Htydjtk

I have a matrix $b$ with elements:
$$b =
begin{pmatrix}
0.01 & 0.02 & cdots & 1 \
0.01 & 0.02 & cdots & 1 \
vdots& vdots & ddots & vdots \
0.01 & 0.02 & cdots & 1 \
end{pmatrix}
$$For which through a series of calculation which is vectorised, $b$ is used to calculate $a$ which is another matrix that has the same dimension/shape as $b$.
$$a =
begin{pmatrix}
3 & 5 & cdots & 17 \
2 & 6 & cdots & 23 \
vdots& vdots & ddots & vdots \
4 & 3 & cdots & 19 \
end{pmatrix}
$$
At this point it is important to note that the elements of $a$ and $b$ have a one to one correspondence. The different row values(let's call it $sigma$) $0.01, 0.02...$ are different parameters for a series of simulations that I'm running. Hence for a fixed value of say $sigma = 0.01$, the length of its column values correspond to the total number of "simulations" I'm running for that particular parameter. If you know python vectorisation then you'll start to understand what I'm doing.

It is known that higher the $sigma$, the more the number of simulations for that particular sigma will have a value higher than 5 i.e. more of the matrix element along a column will have value bigger than 5. Essentially what I'm doing is vectorising $N$(columns) different simulations for $M$(rows) different parameters. Now I wish to find out the value of $sigma$ for which the total number simulation that's bigger than 5, is bigger than 95% of the total simulation.

To put it more concisely, for a $sigma$ of 0.02, each simulation would have results of $$5, 6, ..., 3$$ with say a total simulation of $N$. So let $$kappa = sum{ (text{all the simulations that have values bigger than 5})},$$I wish to find out the FIRST $sigma$ for which
$$frac{kappa}{N} > 0.95*N$$
i.e. the FIRST $sigma$ for which the proportion of total experiment for which its value $>5$ is bigger than 95% of the total number of experiment.

The code that I have written is:

# say 10000 simulations for a particular sigma 

SIMULATION = 10000



# say 100 different values of sigma ranging from 0.01 to 1

# this is equivalent to matrix b in mathjax above

SIGMA = np.ones((EXPERIMENTS,100))*np.linspace(0.01, 1, 100)



def return_sigma(matrix, simulation, sigma):

    """

    My idea here is I put in sigma and matrix and total number of simulation. 

    Each time using np.ndenumerate looping over i and j to compare if the 

    element values are greater than 5. If yes then I add 1 to counter, if no 

    then continue. If the number of experiments with result bigger than 5 is 

    bigger than 95% of total number of experiment then I return that particular 

    sigma.

    """

    counter = 0

    for (i, j), value in np.ndenumerate(matrix):

        if value[i, j] > 5:

            counter+=1

        if counter/experiments > 0.95*simulation:

            break

        return sigma[0, j] # sigma[:, j] should all be the same anyway

"""Now this can be ran by:"""

print(return_sigma(a, SIMULATION, SIGMA))

which doesn't seem to quite work as I'm not well-versed with 2D slicing comprehension so this is quite a challenging problem for me. Thanks in advance.