MATLAB: Calculation of probability using Beta ,lognormal and weibull distribution

beta distributionweibull

Hello,

I have computed three different properties of sample objects and have stored the discrete data values in three vectors.I have fit weibull, lognormal and beta distribution on these three vectors. Now how can i find the individual probability of each of the data values using their respective fitted distributions? I want to multiply these probabilities for each element to find the joint probability.

Thanks

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"if I want to obtain the likelihood of getting a particular combination of values which represent these 3 properties of another cancerous tumor(not from the dataset),i use my fitted distributions for this purpose and i multiply the computed likelihoods to obtain the joint likelihood."

Several comments on this.

First, multiplication is only appropriate here if the 3 properties are independent. If they are at all correlated with one another, the joint likelihood for any given combination of three is not simply the product of the individual ("marginal") likelihoods. You can check this by looking at the pairwise correlations among the three properties across the 200 tumors.

Second, it sounds like you really do want probabilities rather than likelihoods. For that purpose, you would be best to consider each property as falling within a certain numerical range or "bin". For a first pass, I suggest you classify each tumor as falling above vs below the median with respect to each of the three properties. This will give you 8 categories of tumors: 2x2x2, and you can count how many (out of 200) you have in each category. With a sample of only 200, I doubt that you have enough data to get usable estimates of the probabilities in more than 8 bins, but you might try 3x3x3 if you are a daredevil.

Third, it still doesn't sound like you quite have appreciated the difference between likelihoods and probabilities. To maybe make that distinction more meaningful to you, I suggest you do the following: Rescore all of your properties into a different unit of measurement--say, for example, that you divide each property value by 10. I hope you agree that logically this changes nothing. Likewise, if you look at the probabilities in the 2x2x2 bins, you will see that the rescoring has also changed nothing. But if you refit your distributions to the rescored property values, you will see that all of the PDF values (i.e., likelihoods) have increased by a factor of 10. The reason is that the PDF values (likelihoods) must integrate to 1 across the whole range of X, so the PDF values--unlike the corresponding probabilities--depend on the units of measurement for X.

Related Solutions

MATLAB: How to compute the probability of a function of random variables using the Statistics Toolbox 7.4 (R2007a)

The Statistics Toolbox in MATLAB has a number of functions for evaluating PDF's and CDF's of a number of distributions, fitting data to certain distributions and generating random data from a set of distributions. These distributions and functions are listed at,

https://www.mathworks.com/help/stats/supported-distributions.html

Some combination of these functions and Monte Carlo simulations can be used to compute the probability in this application. In all cases, you will need to make certain assumptions on the covariances of B, X1, X2, X3 and S.

Task 1:

If you know the mean and standard deviation of F you can use the function NORMCDF to compute the probability that F > 0 (using 1-NORMCDF(...)). The mean and variance of F may, in some cases, be found analytically in terms of the means and variances of B, X1, X2 and X3 depending on the assumptions you make about those random variables' distributions and covariances. However, the toolbox will not be able to assist you in this task.

In general, MATLAB's strength lies in its ability to generate and handle large amounts of data. Therefore, Monte-Carlo simulation would be the preferred way to solve this problem. You can generate a large number of random numbers from the joint distribution of B, X1, X2, X3 and S. For Gaussians, the MVNRND function can be used for this purpose using the means and covariances of the random variables. You can then compute values of F for all these variables and fit the resulting data to another Gaussian using NORMFIT. You can then compute the probability using NORMCDF. There are also other functions in the toolbox that can help you determine the goodness of the fit.

Task 2:

As before, generate a number of random values drawn from B, X1, X2, X3 and S using their joint multivariate distribution and compute the corresponding values of F. Then instead of fitting a Gaussian, you can simply compute the ratio of positive values of F giving you an estimate of the probability. You can also fit the data to a number of other distributions supported by the toolbox with fitting functions. You can then use the corresponding CDF function to compute the probability.

Task 3:

You can no longer use MVNRND to generate the random data, but you will need to generate it from some other joint distribution. If the variables are independent, this makes it much easier as the random numbers from each random variable can be generated independently. If they are correlated, you may wish to use COPULARND to generate random data from their joint distribution. The rest of the operations are the same.

MATLAB: How to go about finding the standard normal probability based on the z-score

doc normcdf
doc normpdf

When you know what you want but not sure the name, try something like

>> lookfor normal
realmin                        - Smallest positive normalized floating point number.
randn                          - Normally distributed pseudorandom numbers.
sprandn                        - Sparse normally distributed random matrix.
surfnorm                       - Surface normals.
isonormals                     - Isosurface normals.
cde                            - cd elliptic function with normalized complex argument.
sne                            - sn elliptic function with normalized complex argument.
addfreqcsmenu                  - Add a cs menu to switch between linear and normalized frequency
convertfrequnits               - converts between Normalized, Hz, kHz, etc
histfit                        - Histogram with superimposed fitted normal density.
jbtest                         - Jarque-Bera hypothesis test of composite normality.
lhsnorm                        - Generate a latin hypercube sample with a normal distribution
logncdf                        - Lognormal cumulative distribution function (cdf).
lognfit                        - Parameter estimates and confidence intervals for lognormal data.
logninv                        - Inverse of the lognormal cumulative distribution function (cdf).
lognlike                       - Negative log-likelihood for the lognormal distribution.
lognpdf                        - Lognormal probability density function (pdf).
lognrnd                        - Random arrays from the lognormal distribution.
lognstat                       - Mean and variance for the lognormal distribution.
mvncdf                         - Multivariate normal cumulative distribution function (cdf).
mvnpdf                         - Multivariate normal probability density function (pdf).
mvnrnd                         - Random vectors from the multivariate normal distribution.
normcdf                        - Normal cumulative distribution function (cdf).
normfit                        - Parameter estimates and confidence intervals for normal data.
norminv                        - Inverse of the normal cumulative distribution function (cdf).
normlike                       - Negative log-likelihood for the normal distribution.
normpdf                        - Normal probability density function (pdf).
normplot                       - Displays a normal probability plot.
normrnd                        - Random arrays from the normal distribution.
normspec                       - Plots normal density between specification limits.
normstat                       - Mean and variance for the normal distribution.
logn3fit                       - Fit a 3-param lognormal dist'n using cumulative probabilities.
wgtnormfit                     - Fitting example for a weighted normal distribution.
wgtnormfit2                    - Fitting example for a weighted normal distribution (log(sigma) parameterization).
>>

Judicious search terms help but seeing the list of things related to "normal" lets you find the two functions of interest (plus a lot more depending upon which toolboxes are available, maybe) that might be of use/interest...

Best Answer

Related Solutions

MATLAB: How to compute the probability of a function of random variables using the Statistics Toolbox 7.4 (R2007a)

MATLAB: How to go about finding the standard normal probability based on the z-score

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