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ndows and a range of data clustering techniques including vector quantization the most basic form of density estimation is a rescaled histogram example edit estimated density of p glu diabetes 1 red p glu diabetes 0 blue and p glu black estimated probability of p diabetes 1 glu estimated probability of p diabetes 1 glu 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 pima indian heritage and living near phoenix arizona was tested for 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 4 of the r programming language within r 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 bayes rule for brevity diabetes is abbreviated db in this formula p diabetes 1 glu p glu db 1 p db 1 p glu db 1 p db 1 p glu db 0 p db 0 displaystyle p mbox diabetes 1 mbox glu frac p mbox glu mbox db 1 p mbox db 1 p mbox glu mbox db 1 p mbox db 1 p mbox glu mbox db 0 p mbox db 0 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 edit 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 histogram and density function for a gumbel distribution 6 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 edit this section is an excerpt from kernel density estimation edit kernel density estimation of 100 normally distributed random numbers using different smoothing bandwidths in statistics kernel density estimation kde is the application of kernel smoothing for probability density estimation i e a non parametric method to estimate the probability density function of a random variable based on kernels as weights kde answers a fundamental data smoothing problem where inferences about the population are made based on a finite data sample in some fields such as signal processing and econometrics it is also termed the parzen rosenblatt window method after emanuel parzen and murray 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 conditional marginal densities of data when using a naive bayes classifier which can improve its prediction accuracy 12 see also edit frequency distribution 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 edit 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 smith j w everhart j e dickson w c knowler w c and johannes r s 1988 r a greenes ed using the adap learning algorithm to forecast the onset of diabetes mellitus proceedings of the symposium on computer applications in medical care washington 1988 los alamitos ca 261 265 pmc 2245318 cite journal cs1 maint multiple names authors list link support functions and datasets for venables and ripley s mass silverman b w 1986 density estimation for statistics and data analysis chapman and hall isbn 978 0412246203 a calculator for probability distributions and density functions geof h givens 2013 computational statistics wiley p 330 isbn 978 0 470 53331 4 pimentel marco a f clifton david a clifton lei tarassenko lionel 2 january 2014 a review of novelty detection signal processing 99 june 2014 215 249 doi 10 1016 j sigpro 2013 12 026 an illustration of histograms and probability density functions rosenblatt m 1956 remarks on some nonparametric estimates of a density function the annals of mathematical statistics 27 3 832 837 doi 10 1214 aoms 1177728190 parzen e 1962 on estimation of a probability density function and mode the annals of mathematical statistics 33 3 1065 1076 doi 10 1214 aoms 1177704472 jstor 2237880 hastie trevor tibshirani robert friedman jerome h 2001 the elements of statistical learning data mining inference and prediction with 200 full color illustrations new york springer isbn 0 387 95284 5 oclc 46809224 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 external links edit 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 v t e statistics outline index descriptive statistics continuous data center mean arithmetic arithmetic geometric contraharmonic cubic generalized power geometric harmonic heronian heinz lehmer median mode dispersion average absolute deviation coefficient of variation interquartile range percentile range standard deviation variance shape central limit theorem moments kurtosis l moments skewness count data index of dispersion summary tables contingency table frequency distribution grouped data dependence partial correlation pearson product moment correlation rank correlation kendall s τ spearman s ρ scatter plot graphics bar chart biplot box plot control chart correlogram fan chart forest plot histogram pie chart q q plot radar chart run chart scatter plot stem and leaf display violin plot heatmap scatter plot matrix ecdf plot line chart statistical data processing transformations data transformation log transformation power transform box cox transformation yeo johnson transformation variance stabilizing transformation anscombe transform fisher transformation scaling and normalization feature scaling normalization standardization z score min max normalization unit vector normalization data cleaning data cleaning outlier winsorizing truncation missing data data reduction dimensionality reduction principal component analysis factor analysis time series preprocessing differencing detrending seasonal adjustment stationarity transformation data collection study design effect size missing data optimal design population replication sample size determination statistic statistical power survey methodology sampling cluster stratified opinion poll questionnaire standard error controlled experiments blocking factorial experiment interaction random assignment randomized controlled trial randomized experiment scientific control adaptive designs adaptive clinical trial stochastic approximation up and down designs observational studies cohort study cross sectional study natural experiment quasi experiment statistical inference statistical theory population statistic probability distribution sampling distribution order statistic empirical distribution density estimation statistical model model specification l p space parameter location scale shape parametric family likelihood monotone location scale family exponential family completeness sufficiency statistical functional bootstrap u v optimal decision loss function efficiency statistical distance divergence asymptotics robustness frequentist inference point estimation estimating equations maximum likelihood method of moments m estimator minimum distance unbiased estimators mean unbiased minimum variance rao blackwellization lehmann scheffé theorem median unbiased plug in interval estimation confidence interval pivot likelihood interval prediction interval tolerance interval resampling bootstrap jackknife testing hypotheses 1 2 tails power uniformly most powerful test permutation test randomization test multiple comparisons parametric tests likelihood ratio score lagrange multiplier wald specific tests z test normal student s t test f test goodness of fit chi squared g test kolmogorov smirnov anderson darling lilliefors jarque bera normality shapiro wilk likelihood ratio test model selection cross validation aic bic rank statistics sign sample median signed rank wilcoxon hodges lehmann estimator rank sum mann whitney nonparametric anova 1 way kruskal wallis 2 way friedman ordered alternative jonckheere terpstra van der waerden test bayesian inference bayesian probability prior posterior credible interval bayes factor bayesian estimator maximum posterior estimator correlation regression analysis correlation pearson product moment partial correlation confounding variable coefficient of determination regression analysis errors and residuals regression validation mixed effects models simultaneous equations models multivariate adaptive regression splines mars template least squares and regression analysis linear regression simple linear regression ordinary least squares general linear model bayesian regression non standard predictors nonlinear regression nonparametric semiparametric isotonic robust homoscedasticity and heteroscedasticity generalized linear model exponential families logistic bernoulli binomial poisson regressions partition of variance analysis of variance anova anova analysis of covariance multivariate anova degrees of freedom categorical multivariate time series survival analysis categorical cohen s kappa contingency table graphical model log linear model mcnemar s test cochran mantel haenszel statistics multivariate regression manova principal components canonical correlation discriminant analysis cluster analysis classification structural equation model factor analysis multivariate distributions elliptical distributions normal time series general decomposition trend stationarity seasonal adjustment exponential smoothing cointegration structural break granger causality specific tests dickey fuller johansen q statistic ljung box durbin watson breusch godfrey time domain autocorrelation acf partial pacf cross correlation xcf arma model arima model box jenkins autoregressive conditional heteroskedasticity arch vector autoregression var autoregressive model ar frequency domain spectral density estimation fourier analysis least squares spectral analysis wavelet whittle likelihood survival survival function kaplan meier estimator product limit proportional hazards models accelerated failure time aft model first hitting time hazard function nelson aalen estimator test log rank test applications biostatistics bioinformatics clinical trials studies epidemiology medical statistics engineering statistics chemometrics methods engineering probabilistic design process quality control reliability system identification social statistics actuarial science census crime statistics demography econometrics 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