Description Usage Arguments Details References See Also
Kullback-Leibler divergence for each observation in Baysian quantile regression model
1 | bayesKL(y, x, tau, M, burn)
|
y |
vector, dependent variable in quantile regression |
x |
matrix, design matrix in quantile regression. |
tau |
quantile |
M |
the iteration frequancy for MCMC used in Baysian Estimation |
burn |
burned MCMC draw |
Method to address the differences between the posterior distributions from the distinct latent variables in the model, we suggest the use of the Kullback- Leibler divergence as a more precise method of measuring the distance between those latent variables in the Bayesian quantile regression framework. In this posterior information, the divergence is defined as
K(f_{i}, f_{j}) = \int log(\frac{f_{i}(x)}{f_{j}{(x)}})f_{i}(x)dx
where f_{i} could be the posterior conditional distribution of v_{i} and f_{j} the poserior conditional distribution of v_{j}. We should average this divergence for one observation based on the distance from all others, i.e,
KL(f_{i})=\frac{1}{n-1}∑{K(f_{i}, f_{j})}
We expect that when an observation presents a higher value for this divergence, it should also present a high probability value of being an outlier. Based on the MCMC draws from the posterior of each latent vaiable, we estimate the densities using a normal kernel and we compute the integral using the trapezoidal rule.
More details please refer to the paper in references
Santos B, Bolfarine H.(2016)“On Baysian quantile regression and outliers,arXiv:1601.07344
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