np (version 0.60-8)

nptgauss: Truncated Second-order Gaussian Kernels

Description

nptgauss provides an interface for setting the truncation radius of the truncated second-order Gaussian kernel used by np.

Usage

nptgauss(b)

Arguments

b

Truncation radius of the kernel.

Details

nptgauss allows one to set the truncation radius of the truncated Gaussian kernel used by np, which defaults to 3. It automatically computes the constants describing the truncated gaussian kernel for the user.

We define the truncated gaussion kernel on the interval \([-b,b]\) as: $$K = \frac{\alpha}{\sqrt{2\pi}}\left(e^{-z^2/2} - e^{-b^2/2}\right)$$

The constant \(\alpha\) is computed as: $$\alpha = \left[\int_{-b}^{b} \frac{1}{\sqrt{2\pi}}\left(e^{-z^2/2} - e^{-b^2/2}\right)\right]^{-1}$$

Given these definitions, the derivative kernel is simply: $$K' = (-z)\frac{\alpha}{\sqrt{2\pi}}e^{-z^2/2}$$

The CDF kernel is: $$G = \frac{\alpha}{2}\mathrm{erf}(z/\sqrt{2}) + \frac{1}{2} - c_0z$$

The convolution kernel on \([-2b,0]\) has the general form: $$H_- = a_0\,\mathrm{erf}(z/2 + b) e^{-z^2/4} + a_1z + a_2\,\mathrm{erf}((z+b)/\sqrt{2}) - c_0$$

and on \([0,2b]\) it is: $$H_+ = -a_0\,\mathrm{erf}(z/2 - b) e^{-z^2/4} - a_1z - a_2\,\mathrm{erf}((z-b)/\sqrt{2}) - c_0$$

where \(a_0\) is determined by the normalisation condition on H, \(a_2\) is determined by considering the value of the kernel at \(z = 0\) and \(a_1\) is determined by the requirement that \(H = 0\) at \([-2b,2b]\).

Examples

# NOT RUN {
## The default kernel, a gaussian truncated at +- 3
nptgauss(b = 3.0)

# }