# [R] Issues with convolve

From: Vincent Goulet <vincent.goulet_at_act.ulaval.ca>
Date: Thu 21 Jul 2005 - 02:52:16 EST

Our function panjer() (code below) basically computes recursively the probability mass function of a compound Poisson distribution. When the Poisson parameter lambda is very large, the starting value of the recursive scheme --- the mass at 0 --- is 0 and the recursion fails. One way to solve this problem is to divide lambda by 2^n, apply the panjer() function and then convolve the result with itself n times.

We applied the panjer() function with a lambda such that the mass at 0 is just larger than .Machine\$double.xmin. We thus know that once this pmf is convoluted with itself, the first probabilities will be 0 (for the computer).

Here are the two issues we have with convolve():

1. The probabilities we know should be 0 are rather in the vicinity of 1E-19, as if convolve() could not "go lower". Using a hand made convolution function (not given here), we obtained the correct values. When probabilities get around 1E-12, results from convolve() and our home made function are essentially identical.
2. We obtained negative probabilities. More accurately, the same example returns negative probabilities under Windows, but not under Linux. We also obtained negative probabilities for another example under Linux, though.

We understand that convolve() computes the convolutions using fft(), but we are not familiar enough with the latter to assess if the above issues are some sort of bugs or normal behavior. In the latter case, is there is any workaround?

• EXAMPLES === panjer <- function(fx, lambda) { fx0 <- fx[1] fx <- fx[-1] r <- length(fx) cumul <- fs <- exp(-lambda * (1 - fx0)) repeat { x <- length(fs) m <- min(x, r) fs <- c(fs, sum((lambda * 1:m / x) * head(fx, m) * rev(tail(fs, m)))) if ((1-1E-8) < (cumul <- cumul + tail(fs, 1))) break } fs }

### Linux example ###
> R.version

_
platform i386-pc-linux-gnu

```arch     i386
os       linux-gnu
system   i386, linux-gnu
status
major    2
minor    1.0
year     2005
month    04
day      18
language R

> fs <- panjer(rep(0.2, 5), 600)          # compute pmf

> fs[1:10]                                # small values

[1] 3.456597e-209 4.147916e-207 2.530229e-205 1.045690e-203
```
[5] 3.292657e-202 8.422924e-201 1.822768e-199 3.431181e-198  [9] 5.733481e-197 8.637025e-196
> fsc <- convolve(fs, rev(fs), type="o") # convolution
> sum(fsc < 0) # no negatives under Linux
[1] 0
> fsc[1:10] # these should be 0
[1] 4.819870e-19 4.135471e-19 4.511455e-19 3.994485e-19 3.820177e-19  [6] 4.738272e-19 3.885447e-19 3.167399e-19 3.695636e-19 3.701792e-19
> sum(fsc) # no impact on the sum
[1] 1

### Windows example ###
> R.version

_
platform i386-pc-mingw32

```arch     i386
os       mingw32
system   i386, mingw32
status
major    2
minor    1.1
year     2005
month    06
day      20
language R

> fs <- panjer(rep(0.2, 5), 600)          # compute pmf

> fs[1:10]                                # small values

[1] 3.456597e-209 4.147916e-207 2.530229e-205 1.045690e-203
```
[5] 3.292657e-202 8.422924e-201 1.822768e-199 3.431181e-198  [9] 5.733481e-197 8.637025e-196
> fsc <- convolve(fs, rev(fs), type="o") # convolution
> sum(fsc < 0) # negatives under Windows
[1] 39
> fsc[1:10] # these should be 0
[1] 4.049552e-19 3.345584e-19 4.126913e-19 3.351211e-19 2.758626e-19  [6] 3.530111e-19 2.735041e-19 2.376711e-19 2.591287e-19 3.196405e-19
> sum(fsc) # no impact on the sum
[1] 1
```--
Vincent Goulet, Associate Professor
École d'actuariat
Université Laval, Québec
Vincent.Goulet_at_act.ulaval.ca   http://vgoulet.act.ulaval.ca

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