Noisefun: Transition Functions Describing Noise In intrinsicDimension: Intrinsic Dimension Estimation

Description

Transition functions f(s|r) describing the shift in lengths of vectors when Gaussian noise is added. Given a length r, f(s|r) is the probability density for the length after noise is added to one endpoint.

Usage

 ```1 2``` ```dnoiseNcChi(r, s, sigma, k) dnoiseGaussH(r, s, sigma, k) ```

Arguments

 `r` length or vector of lengths of original vector. `s` length or vector of lengths of perturbed vector. `sigma` noise standard deviation. `k` dimension of noise.

Details

`dnoiseNcChi` is the true transition function density when the noise is Gaussian, the other transition functions are approximations of this. `dnoiseGaussH` is the Gaussian approximation used in Haro et al.

If Gaussian noise is added to both endpoints of the vector, `sigma` should be replaced by `sqrt(2)*sigma`.

Value

Vector of probability densities.

Note

Only `r` or `s` can be a vector.

Author(s)

Kerstin Johnsson, Lund University

References

Haro, G., Randall, G. and Sapiro, G. (2008) Translated Poisson Mixture Model for Stratification Learning. Int. J. Comput. Vis., 80, 358-374.

`maxLikPointwiseDimEst`, `maxLikGlobalDimEst`, `maxLikLocalDimEst`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24``` ```# High SNR, high-dim noise sigma <- 0.05 x <- seq(0, 1.5, length.out = 200) y <- dnoiseNcChi(x, s = .5, sigma, k = 20) plot(x, y, type = 'l', main = 'Noise dim = 20') y2 <- dnoiseGaussH(x, s = .5, sigma, k = 20) lines(x, y2, lty = 2) # Low SNR par(mfrow = c(2, 3)) sigma <- 0.2 x <- seq(0, 1.5, length.out = 200) y <- dnoiseNcChi(x, s = .5, sigma, k = 4) plot(x, y, type = 'l', main = 'Noise approximations') y2 <- dnoiseGaussH(x, s = .5, sigma, k) lines(x, y2, lty = 2) # High SNR, low-dim noise sigma <- 0.05 x <- seq(0, 1.5, length.out = 200) y <- dnoiseNcChi(x, s = .5, sigma, k = 4) plot(x, y, type = 'l', main = 'Noise dim = 4') y2 <- dnoiseGaussH(x, s = .5, sigma, k) lines(x, y2, lty = 2) ```