how to define the factorial if data set is non-integer

dear all

I have the data set as attached, and the formula as picture below.

I want to calculat the probabilty certain point using the formula as below.

My problem is how to define my factorial, while my data is non-integer.

clc
clear all
close all

%% TO READ DATA SET,
sz = [128 128 64];
fname = 'jaszak18092023tc10n1bckg10mbq10jutaf.a00';
fid = fopen(fname);
data = fread(fid,'*float'); % assuming uint
fclose(fid);
% this is blindly devectorized 
% may still be transposed, but i can't tell due to symmetry
% note that data is unit-scale single-precision float
data = reshape(data,sz); 


%% TO GET THE MEAN DATA SET
lamda = mean(mean(mean(data)));


% lamda = 0.0223;  % lamda REPRESENT mean data set, I got from my data set
e  = 2.718;  % Euler number (exp)
k = 0.439173; % LET SAY my point to calculate the P (probability) (THE VALUE K FROM MY DATA SET, THE COORDINATE IS , [65 64 1])
j=factorial(k);


P = (lamda^k * exp(-lamda)/factorial(k));

Error using factorial
N must be an array of real non-negative integers.

ANYONE CAN HELP ME?

[Bruno Luong]

The Poisson probability is defined only for integer non negative number k, (it's generate a variable usually interpret as the number of some specific events at frequency lambda occurs in a fixed unit time interval.

If someone tells you "there is 0.439173 answer you your question between 9AM and 10AM" does ot make sense tou you?

https://builtin.com/data-science/poisson-process

[Dyuman Joshi]

Note -

Use

lambda = mean(data,'all');
e  = exp(1);

instead of

lamda = mean(mean(mean(data)));
e  = 2.718; 

[mohd akmal masud]

Dear @Dyuman Joshi,

If use function gamma, still the k value is non-integer

[Dyuman Joshi]

"If use function gamma, still the k value is non-integer"

I don't understand what you are saying.

[DGM]

k is the input to a function which does not modify k, so if k is non-integer, it's not because of the funciton being called.

k = 3; % an integer
kexc = gamma(k+1)

kexc = 6

k % it's still the same

k = 3

k = 3.1; % a non-integer
kexc = gamma(k+1)

kexc = 6.8126

k % it's still the same

k = 3.1000

[dpb]

"LET SAY my point to calculate the P (probability)..."

The probability of what? As @Bruno Luong notes, the Poisson distribution is only defined for integer-valued k so it can't be appropriate for whatever it is you're trying to do.

We would have to know what the data represent and what it is that is a probablilty to compute in order to have a chance of selecting some continuous distribution that might be suitable. As is, there is no solution to the Q?

[mohd akmal masud]

Dear @dbp,

The pixel number represent the number of foton counts

[dpb]

Well, a number of photon(?) counts can't be non-integer, so that doesn't make any sense.

And, you've not told us what it is a probability of that you're trying to calculate...

Remember, we have zero context of what you're trying to do; you know all the background info, we don't know anything except what you tell us.

Don't be so cryptic; expand in full on the problem..."help us help you!"

[Bruno Luong]

The data is 128x128x64 show number between min=0 and max=0.4522. The mean is 0.0223414.

It looks like the data is from some photo detector that is proportional to the energy, thus number of photons, but not directly the number of photons that hits the pixel (which as @dpb remark must be whole number).

[mohd akmal masud]

Dear @Bruno Luong, you are very correct

[Dyuman Joshi]

Both the answers provided here do not understand nor acknowledge the problem with what OP is trying to do.

The comments above by Bruno and dpb have pointed out what the mistake is.

Answer 1

[Akshat Dalal]

Hi Akmal,

The factorial operation is only defined for integral values. However, there are extensions to the concept of factorial for non-integer values. One such extension is the gamma function, which can be used to define the factorial for non-integer values. Mathematically,

You can use the inbuilt ‘gamma’ function in MATLAB to calculate factorials of non-integral values.

For more information on the gamma function, please refer the following documentation - https://en.wikipedia.org/wiki/Gamma_function

For more information on using the gamma function in MATLAB, please refer the following documentation - https://in.mathworks.com/help/releases/R2023b/symbolic/gamma.html

Answer 2

[Francis Mike John Camogao]

The error you are encountering is because the `factorial` function expects the input `k` to be a non-negative integer, but you are providing it with a floating-point value. To calculate the factorial of a floating-point number, you need to modify the code. You can use the `gamma` function, which calculates the factorial of a real number (including floating-point values). Here's how you can modify your code to fix the error:

clc
clear all
close all
%% TO READ DATA SET,
sz = [128 128 64];
fname = 'jaszak18092023tc10n1bckg10mbq10jutaf.a00';
fid = fopen(fname);
data = fread(fid,'*float'); % assuming uint
fclose(fid);
% this is blindly devectorized 
% may still be transposed, but I can't tell due to symmetry
% note that data is unit-scale single-precision float
data = reshape(data, sz);
%% TO GET THE MEAN DATA SET
lamda = mean(mean(mean(data)));
% lamda = 0.0223;  % lamda REPRESENT mean data set, I got from my data set
e = 2.718;  % Euler number (exp)
k = 0.439173; % LET SAY my point to calculate the P (probability) (THE VALUE K FROM MY DATA SET, THE COORDINATE IS , [65 64 1])

% Calculate factorial using gamma function
j = gamma(k + 1);

P = (lamda^k * exp(-lamda) / j);

% Display the result
fprintf('P = %f\n', P);

I also got and implemented the ideas atop my response.

In this code, I replaced the `factorial` function with the `gamma` function to calculate the factorial of `k`, which is a floating-point number. This should resolve the error you were encountering. I as well modifed, added some lines.