Deconvolution
We only see Z = X + Y: a signal X blurred by noise Y. Simulate it, or bring your own data, and estimate the signal and the noise. Everything runs in your browser.
How it works
Pick a signal and a noise, and see what you would observe. Then press Run to estimate them back from the sample. The parametric fit needs you to name both families. The other methods take any shape of signal but use the noise as given.
One sample can't tell every signal from its noise: two normals only show their summed variance. The fit says when that happens. The details are in the docs.
Model
- Sample of Z
- Signal X
- Noise Y
- Observed Z
Estimate
Data
Noise regime
The noise's family is the one in the model above.
Methods
Run
References
References
- Kiefer, J. & Wolfowitz, J. (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Annals of Mathematical Statistics 27(4). The NPMLE: the most likely signal of any shape, which turns out to be a few point masses.
- Koenker, R. & Mizera, I. (2014). Convex optimization, shape constraints, compound decisions, and empirical Bayes rules. Journal of the American Statistical Association 109(506). Made the NPMLE a convex problem on a grid, with a unique answer and a check that it is optimal.
- Efron, B. (2016). Empirical Bayes deconvolution estimates. Biometrika 103(1). g-modeling: a smooth signal from a penalised spline family, fitted by maximum likelihood.
- Narasimhan, B. & Efron, B. (2020). deconvolveR: A g-modeling program for deconvolution and empirical Bayes estimation. Journal of Statistical Software 94(11). The reference implementation behind the g-modeling defaults (5 spline degrees of freedom, c₀ = 1).
- Stefanski, L. A. & Carroll, R. J. (1990). Deconvolving kernel density estimators. Statistics 21(2). The Fourier kernel estimator: divide the data’s characteristic function by the noise’s.
- Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Annals of Statistics 19(3). Why normal noise is so hard to undo: the best possible rate is only logarithmic in n.
- Bovy, J., Hogg, D. W. & Roweis, S. T. (2011). Extreme deconvolution: inferring complete distribution functions from noisy, heterogeneous and incomplete observations. Annals of Applied Statistics 5(2B). The EM updates for a Gaussian-mixture signal under known normal noise.
- Butucea, C. & Matias, C. (2005). Minimax estimation of the noise level and of the deconvolution density in a semiparametric convolution model. Bernoulli 11(2). When the noise’s scale can be estimated too: only if the signal is rougher than the noise.
- Jeffreys, H. (1946). An invariant form for the prior probability in estimation problems. Proceedings of the Royal Society A 186(1007). The prior of the Bayesian fit: √det of the Fisher information, here of the observed f_Z.
- Haario, H., Saksman, E. & Tamminen, J. (2001). An adaptive Metropolis algorithm. Bernoulli 7(2). The sampler of the Bayesian fit: random-walk steps shaped by the draws so far.
- Berger, J. O., Bernardo, J. M. & Sun, D. (2015). Overall objective priors. Bayesian Analysis 10(1). The Dirichlet(1/m) prior of the NPMLE posterior, objective for grids of many cells.
- Delaigle, A. & Hall, P. (2016). Methodology for non-parametric deconvolution when the error distribution is unknown. Journal of the Royal Statistical Society B 78(1). Symmetric noise of unknown shape: the signal is read from the data’s phase.
- Li, T. & Vuong, Q. (1998). Nonparametric estimation of the measurement error model using multiple indicators. Journal of Multivariate Analysis 65(2). Repeated readings recover both the signal and the noise through Kotlarski’s identity.
- Comte, F. & Kappus, J. (2015). Density deconvolution from repeated measurements without symmetry assumption on the errors. Journal of Multivariate Analysis 140. The regularisation that keeps Kotlarski’s divisions stable.