Class prob_dens_mdim_biv_gaussian (o2scl)¶
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template<class vec_t = boost::numeric::ublas::vector<double>>
class prob_dens_mdim_biv_gaussian : public o2scl::prob_dens_mdim<boost::numeric::ublas::vector<double>>¶ A bivariate gaussian probability distribution.
For a two-dimensional gaussian, given a mean \( ( \mu_x, \mu_y ) \) and a covariance matrix
\[\begin{split} \Sigma = \left( \begin{array}{cc} \sigma_x^2 & \rho \sigma_x \sigma_y \\ \rho \sigma_x \sigma_y & \sigma_y^2 \\ \end{array} \right) \end{split}\]the PDF is\[ pdf(x,y) = \left(2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}\right)^{-1} \exp \left\{ - \frac{1}{2 (1-\rho^2)} \left[ \frac{(x-\mu_x)^2}{\sigma_x^2} + \frac{(y-\mu_y)^2}{\sigma_y^2} - \frac{2 \rho (x-\mu_x)(y-\mu_y)}{\sigma_x \sigma_y} \right] \right\} \](taken from the Wikipedia page on the “Multivariate normal
distribution”).
The function o2scl::prob_dens_mdim_biv_gaussian::contour() gives a point on the contour line for a fixed value of the PDF given an angle \( \theta \). In particular, it solves
\[ c = \mathrm{pdf}(r \cos \theta + \mu_x, r \sin \theta + \mu_y ) \]for the radius \( r \) and then stores the values \( r \cos \theta + \mu_x \) and \( r \sin \theta + \mu_y \) in the reference parameters namedxandy. Thus this function can be used to map out the full contour by selecting values for \( \theta \in [0,2 \pi] \).The function o2scl::prob_dens_mdim_biv_gaussian::level_fixed_integral() gives the value of the PDF for which the integral inside the corresponding contour is some fraction of the total integral (which is always 1). Given a fraction \( f \), the argument of the exponential is related to the inverse of the cumulative distribution function for the chi-squared probability distribution for two degrees of freedom for \( 1-f \). For a fraction \( f \), the value \( \chi^2 \) (i.e. the Mahalanobis distance) is \( \chi^2 = -2 \log (1-f) \) and then the value of the PDF for the corresponding contour is \( pdf(x,y) = \left(2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}\right)^{-1} \exp (-\chi^2/2) \) .
Public Functions
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inline prob_dens_mdim_biv_gaussian()¶
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inline virtual ~prob_dens_mdim_biv_gaussian()¶
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inline prob_dens_mdim_biv_gaussian(const prob_dens_mdim_biv_gaussian &pdmbg)¶
Copy constructor.
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inline prob_dens_mdim_biv_gaussian &operator=(const prob_dens_mdim_biv_gaussian &pdmbg)¶
Copy constructor with operator=.
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inline void set(double x_cent, double y_cent, double x_std, double y_std, double covar)¶
Set the properties of the distribution.
Note
If \( |\rho|\geq 1 \) this function will call the error handler.
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inline void get(double &x_cent, double &y_cent, double &x_std, double &y_std, double &covar)¶
Get the properties of the distribution.
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inline virtual double level_fixed_integral(double integral)¶
Return the contour level corresponding to a fixed integral.
This function returns the value of the PDF (as returned by pdf()) for which the integral inside the contour line for that value is equal to the specified integral.
For example, in the standard two-dimensional normal case, if
\[ \int_0^{a}~r dr~\exp\left( - \frac{r^2}{2} \right) \]is equal to the value given inintegral, then this function returns the value\[ \frac{1}{2 \pi} \exp \left( - \frac{a^2}{2} \right) \, . \]
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inline void ellipse_frac_integral(double integral, double &x_cent, double &y_cent, double &x_wid, double &y_wid, double &angle)¶
Return the properties of an ellipse based on the integral.
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inline prob_dens_mdim_biv_gaussian()¶