Methods of Uncertainty Analysis“Persons pretending to forecast the future shall be considered disorderly under subdivision 3, section 901 of the criminal codWhy Worry about Uncertainty?Types of UncertaintyDescribing an Uncertainty QuantityDescribing an Uncertain Quantity IIProbability Density FunctionsCumulative Density FunctionInput distributionsCorrelation between DistributionsMeasuring Uncertainty – Local MeasuresHigh Sensitivity SystemMeasuring Uncertainty - GlobalMonte Carlo SimulationMonte Carlo SimulationMonte Carlo, n=10,100Monte Carlo, n=1000,10000Input Sample, Latin HypercubeThe Greenhouse GambleClick to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level1Methods of Uncertainty Analysis15.023 LectureApril 7th, 2004Marcus C. Sarofim(based on lectures byMort Webster and Ian Sue Wing)Click to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level2“Persons pretending to forecast the future shall be considered disorderly under subdivision 3, section 901 of the criminal code and liable to a fine of $250 and/or six months in prison.”Section 889, New York State Code of Criminal ProcedureClick to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level3Why Worry about Uncertainty?• To identify important factors and assumptions underlying disagreements• To know where more information is needed• To attach a range to model forecasts• To understand attitudes toward risk• To account for learning over time(Attaching uncertainty to predictions might be as natural as attaching cost/benefit to climate targets)Click to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level4Types of Uncertainty• Parametric Uncertainty– Uncertainty in the value of a quantity• Model or Structural Uncertainty– Uncertainty in the form of a model– e.g. Linear vs. Quadratic relationship• Surprise/Indeterminacy– Don’t know what we don’t knowClick to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level5Describing an Uncertainty Quantity•Mean •Variance• Standard Deviation• Covariance][)( xEdxxxfxx==∫∞∞−µ][)()(22xVardxxfx xxx=−=∫∞∞−µσ)(xVarx=σ[]2121),cov( XXEXX∗=(if E[x]=0)Click to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level6Describing an Uncertain Quantity II• Mode: Most likely value (peak)• Median: Value of x such that– Prob (x<x0) = Prob (x>x0) = 0.50• Fractile: The p fractile is the value x0such that– Prob (x<x0) = p• Probability Density Function (PDF)– The integral under a portion of the function is the probability that the event will fall into that range.Click to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level7Probability Density Functions00.511.522.53-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 MeanMedian1 Standard Deviation(68% bounds)2 StandardDeviations(95% bounds)Click to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level8Cumulative Density Function00.20.40.60.811.20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 00.10.20.30.40.50.60.70.80.9100.511.522.533.544.55Click to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level9Input distributions00.10.20.30.40.50.60246 Climate Sensitivity (°C) Probability Density01234567802040Ocean Diffusivity (Kv) (cm2/s)Probability Density (Values x 10-2)00.511.522.50 0.2 0.4 0.6 0.8 1Faer (Aerosol Forcing) (W/m2) Probability DensityMethods of Input Distribution Acquisition:Expert Elicitation, Historical Analysis, Modeling RangesClick to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level10Correlation between Distributions00.10.20.30.40.50.60.70.80.91024681012Climate Sensitivity (°C)Faer (W/m2)IndependenceClick to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level11Measuring Uncertainty – Local Measures• Sensitivity: [dy/dx]x0– (Slope of tangent)• Elasticity: [dy/dx]x0* x0/y0– (Percentage change)• Gaussian (1storder) approximation2220ixXiiydxdyσσ∑⎥⎦⎤⎢⎣⎡≈Click to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level12High Sensitivity SystemClick to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level13Measuring Uncertainty - Global• Range Sensitivity– Varying each x from low to high• Joint Parametric Analysis– Vary lows and highs for multiple variables• Monte Carlo Simulation– Use pdfs of inputs to produce pdfs of outputsClick to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level14Monte Carlo Simulation• Crude Monte Carlo• Stratified Sampling Methods– Latin Hypercube Sampling• Importance Sampling Methods• Response Surface Methods– Probabilistic Collocation MethodClick to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level15Monte Carlo Simulation• Example 1: Throwing Dice• Example 2:– Y = X1+ X2– X1 = N(50,20)– X2 = N(40,25)– Analytic Analysis: Y = N(90,32)– Compare to Monte Carlo samples of size n=10,100,1000,10000Click to edit Master title styleMonte Carlo, n=10,1000.000.000.000.000.010.010.010.010.020.020.020.020.030.030.030.030.040.040.040.040.050.050.050.050.060.060.060.060.070.070.070.070.080.080.080.080.090.090.090.090.100.100.100.10-50-50-50 0 5050 100100100 150150150 200200200Probability DensityProbability DensityProbability DensityProbability DensityProbability DensityProbability DensityProbability DensityProbability DensityProbability DensityProbability DensityProbability DensityProbability DensityProbability DensityProbability DensityProbability DensityProbability DensityProbability DensityProbability DensityProbability DensityResultResultResultResultResultResultMonte Carlo - 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