Citizendia
Your Ad Here

In probability and statistics, density estimation is the construction of an estimate, based on observed data, of an unobservable underlying probability density function. Probability is the likelihood or chance that something is the case or will happen Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Debt AIDS Trade in Africa (or DATA) is a Multinational non-government organization founded in January 2002 in London by U2 's In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability 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.

A variety of approaches to density estimation are used, including Parzen windows and a range of data clustering techniques, including vector quantization. In Statistics, kernel density estimation (or Parzen window method named after Emanuel Parzen) is a non-parametric way of estimating Clustering is the classification of objects into different groups or more precisely the partitioning of a Data set into Subsets (clusters Vector quantization is a classical Quantization technique from Signal processing which allows the modeling of probability density functions by the distribution of

Contents

Example of density estimation

We consider records of the incidence of diabetes. Diabetes mellitus (ˌdaɪəˈbiːtiːz or /ˌdaɪəˈbiːtəs/ /məˈlaɪtəs/ or /ˈmɛlətəs/ often referred to simply as diabetes ( Ancient Greek: grc 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 according to World Health Organization criteria. A data set (or dataset) is a collection of Data, usually presented in tabular form The Akimel O'odham or Pima are a group of American Indians living in an area consisting of what is now central and Southern Arizona ( USA The data were collected by the US National Institute of Diabetes and Digestive and Kidney Diseases. We used the 532 complete records.

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. Blood plasma is the Liquid component of Blood, in which the Blood cells are suspended Glucose (Glc a Monosaccharide (or simple Sugar) also known as grape sugar, is an important Carbohydrate in Biology. Conditional probability is the Probability of some event A, given the occurrence of some other event B. The conditional density estimates are then be used to construct the probability of diabetes conditional on "glu".

The "glu" data were obtained from the MASS package of the R programming language. The R programming language, sometimes described as GNU S, is a programming language and software environment for statistical computing and Within 'R', ?Pima. tr and ?Pima. te give a fuller account of the data.

The mean of "glu" in the diabetes cases is 143. In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean 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.

Estimated density of p(glu | diabetes=1) (red), p(glu | diabetes=0) (blue), and p(glu) (black).

From the density of "glu" conditional on diabetes, we can obtain the probability of diabetes conditional on "glu" via Bayes' rule. In Probability theory, Bayes' theorem (often called Bayes' law after Thomas Bayes) relates the conditional and marginal probabilities of two For brevity, "diabetes" is abbreviated "db. " in this formula.

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.

Estimated probability of p(diabetes=1 | glu).

Script for example

The follow commands of the R programming language will create the figures shown above. The R programming language, sometimes described as GNU S, is a programming language and software environment for statistical computing and These commands can be entered at the command prompt by using cut and paste. 

library (MASS)
data(Pima. tr)

data(Pima. te)

Pima <- rbind (Pima. tr, Pima. te)
glu <- Pima[,'glu']

d0 <- Pima[,'type'] == 'No'
d1 <- Pima[,'type'] == 'Yes'
base. rate. d1 <- sum(d1)/(sum(d1) + sum(d0))

glu. density <- density (glu)
glu. d0. density <- density (glu[d0])
glu. d1. density <- density (glu[d1])

approxfun (glu. d0. density$x, glu. d0. density$y) -> glu. d0. f
approxfun (glu. d1. density$x, glu. d1. density$y) -> glu. d1. f

p. d. given. glu <- function (glu, base. rate. d1)
{
   p1 <- glu. d1. f(glu) * base. rate. d1
   p0 <- glu. d0. f(glu) * (1 - base. rate. d1)
   p1/(p0+p1)
}

x <- 1:250
y <- p. d. given. glu (x, base. rate. d1)
plot (x, y, type='l', col='red', xlab='glu', ylab='estimated p(diabetes|glu)')

plot (density(glu[d0]), col='blue', xlab='glu', ylab='estimate p(glu), 
   p(glu|diabetes), p(glu|not diabetes)', main=NA)
lines (density(glu[d1]), col='red')
lines (density(glu))

See also

References

External links

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
 Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic