how can I use chis-quare - scipy

I have one expected value of C
and I have 20 observed values A and B which I expect A+xB=C
I want to find the best value for x how can I use chi-square to find the best one.
For example, how can I use scipy.stats.chisquare to find the best value for x.

Related

Passing values to a sparse matrix in MATLAB

Might sound too simple to you but I need some help in regrad to do all folowings in one shot instead of defining redundant variables i.e. tmp_x, tmp_y:
X= sparse(numel(find(G==0)),2);
[tmp_x, temp_y] = ind2sub(size(G), find(G == 0));
X(:)=[tmp_x, tmp_y];
(More info: G is a sparse matrix)
I tried:
X(:)=ind2sub(size(G), find(G == 0));
but that threw an error.
How can I achieve this without defining tmp_x, tmp_y?
A couple of comments with your code:
numel(find(G == 0)) is probably one of the worst ways to determine how many entries that are zero in your matrix. I would personally do numel(G) - nnz(G). numel(G) determines how many elements are in G and nnz(G) determines how many non-zero values are in G. Subtracting these both would give you the total number of elements that are zero.
What you are doing is first declaring X to be sparse... then when you're doing the final assignment in the last line to X, it reconverts the matrix to double. As such, the first statement is totally redundant.
If I understand what you are doing, you want to find the row and column locations of what is zero in G and place these into a N x 2 matrix. Currently with what MATLAB has available, this cannot be done without intermediate variables. The functions that you'd typically use (find, ind2sub, etc.) require intermediate variables if you want to capture the row and column locations. Using one output variable will give you the column locations only.
You don't have a choice but to use intermediate variables. However, if you want to make this more efficient, you don't even need to use ind2sub. Just use find directly:
[I,J] = find(~G);
X = [I,J];

Partitioning a number into a number of almost equal partitions

I would like to partition a number into an almost equal number of values in each partition. The only criteria is that each partition must be in between 60 to 80.
For example, if I have a value = 300, this means that 75 * 4 = 300.
I would like to know a method to get this 4 and 75 in the above example. In some cases, all partitions don't need to be of equal value, but they should be in between 60 and 80. Any constraints can be used (addition, subtraction, etc..). However, the outputs must not be floating point.
Also it's not that the total must be exactly 300 as in this case, but they can be up to a maximum of +40 of the total, and so for the case of 300, the numbers can sum up to 340 if required.
Assuming only addition, you can formulate this problem into a linear programming problem. You would choose an objective function that would maximize the sum of all of the factors chosen to generate that number for you. Therefore, your objective function would be:
(source: codecogs.com)
.
In this case, n would be the number of factors you are using to try and decompose your number into. Each x_i is a particular factor in the overall sum of the value you want to decompose. I'm also going to assume that none of the factors can be floating point, and can only be integer. As such, you need to use a special case of linear programming called integer programming where the constraints and the actual solution to your problem are all in integers. In general, the integer programming problem is formulated thusly:
You are actually trying to minimize this objective function, such that you produce a parameter vector of x that are subject to all of these constraints. In our case, x would be a vector of numbers where each element forms part of the sum to the value you are trying to decompose (300 in your case).
You have inequalities, equalities and also boundaries of x that each parameter in your solution must respect. You also need to make sure that each parameter of x is an integer. As such, MATLAB has a function called intlinprog that will perform this for you. However, this function assumes that you are minimizing the objective function, and so if you want to maximize, simply minimize on the negative. f is a vector of weights to be applied to each value in your parameter vector, and with our objective function, you just need to set all of these to -1.
Therefore, to formulate your problem in an integer programming framework, you are actually doing:
(source: codecogs.com)
V would be the value you are trying to decompose (so 300 in your example).
The standard way to call intlinprog is in the following way:
x = intlinprog(f,intcon,A,b,Aeq,beq,lb,ub);
f is the vector that weights each parameter of the solution you want to solve, intcon denotes which of your parameters need to be integer. In this case, you want all of them to be integer so you would have to supply an increasing vector from 1 to n, where n is the number of factors you want to decompose the number V into (same as before). A and b are matrices and vectors that define your inequality constraints. Because you want equality, you'd set this to empty ([]). Aeq and beq are the same as A and b, but for equality. Because you only have one constraint here, you would simply create a matrix of 1 row, where each value is set to 1. beq would be a single value which denotes the number you are trying to factorize. lb and ub are the lower and upper bounds for each value in the parameter set that you are bounding with, so this would be 60 and 80 respectively, and you'd have to specify a vector to ensure that each value of the parameters are bounded between these two ranges.
Now, because you don't know how many factors will evenly decompose your value, you'll have to loop over a given set of factors (like between 1 to 10, or 1 to 20, etc.), place your results in a cell array, then you have to manually examine yourself whether or not an integer decomposition was successful.
num_factors = 20; %// Number of factors to try and decompose your value
V = 300;
results = cell(1, num_factors);
%// Try to solve the problem for a number of different factors
for n = 1 : num_factors
x = intlinprog(-ones(n,1),1:n,[],[],ones(1,n),V,60*ones(n,1),80*ones(n,1));
results{n} = x;
end
You can then go through results and see which value of n was successful in decomposing your number into that said number of factors.
One small problem here is that we also don't know how many factors we should check up to. That unfortunately I don't have an answer to, and so you'll have to play with this value until you get good results. This is also an unconstrained parameter, and I'll talk about this more later in this post.
However, intlinprog was only released in recent versions of MATLAB. If you want to do the same thing without it, you can use linprog, which is the floating point version of integer programming... actually, it's just the core linear programming framework itself. You would call linprog this way:
x = linprog(f,A,b,Aeq,beq,lb,ub);
All of the variables are the same, except that intcon is not used here... which makes sense as linprog may generate floating point numbers as part of its solution. Due to the fact that linprog can generate floating point solutions, what you can do is if you want to ensure that for a given value of n, you could loop over your results, take the floor of the result and subtract with the final result, and sum over the result. If you get a value of 0, this means that you had a completely integer result. Therefore, you'd have to do something like:
num_factors = 20; %// Number of factors to try and decompose your value
V = 300;
results = cell(1, num_factors);
%// Try to solve the problem for a number of different factors
for n = 1 : num_factors
x = linprog(-ones(n,1),[],[],ones(1,n),V,60*ones(n,1),80*ones(n,1));
results{n} = x;
end
%// Loop through and determine which decompositions were successful integer ones
out = cellfun(#(x) sum(abs(floor(x) - x)), results);
%// Determine which values of n were successful in the integer composition.
final_factors = find(~out);
final_factors will contain which number of factors you specified that was successful in an integer decomposition. Now, if final_factors is empty, this means that it wasn't successful in finding anything that would be able to decompose the value into integer factors. Noting your problem description, you said you can allow for tolerances, so perhaps scan through results and determine which overall sum best matches the value, then choose whatever number of factors that gave you that result as the final answer.
Now, noting from my comments, you'll see that this problem is very unconstrained. You don't know how many factors are required to get an integer decomposition of your value, which is why we had to semi-brute-force it. In fact, this is a more general case of the subset sum problem. This problem is NP-complete. Basically, what this means is that it is not known whether there is a polynomial-time algorithm that can be used to solve this kind of problem and that the only way to get a valid solution is to brute-force each possible solution and check if it works with the specified problem. Usually, brute-forcing solutions requires exponential time, which is very intractable for large problems. Another interesting fact is that modern cryptography algorithms use NP-Complete intractability as part of their ciphertext and encrypting. Basically, they're banking on the fact that the only way for you to determine the right key that was used to encrypt your plain text is to check all possible keys, which is an intractable problem... especially if you use 128-bit encryption! This means you would have to check 2^128 possibilities, and assuming a moderately fast computer, the worst-case time to find the right key will take more than the current age of the universe. Check out this cool Wikipedia post for more details in intractability with regards to key breaking in cryptography.
In fact, NP-complete problems are very popular and there have been many attempts to determine whether there is or there isn't a polynomial-time algorithm to solve such problems. An interesting property is that if you can find a polynomial-time algorithm that will solve one problem, you will have found an algorithm to solve them all.
The Clay Mathematics Institute has what are known as Millennium Problems where if you solve any problem listed on their website, you get a million dollars.
Also, that's for each problem, so one problem solved == 1 million dollars!
(source: quickmeme.com)
The NP problem is amongst one of the seven problems up for solving. If I recall correctly, only one problem has been solved so far, and these problems were first released to the public in the year 2000 (hence millennium...). So... it has been about 14 years and only one problem has been solved. Don't let that discourage you though! If you want to invest some time and try to solve one of the problems, please do!
Hopefully this will be enough to get you started. Good luck!

Matlab: Multiple assignment through logical indexing

I am wondering if there is some way, how to multiple assign values to different variables according logical vector.
For example:
I have variables a, b, c and logical vector l=[1 0 1] and vector with values v but just for a and c. Vector v is changing its dimension, but everytime, it has the same size as the number of true in l.
I would like to assign just new values for a and c but b must stay unchanged.
Any ideas? Maybe there is very trivial way but I didn't figure it out.
Thanks a lot.
I think your problem is, that you stored structured data in an unstructured way. You assume a b c to have a natural order, which is pretty obvious but not represented in your code.
Replacing a b c with a vector x makes it a really easy task.
x(l)=v(l);
Assuming you want to keep your variable names, the simplest possibility I know would be to write a function:
function varargout=update(l,v,varargin)
varargout=varargin;
l=logical(l);
varargout{l}=v(l);
end
Usage would be:
[a,b,c]=update(l,v,a,b,c)

using find() in matlab

Say I have a matrix A of dimension Nx3 where N is the number of rows. A stores coordinates x,y,z. Now say I already have a set of known coordinates B = [x' y' z'] which I wanna look up in A. I wanna find out the number of which row index in A stores (x',y',z'). How can I do this? I am guessing I will have to use find()
you can use find, for example
find(A(:,1)==B(1) & A(:,2)==B(2) & A(:,3)==B(3))
will yield the index of the row\rows that match.
Try to get use to reading the documentation of Matlab, it is all there...
by the way, an alternative is to use ismember:
[~,id]=ismember(B,A,'rows')
the variable id will yield the index of the rows where B matched A.

Finding the maximum value of a function under uncertainty

I have three values X,Y and Z. These values have a range of values between 0 and 1 (0 and 1 included).
When I call a function f(X,Y,Z) it returns a value V (value between 0 and 1). My Goal is to choose X,Y,Z so that the returned value V is as close as possible to 1.
The selection Process should be automated and the right values for X,Y,Z are unknown.
Due to my Use Case it is possible to set Y and Z to 1 (the value 1 hasn't any influence on the output) and search for the best value of X.
After that I can replace X by that value and do the same for Y. Same procedure for Z.
How can I find the "maximum of the function"? Is there somekind of "gradient descend" or hill climbing algorithm or something like that?
The whole modul is written in perl so maybe there is an package for perl that can solve that problem?
You can use Simulated Annealing. Its a multi-variable optimization technique. It is also used to get a partial solution for the Travelling Salesperson problem. Its one of the search algorithms mentioned in Peter Norvig's Intro to AI book as well.
Its a hill climbing algorithm which depends on random variables. Also it won't necessarily give you the 'optimal' answer. You can also vary the iterations required by it as per your computational/time needs.
http://en.wikipedia.org/wiki/Simulated_annealing
http://www1bpt.bridgeport.edu/sed/projects/449/Fall_2000/fangmin/chapter2.htm
I suggest you take a look at Math::Amoeba which implements the Nelder–Mead method for finding stationary points on functions.