Density estimation
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In
A variety of approaches to density estimation are used, including
Example
We will consider records of the incidence of diabetes. The following is quoted verbatim from the data set description:
- A population of women who were at least 21 years old, of diabetes mellitus according to World Health Organization criteria. The data were collected by the US National Institute of Diabetes and Digestive and Kidney Diseases. We used the 532 complete records.[2][3]
In this example, we construct three density estimates for "glu" (plasma glucose concentration), one conditional on the presence of diabetes, the second conditional on the absence of diabetes, and the third not conditional on diabetes. The conditional density estimates are then used to construct the probability of diabetes conditional on "glu".
The "glu" data were obtained from the MASS package?Pima.tr
and ?Pima.te
give a fuller account of the data.
The mean of "glu" in the diabetes cases is 143.1 and the standard deviation is 31.26. The mean of "glu" in the non-diabetes cases is 110.0 and the standard deviation is 24.29. From this we see that, in this data set, diabetes cases are associated with greater levels of "glu". This will be made clearer by plots of the estimated density functions.
The first figure shows density estimates of p(glu | diabetes=1), p(glu | diabetes=0), and p(glu). The density estimates are kernel density estimates using a Gaussian kernel. That is, a Gaussian density function is placed at each data point, and the sum of the density functions is computed over the range of the data.
From the density of "glu" conditional on diabetes, we can obtain the probability of diabetes conditional on "glu" via
The second figure shows the estimated posterior probability p(diabetes=1 | glu). From these data, it appears that an increased level of "glu" is associated with diabetes.
Application and purpose
A very natural use of density estimates is in the informal investigation of the properties of a given set of data. Density estimates can give a valuable indication of such features as skewness and multimodality in the data. In some cases they will yield conclusions that may then be regarded as self-evidently true, while in others all they will do is to point the way to further analysis and/or data collection.[5]
An important aspect of statistics is often the presentation of data back to the client in order to provide explanation and illustration of conclusions that may possibly have been obtained by other means. Density estimates are ideal for this purpose, for the simple reason that they are fairly easily comprehensible to non-mathematicians.
More examples illustrating the use of density estimates for exploratory and presentational purposes, including the important case of bivariate data.[7]
Density estimation is also frequently used in anomaly detection or novelty detection:[8] if an observation lies in a very low-density region, it is likely to be an anomaly or a novelty.
- In hydrology the histogram and estimated density function of rainfall and river discharge data, analysed with a probability distribution, are used to gain insight in their behaviour and frequency of occurrence.[9] An example is shown in the blue figure.
Kernel density estimation
See also
- Kernel density estimation
- Mean integrated squared error
- Histogram
- Multivariate kernel density estimation
- Spectral density estimation
- Kernel embedding of distributions
- Generative model
- Application of Order Statistics: Non-parametric Density Estimation
- Probability distribution fitting
References
- ^ Alberto Bernacchia, Simone Pigolotti, Self-Consistent Method for Density Estimation, Journal of the Royal Statistical Society Series B: Statistical Methodology, Volume 73, Issue 3, June 2011, Pages 407–422, https://doi.org/10.1111/j.1467-9868.2011.00772.x
- ^ "Diabetes in Pima Indian Women - R documentation".
- PMC 2245318.)
{{cite journal}}
: CS1 maint: multiple names: authors list (link - ^ "Support Functions and Datasets for Venables and Ripley's MASS".
- ISBN 978-0412246203.
- ^ A calculator for probability distributions and density functions
- ISBN 978-0-470-53331-4.
- .
- ^ An illustration of histograms and probability density functions
- .
- JSTOR 2237880.
- ^ S2CID 216485629.
- OCLC 46809224.
Sources
- Brian D. Ripley (1996). Pattern Recognition and Neural Networks. Cambridge: Cambridge University Press. ISBN 978-0521460866.
- ISBN 0-387-95284-5. (See Chapter 6.)
- Qi Li and Jeffrey S. Racine. Nonparametric Econometrics: Theory and Practice. Princeton University Press, 2007, ISBN 0-691-12161-3. (See Chapter 1.)
- D.W. Scott. Multivariate Density Estimation. Theory, Practice and Visualization. New York: Wiley, 1992.
- ISBN 978-0-412-24620-3
External links
- CREEM: Centre for Research Into Ecological and Environmental Modelling Downloads for free density estimation software packages Distance 4 (from Research Unit for Wildlife Population Assessment "RUWPA") and WiSP.
- UCI Machine Learning Repository Content Summary (See "Pima Indians Diabetes Database" for the original data set of 732 records, and additional notes.)
- MATLAB code for one dimensional and two dimensional density estimation
- libAGF C++ software for variable kernel density estimation.