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Instatistics,probability density estimation or simplydensity estimation is the construction of anestimate, based on observeddata, of an unobservable underlyingprobability density function. The unobservable density function is thought of as the density according to which a large population is distributed; the data are usually thought of as a random sample from that population.^[1]

A variety of approaches to density estimation are used, includingParzen windows and a range ofdata clustering techniques, includingvector quantization. The most basic form of density estimation is a rescaledhistogram.

We will consider records of the incidence ofdiabetes. The following is quoted verbatim from thedata set description:

A population of women who were at least 21 years old, ofPima Indian heritage and living near Phoenix, Arizona, was tested fordiabetes mellitus according toWorld 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" (plasmaglucose concentration), oneconditional 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^[4] of theR programming language. Within R,?Pima.tr and?Pima.te give a fuller account of the data.

Themean 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 ofp(glu | diabetes=1),p(glu | diabetes=0), andp(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" viaBayes' rule. For brevity, "diabetes" is abbreviated "db." in this formula.

${\displaystyle p(\text{diabetes}=1|\text{glu})=\frac{p(\text{glu}|\text{db.}=1)p(\text{db.}=1)}{p(\text{glu}|\text{db.}=1)p(\text{db.}=1)+p(\text{glu}|\text{db.}=0)p(\text{db.}=0)}}$

The second figure shows the estimated posterior probabilityp(diabetes=1 | glu). From these data, it appears that an increased level of "glu" is associated with diabetes.

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 inanomaly detection ornovelty detection:^[8] if an observation lies in a very low-density region, it is likely to be an anomaly or a novelty.

• Inhydrology thehistogram and estimated density function of rainfall and river discharge data, analysed with aprobability distribution, are used to gain insight in their behaviour and frequency of occurrence.^[9] An example is shown in the blue figure.

This section is an excerpt fromKernel density estimation.[edit]

Instatistics,kernel density estimation (KDE) is the application ofkernel smoothing forprobability density estimation, i.e., anon-parametric method toestimate theprobability density function of arandom variable based onkernels asweights. KDE answers a fundamental data smoothing problem where inferences about thepopulation are made based on a finite datasample. In some fields such assignal processing andeconometrics it is also termed the Parzen–Rosenblatt window method, afterEmanuel Parzen andMurray Rosenblatt, who are usually credited with independently creating it in its current form.^[10][11] One of the famous applications of kernel density estimation is in estimating the class-conditionalmarginal densities of data when using anaive Bayes classifier, which can improve its prediction accuracy.^[12]

• 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

Sources

• Brian D. Ripley (1996).Pattern Recognition and Neural Networks. Cambridge: Cambridge University Press.ISBN 978-0521460866.
• Trevor Hastie,Robert Tibshirani, and Jerome Friedman.The Elements of Statistical Learning. New York: Springer, 2001.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.
• B.W. Silverman.Density Estimation. London: Chapman and Hall, 1986.ISBN 978-0-412-24620-3

• CREEM: Centre for Research Into Ecological and Environmental Modelling Downloads for free density estimation software packagesDistance 4 (from Research Unit for Wildlife Population Assessment "RUWPA") andWiSP.
• UCI Machine Learning Repository Content Summary(See "Pima Indians Diabetes Database" for the original data set of 732 records, and additional notes.)
• MATLAB code forone dimensional andtwo dimensional density estimation
• libAGF C++ software forvariable kernel density estimation.

• Estimation of densities
• Nonparametric statistics

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