02.1 - Input Data Analysis - DATS

In stochastic simulations, we must decide on input probability distributions from which to generate random variables:

• Random variables = observation or draws or realizations of a random variables from specified input distributions/processes
• Queueing systems: interarrival and service times

Once those input probability distributions are specified, must have a way to generate random variates from them

Specifying Univariate Input Distributions

• Univariate distribution: a single (scalar) random input
  • May have many across simulation

Usually, we have real-world observed data. We want to fit a probability distribution to the observed data.

Once this is done, we can generate random variates to drive the simulation.

Choosing Probability Distributions

There are many probability distributions, continuous and discrete.

For a comprehensive list of the main distributions, visit: https://www.itl.nist.gov/div898/handbook/eda/section3/eda366.htm

Some common continuous distributions:

• Normal distribution
• Exponential distribution
• Logistic distribution
• Continuous Uniform distribution
• Triangular Distribution - not in the assgm
• Weibull Distribution
• Gamma Distribution
• Pareto distribution
• Lognormal distribution
• Loglogistic distribution
• Erlang k Distribution
• Gumbel Distribution
• etc.

Some common discrete distributions:

• Discrete Uniform Distribution

• Binomial Distribution

• Poisson Distribution

• Geometric Distribution

• Negative Binomial Distribution

• Take a look at the data

  • Continuous vs Discrete
  • Look at the range (negative and positive, just positive,...)

• Make a histogram

  • Match it to a subset of possible distribution of real data

It is still needed to estimate distribution parameters (fit), then test the goodness of fit

Fitting distribution to data

The first step is always to observe the data. Some things to look at are:

• Continuous vs discrete
• Range (also support): finite or infinite on right and left end?

Then, steps to take are:

• Make a histogram
  • Match it to shapes of known probability density functions

With working sample:

• Select a possibile distribution
• Estimate parameters
• Test goodness of fit

With test sample

• Final validation of selected model

There are several distribution-fitting packages in R, provided in MINITAB.

Other resources in R Studio:

• fitidistrplus
• vdc
• moments packages

Coefficient of variation

In practice:

• If $C_X>1⟹$ log-exponential distribution
• If $C_X=1⟹$ exponential distribution
• If $C_X<1⟹$ Gamma family distribution

Descriptive statistics methods in R

• Variate type: continuous or discrete
  • a data sample is always discrete. If there are many many values, it's continuous
• Numerical description:
  • moments
  • mean
  • variance - standard deviation
  • Kurtosis
• Rough description in R:
  • summary(<dataframe>)
• Use library moments in R and compute first fourth sample moments, including skewness and kurtosis
• Check correlation in sample (acf(), pack())
• Remove shift for exponential, gamma, Weibull type data

Auxiliary Graphic Tools

Probability Plot - P Plot

Probability-Probability Plot

If $\hat{F}(x)$ and $\tilde{F}(x)$ are close together, then the P-P plot will also be approximately linear with an intercept 0 and a slope 1.

The linear correlation coefficient of the fit of the PP Plot is a measurement of the goodness of fit of the proposed distribution.

The PP plot graphs 2 funzioni di ripartizione against each other: the theoretical on the x-axis, and the empirical on the y-axis.

Quantile-Quantile Plot - QQ Plot