Methods of Uncertainty Analysis“Persons pretending to forecast the future shall be considered disorderly under subdivision 3, section 901 of the criminal codLecture HighlightsElements of a Decision TreeExample: Biking in the RainDecision tree without MemoryDecision Tree with MemoryEmissions ScenariosClimate ScenariosHandling scientific uncertainty in AR4Why Worry about Uncertainty?Types of UncertaintyDescribing an Uncertainty QuantityDescribing an Uncertain Quantity IIProbability Density FunctionsCumulative Density FunctionInput distributionsCorrelation between DistributionsMeasuring Uncertainty - GlobalMonte Carlo SimulationMonte Carlo SimulationMonte Carlo, n=10,100Monte Carlo, n=1000,10000Webster (2001) GraphThe Greenhouse GambleMIT OpenCourseWare http://ocw.mit.edu15.023J / 12.848J / ESD.128J Global Climate Change: Economics, Science, and PolicySpring 2008For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Click to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level1Methods of Uncertainty Analysis15.023 Lecture9 April 2008Travis FranckEunjee Lee(material from previous lectures by Mort Webster, Ian Sue Wing, Marcus Sarofim, and Travis Franck)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 level3Lecture Highlights• How to use Decision Trees• Understanding mean vs. median, standard deviations, etc.• How to read PDFs (and make one from a histogram)• The role of uncertainty in climate change scienceClick to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level4Elements of a Decision TreeDecision Node: possible actions at a choice pointactBAUUncertainty Node: Shows different possible outcomes for an uncertain event, and the respective probabilities for each outcomep1-pExample: Biking in the RainFigure by MIT OpenCourseWare.Click to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level6Decision tree without MemoryNo connection between periodsHLRPeriod 2HLRDecision OutcomeHLRHLRPeriod 1HLRPeriod 3HLRClick to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level7Depends on the pathDecision Tree with MemoryPeriod 2HLRHLRHLRPeriod 3HLRDecision OutcomeHLRHLRPeriod 1Click to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level8Emissions ScenariosHLRHLRGrowth AEEIEmissionsAEEIHLRHLRHLRHLRGrowthShorthand forClick to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level9Climate ScenariosGrowth AEEI Kv SCollapse into RR, HL, LHHLRLHRLHRHLFaerLHEconomic UncertaintyClimate UncertaintyIn HomeworkProbability distributionClick to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level10Handling scientific uncertainty in AR4• In general, uncertainty ranges for results given in this Summary for Policymakers are 90% uncertainty intervals unless stated otherwise, that is, there is an estimated 5% likelihood that the value could be above the range given in square brackets and 5% likelihood that the value could be below that range. Best estimates are given where available. Assessed uncertainty intervals are not always symmetric about the corresponding best estimate. Note that a number of uncertainty ranges in the Working Group I TAR corresponded to 2 standard deviations (95%), often using expert judgment.(IPCC, 2007: Summary for Policymakers. In: Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change)Click to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level11Why 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 level12Types 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 level13Describing 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 level14Describing 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.• Cumulative Density Function (cdf)Click to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level• Fifth level1500.511.522.53-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9MeanMedian1 Standard Deviation(67% bounds)2 StandardDeviations(95% bounds)Probability Density FunctionsClick to edit Master title style• Click to edit Master text styles• Second level• Third level• Fourth level•