# Using convolutions (S3) vs distributions (S4)

**R-english – Freakonometrics**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

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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))`

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**R-english – Freakonometrics**.

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