Appendix#
This appendix contains lists of all probability density distributions and all kernels that GRAMPC-S supports.
List of probability density functions#
The following list contains all probability density functions that are implemented in GRAMPC-S. Note that it is possible to implement your own probability density functions. These must be declared as derived from the Distribution class.
Name |
Probability density function |
Parameters |
|---|---|---|
Gaussian distribution |
\(p(\vm x) = \left(2 \pi\right)^{-\frac{d}{2}} \det(\vm \Sigma)^{-\frac{1}{2}} \exp \left(-\frac{1}{2} (\vm x - \vm \mu)^T \vm \Sigma^{-1} (\vm x - \vm \mu)\right)\) |
\(\vm \mu \in \mathbb{R}^d\) \(\vm \Sigma \in \mathbb{R}^{d \times d}\) |
Beta distribution |
\(p(x) = \frac{1} {B(p, q)} \, x^{p-1} \,(1 - x)^{q - 1}\) |
\(p>0\,, \; q >0\) |
Chi-squared distribution |
\(p(x) = \frac{x^{\frac{n}{2} - 1} \,\exp\left(-\frac{x}{2}\right)}{\Gamma(\frac{n}{2}) \, 2^{\frac{n}{2}}}\) |
\(n>0\) |
Exponential distribution |
\(p(x) = \lambda \exp(-\lambda \, x)\) |
\(\lambda >0\) |
Extreme value distribution |
\(p(x) = \frac{1}{b} \exp\left(\frac{a - x}{b} - \exp\left(\frac{a - x}{b}\right)\right)\) |
\(a \in \mathbb{R}\,,\; b>0\) |
F-distribution |
\(p(x) = \frac{\Gamma\left(\frac{m+n}{2}\right)}{\Gamma\left(\frac{m}{2} \right) \Gamma\left( \frac{n}{2}\right)} \left(\frac{m}{n}\right)^\frac{m}{2} x^{\frac{m}{2} - 1} (1 + \frac{m}{n} x)^{-\frac{m+n}{2}}\) |
\(m>0\,,\; n >4\) |
Gamma distribution |
\(p(x) = \frac{\exp\left(-\frac{x}{\beta}\right)}{\beta^\alpha \, \Gamma(\alpha)} \, x^{\alpha - 1}\) |
\(\alpha > 0\,,\; \beta > 0\) |
Log-normal distribution |
\(p(x) = \frac{1}{\sigma x \sqrt{2 \pi}} \exp\left(- \frac{(\ln(x)-\mu)^2} {2 \sigma^2}\right)\) |
\(\mu \in \mathbb{R} \,,\; \sigma > 0\) |
Piecewise constant distribution |
Histogram of an arbitrary distribution |
|
Student’s t-distribution |
\(p(x) = \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\sigma \sqrt{\nu \pi} \Gamma\left(\frac{\nu}{2}\right)} \left(1 + \left(\frac{x-\mu}{\sigma}\right)^2 \frac{1}{\nu}\right)^{-\frac{\nu+1}{2}}\) |
\(\mu \in \mathbb{R} \,,\; \sigma > 0\) \(\nu > 2\) |
Uniform distribution |
\(p(x) = \frac{1}{b - a}\) |
\(a < b\) |
Weibull distribution |
\(p(x) = \frac{a}{b} \left(\frac{x}{b}\right)^{a-1} \exp(-\left(\frac{x}{b}\right)^a)\) |
\(a> 0\,,\; b > 0\) |
Product of uncorrelated distributions |
\(\displaystyle p(\vm x) = \prod_{i=1}^d p_i(x_i)\) |
List of kernels for Gaussian processes#
The following list contains all kernels that are implemented in GRAMPC-S.
Note that it is possible to implement your own kernels.
These must be declared as derived from the StationaryKernel class.
Name |
Kernel |
Parameters |
|---|---|---|
Squared exponential kernel |
\(k(\tau) = \sigma^2 \exp\left(-\frac12 \sum\limits_i \left(\frac{\tau_i^2}{l_i^2}\right)\right)\) |
\(\sigma\), \(\vm l\) |
Periodic kernel |
\(k(\tau) = \sigma^2 \exp\left(-2 \sum\limits_i \left(\frac{\sin^2\left(\pi \frac{\tau_i}{p_i}\right)}{l_i^2}\right)\right)\) |
\(\sigma\), \(\vm l\), \(\vm p\) |
Locally periodic kernel |
\(k(\tau) = \sigma^2 \exp\left(-2 \sum\limits_i\left(\frac{\sin^2(\pi \frac{\tau_i}{p_i})}{l_i^2}\right)\right) \exp\left(-\frac12 \sum\limits_i\left(\frac{\tau_i^2}{l_i^2}\right)\right)\) |
\(\sigma\), \(\vm l\), \(\vm p\) |
Matern 3/2 kernel |
\(k(\tau) = \sigma^2 \left(1 + \sqrt{3 \sum\limits_i\left(\frac{\tau_i^2}{l_i^2}\right)} \right) \exp\left(- \sqrt{ 3\sum\limits_i\left(\frac{\tau_i^2}{l_i^2}\right)} \right)\) |
\(\sigma\), \(\vm l\) |
Matern 5/2 kernel |
\(k(\tau) = \sigma^2 \left(1 + \sqrt{5 \sum\limits_i\left(\frac{\tau_i^2}{l_i^2}\right)} + \frac{5}{3} \sum\limits_i\left(\frac{\tau_i^2}{l_i^2}\right)\right) \exp\left(- \sqrt{5 \sum\limits_i\left(\frac{\tau_i^2}{l_i^2}\right)}\right)\) |
\(\sigma\), \(\vm l\) |
Sum of kernels |
\(k(\tau) = k_1(\tau) + k_2(\tau)\) |
|
Product of kernels |
\(k(\tau) = k_1(\tau) k_2(\tau)\) |