## Nonparametric Density Estimation examples

### Histograms:

## Example 1:

wait.times <- faithful$waiting


# using the default number of bins:

hist(wait.times, freq=F, breaks="Scott")

# Scott's method uses the normal rule of thumb with s estimating sigma.

# Printing number of bins:

my.K <- length(hist(wait.times, freq=F, breaks="Scott")$counts); print(my.K)

## Trying different numbers of bins:

hist(wait.times, freq=F, breaks=seq(from=40, to=100, length=8) )  # Smaller K --> larger bin width h

hist(wait.times, freq=F, breaks=seq(from=40, to=100, length=16) )  # Larger K --> smaller bin width h

hist(wait.times, freq=F, breaks=seq(from=40, to=100, length=40) )  # Huge K --> Tiny bin width h


## Example 2:

wind.mph <- airquality$Wind

# using the default number of bins:

hist(wind.mph, freq=F, breaks="Scott")

# Printing number of bins:

my.h <- length(hist(wind.mph, freq=F, breaks="Scott")$counts); print(my.h)

## Trying different numbers of bins:

hist(wind.mph, freq=F, breaks=seq(from=0, to=22, length=8) )  # Smaller K --> larger bin width h

hist(wind.mph, freq=F, breaks=seq(from=0, to=22, length=16) )  # Larger K --> smaller bin width h

hist(wind.mph, freq=F, breaks=seq(from=0, to=22, length=40) )  # Huge K --> Tiny bin width h


######################################################

### Kernel density estimates:

## Example 1:

wait.times <- faithful$waiting

# using the default bandwidth:

plot(density(wait.times, bw="nrd"), main="kde: Waiting Times")  # The "nrd" method gives the rule of thumb 

# The default kernel is normal ("gaussian")

# We could specify another type of kernel:

plot(density(wait.times, kernel = "epanechnikov", bw="nrd"), main="kde: Waiting Times")

# Changing the bandwidth has more effect:
# We can multiply the "rule-of-thumb" bw by the value of "adjust"):

plot(density(wait.times, bw="nrd", adjust=0.6), main="kde: Waiting Times")  # smaller bandwidth


plot(density(wait.times, bw="nrd", adjust=1.4), main="kde: Waiting Times")  # larger bandwidth



plot(density(wait.times, bw="nrd", adjust=0.2), main="kde: Waiting Times")  # MUCH smaller bandwidth -> NOT GOOD!


plot(density(wait.times, bw="nrd", adjust=5), main="kde: Waiting Times")  # MUCH larger bandwidth -> NOT GOOD!


## Example 2:

wind.mph <- airquality$Wind

# using the default bandwidth:

plot(density(wind.mph, bw="nrd"), main="kde: Wind")  # The "nrd" method gives the rule of thumb 

# The default kernel is normal ("gaussian")

# We could specify another type of kernel:

plot(density(wind.mph, kernel = "epanechnikov", bw="nrd"), main="kde: Wind")

# Changing the bandwidth has more effect:
# We can multiply the "rule-of-thumb" bw by the value of "adjust"):

plot(density(wind.mph, bw="nrd", adjust=0.6), main="kde: Wind")  # smaller bandwidth


plot(density(wind.mph, bw="nrd", adjust=1.4), main="kde: Wind")  # larger bandwidth



plot(density(wind.mph, bw="nrd", adjust=0.2), main="kde: Wind")  # MUCH smaller bandwidth -> NOT GOOD!


plot(density(wind.mph, bw="nrd", adjust=5), main="kde: Wind")  # MUCH larger bandwidth -> NOT GOOD!