)

## Loading required package: MASS

## 
## Attaching package: 'MASS'

## The following object is masked from 'package:sm':
## 
##     muscle

## The following object is masked from 'package:dplyr':
## 
##     select

## WaveThresh: R wavelet software, release 4.6.8, installed

## Copyright Guy Nason and others 1993-2016

## Note: nlevels has been renamed to nlevelsWT

wds <- wd(exa$y, filter.number = 1, family = "DaubExPhase")
draw(wds)

plot(wds)

## [1] 6.12868 6.12868 6.12868 6.12868 6.12868 6.12868 6.12868 6.12868

Let’s only retain 3 levels of coefficients.

wtd <- threshold(wds, policy = "manual", value = 9999)
fd <- wr(wtd)
plot(y ~ x, exa, col = gray(0.75))
lines(m ~ x, exa)
lines(fd ~ x, exa, lty = 5, lwd = 2)

Or we can zero out only the small coefficients.

wtd2 <- threshold(wds)
fd2 <- wr(wtd2)
plot(y ~ x, exa, col = gray(0.75))
lines(m ~ x, exa)
lines(fd2 ~ x, exa, lty = 5, lwd = 2)

We may perfer to use continuous wavelet basis functions.

wds <- wd(exa$y, filter.number = 2, bc = "interval")
draw(filter.number = 2, family = "DaubExPhase")

plot(wds)

## [1] 3.3061 3.3061 3.3061 3.3061 3.3061 3.3061 3.3061 3.3061

Now we zero out small coefficients.

wtd <- threshold(wds)
fd <- wr(wtd)

## Warning in wr.wd(wtd): All optional arguments ignored for "wavelets on the
## interval" transform

plot(y ~ x, exa, col = gray(0.75))
lines(m ~ x, exa)
lines(fd ~ x, exa, lty=2)

For the Old Faithful data.

x <- with(faithful, (eruptions - min(eruptions)) / (max(eruptions) - min(eruptions)))
gridof <- makegrid(x, faithful$waiting)
wdof <- irregwd(gridof, bc="symmetric")
wtof <- threshold(wdof)
wrof <- wr(wtof)
plot(waiting ~ eruptions, faithful, col = grey(0.75))
with(faithful, lines(seq(min(eruptions), max(eruptions), len=512), wrof))