### Archive

Posts Tagged ‘code’

## A von Mises variate…

Inspired from a mail that came along the previous random generation post the following question rised :

How to draw random variates from the Von Mises distribution?

First of all let’s check the pdf of the probability rule, it is $f(x):=\frac{e^{b \text{Cos}[y-a]}}{2 \pi \text{BesselI}[0,b]}$, for $-\pi \leq x\leq \pi$.

Ok, I admit that Bessels functions can be a bit frightening, but there is a work around we can do. The solution is a Metropolis algorithm simulation. It is not necessary to know the normalizing constant, because it will cancel in the computation of the ratio. The following code is adapted from James Gentle’s notes on Mathematical Statistics .

n <- 1000
x <- rep(NA,n)
a <-1
c <-3
yi <-3
j <-0

i<-2
while (i < n) {
i<-i+1
yip1 <- yi + 2*a*runif(1)- 1
if (yip1 < pi & yip1 > - pi) {
if (exp(c*(cos(yip1)-cos(yi))) > runif(1)) yi <- yip1
else yi <- x[i-1]
x[i] <- yip1
}
}
hist(x,probability=TRUE,fg = gray(0.7), bty="7")
lines(density(x,na.rm=TRUE),col="red",lwd=2)

Categories: probability, statistics

## The distribution of rho…

There was a post here about obtaining non-standard p-values for testing the correlation coefficient. The R-library

SuppDists

deals with this problem efficiently.

library(SuppDists)

plot(function(x)dPearson(x,N=23,rho=0.7),-1,1,ylim=c(0,10),ylab="density")

legend("topleft", col=c("black","steelblue","red","green"),lty=1,
legend=c("rho=0.7","rho=0","rho=-.2","rho=.9"))</pre>

This is how it looks like,

Now, let’s construct a table of critical values for some arbitrary or not significance levels.

q=c(.025,.05,.075,.1,.15,.2)
xtabs(qPearson(p=q, N=23, rho = 0, lower.tail = FALSE, log.p = FALSE) ~ q )
# q
#     0.025      0.05     0.075       0.1      0.15       0.2
# 0.4130710 0.3514298 0.3099236 0.2773518 0.2258566 0.1842217

We can calculate p-values as usual too…

1-pPearson(.41307,N=23,rho=0)
# [1] 0.0250003
Categories: statistics

## \pi day!

It’s π-day today so we gonna have a little fun today with Buffon’s needle and of course R. A well known approximation to the value of $\pi$ is the experiment tha Buffon performed using a needle of length,$l$. What I do in the next is only to copy from the following file the function estPi and to use an ergodic sample plot… Lame,huh?

estPi<- function(n, l=1, t=2) {
m <- 0
for (i in 1:n) {
x <- runif(1)
theta <- runif(1, min=0, max=pi/2)
if (x < l/2 * sin(theta)) {
m <- m +1
}
}
return(2*l*n/(t*m))
}

So, an estimate would be…

## A quicky..

February 23, 2010 1 comment

If you’re (and you should) interested in principal components then take a good look at this. The linked post will take you by hand to do everything from scratch. If you’re not in the mood then the dollowing R functions will help you.

An example.

# Generates sample matrix of five discrete clusters that have
# very different mean and standard deviation values.
z1 <- rnorm(10000, mean=1, sd=1);
z2 <- rnorm(10000, mean=3, sd=3);
z3 <- rnorm(10000, mean=5, sd=5);
z4 <- rnorm(10000, mean=7, sd=7);
z5 <- rnorm(10000, mean=9, sd=9);
mydata <- matrix(c(z1, z2, z3, z4, z5), 2500, 20, byrow=T,
dimnames=list(paste("R", 1:2500, sep=""), paste("C", 1:20, sep="")))

# Performs principal component analysis after scaling the data.
# It returns a list with class "prcomp" that contains five components:
#   (1) the standard deviations (sdev) of the principal components,
#   (2) the matrix of eigenvectors (rotation),
#   (3) the principal component data (x),
#   (4) the centering (center) and
#   (5) scaling (scale) used.
pca <- prcomp(mydata, scale=T)
Read more...
Categories: statistics

## The truncated Poisson

February 22, 2010 1 comment

A common model for counts data is the Poisson. There are cases however that we only record positive counts, ie there is a truncation of 0. This is the truncated Poisson model.

To study this model we only need the total counts and the sample size. This comes from the sufficient statistic principle as the likelihood is

$logL=-n-\frac{e^{-\lambda } n}{1-e^{-\lambda }}+\frac{T}{\lambda }$,

where $T=\sum _X$. Let’s set $T=160$ and  $n=50$.

sum.x=160
n=50
library(maxLik)

loglik <- function(theta, n, sum.x) {
- n* log(exp(theta) - 1) + sum.x * log(theta)
}

$\frac{e^{-2 \lambda } n}{\left(1-e^{-\lambda }\right)^2}+\frac{e^{-\lambda } n}{1-e^{-\lambda }}-\frac{T}{\lambda ^2}$.

## Uh!

Didn’t know this…

a<-structure(c(25,34,12,5),.Names=c("0","2","4","7+"))
> data
0  2  4 7+
25 34 12  5


It’s becoming clear that I have learned R in the most unstructured way…I always do it in two stages :ashamed:

> data<-c(25,34,12,5)
> names(data)<-c("0","2","4","7+")
> data
0  2  4 7+
25 34 12  5

It’s really useful to wrap it all in a single function.

Attribute Specification

Description:

‘structure’ returns the given object with further attributes
set.

Usage:

structure(.Data, ...)

Arguments:

.Data: an object which will have various attributes attached to it.

...: attributes, specified in ‘tag=value’ form, which will be
attached to data.

Details:

Adding a class ‘"factor"’ will ensure that numeric codes are
given integer storage mode.

For historical reasons (these names are used when deparsing),
attributes ‘".Dim"’, ‘".Dimnames"’, ‘".Names"’,
‘".Tsp"’ and ‘".Label"’ are renamed to ‘"dim"’,
‘"dimnames"’, ‘"names"’, ‘"tsp"’ and ‘"levels"’.

It is possible to give the same tag more than once, in which case
the last value assigned wins.  As with other ways of assigning
attributes, using ‘tag=NULL’ removes attribute ‘tag’ from
‘.Data’ if it is present.

Categories: statistics Tags: , ,

## A merry wolfram xmas!

Christmas coming & time for fun!  Wolfram’s demonstrations give you a sense of the holiday season along some nice demonstrations. I particulary like the “Ornamental Holiday Decoration

Manipulate[
Module[{level0, level1, level2},
level0 = C[spikey, 0];
level1 =
Flatten[daughterPolyhedra[C[spikey, 0], {d1, ω1, s1}, ρ s1]];
level2 =
Flatten[daughterPolyhedra[#, {d2, ω2, s2}, ρ  s1 s2] & /@
Cases[level1, _C]];
Graphics3D[{EdgeForm[],
{Red, egc[level0]}, gg1 = {col1, egc[level1]},
gg2 = {col2, ControlActive[{}, egc[level2]]},
Directive[GrayLevel[0.2], Specularity[colc, 12]],
ecc[{level1, level2}]}, Boxed -> False,
ImageSize -> {400, 400}]],
"layer 1:",
{{d1, 1.5, "distance"}, -3, 3, ImageSize -> Tiny},
{{ω1, 0, "rotation"}, -Pi, Pi, ImageSize -> Tiny},
{{s1, 1/2, "size"}, 0, 1, ImageSize -> Tiny},
{{col1, Yellow, "color"}, Blue, ControlType -> None},
Delimiter,
"layer 2:",
{{d2, 1, "distance"}, -3, 3, ImageSize -> Tiny},
{{ω2, 0, "rotation"}, -Pi, Pi, ImageSize -> Tiny},
{{s2, 1/2, "size"}, 0, 1, ImageSize -> Tiny},
{{col2, Green, "color"}, Green, ControlType -> None},
Delimiter,
"connectors:",
{{ρ, 0.3, "radius"}, 0, 1, ImageSize -> Tiny},
{{colc, Brown, "color"}, Yellow, ControlType -> None},
AutorunSequencing -> {1, 3, 5, 7},
Initialization :> {
spikey =
MapAt[DeveloperToPackedArray,
MapAt[DeveloperToPackedArray, N[PolyhedronData["Spikey"]][[1]],
1], {2, 1}];
mirrorRotateAndShift[gc_GraphicsComplex,
n_, {distance_, angle_, size_}, ρ_] :=
With[{aux =
mirrorRotateAndShiftCF[gc[[1]], gc[[1, n]], distance, angle,
size]}, {C[GraphicsComplex[aux, gc[[2]]], n],
Cylinder[{gc[[1, n]], aux[[n]]}, ρ]}];
mirrorRotateAndShiftCF =
Compile[{{vertices, _Real, 2}, {rPoint, _Real, 1}, distance,
angle, size}, Read more...
Categories: infos