Functions for estimating spatial varying coefficient models, mixed models, and other spatial regression models for Gaussian and non-Gaussian data. Moran eigenvectors are used to an approximate spatial Gaussian processes. These processes are used for modeling the spatial processes in residuals and regression coefficients. For details see Murakami (2021) <arXiv:1703.04467>.

Identify statistically significant flow clusters using the local spatial network autocorrelation statistic G_ij* 
    proposed by 'Berglund' and 'Karlström' (1999) <doi:10.1007/s101090050013>. 
    The metric, an extended statistic of 'Getis/Ord' G ('Getis' and 'Ord' 1992) <doi:10.1111/j.1538-4632.1992.tb00261.x>, 
    detects a group of flows having similar traits in terms of directionality. 
    You provide OD data and the associated polygon to get results 
    with several parameters, some of which are defined by spdep package.

Perform spatial analysis on network.
    Implement several methods for spatial analysis on network: Network Kernel Density estimation, 
    building of spatial matrices based on network distance ('listw' objects from 'spdep' package), K functions estimation 
    for point pattern analysis on network, k nearest neighbours on network, reachable area calculation, and graph generation
    References: Okabe et al (2019) <doi:10.1080/13658810802475491>;
    Okabe et al (2012, ISBN:978-0470770818);Baddeley et al (2015, ISBN:9781482210200).

Non-negative Matrix Factorization(NMF) is a powerful tool for identifying the key features of microbial communities and a dimension-reduction method. When we are interested in the differences between the structures of two groups of communities, supervised NMF(Yun Cai, Hong Gu and Tobby Kenney (2017),<doi:10.1186/s40168-017-0323-1>) provides a better way to do this, while retaining all the advantages of NMF -- such as interpretability, and being based on a simple biological intuition.

Scale invariant version of the original PNN proposed by Specht (1990) <doi:10.1016/0893-6080(90)90049-q> with the added functionality of allowing for smoothing along multiple dimensions while accounting for covariances within the data set. It is written in the R statistical programming language. Given a data set with categorical variables, we use this algorithm to estimate the probabilities of a new observation vector belonging to a specific category. This type of neural network provides the benefits of fast training time relative to backpropagation and statistical generalization with only a small set of known observations.