# Need to solve the following equation with three knowns and 2 unknowns.

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Prabhath Manuranga
on 21 May 2024 at 6:00

Commented: Prabhath Manuranga
on 30 May 2024 at 23:18

W0 = Wi + (1 - L)*Hi*gi

W0 and L is an unknowns.

Wi, Hi, and gi are the knowns with 172 values per each.

I want to find W0 and L and their standard deviations as well.

I want to get each values comes for W0 and L when the program runs.

But when this code runs i got 0 for standard deviation of both W10 and L. Is this code is right? Can anyone help me to solve this?

The following code was written.

Wi = Wi;

Hi = H_BM;

gi = g_mean;

% Define the objective function

fun = @(x) norm(Wi + (1 - x(1))*Hi.*gi - x(2));

% Set initial guess for L and W0

x0 = [0, 0];

% Solve the optimization problem

x = fminsearch(fun, x0);

% Extract the values of L and W0

L = x(1);

L

W0 = x(2);

W0

% Calculate standard deviation of W0 and L

std_W0 = std(W0);

std_W0

std_L = std(L);

std_L

##### 4 Comments

Rik
on 21 May 2024 at 8:40

### Answers (2)

Torsten
on 21 May 2024 at 8:51

H_BM = rand(100,1);

g_mean = rand(100,1);

Wi = rand(100,1);

Hi = H_BM;

gi = g_mean;

x = Hi.*gi;

y = Wi + Hi.*gi;

fitlm(x,y)

##### 20 Comments

Torsten
on 30 May 2024 at 10:00

Edited: Torsten
on 30 May 2024 at 12:34

Rename your variable "sum". "sum" is an inbuilt MATLAB function to sum elements, and you overwrite this function in your code.

If you are sure that sigma_e^2(V_n) equals what you compute as sum(n) (I doubt it because you don't sum anything when computing sum(n)), your Dw should be computed as

Dw = sqrt(sum(s(2:78)))

Here, I renamed you variable "sum" to "s" and used the MATLAB function "sum" to sum elements in an array.

According to the mathematical formulae, the sigma1 and beta1 are lower triangular matrices, and your "sum" variable is computed by summing both sigma1 and beta1 over their columns, adding the result and multiply it by (GM/alpha)^2.

Rupesh
on 23 May 2024 at 4:44

Hi Prabhat,

I understand that you want to calculate the values of W0 and L, and their standard deviations, from your given data using an optimization approach. The initial code calculates W0 and L but results in zero standard deviations because it operates on single scalar values instead of distributions (set of values) and single scalar values don’t have standard deviation associated with them.

To solve this, we can use a bootstrapping method. By repeatedly sampling your data and performing the optimization for each sample, we can obtain distributions of W0 and L. From these distributions, we can calculate the standard deviations.

Wi = Wi; % Your known values

Hi = H_BM; % Your known values

gi = g_mean; % Your known values

% Number of bootstrap samples

num_samples = 100;

% Arrays to store the results

L_values = zeros(1, num_samples);

W0_values = zeros(1, num_samples);

% Perform bootstrap sampling

for i = 1:num_samples

% Randomly sample indices with replacement

sample_indices = randsample(length(Wi), length(Wi), true);

% Sample the data

Wi_sample = Wi(sample_indices);

Hi_sample = Hi(sample_indices);

gi_sample = gi(sample_indices);

% Define the objective function for this sample

fun = @(x) norm(Wi_sample + (1 - x(1))*Hi_sample.*gi_sample - x(2));

% Set initial guess for L and W0

x0 = [0, 0];

% Solve the optimization problem for this sample

x = fminsearch(fun, x0);

% Store the results

L_values(i) = x(1);

W0_values(i) = x(2);

end

% Calculate mean and standard deviation of L and W0

L_mean = mean(L_values);

L_std = std(L_values);

W0_mean = mean(W0_values);

W0_std = std(W0_values);

% Display the results

disp(['L mean: ', num2str(L_mean)]);

disp(['L std: ', num2str(L_std)]);

disp(['W0 mean: ', num2str(W0_mean)]);

disp(['W0 std: ', num2str(W0_std)]);

This script repeatedly samples your data and runs the optimization to build up distributions for W0 and L, allowing for the calculation of their standard deviations. You can refer to below documentation to get clear understanding of inbuilt sample bootstrapping.

Hope it helps!

Thanks

##### 0 Comments

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