I want to simulate some random variables distributed as a Variance Gamma.
I know the pdf ( http://en.wikipedia.org/wiki/Variance-gamma_distribution ) but I don't know the inverse of the cumulative function F: so I can't generate a random uniform variable U and compute x=F^(-1)(U).
I have to do this in MATLAB.
Thank you!
Stefano
The next natural alternative to look into is Von Neumann's "acceptance-rejection method".
If you can find a density g defined on the same space as your f such that
you know how to generate samples from g, and
f(x) <= cg(x), for some c, for all x,
then you are good to go.
If you search the literature, people must have done this. The VG is widely used in pricing options.
Following #Drake 's idea: for the first step you can use Marsaglia and Tsang’s Method from here.
This is the code to generate gamma random numbers:
function x=gamrand(alpha,lambda)
% Gamma(alpha,lambda) generator using Marsaglia and Tsang method
% Algorithm 4.33
if alpha>1
d=alpha-1/3; c=1/sqrt(9*d); flag=1;
while flag
Z=randn;
if Z>-1/c
V=(1+c*Z)^3; U=rand;
flag=log(U)>(0.5*Z^2+d-d*V+d*log(V));
end
end
x=d*V/lambda;
else
x=gamrand(alpha+1,lambda);
x=x*rand^(1/alpha);
end
Related
I have some troubles to understand how to implement the following MIQP (Mixed Integer Quadratic Programming) with linear constraints in Matlab calling Gurobi.
Let me explain in a schematic way my setting.
(1) x is the unknown and it is a column vector with size 225x1.
(2) The objective function (which should be minimised wrto x) looks like
which can be rewritten as
I have a Matlab script computing alpha, Q,c (Q,c sparse) when some_known_parameters1 are given:
function [alpha, Q,c]=matrix_objective_function(some_known_parameters1)
%...
end
(3) The constraints are linear in x, include equalities and inequalities, and are written in the form
I have a Matlab script computing Aeq,beq,Aineq,bineq (Aeq,Aineq sparse) when some_known_parameters2 is given:
function [Aeq,beq,Aineq,bineq]=constraints(some_known_parameters2)
%...
end
(4) Some components of x are restricted to be in {0,1}. I have a Matlab script producing a string of letters B (binary), C (continous) when some_known_parameters3 is given:
function type=binary_continuous(some_known_parameters3)
%...
end
Now, I need to put together (1)-(4) using Gurobi. I am struggling to understand how. I found this example but it looks very cryptic to me. Below I report some lines I have attempted to write, but they are incomplete and I would like your help to complete them.
clear
rng default
%Define some_known_parameters1,
some_known_parameters2,some_known_parameters3 [...]
%1) generate alpha,Q,c,Aeq,beq,Aineq,bineq,type with Q,c,Aeq, Aineq sparse
[alpha, Q,c]=matrix_objective_function(some_known_parameters1)
[Aeq,beq,Aineq,bineq]=constraints(some_known_parameters2)
type=binary_continuous(some_known_parameters3)
%2) Set up Gurobi
clear model;
model.A=[Aineq; Aeq];
model.rhs=full([bineq(:); beq(:)]);
model.sense=[repmat('<', size(Aineq,1),1); repmat('=', size(Aeq,1),1)];
model.Q=Q; %not sure?
model.alpha=alpha; %not sure?
model.c=c; %not sure?
model.vtype=type;
result=gurobi(model); %how do I get just the objective function here without the minimiser?
Questions:
(1) I'm not sure about
model.Q=Q;
model.alpha=alpha;
model.c=c;
I'm just trying to set the matrices of the objective function using the letters provided here but it gives me error. The example here seems to me doing
model.Q=Q;
model.obj=c;
But then how do I set alpha? Is it ignoring it because it does not change the set of solutions?
(2) How do I get as output stored in a matrix just the minimum value of the objective function without the corresponding x?
(1) You're right, there's no need to pass the constant alpha since it doesn't affect the optimal solution. Gurobi's MATLAB API only accepts sparse matrices. Furthermore model.obj is always the c vector in the problem statement:
model.Q = sparse(Q);
model.obj = c;
(2) To get the optimal objective value, you first need to pass your model to gurobi and solve it. Then you can access it via the objval attribute:
results = gurobi(model);
val = results.objval + alpha
Incredibly simple question, but I think I'm unable to come up with the correct terminology to google search it.
If I have a snippet of code that relies on three independent variables:
code(x,y,z)
That produces two values, i.e.:
output1, output2
How do I go about iterating like so (pseudocode):
for x
for y
for z
code(x,y,z)
end
end
end
And have data I can parse to generate 3D graphs such as
surf(x,y,output1)
A naive solution I came up with was just to create a bin of n length and then iterating one variable n times to come up with a 2D graph, i.e:
x_axis = zeros(1,25)
for m = 1:25
xm = x + 1
x_axis(m) = xm
code(x,y,z)
Even a referral to some documentation would be extremely helpful.
Thanks!
Brute force approach:
for x=[1:50]
for y=[1:50]
for z=[1:50]
result(y,x,z)=code(x,y,z);
end
end
end
More paradigmatic approach (in MATLAB) is to meshgrid it, and pump those in.
[XX,YY,ZZ]=meshgrid([1:50],[1:50],[1:50]);
result=code(XX,YY,ZZ);
we have measured data that we managed to determine the distribution type that it follows (Gamma) and its parameters (A,B)
And we generated n samples (10000) from the same distribution with the same parameters and in the same range (between 18.5 and 59) using for loop
for i=1:1:10000
tot=makedist('Gamma','A',11.8919,'B',2.9927);
tot= truncate(tot,18.5,59);
W(i,:) =random(tot,1,1);
end
Then we tried to fit the generated data using:
h1=histfit(W);
After this we tried to plot the Gamma curve to compare the two curves on the same figure uing:
hold on
h2=histfit(W,[],'Gamma');
h2(1).Visible='off';
The problem s the two curves are shifted as in the following figure "Figure 1 is the generated data from the previous code and Figure 2 is without truncating the generated data"
enter image description here
Any one knows why??
Thanks in advance
By default histfit fits a normal probability density function (PDF) on the histogram. I'm not sure what you were actually trying to do, but what you did is:
% fit a normal PDF
h1=histfit(W); % this is equal to h1 = histfit(W,[],'normal');
% fit a gamma PDF
h2=histfit(W,[],'Gamma');
Obviously that will result in different fits because a normal PDF != a gamma PDF. The only thing you see is that for the gamma PDF fits the curve better because you sampled the data from that distribution.
If you want to check whether the data follows a certain distribution you can also use a KS-test. In your case
% check if the data follows the distribution speccified in tot
[h p] = kstest(W,'CDF',tot)
If the data follows a gamma dist. then h = 0 and p > 0.05, else h = 1 and p < 0.05.
Now some general comments on your code:
Please look up preallocation of memory, it will speed up loops greatly. E.g.
W = zeros(10000,1);
for i=1:1:10000
tot=makedist('Gamma','A',11.8919,'B',2.9927);
tot= truncate(tot,18.5,59);
W(i,:) =random(tot,1,1);
end
Also,
tot=makedist('Gamma','A',11.8919,'B',2.9927);
tot= truncate(tot,18.5,59);
is not depending in the loop index and can therefore be moved in front of the loop to speed things up further. It is also good practice to avoid using i as loop variable.
But you can actually skip the whole loop because random() allows to return multiple samples at once:
tot=makedist('Gamma','A',11.8919,'B',2.9927);
tot= truncate(tot,18.5,59);
W =random(tot,10000,1);
I have 2 systems [H]: y(n) = x(2n), [G]: y(n)=x(n).x(n-1).x(n-2) – 2y(n-1)
1. How can i check if whether 2 systems above is LTI or not? i can't use num,den, and filter function for those function.
2. How can i simulate the output signal with input x(n) = (0.5).^ .*u(n) ?
Thank you for helping me.
And how can i simulate the impulse response of G and H ?
Please explain "." in G and ".^.*" in x. It is not really clear.
1.Question: See https://ccrma.stanford.edu/~jos/fp/Showing_Linearity_Time_Invariance.html
2.Question: The simulation is not difficult, because it is normal algebra. Firstly, the input signal is created:
u=rand(1,50); % 50 random values between 0 and 1
Then, x can be calculated.
x = 0.5.^u;
For H, the command is very simple:
y = x(1:2:end);
For G it is a little bit more difficult. The simplest solution would be a loop (or a function). An idea would be:
y2=zeros(1,length(x));
for n=3:length(x)
y2(n)=x(n)*x(n-1)*x(n-2)-2*y2(n-1);
end
An alternative solution would be with a Simulink model.
Let me ask whether using Matlab for my particular problem is nonsense or some people do the similar.
I have an initial sequence S(1), where each term is a 2D point.
I create a new sequence S(2) by inserting a new term point p
between each consecutive 2 term points p(i) and p(i+1).
Where p is a function f of 4 term points of nearest indices on S(2).
Namely,
p= f( p(i-1),p(i),p(i+1),p(i+2) )
And the function f is written in a C like style
but not in the pure style of matrix language.
In the same way , I repeat generating the new longer sequence S(i+1) up to S(m).
The above may be vague for you, but please give some advice.
I do not ask whether Matlab is the best choice for the problem , but whether no expert will use Matlab for such a problem or some will.
Thank you in advance.
It heavily depends on f. If f could be coded efficiently in Matlab or you are willing to spend the time to MEX it (Matlab C extension), then Matlab will perform efficiently.
The code could be vectorized like this:
f = #(x) mean(x,3);
m=3;
S{1}=[1,2,3;4,5,6];
for i=2:m
S{i} = cat(3,...
[[0;0] S{i-1}(:,1:end-2)],...
S{i-1}(:,1:end-1),...
S{i-1}(:,2:end),...
[S{i-1}(:,3:end) [0;0]]);
S{i} = [f(S{i}) [0;0]];
S{i} = cat(3,S{i-1},S{i});
S{i} = permute(S{i},[1 3 2]);
S{i} = S{i}(:,:);
S{i}(:,end)=[];
end
Yes, Matlab seems to be suitable for such a task. For the data structure of your list of sequences, consider using cell arrays. You could have S as a cell array, and S{1} would correspond to your S(1), and could again be a cell array of points, or a usual matrix if points are just pairs or triples of numbers.
As an alternative, Python in my opinion is particulary strong when it comes to all kind of sequences.