[778] | 1 | /*! |
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| 2 | \page userguide_pdf BDM Use - Probability density functions |
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| 3 | |
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| 4 | This section serves as an introduction to basic elements of the BDM: probability density functions, pdfs. |
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| 5 | |
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| 6 | The tutorial is written in for the BDM toolbox, if you are interested in use of C++ classes see class reference pages. |
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| 7 | |
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| 8 | \section ugpdf_create Creation of pdfs |
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| 9 | |
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| 10 | In BDM toolbox, a pdf is specified by matlab structure, e.g. |
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| 11 | \code |
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| 12 | Nab.class= 'enorm<ldmat>'; |
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| 13 | Nab.mu = [3,2]; |
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| 14 | Nab.R = eye(2); |
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| 15 | Nab.rv = RV({'a','b'}); |
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| 16 | \endcode |
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| 17 | Which encodes information \f$ f(a,b) = \mathcal{N}(mu=[3;2],R=eye(2))\f$. |
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| 18 | \li the keyword "enorm\<ldmat\>" means "Unconditional Normal distribution with covariance matrix in L'DL form", othe rpossibilities are: |
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| 19 | "enorm\<chmat\>" for Choleski decomposition, and "enorm\<fsqmat\>" for full (non-decomposed) matrices. |
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| 20 | \li mu denotes mean value |
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| 21 | \li R denotes variance (written in full matrix regardles of the used decomposition), |
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| 22 | \li rv denotes names assigned to the variables. RV is more complicated structure, but here it is sufficient to use default values. |
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| 23 | \li rv is an optional parameter, some operations do not need it, such as sampling or evaluation of moments |
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| 24 | |
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| 25 | For generating samples try: |
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| 26 | \code |
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| 27 | >> M=epdf_sample_mat(Nab,4); |
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| 28 | \endcode |
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| 29 | which should return 4 samples of the Nab distribution. |
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| 30 | |
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| 31 | For evaluation of mean and variance: |
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| 32 | \code |
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| 33 | >> Nab_m=epdf_mean(Nab); |
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| 34 | >> Nab_v=epdf_variance(Nab); |
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| 35 | \endcode |
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| 36 | |
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| 37 | Other distributions are created analogously, see ??? for their list and parameters??? |
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| 38 | Sampling and evaluation of moments are done by exactly the same functions as for the normal density. |
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| 39 | |
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| 40 | \section ug_pdf_marg Marginalization and conditioning |
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| 41 | |
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| 42 | Basic operations on pdfs are marginalization and conditioning, which are provided by mex functions edpf_marginal and epdf_condition, respectively. |
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| 43 | |
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| 44 | This operation does require the rv parametetr to be fully specified. If it isn't, it will fail with the following message |
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| 45 | \code |
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| 46 | --- fill in the message ---- |
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| 47 | \endcode |
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| 48 | |
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| 49 | If rv is correctly specified, marginal pdf of Nab on variable "a" is obtained by: |
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| 50 | \code |
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| 51 | Na = epdf_marginal(Nab,RV('a')); |
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| 52 | \endcode |
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| 53 | |
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| 54 | Similarly for conditional: |
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| 55 | \code |
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| 56 | Na_b = epdf_condition(Nab,RV('a')); |
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| 57 | Nb_a = epdf_condition(Nab,RV('b')); |
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| 58 | \endcode |
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| 59 | |
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| 60 | \section ugpdf_cond Conditioned densities |
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| 61 | Note that the result of conditioning is of type "mlnorm\<ldmat\>" which is a special case of pdf with variables in condition, specifically |
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| 62 | \f[ f(a|b) = \mathcal{N}(A*b+const, R)\f] |
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| 63 | i.e. "Normal distributed pdf with mean value as linear function of variable b". |
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| 64 | |
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| 65 | This type of pdfs differ from previously used type is the way of use. For example, it is not possible to sample directly form such density, |
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| 66 | it is necessary to specify what is the value of variable in condition. |
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| 67 | |
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| 68 | That is why a different function is used: |
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| 69 | \code |
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| 70 | Smp=pdf_samplecond_mat(Na_b, 10) |
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| 71 | \endcode |
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| 72 | |
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| 73 | The conditioned and Unconditioned pdf may be combined together in the chain rule. The chain rule can be of two different types: conditioned or unconditioned, i.e.: |
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| 74 | \f[ f(a,b)=f(a|b)f(b), OR, f(a,b|c)=f(a|b)f(b|c)\f] |
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| 75 | Thus it is differently encoded as: |
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| 76 | \code |
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| 77 | fab.class = 'eprod'; |
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| 78 | fab.pdfs = {fa_b, fb}; |
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| 79 | |
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| 80 | fab_c.class = 'mprod'; |
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| 81 | fab_c.pdfs = {fa_b, fb_c}; |
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| 82 | \endcode |
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| 89 | |
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| 90 | */ |
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