CODES / reliability / mc

Monte-Carlo

Syntax
Description
Parameters
Example
Mini Tutorial
References
See also

Syntax

res=CODES.reliability.mc(g,dim) compute a Crude Monte-Carlo estimate of the probability that $g(\mathbf{X})\leq 0$ . The dim random variables $\mathbf{X}$ are independent standard gaussian.
res=CODES.reliability.mc(g,x_mc) uses the user-defined Monte-Carlo sample x_mc.
res=CODES.reliability.mc(...,param,value) uses a list of parameters param and values value (see, parameter table).

Description

For a given limit state function $g$ and a joint PDF $\mathbf{f}_\mathbf{X}$ , the probability of failure is:

$P_f=\int_\Omega I[g(\mathbf{x})\leq 0]\mathbf{f}_\mathbf{X}(\mathbf{x})\mathrm d\mathbf{x}$

A Crude Monte Carlo simulation with $n$ samples approximates the integral as:

$\displaystyle P_f=E[I[g(\mathbf{x})\leq 0]]\approx\displaystyle\frac{1}{n}\sum_{i=1}^nI\left[g(\mathbf{x}^{(i)})\leq0\right]$

In general, the probability of failure is dependent on distribution hyper-parameters $\theta$ and deterministic variables $\mathbf{z}$ :

$P_f=\int I[g(\mathbf{x},\mathbf{z})\leq 0]\mathbf{f}_\mathbf{X}(\mathbf{x}|\theta)\mathrm d\mathbf{x}$

Derivatives of the probability of failure can be derived for hyper-parameters:

$\begin{array}{rcl}\displaystyle\frac{dP_f}{d\theta}&=&\displaystyle\frac{d}{d\theta}\int I[g(\mathbf{x},\mathbf{z})\leq 0]\mathbf{f}_\mathbf{X}(\mathbf{x}|\theta)d\mathbf{x}\\&=&\displaystyle\int I[g(\mathbf{x},\mathbf{z})\leq 0]\frac{d\ln \mathbf{f}_\mathbf{X}}{d\theta}f_\mathbf{X}(\mathbf{x}|\theta)d\mathbf{x}\\&\approx&\displaystyle\frac{1}{n}\sum_{i=1}^n I[g(\mathbf{x}^{(i)},\mathbf{z})\leq 0]\left.\frac{d\ln \mathbf{f}_\mathbf{X}}{d\theta}\right|_{\mathbf{x}^{(i)}}\end{array}$

and for deterministic variables (Lacaze et al., 2015):

$\begin{array}{rcl}\displaystyle\frac{dP_f}{d\mathbf{z}}&=&\displaystyle\frac{d}{d\mathbf{z}}\int I[g(\mathbf{x},z)\leq 0]f_\mathbf{X}(\mathbf{x}|\theta)d\mathbf{x}\\&=&\displaystyle-\int\frac{dg}{d\mathbf{z}}\delta[g(\mathbf{x},\mathbf{z})]f_\mathbf{X}(\mathbf{x}|\theta)d\mathbf{x}\\&\approx&\displaystyle-\frac{1}{n}\sum_{i=1}^n\tilde{\delta}[g(\mathbf{x}^{(i)},\mathbf{z})]\left.\frac{dg}{d\mathbf{z}}\right|_{\mathbf{x}^{(i)}}\end{array}$

Note that these sensitivities are cross-products of the probability of failure estimation and do not require any additional function call.

Parameters

param value Description

'sampler' function_handle, { [ ] } A function that returns realizations according to the desired distribution. For example, the default is @(N)normrnd(0,1,N,dim).

'Tinv' function_handle, { [ ] } An inverse transformation function that transform realizations from a standard gaussian space into the desired space. For example, for an exponential space Tinv=@(u)expinv(normcdf(u),1).

'CoV' positive, {Inf} Coefficient of variation to be achieved (in %), see Description.

'memory' positive integer, {1e6} When functions are vectorial in matlab, computing time is improved in exchange for memory use. To avoid memory crashes, no more than 'memory' samples are evaluated at once.

'alpha' positive, {0.05} Significance level for (1-alpha) confidence interval.

'limit' positive integer, {1e9} Maximum limit of samples allowed to be evaluated (in case CoV requested or actual Pf are too low).

'vectorial' logical, {false} Whether the limit state function g is vectorial. Vectorization should always be used if possible and 'vectorial' set to true.

'verbose' logical, {false} Defines the verbose level.

'store' logical, {false} Whether the Monte-Carlo sample should be stored and returned.

'lnPDF' function_handle Log of joint PDF as a function of (x,theta) for dPfdtheta (see Mini Tutorial for an example).

'dlnPDF' function_handle Derivative of log of joint PDF with respect to theta as a function of (x,theta)for dPfdtheta (see Mini Tutorial for an example).

'theta' real value Theta value for dPfdtheta (see Mini Tutorial for an example).

'nz' positive integer Number of deterministic variables for dPfdz. g is expected to return two outputs, g values and g gradients with respect to z (see Mini Tutorial for an example).

'frac' positive integer Alpha fraction for dPfdz (see Mini Tutorial for an example and Lacaze et al., 2015, for details).

'dirac' {'gauss'}, 'tgauss', 'poisson', 'sinc', 'bump' Dirac approximation function (see Mini Tutorial for an example and Lacaze et al., 2015, for details).

`param`	`value`	Description
'sampler'	`function_handle`, { [ ] }	A function that returns realizations according to the desired distribution. For example, the default is `@(N)normrnd(0,1,N,dim)`.
'Tinv'	`function_handle`, { [ ] }	An inverse transformation function that transform realizations from a standard gaussian space into the desired space. For example, for an exponential space `Tinv=@(u)expinv(normcdf(u),1)`.
'CoV'	positive, {`Inf`}	Coefficient of variation to be achieved (in %), see Description.
'memory'	positive integer, {1e6}	When functions are vectorial in matlab, computing time is improved in exchange for memory use. To avoid memory crashes, no more than 'memory' samples are evaluated at once.
'alpha'	positive, {0.05}	Significance level for (1-alpha) confidence interval.
'limit'	positive integer, {1e9}	Maximum limit of samples allowed to be evaluated (in case CoV requested or actual Pf are too low).
'vectorial'	logical, {`false`}	Whether the limit state function `g` is vectorial. Vectorization should always be used if possible and 'vectorial' set to `true`.
'verbose'	logical, {`false`}	Defines the verbose level.
'store'	logical, {`false`}	Whether the Monte-Carlo sample should be stored and returned.
'lnPDF'	`function_handle`	Log of joint PDF as a function of (x,theta) for dPfdtheta (see Mini Tutorial for an example).
'dlnPDF'	`function_handle`	Derivative of log of joint PDF with respect to theta as a function of (x,theta)for dPfdtheta (see Mini Tutorial for an example).
'theta'	real value	Theta value for dPfdtheta (see Mini Tutorial for an example).
'nz'	positive integer	Number of deterministic variables for dPfdz. g is expected to return two outputs, g values and g gradients with respect to z (see Mini Tutorial for an example).
'frac'	positive integer	Alpha fraction for dPfdz (see Mini Tutorial for an example and Lacaze et al., 2015, for details).
'dirac'	{'gauss'}, 'tgauss', 'poisson', 'sinc', 'bump'	Dirac approximation function (see Mini Tutorial for an example and Lacaze et al., 2015, for details).

Example

Compute and plot a generalized "max-min" sample

g=@CODES.test.lin;
res=CODES.reliability.mc(g,2,'vectorial',true);
disp(res)

          Pf: 0.0013
        beta: 3.0002
    LS_count: 1000000
         CoV: 2.7208
       CI_Pf: [0.0013 0.0014]
     CI_beta: [2.9843 3.0169]

Mini Tutorial

A mini tutorial of the capabilities of the mc function.

References

Lacaze et al. (2015): Lacaze, S., Brevault, L., Missoum, S., Balesdent, M. (2015). Probability of failure sensitivity with respect to decision variables. Structural and Multidisciplinary Optimization, First Published Online. DOI