# Model design

See also: Building models that are easy to check and modify, Building models that are efficient, Colour coding models for clarity

Risk analysis is about supporting decisions by answering questions about risk. A good risk analyst provides qualitative, and where time and knowledge permit, quantitative, information to decision-makers that are pertinent to their questions. Inevitably, decision-makers must deal with other factors that may not be quantified in a risk analysis, which can be frustrating for a risk analyst when they see their work being 'ignored'. Don't let it be: the best risk analysts remain professionally neutral to the decisions that are made from their work.

The first step to designing a good model is to put yourself in the position of the decision-maker by understanding how the information you might provide connects to the questions they are asking. A decision-maker often does not appreciate all that comes with asking a question in a certain way, and may not initially have worked out all the possible options for handling the risk (or opportunity).

When you believe that you properly understand the risk question or questions that need(s) answering, it is time to brainstorm with colleagues, stakeholders, and the managers about how you might put an analysis together that satisfies the managers' needs. Effort put into the brainstorm stage pays back ten fold: everyone is clear on the purpose of your analysis; the participants will be more co-operative in providing information and estimates; and you can discuss the feasibility of any risk analysis approach.

We recommend you think of mapping out your ideas with Venn diagrams and event trees. Then look at the data (and perhaps expertise for subjective estimates) you believe are available to populate the model. If there are data gaps (there usually are), consider whether you will be able to get the necessary data to fill the gap, and quickly enough to be able to produce an analysis within the decision-maker's time frame. If the answer is 'no', look for other ways to produce an analysis that will meet the decision maker's needs, or perhaps a sub-set of those needs.

But whatever you do, don't embark on a risk analysis where you know that data gaps will remain and your decision maker is left with no useful support. Some scientists argue that risk analysis can also be for research purposes - to determine where the data gaps lie. We see the value in that determination, of course, but if that is your purpose, state it clearly and don't leave any expectation from the managers that will be unfulfilled.

The following topics offer some quality control tips to help you produce an efficient model that is the most easy to understand, modify and check:

There is a tendency to settle on the form that a risk analysis model will take too early on in the risk analysis process. In part that will be because of a limited knowledge of the available options, but also because people tend not to take a step back and ask themselves what the purpose of the analysis is, and also how it might evolve over time. In this topic we give a short guide to various types of model structure used in risk analysis.

### Choice of model structure

##### Spreadsheet modeling

Spreadsheets, and by that we mean Excel these days, are the most natural and first choice for most people because it is perceived that relatively little additional knowledge is required to produce a risk analysis model. Monte Carlo add-ins for Excel have made adding uncertainty into a spreadsheet as simple as clicking a few buttons. You can run a simulation and look at the distribution results in a few seconds and a few more button clicks. Monte Carlo simulation software tools for Excel have focused very much on the graphical interfaces to make risk analysis modelling easy: combine that with the ability to track formulae across spreadsheets, imbed graphs, format sheets in many ways, VBA and data importing capabilities and we can see why Excel is so popular. I have even seen a whole trading floor run on Excel using VBA, and not a single recognisable spreadsheet appeared on any dealer's screen.

But Excel has its limitations. ModelRisk overcomes many of them for high level financial and insurance modelling, but there are many types of problems for which Excel is not suitable. Project cost and schedule risk analysis can be done in spreadsheets at a crude level, and a crude level is often enough for large scale risk analysis since we are rarely interested in the minutia that can be built into a project planning model (like you might make with Primevera or Microsoft Project). However, a risk register is better constructed in an electronic database with various levels of access. The problem with building a project plan in a spreadsheet is that expanding the model into greater detail becomes mechanically very awkward, whilst it is a simple matter in project planning software.

In other areas, risk analysis models with spreadsheets have a number of limitations:

1. They scale very badly meaning that spreadsheets can become really huge when one has a lot of data, or when one is performing repetitive calculations that could be succinctly written in another language (e.g. a looping formula), although one can get round this to some degree with Visual Basic. Our company reviews many risk models built in spreadsheets and they can be vast, often unnecessarily so because there are shortcuts to achieving the same result if one knows a bit of probability mathematics. The next version of Excel will handle even bigger sheets so I predict this problem will only get worse;

2. They are limited to the two dimensions of a grid, three at a push if one uses sheets as a third dimension: if you have a multidimensional problem you should really think hard before deciding on a spreadsheet. There are a lot of other modelling environments one could use: C++ is highly flexible, but opaque to anyone who is not a C++ programmer. Matlab and, to a lesser extent, Mathematica and Maple, are highly sophisticated mathematical modelling software with very powerful built-in modelling capabilities that will handle many dimensions and can also perform simulations;

3. They are really slow. Running a simulation in Excel will take hundreds or more times longer than specialised tools. That's a problem if you have a huge model, or if you need to achieve a high level of precision (i.e. require many iterations);

4. Simulation models built in spreadsheets calculate in one direction meaning that if one acquires new data that can be matched to a forecast in the model, the data cannot be integrated into the model to update the estimates of parameters on which the model was based and therefore produce a more accurate forecast.

5. Spreadsheets cannot easily handle modelling dynamic systems. There are a number of flexible and user-friendly tools like Simul8 which give very good approximations to continuously varying stochastic systems with many interacting components. I give an example later in this topic. Attempting to achieve the same in Excel is not worth the pain.

##### Alternatives to spreadsheet modeling

Influence diagramsInfluence diagrams are quite popular - they essentially replicate the mathematics you can build in a spreadsheet, but the modelling environment is quite different. Analytica is the most popular influence diagram tool. Variables (called nodes) are represented as graphical objects (circles, squares, etc) and connected together with arrows (called arcs) which show the direction of interaction between these variables. The visual result is a network that shows the viewer which variables affect which, but you can imagine that such a diagram quickly becomes overly complex, so one builds sub-models. Click on a model object and it opens another view to show a lower level of interaction. A big drawback is that the mathematics and data behind the model are hard to get to, but some people love them. They are certainly very visual.

**Event trees**

Event trees offer a way to describe a sequence of probabilistic events, together with their probabilities and impacts. They are perhaps the most useful of all the methods for depicting a probabilistic sequence, because they are very intuitive, the mathematics to combine the probabilities is simple and the diagram helps ensure the necessary discipline. Event trees are built out of nodes (boxes) and arcs (arrows):

Example of a simple event tree

The tree starts from the left with a node (in the above diagram, Select Animal to denote the random selection of an animal from some population) and arrows to the right indicated possible outcomes (here whether the animal is infected with some particular disease agent, or not) and their probabilities (p, which would be the prevalence of infected animals in the population, and (1-p) respectively). Branching out from these boxes are arrows to the next probability event (the testing of an animal for the disease), and attached to these arrows are the conditional probabilities of the next level of event occurring. The conditional nature of the probabilities in an event tree are extremely important to underline. In this example:

Se = P(Test positive for disease *given* the animal is infected); and

Sp = P(Test negative for disease *given* the animal is not infected)

Thus, following the rules of conditional probability algebra, we can say, for example:

P(animal is infected and tests positive) = p*Se

P(animal is infected and tests negative) = p*(1-Se)

P(animal tests positive) = p*Se + (1-p)*(1-Sp)

Event trees are very useful for building up your probability thinking, although they will get quite complex rather quickly. We use them a great deal to help understand and communicate a problem.

**Decision trees**

Decision trees are like event trees but add possible decision options. They have a role in risk analysis and in fields like petroleum exploration are very popular. They sketch the possible decisions that one might make and the outcomes that might result. Decision tree software (which can also produce event trees) can also calculate the best option to take under the assumption of some user-defined utility function. It often is difficult for decision-makers to be comfortable with defining a utility curve, but they are useful for communicating the logic of a problem.

Example of a simple decision tree. The decision options are to make either of two investments or do nothing with associated revenues as a result. More involved decision trees would include two or more sequential decisions depending on how well the investment went.

**Fault trees**

Fault trees start from the reverse approach to an event tree. An event tree looks forward from a starting point and considers the possible future outcomes. A fault tree starts with the outcome and looks at the ways it could have arisen. A fault tree is therefore constructed from the right with the outcome and moves to the left with the possible immediate events that could have made that outcome arise, and continues backwards with the possible events that could have made the first set of events arise, etc.

Fault trees are very useful for focusing attention on what might go wrong and why. It has been used in reliability engineering for a long time, but also has applications in areas like terrorism. For example, one might start with the risk of deliberate contamination of a city's drinking water supply and then consider routes that the terrorist could use (pipeline, treatment plant, reservoir, etc) and the probabilities of being able to do that given the security in place.

**Discrete event simulation**

Discrete Event Simulation (DES) differs from Monte Carlo simulation mainly in that it models the evolution of a (usually stochastic) system over time. It does this by allowing the user to define equations for each element in the model for how it changes, moves and interacts with other elements. Then it steps the system through small time increments and keeps track of where all elements are at any time (e.g. parts in a manufacturing system, passengers in an airport or ships in a harbour). More sophisticated tools can increase the clock steps when nothing is happening, then decrease again to get a more accurate approximation to the continuous behavior it is modelling.

### Monte Carlo Simulation

See also: Monte Carlo simulation introduction

This technique involves the random sampling of each probability distribution within the model to produce hundreds or even thousands of scenarios (also called iterations or trials). Each probability distribution is sampled in a manner that reproduces the distribution's shape. The distribution of the values calculated for the model outcome therefore reflects the probability of the values that could occur. Monte Carlo simulation offers many advantages over the other techniques presented above:

· The distributions of the model's variables do not have to be approximated in any way.

· Correlation and other inter-dependencies can be modelled.

· The level of mathematics required to perform a Monte Carlo simulation is quite basic.

· The computer does all of the work required in determining the outcome distribution.

· Software is commercially available to automate the tasks involved in the simulation.

· Complex mathematics can be included (e.g. power functions, logs, IF statements, etc.) with no extra difficulty.

· Monte Carlo simulation is widely recognised as a valid technique so its results are more likely to be accepted.

· The behaviour of the model can be investigated with great ease.

· Changes to the model can be made very quickly and the results compared with previous models.

Monte Carlo simulation is often criticised as being an approximate technique. However, in theory at least, any required level of precision can be achieved by simply increasing the number of iterations in a simulation. The limitations are in the number of random numbers that can be produced from a random number generating algorithm and, more commonly, the time a computer needs to generate the iterations. For a great many problems, these limitations are irrelevant or can be avoided by structuring the model into sections.

We can conclude that, except for the situations described above, that spreadsheet modeling combined with Monte Carlo simulation is the best approach in the majority of risk analyses. Monte Carlo simulation is explained in more depth in the Monte Carlo simulation section.

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**Navigation**

- Risk management
- Risk management introduction
- What are risks and opportunities?
- Planning a risk analysis
- Clearly stating risk management questions
- Evaluating risk management options
- Introduction to risk analysis
- The quality of a risk analysis
- Using risk analysis to make better decisions
- Explaining a models assumptions
- Statistical descriptions of model outputs
- Simulation Statistical Results
- Preparing a risk analysis report
- Graphical descriptions of model outputs
- Presenting and using results introduction
- Statistical descriptions of model results
- Mean deviation (MD)
- Range
- Semi-variance and semi-standard deviation
- Kurtosis (K)
- Mean
- Skewness (S)
- Conditional mean
- Custom simulation statistics table
- Mode
- Cumulative percentiles
- Median
- Relative positioning of mode median and mean
- Variance
- Standard deviation
- Inter-percentile range
- Normalized measures of spread - the CofV

- Graphical descriptionss of model results
- Showing probability ranges
- Overlaying histogram plots
- Scatter plots
- Effect of varying number of bars
- Sturges rule
- Relationship between cdf and density (histogram) plots
- Difficulty of interpreting the vertical scale
- Stochastic dominance tests
- Risk-return plots
- Second order cumulative probability plot
- Ascending and descending cumulative plots
- Tornado plot
- Box Plot
- Cumulative distribution function (cdf)
- Probability density function (pdf)
- Crude sensitivity analysis for identifying important input distributions
- Pareto Plot
- Trend plot
- Probability mass function (pmf)
- Overlaying cdf plots
- Cumulative Plot
- Simulation data table
- Statistics table
- Histogram Plot
- Spider plot
- Determining the width of histogram bars
- Plotting a variable with discrete and continuous elements
- Smoothing a histogram plot

- Risk analysis modeling techniques
- Monte Carlo simulation
- Monte Carlo simulation introduction
- Monte Carlo simulation in ModelRisk
- Filtering simulation results
- Output/Input Window
- Simulation Progress control
- Running multiple simulations
- Random number generation in ModelRisk
- Random sampling from input distributions
- How many Monte Carlo samples are enough?

- Probability distributions
- Distributions introduction
- Probability calculations in ModelRisk
- Selecting the appropriate distributions for your model
- List of distributions by category
- Distribution functions and the U parameter
- Univariate continuous distributions
- Beta distribution
- Beta Subjective distribution
- Four-parameter Beta distribution
- Bradford distribution
- Burr distribution
- Cauchy distribution
- Chi distribution
- Chi Squared distribution
- Continuous distributions introduction
- Continuous fitted distribution
- Cumulative ascending distribution
- Cumulative descending distribution
- Dagum distribution
- Erlang distribution
- Error distribution
- Error function distribution
- Exponential distribution
- Exponential family of distributions
- Extreme Value Minimum distribution
- Extreme Value Maximum distribution
- F distribution
- Fatigue Life distribution
- Gamma distribution
- Generalized Extreme Value distribution
- Generalized Logistic distribution
- Generalized Trapezoid Uniform (GTU) distribution
- Histogram distribution
- Hyperbolic-Secant distribution
- Inverse Gaussian distribution
- Johnson Bounded distribution
- Johnson Unbounded distribution
- Kernel Continuous Unbounded distribution
- Kumaraswamy distribution
- Kumaraswamy Four-parameter distribution
- Laplace distribution
- Levy distribution
- Lifetime Two-Parameter distribution
- Lifetime Three-Parameter distribution
- Lifetime Exponential distribution
- LogGamma distribution
- Logistic distribution
- LogLaplace distribution
- LogLogistic distribution
- LogLogistic Alternative parameter distribution
- LogNormal distribution
- LogNormal Alternative-parameter distribution
- LogNormal base B distribution
- LogNormal base E distribution
- LogTriangle distribution
- LogUniform distribution
- Noncentral Chi squared distribution
- Noncentral F distribution
- Normal distribution
- Normal distribution with alternative parameters
- Maxwell distribution
- Normal Mix distribution
- Relative distribution
- Ogive distribution
- Pareto (first kind) distribution
- Pareto (second kind) distribution
- Pearson Type 5 distribution
- Pearson Type 6 distribution
- Modified PERT distribution
- PERT distribution
- PERT Alternative-parameter distribution
- Reciprocal distribution
- Rayleigh distribution
- Skew Normal distribution
- Slash distribution
- SplitTriangle distribution
- Student-t distribution
- Three-parameter Student distribution
- Triangle distribution
- Triangle Alternative-parameter distribution
- Uniform distribution
- Weibull distribution
- Weibull Alternative-parameter distribution
- Three-Parameter Weibull distribution

- Univariate discrete distributions
- Discrete distributions introduction
- Bernoulli distribution
- Beta-Binomial distribution
- Beta-Geometric distribution
- Beta-Negative Binomial distribution
- Binomial distribution
- Burnt Finger Poisson distribution
- Delaporte distribution
- Discrete distribution
- Discrete Fitted distribution
- Discrete Uniform distribution
- Geometric distribution
- HypergeoM distribution
- Hypergeometric distribution
- HypergeoD distribution
- Inverse Hypergeometric distribution
- Logarithmic distribution
- Negative Binomial distribution
- Poisson distribution
- Poisson Uniform distribution
- Polya distribution
- Skellam distribution
- Step Uniform distribution
- Zero-modified counting distributions
- More on probability distributions

- Multivariate distributions
- Multivariate distributions introduction
- Dirichlet distribution
- Multinomial distribution
- Multivariate Hypergeometric distribution
- Multivariate Inverse Hypergeometric distribution type2
- Negative Multinomial distribution type 1
- Negative Multinomial distribution type 2
- Multivariate Inverse Hypergeometric distribution type1
- Multivariate Normal distribution

- More on probability distributions
- Approximating one distribution with another
- Approximations to the Inverse Hypergeometric Distribution
- Normal approximation to the Gamma Distribution
- Normal approximation to the Poisson Distribution
- Approximations to the Hypergeometric Distribution
- Stirlings formula for factorials
- Normal approximation to the Beta Distribution
- Approximation of one distribution with another
- Approximations to the Negative Binomial Distribution
- Normal approximation to the Student-t Distribution
- Approximations to the Binomial Distribution
- Normal_approximation_to_the_Binomial_distribution
- Poisson_approximation_to_the_Binomial_distribution

- Normal approximation to the Chi Squared Distribution
- Recursive formulas for discrete distributions
- Normal approximation to the Lognormal Distribution
- Normal approximations to other distributions

- Approximating one distribution with another

- Correlation modeling in risk analysis
- Common mistakes when adapting spreadsheet models for risk analysis
- More advanced risk analysis methods
- SIDs
- Modeling with objects
- ModelRisk database connectivity functions
- PK/PD modeling
- Value of information techniques
- Simulating with ordinary differential equations (ODEs)
- Optimization of stochastic models
- ModelRisk optimization extension introduction
- Optimization Settings
- Defining Simulation Requirements in an Optimization Model
- Defining Decision Constraints in an Optimization Model
- Optimization Progress control
- Defining Targets in an Optimization Model
- Defining Decision Variables in an Optimization Model
- Optimization Results

- Summing random variables
- Aggregate distributions introduction
- Aggregate modeling - Panjer's recursive method
- Adding correlation in aggregate calculations
- Sum of a random number of random variables
- Moments of an aggregate distribution
- Aggregate modeling in ModelRisk
- Aggregate modeling - Fast Fourier Transform (FFT) method
- How many random variables add up to a fixed total
- Aggregate modeling - compound Poisson approximation
- Aggregate modeling - De Pril's recursive method

- Testing and modeling causal relationships
- Stochastic time series
- Time series introduction
- Time series in ModelRisk
- Autoregressive models
- Thiel inequality coefficient
- Effect of an intervention at some uncertain point in time
- Log return of a Time Series
- Markov Chain models
- Seasonal time series
- Bounded random walk
- Time series modeling in finance
- Birth and death models
- Time series models with leading indicators
- Geometric Brownian Motion models
- Time series projection of events occurring randomly in time

- Simulation for six sigma
- ModelRisk's Six Sigma functions
- VoseSixSigmaCp
- VoseSixSigmaCpkLower
- VoseSixSigmaProbDefectShift
- VoseSixSigmaLowerBound
- VoseSixSigmaK
- VoseSixSigmaDefectShiftPPMUpper
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- VoseSixSigmaYield
- VoseSixSigmaUpperBound
- VoseSixSigmaZupper
- VoseSixSigmaZmin
- VoseSixSigmaZlower

- Modeling expert opinion
- Modeling expert opinion introduction
- Sources of error in subjective estimation
- Disaggregation
- Distributions used in modeling expert opinion
- A subjective estimate of a discrete quantity
- Incorporating differences in expert opinions
- Modeling opinion of a variable that covers several orders of magnitude
- Maximum entropy

- Probability theory and statistics
- Probability theory and statistics introduction
- Stochastic processes
- Stochastic processes introduction
- Poisson process
- Hypergeometric process
- The hypergeometric process
- Number in a sample with a particular characteristic in a hypergeometric process
- Number of hypergeometric samples to get a specific number of successes
- Number of samples taken to have an observed s in a hypergeometric process
- Estimate of population and sub-population sizes in a hypergeometric process

- The binomial process
- Renewal processes
- Mixture processes
- Martingales
- Estimating model parameters from data

- The basics
- Probability equations
- Probability theorems and useful concepts
- Probability parameters
- Probability rules and diagrams
- The definition of probability
- The basics of probability theory introduction

- Fitting probability models to data
- Fitting time series models to data
- Fitting correlation structures to data
- Fitting in ModelRisk
- Fitting probability distributions to data
- Fitting distributions to data
- Method of Moments (MoM)
- Check the quality of your data
- Kolmogorov-Smirnoff (K-S) Statistic
- Anderson-Darling (A-D) Statistic
- Goodness of fit statistics
- The Chi-Squared Goodness-of-Fit Statistic
- Determining the joint uncertainty distribution for parameters of a distribution
- Using Method of Moments with the Bootstrap
- Maximum Likelihood Estimates (MLEs)
- Fitting a distribution to truncated censored or binned data
- Critical Values and Confidence Intervals for Goodness-of-Fit Statistics
- Matching the properties of the variable and distribution
- Transforming discrete data before performing a parametric distribution fit
- Does a parametric distribution exist that is well known to fit this type of variable?
- Censored data
- Fitting a continuous non-parametric second-order distribution to data
- Goodness of Fit Plots
- Fitting a second order Normal distribution to data
- Using Goodness-of Fit Statistics to optimize Distribution Fitting
- Information criteria - SIC HQIC and AIC
- Fitting a second order parametric distribution to observed data
- Fitting a distribution for a continuous variable
- Does the random variable follow a stochastic process with a well-known model?
- Fitting a distribution for a discrete variable
- Fitting a discrete non-parametric second-order distribution to data
- Fitting a continuous non-parametric first-order distribution to data
- Fitting a first order parametric distribution to observed data
- Fitting a discrete non-parametric first-order distribution to data

- Fitting distributions to data
- Technical subjects
- Comparison of Classical and Bayesian methods
- Comparison of classic and Bayesian estimate of Normal distribution parameters
- Comparison of classic and Bayesian estimate of intensity lambda in a Poisson process
- Comparison of classic and Bayesian estimate of probability p in a binomial process
- Which technique should you use?
- Comparison of classic and Bayesian estimate of mean "time" beta in a Poisson process

- Classical statistics
- Bayesian
- Bootstrap
- The Bootstrap
- Linear regression parametric Bootstrap
- The Jackknife
- Multiple variables Bootstrap Example 2: Difference between two population means
- Linear regression non-parametric Bootstrap
- The parametric Bootstrap
- Bootstrap estimate of prevalence
- Estimating parameters for multiple variables
- Example: Parametric Bootstrap estimate of the mean of a Normal distribution with known standard deviation
- The non-parametric Bootstrap
- Example: Parametric Bootstrap estimate of mean number of calls per hour at a telephone exchange
- The Bootstrap likelihood function for Bayesian inference
- Multiple variables Bootstrap Example 1: Estimate of regression parameters

- Bayesian inference
- Uninformed priors
- Conjugate priors
- Prior distributions
- Bayesian analysis with threshold data
- Bayesian analysis example: gender of a random sample of people
- Informed prior
- Simulating a Bayesian inference calculation
- Hyperparameters
- Hyperparameter example: Micro-fractures on turbine blades
- Constructing a Bayesian inference posterior distribution in Excel
- Bayesian analysis example: Tigers in the jungle
- Markov chain Monte Carlo (MCMC) simulation
- Introduction to Bayesian inference concepts
- Bayesian estimate of the mean of a Normal distribution with known standard deviation
- Bayesian estimate of the mean of a Normal distribution with unknown standard deviation
- Determining prior distributions for correlated parameters
- Improper priors
- The Jacobian transformation
- Subjective prior based on data
- Taylor series approximation to a Bayesian posterior distribution
- Bayesian analysis example: The Monty Hall problem
- Determining prior distributions for uncorrelated parameters
- Subjective priors
- Normal approximation to the Beta posterior distribution
- Bayesian analysis example: identifying a weighted coin
- Bayesian estimate of the standard deviation of a Normal distribution with known mean
- Likelihood functions
- Bayesian estimate of the standard deviation of a Normal distribution with unknown mean
- Determining a prior distribution for a single parameter estimate
- Simulating from a constructed posterior distribution

- Bootstrap

- Comparison of Classical and Bayesian methods
- Analyzing and using data introduction
- Data Object
- Vose probability calculation
- Bayesian model averaging

- Miscellaneous
- Excel and ModelRisk model design and validation techniques
- Using range names for model clarity
- Color coding models for clarity
- Compare with known answers
- Checking units propagate correctly
- Stressing parameter values
- Model Validation and behavior introduction
- Informal auditing
- Analyzing outputs
- View random scenarios on screen and check for credibility
- Split up complex formulas (megaformulas)
- Building models that are efficient
- Comparing predictions against reality
- Numerical integration
- Comparing results of alternative models
- Building models that are easy to check and modify
- Model errors
- Model design introduction

- About array functions in Excel

- Excel and ModelRisk model design and validation techniques

- Monte Carlo simulation
- RISK ANALYSIS SOFTWARE
- Risk analysis software from Vose Software
- ModelRisk - risk modeling in Excel
- ModelRisk functions explained
- VoseCopulaOptimalFit and related functions
- VoseTimeOptimalFit and related functions
- VoseOptimalFit and related functions
- VoseXBounds
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- VoseRawMoments
- VoseSkewness
- VoseMoments
- VoseKurtosis
- VoseAggregatePanjer
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- VoseCombined
- VoseCopulaBiGumbel
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- VosePrincipleEsscher
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- VosePBounds
- VoseAIC
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- VoseNBootStdev
- VoseNBoot
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- VoseAggregateMC
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- VoseDeduct
- VoseExpression
- VoseLargestSet
- VoseKthSmallest
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- VoseNBootCofV
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- VoseIntegrate
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- VoseCopulaMultiGumbel
- VoseCopulaMultiT
- VoseAggregateMultiMC
- VoseCopulaMultiFrank
- VoseTimeMultiMA1
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- VoseTimeMultiGBM
- VoseTimeMultBEKK
- VoseAggregateDePril
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- VoseTimeMultiAR2
- VoseRuinFlag
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- VoseDepletionShortfall
- VoseDepletion
- VoseDepletionFlag
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- VosejProduct
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- VoseDataMin
- VoseDataMax
- VoseTimeSME2Perc
- VoseTimeSMEUniform
- VoseTimeSMESaturation
- VoseOutput
- VoseInput
- VoseTimeSMEPoisson
- VoseTimeBMAObject
- VoseBMAObject
- VoseBMAProb10
- VoseBMAProb
- VoseCopulaBMA
- VoseCopulaBMAObject
- VoseTimeEmpiricalFit
- VoseTimeBMA
- VoseBMA
- VoseSimKurtosis
- VoseOptConstraintMin
- VoseSimProbability
- VoseCurrentSample
- VoseCurrentSim
- VoseLibAssumption
- VoseLibReference
- VoseSimMoments
- VoseOptConstraintMax
- VoseSimMean
- VoseOptDecisionContinuous
- VoseOptRequirementEquals
- VoseOptRequirementMax
- VoseOptRequirementMin
- VoseOptTargetMinimize
- VoseOptConstraintEquals
- VoseSimVariance
- VoseSimSkewness
- VoseSimTable
- VoseSimCofV
- VoseSimPercentile
- VoseSimStDev
- VoseOptTargetValue
- VoseOptTargetMaximize
- VoseOptDecisionDiscrete
- VoseSimMSE
- VoseMin
- VoseMin
- VoseOptDecisionList
- VoseOptDecisionBoolean
- VoseOptRequirementBetween
- VoseOptConstraintBetween
- VoseSimMax
- VoseSimSemiVariance
- VoseSimSemiStdev
- VoseSimMeanDeviation
- VoseSimMin
- VoseSimCVARp
- VoseSimCVARx
- VoseSimCorrelation
- VoseSimCorrelationMatrix
- VoseOptConstraintString
- VoseOptCVARx
- VoseOptCVARp
- VoseOptPercentile
- VoseSimValue
- VoseSimStop
- Precision Control Functions
- VoseAggregateDiscrete
- VoseTimeMultiGARCH
- VoseTimeGBMVR
- VoseTimeGBMAJ
- VoseTimeGBMAJVR
- VoseSID
- Generalized Pareto Distribution (GPD)
- Generalized Pareto Distribution (GPD) Equations
- Three-Point Estimate Distribution
- Three-Point Estimate Distribution Equations
- VoseCalibrate

- ModelRisk interfaces
- Integrate
- Data Viewer
- Stochastic Dominance
- Library
- Correlation Matrix
- Portfolio Optimization Model
- Common elements of ModelRisk interfaces
- Risk Event
- Extreme Values
- Select Distribution
- Combined Distribution
- Aggregate Panjer
- Interpolate
- View Function
- Find Function
- Deduct
- Ogive
- AtRISK model converter
- Aggregate Multi FFT
- Stop Sum
- Crystal Ball model converter
- Aggregate Monte Carlo
- Splicing Distributions
- Subject Matter Expert (SME) Time Series Forecasts
- Aggregate Multivariate Monte Carlo
- Ordinary Differential Equation tool
- Aggregate FFT
- More on Conversion
- Multivariate Copula
- Bivariate Copula
- Univariate Time Series
- Modeling expert opinion in ModelRisk
- Multivariate Time Series
- Sum Product
- Aggregate DePril
- Aggregate Discrete
- Expert

- ModelRisk introduction
- Building and running a simple example model
- Distributions in ModelRisk
- List of all ModelRisk functions
- Custom applications and macros

- ModelRisk functions explained
- Tamara - project risk analysis
- Introduction to Tamara project risk analysis software
- Launching Tamara
- Importing a schedule
- Assigning uncertainty to the amount of work in the project
- Assigning uncertainty to productivity levels in the project
- Adding risk events to the project schedule
- Adding cost uncertainty to the project schedule
- Saving the Tamara model
- Running a Monte Carlo simulation in Tamara
- Reviewing the simulation results in Tamara
- Using Tamara results for cost and financial risk analysis
- Creating, updating and distributing a Tamara report
- Tips for creating a schedule model suitable for Monte Carlo simulation
- Random number generator and sampling algorithms used in Tamara
- Probability distributions used in Tamara
- Correlation with project schedule risk analysis

- Pelican - enterprise risk management