Bernstein Polynomials

The n+1 Bernstein polynomials that comprise a basis are: b(nu, n, x) = choose (n, nu) * x^nu * (1-x)^(n-nu)

Bernstein polynomials are a partition of unity.

Sum(from nu=0 to nu=n, b(nu,n,x)) = 1 ; for an arbitraty (!!!) x

this can be used to assess the error of calculations of the polynomials themself.

Bernstein expansion of an arbitraty function is:

B(n, f) at x is equal to Sum(from nu=0 to nu=n, f(nu/n) * b(nu,n,x)

Let’s define the function

## Bernstein polynomial Bern(nu, n, x)

# defined for nu <n and 0<x<1 ## add ifelse sometime

Bern <- function (nu, n, x) {
  f <- choose (n, nu) * x^nu * (1-x)^(n-nu) 
}

#it's just dbinom(nu, n, x, log = FALSE)

Let’s stage the scene and plot the first polynomials, that are well known

require('ggplot2')

# is a vector of a 101 values between 0 and 1 including the end points

x <- seq(0,1, .01)

n <- 0; nu <- 0
y <- Bern(nu,n, x)
z <- dbinom(nu, n, x, log = FALSE)

qplot(x, y, color = I("red"), 
      size = I(1), alpha = I(1/2), geom = c("point"))

TRUE. b(0,0,x) is identically equal to 1 (because there is just one polynomial and the partition of unity principle stands)

For n=1 (order one) the TWO polynomials must be x and 1-x (for the same reason)

n <- 1; nu <- 0  #change to nu = 1 if you want to see the second one
y <- Bern(nu,n, x)

qplot(x, y, color = I("red"), 
      size = I(1), alpha = I(1/2), geom = c("point"))

Order 2 (with THREE polynomials: (1-x)^2 ; 2x(1-x) ; x^2)

n <- 2; nu <- 2 #change to nu = 1,2 if you want to see the second one
y <- Bern(nu,n, x)
z <- dbinom(nu, n, x, log = FALSE)

qplot(x, y, color = I("brown"), 
      size = I(1), alpha = I(1/2), geom = c("point"))

Let’ check the precision at the end points

z <- c(Bern(0,2,0), Bern(1,2,0), Bern(2,2,0))
z
u <- c(Bern(0,2,1), Bern(1,2,1), Bern(2,2,1))
u

So far so good.

Now something more elaborate

n <- 10; nu <- 5 # similar to a binomial with size=10 and p=0.5
y <- Bern(nu,n, x)

qplot(x, y, color = I("blue"), 
      size = I(1), alpha = I(1/2), geom = c("point", "line"))

Very understandable. We are changing probabilities for the same structure of a discrete relationship.