If you want to choose randomly your next holidays destination, you are likely to process in a way which is certainly biased. Especially if you choose randomly the latitude and the longitude. A bit like they do in this lovely advertising (For those of you who do not speak French, this is about a couple who have won the national gamble prize and have to decide their next travel. The husband randomly picks Australia and the wife is complaining : "Not again!"). So let me help you to choose your next travel!


If we were able to generate uniformly distributed variables on [0 ; 1], we could easily generate variables on other spaces such as [0 ; 1] x [0 ; 1], which is a square of side 1. It can be simply done by using two independant variables X1 and X2, uniformly distributed on [0 ; 1], and by considering X = (X1 , X2). Then X is a random variable evenly distributed on the square [0 ; 1] x [0 ; 1]

However, this generation may be a bit more complicated if we work with more complicated shapes, such as a sphere for example. Indeed, for a sphere of a given radius, say R = 1, it is possible to project the variables of a square on the sphere.

The first method I thought of, is wrong. I was tempted to generate two variables Y1 and Y2 where Y1 follows a uniform on  [0 ; Pi] and Y2 follows a uniform on [-Pi ; Pi]. In other words :
- Y1 = Pi * X1
- Y2 = 2Pi * (X2 - 0.5).
 Then I wanted to consider (Y1, Y2) as the spherical coordinated of the sphere.

However, this method does NOT generate a uniform on the sphere. Indeed, it has a tendency to over generate north and south poles. The reason is simple, this method generates, in average the same amount of variable in each latitude. North and South poles are of smaller area than Equator. Therefore, the closer we are from the Equator latitude the less variables there are.

On the following graph, we observe a high density of point in the "middle" of the sphere, this is the North Pole. It shows that this method does not offer a uniform distribution. By the way, the graph is computed with the library rgl, which provides a display device for 3 dimensions objects. And then a little code provided in the rgl help allows to move automatically the device and take a snapshot for each view. You can eventually generate a GIF on gifmake (like for the map in Spatial segregation in cities - An explanation by a neural network model).
A wrong method to generate evenly distributed variables on the sphere. The North Pole and the South Pole are over represented.

Many methods have been proposed to generate evenly distributed random variables on a sphere. We propose one of them here. We consider the couple z = (u,v) defined as :

- u = 2 * Pi * X1
- v = arc-cos(2 * (X2 - 0.5))

In this case, a theorem shows that z = (u , v) generates evenly distributed variables. It can be observed on the next graph. There is no irregularity in the distribution of the random variables.
A correct simulation of a uniform distribution on a sphere. There is no over represented area.

Other methods exist. I like this one since it is really simple and uses a uniform distribution at the beginning. The idea of this post is to show that generating a uniform distribution can be adapted to many shapes and cases. However, to do so, a previous analytical study has to be done to find the correct transformation. 

So this method would help you to avoid being too many times in the chilly places such as North Pole and South Pole since it does not overrepresent the extreme latitudes.

The program (R) :

# import package to plot in 3D
install.packages("rgl", dependencies = TRUE)
library(rgl)

################################################################
# Uniform distribution in a square
################################################################

size = 10000
x1 = runif(size)
x2 = runif(size)

# the option pch = '.' change the symbol for the graph into dot.
# cex = 2 doubles the size of the dots
plot(x1,x2, col = 'blue', pch = '.', cex = 2)

################################################################
# Wrong solution for the sphere
################################################################

y1 = pi * x1
y2 = 2* pi * (x2-0.5)
y = matrix(0, nrow = 2, ncol = size)
y[1,] = y1
y[2,] = y2
plot(y1, y2)

# This function transform the spherical coordinates into cartesian coordinates
sphereToCartesian = function(matrice){
  x= matrix(0,nrow = 3, ncol = length(matrice[1,]))
  x[1,] = sin(matrice[2,]) * cos(matrice[1,])
  x[2,] = sin(matrice[2,]) * sin(matrice[1,])
  x[3,] = cos(matrice[2,])
  return(x)
}

a = sphereToCartesian(y)
plot3d(x = a[1,], y = a[2,], z = a[3,])

#you should enlarge the device window, before running this, if you want to have a meaningful graph
rgl.bringtotop()
rgl.viewpoint(0,20)

for (i in 1:45) {
  rgl.viewpoint(i,20)
  filename <- paste("pic",i,".png",sep="")
  rgl.snapshot(filename, fmt="png")
}


################################################################
# Correct solution for the sphere
################################################################

uniformSphere = function(length){
  x1 = runif(length, 0,1)
  x2 = runif(length, 0,1)
  u = 2*pi*x1
  v = acos(2*x2- 1)
  z =matrix(0, ncol = length, nrow = 2)
  z[1,] = u
  z[2,] = v
  return(z)
}

z = uniformSphere(size)
b = sphereToCartesian(z)
plot3d(x = b[1,], y = b[2,], z = b[3,])


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Labels: random randomness simulation sphere spherical uniform distribution
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The financial market is not only made of stock options. Other financial products enable market actors to target specific aims. For example, an oil buyer like a flight company may want to cover the risk of increase in the price of oil. In this case it is possible to buy on the financial market what is known as a "Call" or a "Call Option".

A Call Option is a contract between two counterparties (the flight company and a financial actor). The buyer of the Call has the opportunity but not the obligation to buy a certain  quantity of a certain product (called the underlying) at a certain date (the maturity) for a certain price (the strike).

I found a golden website. The blog of Esteban Moro. He uses R to work on networks. In particular he has done a really nice code to make some great videos of networks. This post is purely a copy of his code. I just changed a few arguments to change colors and to do my own network.

To create the network, I used the  Barabási-Albert algorithm that you can find at the end of the post on the different algorithms for networks. Igraph is the library which has been used.
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As you have certainly seen now, I like working on artificial neural networks. I have written a few posts about models with neural networks (Models to generate networks, Want to win to Guess Who and Study of spatial segregation).

Unfortunately, I missed so far a nice and pleasant aspect of networks : its graphical approach. Indeed, plots of neural networks are often really nice and really useful to understand the network.

Sometimes such a graph can point out some characteristics of the network.
1

I already talked about networks a few times in this blog. In particular, I had this approach to explain spatial segregation in a city or to solve the Guess Who? problem. However, one of the question is how to generate a good network. Indeed, I aim to study strategy to split a network, but I need first to work with a realistic neural network. I could have downloaded data of a network, but I'd rather study the different models proposed to generate neural networks.

The function apply() is certainly one of the most useful function. I was scared of it during a while and refused to use it. But it makes the code so much faster to write and so efficient that we can't afford not using it. If you are like me, that you refuse to use apply because it is scary, read the following lines, it will help you. You want to know how to use apply() in general, with a home-made function or with several parameters ? Then, go to see the following examples.
1

Have you ever played the board game "Guess who?". For those who have not experienced childhood (because it might be the only reason to ignore this board game), this is a game consisting in trying to guess who the opponent player is thinking of among a list of characters - we will call the one he chooses the "chosen character". These characters have several characteristics such as gender, having brown hair or wearing glasses.

If you want to choose randomly your next holidays destination, you are likely to process in a way which is certainly biased. Especially if you choose randomly the latitude and the longitude. A bit like they do in this lovely advertising (For those of you who do not speak French, this is about a couple who have won the national gamble prize and have to decide their next travel. The husband randomly picks Australia and the wife is complaining : "Not again!").
4

My previous post is about a method to simulate a Brownian motion. A friend of mine emailed me yesterday to tell me that this is useless if we do not know how to simulate a normally distributed variable.

My first remark is: use the rnorm() function if the quality of your simulation is not too important (Later, I'll try to explain you why the R "default random generation" functions are not perfect). However, it may be fun to generate a normal distribution from a simple uniform distribution.

The Brownian motion is certainly the most famous stochastic process (a random variable evolving in the time). It has been the first way to model a stock option price (Louis Bachelier's thesis in 1900).

The reason why is easy to understand, a Brownian motion is graphically very similar to the historical price of a stock option.
1

The merge of two insurance companies enables to curb the probability of ruin by sharing the risk and the capital of the two companies.

For example, we can consider two insurance companies, A and B. A is a well known insurance company with a big capital and is dealing with a risk with a low variance. We will assume that the global risk of all its customers follow a chi-square distribution with one degree of freedom.
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