4 Maximum likelihood method
========================================




4.1 Principle
---------------

Let's note :math:`{p}_{f}({\sigma }_{w})` the probability density associated with the cumulative rupture probability :math:`{P}_{f}({\sigma }_{w})`:

:math:`{p}_{f}({\sigma }_{w})=\frac{m}{{\sigma }_{u}}{(\frac{{s}_{w}}{{\sigma }_{u}})}^{m-1}\text{exp}\left[-{(\frac{{\sigma }_{w}}{{\sigma }_{u}})}^{m}\right]`

The quantity :math:`{p}_{f}({\sigma }_{w})d{\sigma }_{W}` is equal to the probability of breaking a test piece subjected to a stress corresponding to a stress of WEIBULL included in the interval :math:`\left[{\sigma }_{W},{\sigma }_{W}+d{\sigma }_{W}\right]`. The probability that all the test tubes in the base have broken is therefore:


.. _RefEquation 4.1-1:

:math:`p(m,{\sigma }_{u})d{\sigma }_{w}=\prod _{i}{p}_{f}({\sigma }_{W}^{i})d{\sigma }_{w}`, eq 4.1-1

p being the likelihood function. The maximum likelihood method then consists in choosing the parameters of the WEIBULL model so that the likelihood function defined by [:ref:`éq 4.1-1 <éq 4.1-1>`] (in practice rather its natural logarithm) is maximum.




4.2 Resolution
-----------------

An iterative method is used again. Again, at iteration :math:`(k)`, (:math:`{m}_{k},{\sigma }_{u(k)}`) as well as the :math:`{\sigma }_{W(k)}^{i}` are known. For these fixed WEIBULL constraint values, the maximization of :math:`\text{Log}(p)` leads to a new pair (:math:`{m}_{k+1},{\sigma }_{u(k+1)}`) given by:


.. _RefEquation 4.2-1:

:math:`f({m}_{k+1})=\frac{N}{{m}_{k+1}}+\sum _{i=1}^{i=N}\text{Log}({\mathrm{\sigma }}_{W(k)}^{i})-N\frac{\sum _{i=1}^{i=N}({\mathrm{\sigma }}_{W(k)}^{i}{)}^{{m}_{k+1}}\text{Log}({\mathrm{\sigma }}_{W(k)}^{i})}{\sum _{i=1}^{i=N}{({\mathrm{\sigma }}_{W(k)}^{i})}^{{m}_{k+1}}}=0` eq 4.2-1

:math:`{\sigma }_{(k+1)}=\sqrt[{m}_{k+1}]{\frac{1}{N}\sum _{i=1}^{i=N}({\sigma }_{W(k)}^{i}{)}^{{m}_{k+1}}}`. eq 4.2-2

At each step, the resolution of [:ref:`éq 4.2-1 <éq 4.2-1>`] can be achieved using Newton's method, the :math:`f(m)` gradient being given by:

:math:`\frac{\text{df}}{\text{dm}}(m)\text{=-}N\left(\frac{1}{{m}^{2}}+\frac{\left(\sum _{i=1}^{i=N}({\mathrm{\sigma }}_{W}^{i}{)}^{m}{\text{Log}}^{2}({\mathrm{\sigma }}_{W}^{i})\right)\left(\sum _{i=1}^{i=N}({\mathrm{\sigma }}_{W}^{i}{)}^{m}\right)-{\left(\sum _{i=1}^{i=N}({\mathrm{\sigma }}_{W}^{i}{)}^{m}\text{Log}({\mathrm{\sigma }}_{W}^{i})\right)}^{2}}{{\left(\sum _{i=1}^{i=N}({\mathrm{\sigma }}_{W}^{i}{)}^{m}\right)}^{2}}\right)`.

**Note**:

*If* :math:`m` *is set,* :math:`{\sigma }_{u(k+1)}` is given by [:ref:`4.2-2 <4.2-2>`]. On the other hand, if :math:`{\sigma }_{u}` *is fixed,* :math:`{m}_{k+1}` is no longer a solution to [:ref:`4.2-1 <4.2-1>`] but to:

:math:`f({m}_{k+1})=\frac{N}{{m}_{k+1}}+\sum _{i=1}^{i=N}\text{Log}(\frac{{\sigma }_{W(k)}^{i}}{{\sigma }_{u}})(1-(\frac{{\sigma }_{W(k)}^{i}}{{\sigma }_{u}}{)}^{{m}_{k+1}})=0` *.*

*This equation can be solved again using Newton's method, the gradient now being given by:*

:math:`\frac{\text{df}}{\text{dm}}(m)\text{=-}\frac{N}{{m}^{2}}-\sum _{i=1}^{i=N}(\frac{{\sigma }_{W}^{i}}{{\sigma }_{u}}{)}^{m}{\text{Log}}^{2}(\frac{{\sigma }_{W}^{i}}{{\sigma }_{u}})` *.*