1d:T797,# Transition Functions Describing Noise 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. ```r 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`. ## Returns Vector of probability densities. ## References Haro, G., Randall, G. and Sapiro, G. (2008) Translated Poisson Mixture Model for Stratification Learning. **Int. J. Comput. Vis.**, 80 , 358-374. ## Author(s) Kerstin Johnsson, Lund University ## Note Only `r` or `s` can be a vector. ## See Also `maxLikPointwiseDimEst`, `maxLikGlobalDimEst`, `maxLikLocalDimEst` ## Examples ```r # 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) ```

Noisefun function