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remove rho=NULL in pk.test add best practice add cases in which use pk.test
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| #' - **Goodness-of-Fit Tests**: The software implements one, two, and | ||
| #' *k*-sample tests for goodness of fit, offering an efficient and | ||
| #' mathematically sound way to assess the fit of probability distributions. | ||
| #' Expanded capabilities include supporting tests for uniformity on the | ||
| #' *d*-dimensional Sphere based on Poisson kernel densities. Our tests are | ||
| #' Our tests are | ||
| #' particularly useful for large, high dimensional data sets where the | ||
| #' assessment of fit of probability models is of interest. Specifically, we | ||
| #' offer tests for normality, as well as two- and *k*-sample tests, where | ||
| #' testing equality of two or more distributions is of interest, that is | ||
| #' \eqn{H_0: F_1 = F_2} and \eqn{H_0: F_1 = \ldots = F_k} respectively. | ||
| #' The proposed tests perform well in terms of level and power for contiguous | ||
| #' alternatives, heavy tailed distributions and in higher dimensions. | ||
| #' alternatives, heavy tailed distributions and in higher dimensions. \cr | ||
| #' Expanded capabilities include supporting tests for uniformity on the | ||
| #' *d*-dimensional Sphere based on the Poisson kernel, exhibiting excellent | ||
| #' results especially in the case of multimodal distributions. | ||
| #' - **Poisson kernel-based distribution (PKBD)**: the package offers | ||
| #' functions for computing the density value and for generating random samples | ||
| #' from a PKBD. The Poisson kernel-based densities are based on the normalized | ||
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@@ -42,8 +44,6 @@ | |
| #' algorithm leverages a mixture of Poisson kernel-based densities on the | ||
| #' Sphere, enabling effective clustering of spherical data or data that has | ||
| #' been spherically transformed. | ||
| #' The package also provides the functions for density evaluation and random | ||
| #' sampling from the Poisson kernel-based distribution. | ||
| #' - **Additional Features**: Alongside these functionalities, the software | ||
| #' includes additional graphical functions, aiding users in validating and | ||
| #' representing the cluster results as well as enhancing the interpretability | ||
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@@ -64,7 +64,7 @@ | |
| #' @author | ||
| #' Giovanni Saraceno, Marianthi Markatou, Raktim Mukhopadhyay, Mojgan Golzy | ||
| #' | ||
| #' Mantainer: Giovanni Saraceno \email{[email protected]} | ||
| #' Maintainer: Giovanni Saraceno \email{[email protected]} | ||
| #' | ||
| #' | ||
| #' @references | ||
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| `QuadratiK` provides the first implementation, in R and Python, of a comprehensive set of goodness-of-fit tests and a clustering technique for $d$-dimensional spherical data $d \ge 2$ using kernel-based quadratic distances. It includes: | ||
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| - **Goodness-of-Fit Tests**: The software implements one, two, and *k*-sample tests for goodness of fit, offering an efficient and mathematically sound way to assess the fit of probability distributions. Expanded capabilities include supporting tests for uniformity on the *d*-dimensional Sphere based on Poisson kernel densities. Our tests are particularly useful for large, high dimensional data sets where the assessment of fit of probability models is of interest. Specifically, we offer tests for normality, as well as two- and *k*-sample tests, where testing equality of two or more distributions is of interest, that is $H_0: F_1 = F_2$ and $H_0: F_1 = \ldots = F_k$ respectively. The proposed tests perform well in terms of level and power for contiguous alternatives, heavy tailed distributions and in higher dimensions. | ||
| - **Goodness-of-Fit Tests**: The software implements one, two, and *k*-sample tests for goodness of fit, offering an efficient and mathematically sound way to assess the fit of probability distributions. Our tests are particularly useful for large, high dimensional data sets where the assessment of fit of probability models is of interest. Specifically, we offer tests for multivariate normality, as well as two- and *k*-sample tests, where testing equality of two or more distributions is of interest, that is $H_0: F_1 = F_2$ and $H_0: F_1 = \ldots = F_k$ respectively. The proposed tests perform well in terms of level and power for contiguous alternatives, heavy tailed distributions and in higher dimensions.\ | ||
| Expanded capabilities include supporting tests for uniformity on the *d*-dimensional Sphere based on the Poisson kernel, exhibiting excellent results especially in the case of multimodal distributions. | ||
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| - **Poisson kernel-based distribution (PKBD)**: the package offers functions for computing the density value and for generating random samples from a PKBD. The Poisson kernel-based densities are based on the normalized Poisson kernel and are defined on the $d$-dimensional unit sphere. Given a vector $\mu \in \mathcal{S}^{d-1}$, and a parameter $\rho$ such that $0 < \rho < 1$, the probability density function of a $d$-variate Poisson kernel-based density is defined by: $$f(\mathbf{x}|\rho, \mathbf{\mu}) = \frac{1-\rho^2}{\omega_d ||\mathbf{x} - \rho \mathbf{\mu}||^d},$$ where $\mu$ is a vector orienting the center of the distribution, $\rho$ is a parameter to control the concentration of the distribution around the vector $\mu$ and it is related to the variance of the distribution. Furthermore, $\omega_d = 2\pi^{d/2} [\Gamma(d/2)]^{-1}$ is the surface area of the unit sphere in $\mathbb{R}^d$ (see Golzy and Markatou, 2020). | ||
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| - **Clustering Algorithm for Spherical Data**: the package incorporates a unique clustering algorithm specifically tailored for $d$-dimensional spherical data and it is especially useful in the presence of noise in the data and the presence of non-negligible overlap between clusters. This algorithm leverages a mixture of Poisson kernel-based densities on the $d$-dimensional Sphere, enabling effective clustering of spherical data or data that has been spherically transformed. The package also provides the functions for density evaluation and random sampling from the Poisson kernel-based distribution. | ||
| - **Clustering Algorithm for Spherical Data**: the package incorporates a unique clustering algorithm specifically tailored for $d$-dimensional spherical data and it is especially useful in the presence of noise in the data and the presence of non-negligible overlap between clusters. This algorithm leverages a mixture of Poisson kernel-based densities on the $d$-dimensional Sphere, enabling effective clustering of spherical data or data that has been spherically transformed. | ||
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| - **Additional Features**: Alongside these functionalities, the software includes additional graphical functions, aiding users in validating and representing the cluster results as well as enhancing the interpretability and usability of the analysis. | ||
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| For an introduction to `QuadratiK` see the vignette [Introduction to the QuadratiK Package](https://giovsaraceno.github.io/QuadratiK-package/articles/Introduction.html). | ||
| For an introduction to the usage of `QuadratiK` see the vignette [Introduction to the QuadratiK Package](https://giovsaraceno.github.io/QuadratiK-package/articles/Introduction.html). | ||
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| ## Installation | ||
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| ## Authors | ||
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| Giovanni Saraceno, Marianthi Markatou, Raktim Mukhopadhyay, Mojgan Golzy\ | ||
| Mantainer: Giovanni Saraceno \<[gsaracen\@buffalo.edu](mailto:[email protected])\> | ||
| Maintainer: Giovanni Saraceno \<[gsaracen\@buffalo.edu](mailto:[email protected])\> | ||
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| ## Citation | ||
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