* Added integration with sympy and adjusted variables.
* Integration now works.
* Add 4th and second order moments.
* check equation 56 with 20, 11 and 4
* check equation 56 with 20, 11 and 4
* remove unnecessary symbols
* Move documents and rename
* Rename document
* Add files and start to implement the analytic integral
* Add functions
* Add symbols and functions. Tried analytic integration with simpler functions but the triple integral still does not work. For integrating numerically I like to use lambdify and then numpy but therefore the symbols have to be renamed because lambdify does not like backslashes.
* Numerical evaluation of the first, second, third, and fourth order moments of the triple Gaussian with limits -10 to 10.
* Added mixture fractions to the Gaussian.
* Convert results to dataframe for better readability.
* Check results for third Gaussian, also adjusted the third Gaussian.
* Tidy up code.
* Changed name of file. Added function for evaluating the fraction w_prime_4_bar divided by w_prime_2_bar_squared which is parameterized.
* Added columns to the dataframe to check if the numerical evaluation of the 4th order moment is equal to the parameterized evaluation of the 4th order moment.
* Adjusted symbols and added the constant for lambda_w.
* Added documentation.
* Added check for w_prime_4_bar calculated just with the parameters.
* Improved functions and improved calculating the moments from a dataframe.
* Tidied up code
* Tidied up code
* Reordered document
* Added some files and restructured the document.
* Added a file to compute another function
* Added a file for computing the 4th order moment based on the parameters and checking this with the numeric integrals.
* fixed notebook
* Checked some equations analytically.
* Added document for checking eq_26
* Added document for checking eq_24
* Added document for checking eq_33
* Fixed some parameters
* finished document for computing the moments numerically.
* Verified(?) eq 75
* Changed values for ?verifying? eq 75
* Adjusted alphas and deltas but did not run it to the end yet.
* Ran the code with the adjusted alphas and deltas.
* Calculated the integral based on the marginal of G_3
* Ran the notebook again.
* Added Checking for eq 80. Not done yet.
* Added new document.
* Calculated w_bar
* Finished 1 dim integral.
* Added calculation for sigma_lambda_w and lambda_w.
* Added calculation for sigma_lambda_w and lambda_w.
* Added third Gaussian.
* Calculated 4th order moment.
* Added files and split up evaluation of the moments for w
* Verified all moments for w.
* Fixed a typo.
* Set up document for another integral.
* Working on the integral for w_prime_theta_l_prime_bar.
* Working on the integral for w_prime_theta_l_prime_bar.
* Did the check for w_prime_theta_l_prime_bar.
* Did the check for w_prime_theta_l_prime_bar.
* Did the check for w_prime_theta_l_prime_bar with rational numbers.
* Add numerical difference.
* Add numerical difference.
* Finished the check for w_prime_theta_l_prime
* Finished the check for w_prime_squared_theta_l_prime
* Finished the check for theta_l_prime_3_bar
* Finished the check for w_prime_theta_l_prime_squared_bar
* Did the checks again with arbitrary distinct values.
* Did the checks again with arbitrary distinct values.
* Add check for theta_l_prime_2_bar.
* Calculated $\beta$.
* Corrected a minor mistake.
* Backed out beta from the theta_prime_l_bar equation.
* Checked the beta equation numerically.
* Printed all NaN values.
* Reformat Check.
* Reformat Check.
* Reformat Check.
* Reformat Check.
* Reformat Check.
* Reformat Check.
* Reformat Check.
* Reformat Check.
* Reformat Check.
* Changed Heading.
* Changed the limits of integration.
* Delete doc/.ipynb_checkpoints directory
* Rearrange symbols
* Added beta to the dataframe.
* Added diagnostic values to the dataframe.
* Fixed typo.
* rearranged dataframe
* rearranged dataframe
* rearranged dataframe
* rearranged dataframe
* rearranged dataframe
* Computed the triple integral again.
* Rearranged symbols.
* Working on the trivariate Normal.
* Fixed numerical integration for triple gaussian
* plot relationship between delta and lambda_w
* plotted proposed relationship
* Rearranged files and computed everything again.
* Added README.md
* Added file to README.md
* Add function for plotting
* Finished numerical check for w_prime_q_t_prime_theta_l_prime_bar_beta.ipynb
* Finished numerical check for w_prime_q_t_prime_theta_l_prime_bar_beta.ipynb
* Finished numerical check for theta_l_prime_q_t_prime_bar.ipynb
* Worked on plotting
* Worked on README.md
* replace q_t with r_t
* replace q_t with r_t
* replace q_t with r_t
* Add explanation
* Reformat code
* Corrected a file
* Corrected a file
* Worked on checking w_prime_r_t_prime_theta_l_prime_bar_beta.ipynb
* Simplified creating of dataframe.
* - Added calculation for w_prime_r_t_prime_theta_l_prime_bar_E.ipynb and improved diagnostics output.
* Changed all sigma_lamda to sigma_3 and plotted \lambda vs \delta
* Plotted \lambda vs \delta with the "initial condition"
* Plotted \lambda vs \delta with the "initial condition"
* Update README.md
* Reran jupyter notebooks
* Refactored code
* Reran notebook
* Rearranged README.md
* Rearranged README.md
* Remove files
---------
Co-authored-by: Vincent Larson <vlarson@users.noreply.github.com>
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