Least Squares Linear Regression
See also: The basics of probability theory introduction, Estimating model parameters from data
The purpose of least squares linear regression is to represent the relationship between one or more independent variables x1, x2, and a variable y that is dependent upon them in the following form:
where
= the ith observed value of the independent variable xj
= the ith observed value of the dependent variable y
= the error term or residual (i.e. the difference between the observed y values and that predicted by the model)
= the regression slope for the variable xj and
= the yaxis intercept
Simple least squares linear regression assumes that there is only one independent variable x . If we assume that the error terms are Normally distributed, the equation reduces to:
yi = Normal(m*xi + c, s)
where m is the slope of the line and c is the yaxis intercept and s is the standard deviation of the variation of y about this line.
Simple least squares linear regression is a very standard statistical analysis techniques, particularly when one has little or no idea of the relationship between the x and y variables. It is probably particularly common because the analysis mathematics are simple (because of the Normality assumption), rather than it being a very common rule for the relationship between variables. LSR makes four important assumptions:

Individual y values are independent

For each xi, there are an infinite number of possible values of y, which are Normally distributed

The distribution of y given a value of x has equal standard deviation for all x values and is centred about the least squares regression line

The means of the distribution of y at each x value can be connected by a straight line y = mx + c.
Assumptions behind least squares regression analysis
Statisticians often make transformations of the data (e.g. Log(Y), √X) to force a linear relationship. That greatly extends the applicability of the regression model but one must be particularly careful that the errors are reasonably Normal, and one runs an enormous risk in using the regression equations to make predictions outside the range of observations.
Estimation of parameters
The simple least squares regression model determines the straight line that minimizes the sum of the square of the ei errors. It can be shown that this occurs when:
where are the mean of the observed x and y data and n is the number of data pairs (xi,yi).
The fraction of the total variation in the dependent variable that is explained by the independent variable is known as the coefficient of determination R2, which is calculated as:
where SSE, the sum of squares errors, is given by:
and TSS, the total sum of squares, is given by:
and where are the predicted y values at each xi:
For simple least squares regression (i.e. only one independent variable), the square root of R2 is equivalent to the simple correlation coefficient r
sqrt(R) = r
r may alternatively be calculated as:
r provides a quantitative measure of the linear relationship between x and y. It ranges from 1 to +1: a value of r = 1 or +1 indicates a perfect linear fit, and r = 0 indicates no linear relationship exists at all. As
,
the sum of squared errors between the observed and predicted yvalues, tends to zero, so r2 tends to 1 and therefore r tends to 1 or +1, its sign depending on whether m is negative or positive respectively.
The value of r is used to determine the statistical significance of the fitted line, by first calculating the test statistic t as:
The t statistic follows a tdistribution with (n2) degrees of freedom (provided the linear regression assumptions of Normally distributed variation of y about the regression line hold) which is used to determine whether the fit should be rejected or not at the required level of confidence.
The standard error of the yestimate Syx is calculated as:
This is equivalent to the standard deviation of the error terms ei. These errors reflect the true variability of the dependent variable y from the least squares regression line. The denominator (n2) is used, instead of the (n1) we have seen before for sample standard deviation calculations, because two values m and c have been estimated from the data to determine the equation values, and we have therefore lost 2 degrees of freedom instead of the 1 degree of freedom usually lost in determining the mean.
The equation of the regression line equation and the Syx statistic can be used together to produce a stochastic model of the relationship between X and Y, as follows:
Y = Normal(m*X+c,Syx)
Some caution is needed in using such a model. The regression model is intended to work within the range of the independent variable x for which there have been observations. Using the model outside this range can produce very significant errors if the relationship between x and y deviates from this linear relationship. This is also purely a model of variability, i.e. we are assuming that the linear relationship is correct and that the parameters are known. We should also include our uncertainty about the parameters, and perhaps about whether the linear relationship is even appropriate.
Example
Consider the data set in Table 1 that shows the result of a survey of 30 people. They were asked to provide details of their monthly net income {xi} and the amount they spent on food each month {yi}.
The values of m, c, r and Syx were calculated using the Excel functions:
m = SLOPE({yi},{xi}) = 0.1356
c = INTERCEPT({yi},{xi}) = 167.8
r2 = RSQ({yi},{xi}) = 0.8831
Syx = STEYX({yi},{xi}) = 54.86
The line is plotted against the data points in Figure 2.
Figure2.
The error terms are shown in Figure 3:
Figure 3.
A distribution fit of these ei values shows that they are approximately Normally distributed. A test of significance of r also shows that, for 28 degrees of freedom (n2), there is only about a 5.1011 chance that such a high value of r could have been observed from purely random data. We would therefore feel confident in modelling the relationship between any net monthly income value N (between the values 505 and 1581) and monthly expenditure on food F using:
F = Normal(0.1356*N+167.8, 54.86)
Uncertainty about least squares regression parameters
The parameters m, c, and Syx for the least squares regression represent the best estimate of the variability model where we are assuming some stochastically linear relationship between x and y. However, since we will have only a limited number of observations (i.e. {x,y} pairs), we do not have perfect knowledge of the stochastic system and there is therefore some uncertainty about the regression parameters. The ttest tells us whether the linear relationship might exist at some level of confidence. More useful, however, from a risk analysis perspective is that we can readily determine distributions of uncertainty about these parameters using the Bootstrap.
Read on: Taylor series
ModelRisk
Monte Carlo simulation in Excel. Learn more
Tamara
Adding risk and uncertainty to your project schedule. Learn more
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
 Semivariance and semistandard 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
 Interpercentile 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
 Riskreturn 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
 Fourparameter 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
 HyperbolicSecant distribution
 Inverse Gaussian distribution
 Johnson Bounded distribution
 Johnson Unbounded distribution
 Kernel Continuous Unbounded distribution
 Kumaraswamy distribution
 Kumaraswamy Fourparameter distribution
 Laplace distribution
 Levy distribution
 Lifetime TwoParameter distribution
 Lifetime ThreeParameter distribution
 Lifetime Exponential distribution
 LogGamma distribution
 Logistic distribution
 LogLaplace distribution
 LogLogistic distribution
 LogLogistic Alternative parameter distribution
 LogNormal distribution
 LogNormal Alternativeparameter 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 Alternativeparameter distribution
 Reciprocal distribution
 Rayleigh distribution
 Skew Normal distribution
 Slash distribution
 SplitTriangle distribution
 Studentt distribution
 Threeparameter Student distribution
 Triangle distribution
 Triangle Alternativeparameter distribution
 Uniform distribution
 Weibull distribution
 Weibull Alternativeparameter distribution
 ThreeParameter Weibull distribution
 Univariate discrete distributions
 Discrete distributions introduction
 Bernoulli distribution
 BetaBinomial distribution
 BetaGeometric distribution
 BetaNegative 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
 Zeromodified 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 Studentt 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
 VoseSixSigmaDefectShiftPPMLower
 VoseSixSigmaDefectShiftPPM
 VoseSixSigmaCpm
 VoseSixSigmaSigmaLevel
 VoseSixSigmaCpkUpper
 VoseSixSigmaCpk
 VoseSixSigmaDefectPPM
 VoseSixSigmaProbDefectShiftLower
 VoseSixSigmaProbDefectShiftUpper
 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 subpopulation 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
 KolmogorovSmirnoff (KS) Statistic
 AndersonDarling (AD) Statistic
 Goodness of fit statistics
 The ChiSquared GoodnessofFit 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 GoodnessofFit 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 nonparametric secondorder distribution to data
 Goodness of Fit Plots
 Fitting a second order Normal distribution to data
 Using Goodnessof 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 wellknown model?
 Fitting a distribution for a discrete variable
 Fitting a discrete nonparametric secondorder distribution to data
 Fitting a continuous nonparametric firstorder distribution to data
 Fitting a first order parametric distribution to observed data
 Fitting a discrete nonparametric firstorder 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 nonparametric 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 nonparametric 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: Microfractures 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
 VoseCLTSum
 VoseAggregateMoments
 VoseRawMoments
 VoseSkewness
 VoseMoments
 VoseKurtosis
 VoseAggregatePanjer
 VoseAggregateFFT
 VoseCombined
 VoseCopulaBiGumbel
 VoseCopulaBiClayton
 VoseCopulaBiNormal
 VoseCopulaBiT
 VoseKendallsTau
 VoseRiskEvent
 VoseCopulaBiFrank
 VoseCorrMatrix
 VoseRank
 VoseValidCorrmat
 VoseSpearman
 VoseCopulaData
 VoseCorrMatrixU
 VoseTimeSeasonalGBM
 VoseMarkovSample
 VoseMarkovMatrix
 VoseThielU
 VoseTimeEGARCH
 VoseTimeAPARCH
 VoseTimeARMA
 VoseTimeDeath
 VoseTimeAR1
 VoseTimeAR2
 VoseTimeARCH
 VoseTimeMA2
 VoseTimeGARCH
 VoseTimeGBMJDMR
 VoseTimePriceInflation
 VoseTimeGBMMR
 VoseTimeWageInflation
 VoseTimeLongTermInterestRate
 VoseTimeMA1
 VoseTimeGBM
 VoseTimeGBMJD
 VoseTimeShareYields
 VoseTimeYule
 VoseTimeShortTermInterestRate
 VoseDominance
 VoseLargest
 VoseSmallest
 VoseShift
 VoseStopSum
 VoseEigenValues
 VosePrincipleEsscher
 VoseAggregateMultiFFT
 VosePrincipleEV
 VoseCopulaMultiNormal
 VoseRunoff
 VosePrincipleRA
 VoseSumProduct
 VosePrincipleStdev
 VosePoissonLambda
 VoseBinomialP
 VosePBounds
 VoseAIC
 VoseHQIC
 VoseSIC
 VoseOgive1
 VoseFrequency
 VoseOgive2
 VoseNBootStdev
 VoseNBoot
 VoseSimulate
 VoseNBootPaired
 VoseAggregateMC
 VoseMean
 VoseStDev
 VoseAggregateMultiMoments
 VoseDeduct
 VoseExpression
 VoseLargestSet
 VoseKthSmallest
 VoseSmallestSet
 VoseKthLargest
 VoseNBootCofV
 VoseNBootPercentile
 VoseExtremeRange
 VoseNBootKurt
 VoseCopulaMultiClayton
 VoseNBootMean
 VoseTangentPortfolio
 VoseNBootVariance
 VoseNBootSkewness
 VoseIntegrate
 VoseInterpolate
 VoseCopulaMultiGumbel
 VoseCopulaMultiT
 VoseAggregateMultiMC
 VoseCopulaMultiFrank
 VoseTimeMultiMA1
 VoseTimeMultiMA2
 VoseTimeMultiGBM
 VoseTimeMultBEKK
 VoseAggregateDePril
 VoseTimeMultiAR1
 VoseTimeWilkie
 VoseTimeDividends
 VoseTimeMultiAR2
 VoseRuinFlag
 VoseRuinTime
 VoseDepletionShortfall
 VoseDepletion
 VoseDepletionFlag
 VoseDepletionTime
 VosejProduct
 VoseCholesky
 VoseTimeSimulate
 VoseNBootSeries
 VosejkProduct
 VoseRuinSeverity
 VoseRuin
 VosejkSum
 VoseTimeDividendsA
 VoseRuinNPV
 VoseTruncData
 VoseSample
 VoseIdentity
 VoseCopulaSimulate
 VoseSortA
 VoseFrequencyCumulA
 VoseAggregateDeduct
 VoseMeanExcessP
 VoseProb10
 VoseSpearmanU
 VoseSortD
 VoseFrequencyCumulD
 VoseRuinMaxSeverity
 VoseMeanExcessX
 VoseRawMoment3
 VosejSum
 VoseRawMoment4
 VoseNBootMoments
 VoseVariance
 VoseTimeShortTermInterestRateA
 VoseTimeLongTermInterestRateA
 VoseProb
 VoseDescription
 VoseCofV
 VoseAggregateProduct
 VoseEigenVectors
 VoseTimeWageInflationA
 VoseRawMoment1
 VosejSumInf
 VoseRawMoment2
 VoseShuffle
 VoseRollingStats
 VoseSplice
 VoseTSEmpiricalFit
 VoseTimeShareYieldsA
 VoseParameters
 VoseAggregateTranche
 VoseCovToCorr
 VoseCorrToCov
 VoseLLH
 VoseTimeSMEThreePoint
 VoseDataObject
 VoseCopulaDataSeries
 VoseDataRow
 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
 ThreePoint Estimate Distribution
 ThreePoint 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
 ModelRisk Cloud system
 ModelRisk Cloud introduction
 Getting your software ready
 Starting ModelRisk Cloud
 Uploading a risk analysis model
 Creating a new scenario for the risk analysis model
 Running a Monte Carlo simulation of the model
 Uploading a SID (Simulation Imported Data file)
 Building a risk analysis model that uses SIDs
 Viewing the Monte Carlo results from a simulation run
 Administrator's use of ModelRisk Cloud
 Preparing a risk analysis model for upload to ModelRisk Cloud
 ModelRisk Result Viewer
Enterprise Risk Management software (ERM)
Learn more about our enterprise risk analysis management software tool, Pelican