Usually, to illustrate the difference between S3 and S4 classes in R, I mention glm (from base) and vglm (from VGAM) that provide similar outputs, but one is based on S3 codes, while the second one is based on S4 codes. Another way to illustrate is to manipulate distributions.

Consider the case where we want to sum (independent) random variables. For instance two lognormal distribution. Let us try to compute the median of the sum.

The distribution function of the sum of two independent (positive) random variables is F_{S_2}(x)=\int_0^x F_{X_1}(x-y)dF_{X_2}(x)

pSum2 = function(x) integrate(function(y) plnorm(x-y,1,2)*dlnorm(y,2,1),0,x)$value |

Let us visualize that cumulative distribution function

vx=seq(0.1,50,by=.1) vy=Vectorize(pSum2)(vx) plot(vx,vy,type="l",ylim=c(0,1)) abline(h=.5,lty=2) |

Let us find an upper bound to compute (in a decent time) quantiles

pSum2(350) [1] 0.99195 |

and then use the uniroot function to inverse that function

qSum = function(u) uniroot(function(x) Vectorize(pSum2)(x)-u, interval=c(0,350))$root vu=seq(.01,.99,by=.01) vv=Vectorize(qSum)(vu) |

The median is here

qSum(.5) [1] 14.155 |

Why not consider the sum of three (independent) distributions ? Its cumulative distribution function can be writen using our previous function F_{S_3}(x)=\int_0^x F_{S_2}(x-y)dF_{X_3}(x)

pSum3 = function(x) integrate(function(y) pSum2(x-y)*dlnorm(y,2,2),0,x)$value |

If we look at some values we good

pSum3(4) [1] 0.015624 |

pSum3(5) Error in integrate(function(y) plnorm(x - y, 1, 2) * dlnorm(y, 2, 1), : maximum number of subdivisions reached |

So obviously, there are computational issues here.

Let us consider the following alternative expression F_{S_3}(x)=\int_0^x F_{X_3}(x-y)dF_{S_2}(x). Of course, it is necessary here to compute the density of the sum of two variables

dSum2 = function(x) integrate(function(y) dlnorm(x-y,1,2)*dlnorm(y,2,1),0,x)$value pSum3 = function(x) integrate(function(y) dlnorm(x-y,2,2)*dSum2(y),0,x)$value |

Again, let us compute some values

pSum3(4) [1] 0.0090285 pSum3(5) [1] 0.01186 |

This one seems to work quite well. But it is just an illusion.

pSum3(9) Error in integrate(function(y) dlnorm(x - y, 1, 2) * dlnorm(y, 2, 1), : maximum number of subdivisions reached |

Clearly, with those S3-type functions, it wlll be complicated to run computations with 3 variables, or more.

Let us consider distributions in the S4-type format of the following package

library(distr) X1 = Lnorm(mean=1,sd=2) X2 = Lnorm(mean=2,sd=1) S2 = X1+X2 |

To compute the median, we simply have to use

distr::q(S2)(.5) [1] 14.719 |

We can also visualize it easily

plot(q(S2)) |

which looks (very) close to what we got, manually. But here, it is also possible to work with the sum of 3 (independent) random variables

X3 = Lnorm(mean=2,sd=2) S3 = X1+X2+X3 |

To compute the median, use

distr::q(S3)(.5) [1] 33.208 |

The function is here

plot(q(S3)) |

Cite this blog post

Arthur Charpentier (2018, January 24). Using convolutions (S3) vs distributions (S4). *Freakonometrics*. Retrieved June 23, 2024, from https://doi.org/10.58079/ov9j