Kernel density estimation

### 20.4.22 Kernel density estimation

The kernel_density or kde command performs kernel density estimation (KDE)1. kernel_density takes a sample, optionally restricted to an interval [a,b], and obtains an estimate f of the (unknown) probability density function f from which the samples are drawn. The function f is defined by:

f(x)= |
1
n h
n
∑
i=1
| K | ⎛ ⎜ ⎜ ⎝ |
x−Xi
h
| ⎞ ⎟ ⎟ ⎠ | , (13)

where K is the Gaussian kernel

K(u)= |
1
√ |
2 π
| exp | ⎛ ⎜ ⎜ ⎝ | − |
1
2
| u2 | ⎞ ⎟ ⎟ ⎠

and h is the positive real parameter called the bandwidth.

- kernel_density takes one mandatory argument and a sequence of optional arguments:
  - L, a list of samples L=[X1,X2,…,Xn].
  - Optionally, a sequence of options from:
    - output=type (or Output=type to specify the form of the return value f, where type can be one of:
      - exact, to return f as the sum of Gaussian kernels, i.e. as the right side of (13), which is usable only when the number of samples is relatively small (up to few hundreds).
      - piecewise, to return f as a piecewise expression obtained by the spline interpolation of the specified degree (by default, the interpolation is linear) on the interval [a,b] segmented to the specified number of bins.
      - list (the default), to return f in discrete form, as a list of values f(a+k b−a/M−1) for k=0,1,…,M, where M is the number of bins.
    - bandwidth=value, to specify the bandwidth. value can be one of:
      - h, a positive real number.
      - select (the default), to have the bandwidth selected using a direct plug-in method,
      - gauss (or normal or normald) to use Silverman’s rule of thumb for selecting bandwidth (this method is fast but the results are close to optimal ones only when f is approximately normal).
    - bins=n for a positive integer n (by default 100), the number of bins for simplifying the input data. Only the number if samples in each bin is stored. Bins represent the elements of an equidistant segmentation of the interval S on which KDE is performed. This allows evaluating kernel summations using convolution when output is set to piecewise or list, which significantly lowers the computational burden for large values of n (say, few hundreds or more). If output is set to exact, this option is ignored.
    - a..b or range=[a,b] or x=a..b for real numbers a and b, to specify the interval [a,b] on which KDE is performed. If an identifier x is specified, it is used as the variable of the output. If the range endpoints are not specified, they are set to a=min1≤ i≤ n Xi−3 h and b=max1≤ i≤ nXi+3 h (unless output is set to exact, in which case this option is ignored).
    - interp=n for an integer n (by default 1), which specifies the degree of the spline interpolation, ignored unless output is set to piecewise.
    - spline=n for an integer n, which sets option to piecewise and interp to n.
    - eval=x0 to only return the value f(x0) (this cannot be used with output set to list).
    - x, an unassigned identifier (by default x) to use as the variable of the output.
    - exact, the same as output=exact.
    - piecewise, the same as output=piecewise.
- kernel_density(L ⟨,options ⟩) returns the function f given in (13).

#### Examples

kernel_density([1,2,3,2],bandwidth=1/4,exact)

e |
− |
⎛ ⎝ | x−1.0 | ⎞ ⎠ | 2
0.125
| +e |
− |
⎛ ⎝ | x−2.0 | ⎞ ⎠ | 2
0.125
| +e |
− |
⎛ ⎝ | x−3.0 | ⎞ ⎠ | 2
0.125
| +e |
− |
⎛ ⎝ | x−2.0 | ⎞ ⎠ | 2
0.125
2.50662827463

f:=unapply(normald(4,1,x)/2+normald(7,1/2,x)/2,x)

x↦ |
1
√ |
2 π
| e |
− |
⎛ ⎝ | x−4 | ⎞ ⎠ | 2
2
2
| + |
1
√
2 π
4
| e |
−2 | ⎛ ⎝ | x−7 | ⎞ ⎠ | 2
2

X:=randvar(f,range=0..10,1000):; S:=sample(X,1000):; F:=kernel_density(S,piecewise):; plotfunc([f(x),F],x=0..10,color=[red,blue])

The exact density is drawn in red.

kernel_density(S,bins=50,spline=3,eval=4.75)

0.14655478136 |  |  |  |  |  |  |  |  |  |

time(kernel_density(sample(X,1e5),piecewise))

| ⎡ ⎣ | 0.17,0.1653323 | ⎤ ⎦

S:=sample(X,5000):; sqrt(int((f(x)-kde(S,piecewise))^2,x=0..10))

0.0269841239243 |  |  |  |  |  |  |  |  |  |

S:=sample(X,25000):; sqrt(int((f(x)-kde(S,bins=150,piecewise))^2,x=0..10))

0.0144212781377 |  |  |  |  |  |  |  |  |  |