Density function estimation underlying distribution

FIGURE 21 Density function (a) underlying exponential distribution, (b) for the estimator of size 2. 

The knowledge required to implement Bayes formula is daunting in that a priori as well as class conditional probabilities must be known. Some reduction in requirements can be accomplished by using joint probability distributions in place of the a priori and class conditional probabilities. Even with this simplification, few interpretation problems are so well posed that the information needed is available. It is possible to employ the Bayesian approach by estimating the unknown probabilities and probability density functions from exemplar patterns that are believed to be representative of the problem under investigation. This approach, however, implies supervised learning where the correct class label for each exemplar is known. The ability to perform data interpretation is determined by the quality of the estimates of the underlying probability distributions. 

A set of observed data points is assumed to be available as samples from an unknown probability density function. Density estimation is the construction of an estimate of the density function from the observed data. In parametric approaches, one assumes that the data belong to one of a known family of distributions and the required function parameters are estimated. This approach becomes inadequate when one wants to approximate a multi-model function, or for cases where the process variables exhibit nonlinear correlations [127]. Moreover, for most processes, the underlying distribution of the data is not known and most likely does not follow a particular class of density function. Therefore, one has to estimate the density function using a nonparametric (unstructured) approach. 

Suppose the distribution has mean, variance, and density function of /t, and f x). To illustrate the mathematical formulation of the estimator, we will do a detailed derivation that the sample mean is an unbiased estimator for the mean of the underlying distribution. We have... 

For emphasis, aU of these results about estimators arrive fi om the formulation of a sample as a collection of independent and identically distributed random variables. Their joint distribution is derived from the -fold product of the density function of the underlying distribution. Once these ideas are expressed as multiple integrals, deriving results in statistics are exercises in calculus and analysis. 

We will consider a very simple example for our favorite estimator—the sample average. Suppose the underlying distribution is a constant rate process with the density function /(z) = e. Suppose the estimator is the average of a sample of size two z = (x + y)jl. The distribution function for z is G(t) = Pr[x + y < 2t]. The integration iscarryed out over the shaded area in Fig. 21.7 bounded by the line y = —x + 2t and the axes. 

Figure 21.8 displays the density functions for the underlying exponential distribution and for the sample average of size 2. Obviously, we want the density function for a sample of size n, we want the density function of the estimator for other underlying distributions, and we want the density function for estimators other than the sample average. Statisticians have expended considerable effort on this topic, and this material... 

As an example of what can be done with more information about the underlying distribution available, consider the exponential (constantrate) failure distribution with density function fit) = ke K Suppose the requirement is to estimate the mean-time-to-failure within 10% with a 95% confidence... 

The frequency distribution can also be used to estimate the type of probability distribution that applies to a data set. For a continuous function, the probability density function (pdf) is the probability that the variate has the value X, thus the probability of a parameter exceeding a particular value can be estimated. For example the probability that gas emissions overwhelm an under floor venting system can be estimated. 

The distribution of laser points in the image is discrete and it is difficult to locate points directly with a maximum density around. The mean shift presented by Comaniciu and Meer [COM 99] is a powerful non-parametric algorithm that allows us to find the local maximum of a function of the underlying density. It estimates the feature space as an empirical probability density function. For each data point, the mean shift defines a window and calculates the average of data within the window. Then it moves the center of the window to the average in the direction of the... 

On the other hand t and A r contain the rate constants of the equilibrium under study and, for bimolecular reactions, the concentrations of the species involved in the reaction [4b, 8]. Had counter-ion site binding been a one step process the determination of t would have provided a direct estimation of the lifetime of the bound counter-ions and, therefore, of the exchange rate between bound and free counterions. On the other hand the study of as a function of concentration would have permitted to obtain informations on the distribution of bound counterions between those bound with and without dehydration. Unfortunately, as will be shown in Section 4, site binding is a multistep process involving at least two equilibria. All of the unknown quantities involved in such a process (four rate constants, two volume changes and the concentrations of the species) cannot be obtained from ultrasonic absorption data alone. Independent measurements become necessary. For this purpose we have measured the density d of the polyelectrolyte solutions from which can be obtained the apparent molal volume FcP of the polyelectrolyte CP (C counterion, P polyion) according to ... 

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