Examples of "gaussians"
These Gaussians are plotted in the accompanying figure.
which illustrates the above time evolution of Gaussians.
The above result could also be generalized in formula_145mixture of Gaussians.
The sum in formula_123 is then just a sum of Fourier Transforms of Gaussians, and so
The difference of Gaussians (DoG) of the image formula_2 is the function
in two dimensions, it is the product of two such Gaussians, one in each dimension:
In two dimensions, it is the product of two such Gaussians, one per direction:
The sum in formula_29 is then just a sum of Fourier Transforms of Gaussians, and so
Financial return distributions in the New York Stock Exchange, NASDAQ and elsewhere have been interpreted as q-Gaussians.
which represents an image convoluted to the difference of two Gaussians, which approximates a Mexican Hat function.
Moreover, it could represent even more complex behavior when the output of the states is represented as mixture of two or more Gaussians, in which case the probability of generating an observation is the product of the probability of first selecting one of the Gaussians and the probability of generating that observation from that Gaussian.
The solution of Eqs. ()-() is a complete set of permutations of all initial coordinates appearing in the Gaussians,
many probabilities formula_35 we care about, and thus will not be able to avoid its estimation. In this case formula_35 is proportional to the squared absolute value of the permanent of the "M×M" matrix formula_45 of i.i.d. Gaussians, smuggled inside formula_46 These arguments bring us to the first conjecture of the hardness proof of approximate boson sampling problem – the permanent-of-Gaussians conjecture:
For reasons of convenience, many quantum chemistry programs work in a basis of Cartesian Gaussians even when spherical Gaussians are requested, as integral evaluation is much easier in the cartesian basis, and the spherical functions can be simply expressed using the cartesian functions.
where formula_4 and formula_5 are constants and {formula_6} are independent standard Gaussians. Volatility is driven by the first-order latent Markov state vector:
Because we are using Gaussians, the expected value is equivalent to the maximum likely value, and so this is also a form of maximum likelihood estimation.
The relation between the difference of Gaussians operator and the Laplacian of the Gaussian operator (the Mexican hat wavelet) is explained in appendix A in Lindeberg (2015).
The aim is to estimate the unknown parameters representing the "mixing" value between the Gaussians and the means and covariances of each:
1) PM3 uses two Gaussian functions for the core repulsion function, instead of the variable number used by AM1 (which uses between one and four Gaussians per element);
Case 1: A match is found with one of the K Gaussians. For the matched component, the update is done as follows