Probability, Statistics and Life Cycle Assessment

Preface
	References
Contents
1 Introduction
	1.1 Life Cycle Assessment
	1.2 Uncertainty in LCA
	1.3 Sensitivity in LCA
	1.4 Uncertainty and Sensitivity in LCA: Past, Present and Future
		1.4.1 Early Recommended Practice
		1.4.2 Developments
		1.4.3 Application by Case Studies
		1.4.4 Review Papers
		1.4.5 Current Recommended Practice
		1.4.6 Current Practice
		1.4.7 The Future
		1.4.8 The Role of Software
		1.4.9 Fixing the Errors
		1.4.10 This Book
	1.5 Terminology and Notation
	1.6 Outline
	References
Part I A Primer
2 Probability 1: Basics
	2.1 Probability and Random Variables
		2.1.1 Probability Defined
		2.1.2 Random Variables
		2.1.3 Infinite and Finite Populations
	2.2 Probability Distributions and Probability Distribution Functions
		2.2.1 Probability Distributions
		2.2.2 Discrete Probability Distribution Functions
		2.2.3 Continuous Probability Distribution Functions
	2.3 Cumulative Distribution Functions
		2.3.1 Discrete Cumulative Distribution Functions
		2.3.2 Continuous Cumulative Distribution Functions
		2.3.3 Inverse Cumulative Distribution Functions
		2.3.4 Complementary Cumulative Distribution Functions
	2.4 Probability Distributions, Dimensions and Units
		2.4.1 A Brief Digression on Dimensions and Units
		2.4.2 Dimensions and Units for Probability Theory
	2.5 Moments and Other Functions of a Distribution
		2.5.1 Expected Value
		2.5.2 Variance
		2.5.3 Raw, Central, Higher and Standardized Moments
		2.5.4 Other Functions of a Distribution
		2.5.5 Chebyshev's Inequality
	2.6 Model Distributions
		2.6.1 The Binomial and Bernoulli Distributions
		2.6.2 The Multinomial Distribution
		2.6.3 The Normal Distribution
		2.6.4 The Standard Normal Distribution, the Standardization Procedure and the Interpercentile Interval
		2.6.5 The Lognormal Distribution
		2.6.6 The Two Triangular Distributions
		2.6.7 The Beta Distribution
		2.6.8 The Gamma Distribution
		2.6.9 The PERT Distribution
		2.6.10 The t-distribution
		2.6.11 The chi2-distribution
		2.6.12 The F-distribution
		2.6.13 The Two Uniform Distributions
		2.6.14 Summary of Main Continuous Probability Distributions
		2.6.15 A Few Relationships Between Probability Distributions
		2.6.16 Other Distributions
	2.7 More Than One Random Variable
		2.7.1 Probabilities Relating to Multiple Events
		2.7.2 Combining Random Variables: The Sum
		2.7.3 Dependence and Covariance
		2.7.4 Comparing Two Random Variables
		2.7.5 The Correlation Between Two Random Variables
		2.7.6 Simple Regression
		2.7.7 Multiple Regression
		2.7.8 Bivariate Probability Distributions and Copulas
		2.7.9 Multivariate Probability Distributions
		2.7.10 Sums of Several Random Variables
		2.7.11 Functions of Probability Distributions
	References
3 Probability 2: Alternatives
	3.1 Alternative Interpretations of Probability
		3.1.1 Bayesianism
		3.1.2 The Classical View
		3.1.3 Other Extreme Points of View
		3.1.4 Imprecise Probability
		3.1.5 Second-Order Uncertainty
		3.1.6 Where is the Stochastic Process?
	3.2 Alternatives to Probability
		3.2.1 Possibility Theory and Fuzzy Numbers
		3.2.2 Dempster–Shafer Theory
		3.2.3 Rough Sets
		3.2.4 Grey Relational Analysis
		3.2.5 Set Pair Analysis
		3.2.6 Interval Arithmetic
		3.2.7 Information Theory
	3.3 Combination of Paradigms
	References
4 Statistics 1: Descriptive
	4.1 Data
		4.1.1 The Data Matrix
		4.1.2 Some Issues of Notation
		4.1.3 Data and Summaries
		4.1.4 Data and Samples
		4.1.5 Statistics and Their Symbols
		4.1.6 Order Statistics and Ranks
	4.2 Measures of Centrality
		4.2.1 The Mean
		4.2.2 Other Measures of Central Tendency
	4.3 Measures of Dispersion
		4.3.1 The Absolute Deviation
		4.3.2 The Variance and the Standard Deviation
		4.3.3 Other Measures of Dispersion
	4.4 Other Univariate Descriptive Statistics and Visualizations
		4.4.1 The Five Quartiles and the Box Plot
		4.4.2 The Empirical Distribution Function
		4.4.3 Kernel Density Estimation
		4.4.4 Histogram, Frequency Polygon and Ogive
		4.4.5 The Q-Q Plot and P-P Plot
		4.4.6 Standardized Scores and Other Transformations
		4.4.7 Measures of Shape
	4.5 Univariate Statistics for Categorical Data
		4.5.1 Frequencies, Proportions and Odds
		4.5.2 Bar Charts and Pie Charts
		4.5.3 The Case of k=2 Levels
		4.5.4 Bar Charts and Pie Charts for Numerical Data
	4.6 Bivariate and Multivariate Descriptive Statistics
		4.6.1 Dependent and Independent Variables
		4.6.2 Two Numerical Variables
		4.6.3 One Numerical and One Categorical Variable
		4.6.4 Two Categorical Variables
		4.6.5 Multivariate Analysis
	References
5 Statistics 2: Inferential
	5.1 The Estimation Problem
		5.1.1 Estimators, Estimates and Estimands
		5.1.2 Estimation Principles
		5.1.3 The Distribution of Estimates
	5.2 The Central Limit Theorem
		5.2.1 An Introduction to the Central Limit Theorem
		5.2.2 Statement of the Central Limit Theorem
	5.3 The Sampling Distribution
		5.3.1 The Sampling Distribution of the Mean
		5.3.2 The Sampling Distribution of the Mean in Case sigma is Unknown
		5.3.3 The Sampling Distribution of the Standard Deviation
		5.3.4 The Sampling Distribution of the Proportion
		5.3.5 Other Sampling Distributions
	5.4 Point Estimates and the Standard Error of the Estimate
		5.4.1 Point Estimates
		5.4.2 The Standard Error of the Estimate
		5.4.3 The Standard Error of the Mean
		5.4.4 The Standard Error of Other Statistics
		5.4.5 The Relative Standard Error
	5.5 Confidence Intervals
		5.5.1 The Idea of a Confidence Interval
		5.5.2 Interpretation of a Confidence Interval
		5.5.3 The Confidence Interval of the Mean
		5.5.4 The Confidence Interval of the Mean in Case sigma  is Unknown
		5.5.5 The Confidence Interval of the Variance
		5.5.6 The Confidence Interval of the Standard Deviation
		5.5.7 The Confidence Interval of the Proportion
		5.5.8 Confidence Intervals of the Correlation Coefficient
		5.5.9 Confidence Intervals of the Regression Coefficient
		5.5.10 Confidence Intervals of Other Statistics
	5.6 Estimating Probability Distribution Functions
		5.6.1 Selecting an Appropriate Probability Distribution Functions
		5.6.2 Tails or Central Part?
		5.6.3 Estimating Parameters of a Probability Distribution Functions
	5.7 Hypothesis Tests
		5.7.1 Hypothesis Tests in Science
		5.7.2 Null and Alternative Hypothesis, One-Sided and Two-Sided Hypothesis
		5.7.3 The Test Statistic
		5.7.4 Tails and Sides
		5.7.5 Significance Level, Critical Region, Critical Value,  and Null Distribution
		5.7.6 The Realized Value of the Standardized Test Statistic
		5.7.7 A Worked Example
		5.7.8 The Statistical Decision
		5.7.9 The p-Value
		5.7.10 The Significance Level alpha and Type I Errors
		5.7.11 Type II Errors and Power
		5.7.12 Asterisks (*, **, ***) and Other Conventions  to Denote Significance
		5.7.13 Further Requirements and Assumptions
		5.7.14 Non-parametric Tests
		5.7.15 Bootstrap Tests
		5.7.16 A Catalogue of Null Hypothesis Significance Tests
	5.8 Univariate Hypothesis Tests
		5.8.1 Hypothesis Tests for the Mean (mu)
		5.8.2 Hypothesis Tests for the Median (nu)
		5.8.3 Hypothesis Tests for the Standard Deviation (sigma)  and the Variance (sigma2)
		5.8.4 Hypothesis Tests for the Proportion (pi)
		5.8.5 Goodness-of-Fit Tests
		5.8.6 Other Hypothesis Tests
	5.9 Bivariate Hypothesis Tests
		5.9.1 Independent Samples Tests
		5.9.2 Dependent Samples Tests
		5.9.3 Comparing Two Distributions
		5.9.4 The Association Between Two Numerical Variables
		5.9.5 The Association Between Two Categorical Variables
	5.10 Multivariate Hypotheses Tests
		5.10.1 The Problem of Multiple Testing
		5.10.2 Comparing Several Numerical Variables
		5.10.3 Hypothesis Tests for Multiple Regression Analysis
		5.10.4 Multi-way Analyses
		5.10.5 Comparing Several Categorical Variables
		5.10.6 Other Multivariate Methods
	References
6 LCA
	6.1 The Mathematical Model for LCA
		6.1.1 The Mathematical Structure of a Model
		6.1.2 LCA Without Maths?
		6.1.3 Parameters, Arguments, Inputs and Outputs
		6.1.4 Black Box or Not?
		6.1.5 Goal and Scope Definition
		6.1.6 Inventory Analysis
		6.1.7 Impact Assessment
		6.1.8 Interpretation
	6.2 More Refined LCA
		6.2.1 Regional/spatial Differentiation
		6.2.2 Temporal/dynamic Differentiation
	6.3 Alternative Methods to Calculate LCA Results
		6.3.1 Matrix-Based LCI Versus Other Approaches
		6.3.2 Matrix-Inversion-Based LCI Versus Other Matrix-Based Approaches
		6.3.3 Process-Based LCI Versus IO-Based LCI
		6.3.4 Matrix-Based LCIA
	6.4 Data in LCA
		6.4.1 Types of Data
		6.4.2 Data for LCI
		6.4.3 Characterization Data
		6.4.4 Linear and Non-linear LCI and LCIA
		6.4.5 Data Defined by Scenarios
	6.5 Derivatives of the Mathematical Model
		6.5.1 Derivatives at the Inventory Level
		6.5.2 Derivatives at the Characterization Level
		6.5.3 Derivatives at the Normalization Level
		6.5.4 Derivatives at the Weighting Level
		6.5.5 Higher Order Derivatives
		6.5.6 Numerical Derivatives
	6.6 Dealing with Uncertainty and Sensitivity in LCA
		6.6.1 What Are the Issues?
		6.6.2 Frameworks
		6.6.3 Estimation of Probability Distributions
		6.6.4 Software and Databases for Including Uncertainty and Sensitivity in LCA
		6.6.5 Representing Statistical Information in LCA Databases
		6.6.6 From Format to Calculations
		6.6.7 Probability Distributions in LCA
		6.6.8 From Statistics to Decisions
	6.7 Uncertainty and Sensitivity in IOA and Related Models
		6.7.1 The Case of IOA
		6.7.2 The Case of EIOA and IO-Based LCA
		6.7.3 Is IO-Based LCA More Precise Than Process-Based LCA?
	6.8 Visualization in LCA
		6.8.1 Graphs for Visualizing Uncertainty
		6.8.2 Graphs for Visualizing Sensitivity
		6.8.3 Miscellaneous Graphs
	References
7 Error and Quality
	7.1 Measurements and Their Errors
		7.1.1 Terminology of Measurements
		7.1.2 Terminology of Errors
		7.1.3 Error and Variability
		7.1.4 Measurement Errors Across the Sciences
		7.1.5 Numerical and Categorical Measurements
		7.1.6 What is a Measurement?
		7.1.7 Models and Their Errors
		7.1.8 Alternative Categorizations of Errors
		7.1.9 Uncertainty > Error
	7.2 Quantification of Error
		7.2.1 Quantification of Random Error
		7.2.2 Quantification of Systematic Error
		7.2.3 Quantification of Total Error
	7.3 Error Propagation
		7.3.1 Notation
		7.3.2 The Distribution of the Output Error
		7.3.3 Methods for Error Propagation
	7.4 Gaussian Error Propagation
		7.4.1 First-Order Gaussian Error Propagation
		7.4.2 Proof of the Gaussian Error Propagation Formula
		7.4.3 Second-Order Correction
		7.4.4 Larger Errors
		7.4.5 Correlated Errors
		7.4.6 Estimating the Partial Derivatives
	7.5 Sampling-Based Error Propagation
		7.5.1 Monte Carlo for Error Propagation
		7.5.2 More Refined Sampling Methods
		7.5.3 The Number of Replications for Sampling Approaches
	7.6 Some Other Techniques for Error Propagation
		7.6.1 Naive Numerical Error Propagation
		7.6.2 Error Propagation Using Fuzzy Numbers
		7.6.3 Error Propagation Using Interval Arithmetic
		7.6.4 Error Propagation Using Scenarios
		7.6.5 Miscellaneous and Combined Methods
	7.7 Comparisons of Different Propagation Methods
		7.7.1 Comparison of Conclusions
		7.7.2 Comparison of Requirements
		7.7.3 Combining the Strengths
	7.8 Suspicious and Missing Measurements
		7.8.1 Detecting Suspicious Measurements
		7.8.2 Managing Suspicious Measurements
		7.8.3 Missing Observations and Data Gaps
		7.8.4 Data Estimation
		7.8.5 Data Reconciliation
		7.8.6 Estimation of Systematic Error
		7.8.7 Outliers, Once More
		7.8.8 The Role of Suspicious and Missing Measurements in LCA
	7.9 Truncation, Aggregation and Approximation Errors
		7.9.1 Cut-off
		7.9.2 Truncation Error
		7.9.3 Streamlining, LCA for Design and Prospective LCA
		7.9.4 Aggregation Error
	7.10 Other Types of Error
		7.10.1 Sampling Error
		7.10.2 Errors in Computation
		7.10.3 Power Series Expansions
		7.10.4 Database Errors
		7.10.5 Software Implementation Errors
		7.10.6 Model Errors
		7.10.7 Trade-off of Errors
	7.11 Validity, Reliability, Repeatability and Reproducibility
		7.11.1 Validity and Reliability in the Social Sciences
		7.11.2 Repeatability, Reproducibility and Reliability in the Quality Management and Production Engineering
		7.11.3 Data Quality in LCA
	7.12 Significant Digits
	7.13 Data Quality Indicators
		7.13.1 The NUSAP Scheme and the Pedigree
		7.13.2 DQIs in LCA
		7.13.3 Conversion of DQIs into Overall Data Quality Scores
		7.13.4 Conversion of DQIs into Probability Distributions
		7.13.5 Conversion of DQIs into Possibilistic Information
		7.13.6 System-Wide DQIs
		7.13.7 The `Basic Uncertainty'
		7.13.8 Assessment Factors
	References
8 Uncertainty, Risk and Decisions
	8.1 Decisions
		8.1.1 The Decision Matrix
		8.1.2 Expected Value
		8.1.3 Expected Utility
		8.1.4 Value of Information
	8.2 Uncertainty
		8.2.1 Uncertainty in Daily Life and Society
		8.2.2 Uncertainty in Science and Engineering
		8.2.3 Uncertainty in Policy and the Post-normal Science Movement
		8.2.4 Uncertainty = Error + Variability
		8.2.5 Is Uncertainty Quantifiable?
		8.2.6 Types of Uncertainty
	8.3 Modeling Uncertainty
		8.3.1 Uncertainty = Error + Variability: A Closer Look
		8.3.2 Combining Variability and Measurement Error
		8.3.3 Inferring the Underlying Measurand
		8.3.4 Non-normal Distributions
	8.4 Risk
		8.4.1 What Is Risk?
		8.4.2 Risk and Decisions
		8.4.3 The Role of Risk in LCA
	8.5 Multi-criteria Methods
		8.5.1 Basics of Multi-criteria Decision Analysis
		8.5.2 Multi-criteria Decision Analysis and Uncertainty
		8.5.3 Use of Multi-criteria Decision Analysis in LCA
		8.5.4 Use of Multi-criteria Decision Analysis in Uncertain LCA
	8.6 Decisions in LCA
		8.6.1 From Numbers to Decisions
		8.6.2 Decision Situations and Application Areas
		8.6.3 LCA as an Optimization Problem
	8.7 Single-Product Decisions
		8.7.1 Provision of Product Information
		8.7.2 Benchmarking a Product
		8.7.3 Benchmarking the Mean Product
		8.7.4 Benchmarking `all' Products
		8.7.5 Benchmarking a `Significant' Share of Products
		8.7.6 The Problem of Defining Benchmarks
		8.7.7 Product Improvement with a Contribution Analysis
		8.7.8 Product Improvement with a Sensitivity Analysis
		8.7.9 Product Improvement with Mathematical Optimization
	8.8 Multi-product Decisions
		8.8.1 Pairwise Comparisons
		8.8.2 The Best Product Alternative
		8.8.3 The Product Alternative with the Lowest Mean Impact
		8.8.4 The Product Alternative with the Significantly Lowest Mean Impact—Two Alternatives
		8.8.5 Dependent and Independent Comparisons
		8.8.6 Issues with `Significance'
		8.8.7 Effect Size
		8.8.8 `Modified' Significance Tests
		8.8.9 The Product Alternative with the Significantly Lowest Mean Impact—Three or More Alternatives
		8.8.10 Medians Instead of Means
		8.8.11 Ranking a Set of Product Alternatives
		8.8.12 Rank-Based Procedures
		8.8.13 Pareto Fronts and Dominated and Non-dominated Products
		8.8.14 Overlap
		8.8.15 Discernibility and the Comparison Indicator
		8.8.16 Probabilities
		8.8.17 A Tableau of Pairwise Comparisons
		8.8.18 Multiple Products and Multiple Impacts
		8.8.19 Other Types of LCA Research
	References
9 Sensitivity
	9.1 Defining Sensitivity Analysis
		9.1.1 Sensitivity Versus Uncertainty
		9.1.2 Local Versus Global Sensitivity Analysis
		9.1.3 One-at-a-Time Versus All-at-a-Time
		9.1.4 OAT and LSA Are Not the Same, and Neither Are AAT and GSA Synonyms
		9.1.5 Sensitivity, Uncertainty and Influence
		9.1.6 Uncertainty Apportioning
		9.1.7 Discrete Changes and Scenarios
		9.1.8 Subdividing the Field
		9.1.9 Nominal Value Versus Probability Distribution
		9.1.10 Is Sensitivity Analysis Probabilistic?
		9.1.11 A Note on Notation
		9.1.12 Visualizing Sensitivity
		9.1.13 Screening
	9.2 Local Sensitivity Analysis, One-at-a-Time
		9.2.1 The Basic Idea of LSA-OAT
		9.2.2 Derivatives, Multipliers, the Gradient and the Jacobian Matrix
		9.2.3 Non-linear Effects and the Second Derivative
		9.2.4 Derivative-Based Measures of Sensitivity
		9.2.5 Data Requirements for Local Sensitivity Analysis
		9.2.6 The Choice of the Nominal Value
	9.3 Local Sensitivity Analysis, All-at-a-Time
		9.3.1 Two-at-a-Time
		9.3.2 All-at-a-Time
	9.4 Discrete Sensitivity Analysis
		9.4.1 Sensitivity for Choices
		9.4.2 Sensitivity for Changes in Discrete Variables
		9.4.3 Sensitivity for Changes in Discretized Continuous Variables
		9.4.4 Sensitivity for Arbitrary Changes
		9.4.5 All-at-a-Time Choices: Scenarios
		9.4.6 Visualizing DSA
	9.5 Global Sensitivity Analysis, One-at-a-Time
		9.5.1 Sensitivity Functions and Their Visualization
		9.5.2 Numerical Indicators
		9.5.3 Stability Setting
		9.5.4 Reliability Theory
	9.6 Global Sensitivity Analysis, All-at-a-Time
		9.6.1 Scatter Plots, Correlation and Regression
		9.6.2 Some Other Approaches
		9.6.3 Discretized Continuous Variables: Factorial Design
		9.6.4 Discretized, Two-at-a-Time: Visualization
		9.6.5 Morris' Elementary Effects
		9.6.6 Screening Methods
	9.7 Uncertainty Apportioning
		9.7.1 The Basic Idea of UA
		9.7.2 Regression-Based UA
		9.7.3 Variance-Based UA
		9.7.4 Moment-Independent UA
		9.7.5 Estimating Conditional Variances in Variance-Based UA
		9.7.6 Probing the Input Space
		9.7.7 Other Variations
		9.7.8 When is a Contribution Large?
	9.8 Computational Aspects
		9.8.1 Change of One Element
		9.8.2 Change of One Row or Column
		9.8.3 Change of Several Rows or Columns
		9.8.4 Change of a Block of a Partitioned Matrices
	References
Part II A Critique
10 Statistical Concepts, Terminology and Notation
	10.1 Parameters, `true' Values and `deterministic' Values
		10.1.1 Parameter Uncertainty Does Not Exist
		10.1.2 What If the Parameters of the Input Data Were Uncertain?
		10.1.3 The Double Meaning of `parameter'
		10.1.4 The `true' Value
		10.1.5 Is the Central Value the `true' Value?
		10.1.6 The Roles of Error and Variability
		10.1.7 The `deterministic' Value
		10.1.8 What Is the `deterministic' Value?
		10.1.9 Deterministic Input Value Versus Deterministic Output Result
		10.1.10 The Case of Non-sampling Methods
		10.1.11 Do We Need a `deterministic' Value? Yes, But Let's Call it the `nominal' Value
	10.2 The Use of Confidence Intervals
		10.2.1 Statistical Intervals: Some Theory
		10.2.2 A Short History of Confidence Intervals in LCA
		10.2.3 Terminology Related to `confidence'
	10.3 Uncertainty Factors
		10.3.1 Qualitative Interpretations
		10.3.2 Uncertainty Factors Related to the Geometric Standard Deviation
		10.3.3 Uncertainty Factors Related to a Specific Probability Range
		10.3.4 Uncertainty Factors that Change the Central Value
		10.3.5 Unspecified or Unclear Uncertainty Factors and Other Variations
	10.4 Means, Variances and Standard Deviations
		10.4.1 The Many Faces of Means, Variances and Standard Deviations
		10.4.2 Issues of Notation
		10.4.3 Expected Values and Other Central Values
		10.4.4 Standard Deviations and Other Measures of Dispersion
	10.5 Hypotheses and Significance
		10.5.1 Hypotheses Without a Hypothesis Test
		10.5.2 Hypotheses with a Hypothesis Test
		10.5.3 Significance Without a Hypothesis Test
		10.5.4 Significance with a Hypothesis Test
		10.5.5 Hypothesis Tests Without Significance
	10.6 Correlation, Dependence and Regression
		10.6.1 Correlation Versus Regression
		10.6.2 Significant Correlations
		10.6.3 Dependence and Association
		10.6.4 Other Meanings of `dependence'
	10.7 Numbers, Graphs and Notation
		10.7.1 Numbers and Units
		10.7.2 Graphs
		10.7.3 The Use of x+-sigma
		10.7.4 Mathematical Notation
	10.8 Some Other Issues
		10.8.1 Sensitivity Analysis
		10.8.2 Misuse of Terms
		10.8.3 The Role of the Central Limit Theorem
		10.8.4 The Use of the t-Distribution for Input Variables
		10.8.5 The Ostrich Phenomenon
		10.8.6 More Curiosities
	10.9 Conclusion
	References
11 The Lognormal Distribution in LCA
	11.1 Parametrizations
		11.1.1 Parametrization Ia: Abstract Definition
		11.1.2 Parametrization Ib: Definition from an Underlying Normal Distribution
		11.1.3 Parametrization IIa: Definition with Geometric Parameters
		11.1.4 Parametrization IIb: Definition in Logarithmic Units
		11.1.5 Relation Between the Different Forms
		11.1.6 Other Definitions
		11.1.7 Implementation in LCA Databases
	11.2 Some Properties of the Lognormal Distribution
		11.2.1 The Cumulative Distribution Function
		11.2.2 Graphs of the pdf and cdf
		11.2.3 Moments and Other Properties
		11.2.4 Is the Median Equal to the Geometric Mean?
		11.2.5 Interpercentile Intervals
		11.2.6 Combining Lognormal Distributions
		11.2.7 In Search of a `most representative' Value
	11.3 Estimation of the Parameters of the Lognormal Distribution
		11.3.1 Maximum Likelihood Estimation
		11.3.2 Method of Moments Estimation
		11.3.3 Example of Parameter Estimation
		11.3.4 Other Estimation Methods
		11.3.5 Lognormal Estimation in LCA
		11.3.6 Confidence Intervals for the Parameters
	11.4 Alleged Confidence Intervals, the Dispersion Factor …
		11.4.1 Lognormal Distributions and `confidence intervals' in LCA
		11.4.2 The Dispersion Factor
		11.4.3 The Role of GSD2
		11.4.4 Why Not Another Power of GSD?
	11.5 The Issue of Dimensions and Units
		11.5.1 Dimensions and the Normal Distribution
		11.5.2 Units and the Normal Distribution
		11.5.3 Dimensions and the Lognormal Distribution
		11.5.4 Units and the Lognormal Distribution
		11.5.5 Implications for Parameters
	11.6 Variants of the Lognormal Distribution
		11.6.1 The Lognormal Distribution for Negative Numbers
		11.6.2 The 3-Parameter Lognormal Distribution
		11.6.3 The 4-Parameter Lognormal Distribution
		11.6.4 Other Variations
	11.7 Comparison with the Beta Distribution
		11.7.1 Comparison of Lognormal and Beta
		11.7.2 Dimensions and Units for the Beta Distribution
	11.8 Is the Lognormal Distribution a Natural Choice for LCA?
		11.8.1 The Case of Input Data
		11.8.2 The Case of Output Results
	11.9 Conclusion
	References
12 The Quantitative Pedigree Approach
	12.1 The NUSAP Scheme and the (Qualitative) Pedigree
	12.2 The Pedigree in LCA
		12.2.1 NUSAP or Pedigree?
		12.2.2 Pedigree, Pedigree Matrix, or Pedigree Approach?
		12.2.3 Weidema and Wesnæs (ch12Weidema1996)
		12.2.4 Meier (ch12Meier1997)
		12.2.5 Frischknecht et al. (ch12Frischknecht2004/ch12Frischknecht2005b/ch12Frischknecht2007a)
		12.2.6 GHG Protocol (ch12GreenhouseGasProtocol2011)
		12.2.7 Weidema et al. (ch12Weidema2013)
		12.2.8 Muller et al. (ch12Muller2016a) and Ciroth et al. (ch12Ciroth2016)
		12.2.9 Muller et al. (ch12Muller2016b)
		12.2.10 Adoption by Others
	12.3 Alternative Quantitative DQIs
		12.3.1 Kennedy et al. (ch12Kennedy1996/ch12Kennedy1997)
		12.3.2 Van den Berg et al. (ch12VanDenBerg1999)
		12.3.3 Rousseaux et al. (ch12Rousseaux2001)
		12.3.4 Ardente et al. (ch12Ardente2004)
		12.3.5 Chen and Lee (ch12Chen2021b)
		12.3.6 Zheng et al. (ch12Zheng2019)
		12.3.7 Other Reviews and Modifications
	12.4 Critical Observations
		12.4.1 The Combination Rules
		12.4.2 A Proof of the Addition Rule for Squared Geometric Standard Deviations—With a Few Side Notes
		12.4.3 Adding GSD Unsquared?
		12.4.4 The Role of Sample Size
		12.4.5 What Is the Basic Uncertainty?
		12.4.6 Comparison with Observed Data
		12.4.7 Some Other Objections
	12.5 The Quantitative Pedigree Approach in Practice
		12.5.1 Use of the Quantitative Pedigree
		12.5.2 Perception of the Quantitative Pedigree
	12.6 Conclusion
	References
13 Statistical Analysis of Non-stochastic LCA
	13.1 Univariate Analysis
	13.2 Correlation and Regression Analysis
	13.3 Multivariate Statistics and Machine Learning
		13.3.1 Principal Component Analysis
		13.3.2 Other Multivariate Methods
		13.3.3 Machine Learning
	13.4 Meta-analysis
		13.4.1 What is Meta-analysis?
		13.4.2 Meta-analysis in LCA
	13.5 Conclusion
	References
Part III A Guidance
14 Including Uncertainty and Sensitivity in LCA
	14.1 The LCA Model Equations as a Basis for Uncertainty Modeling
		14.1.1 A Concrete Example of phi(.)
		14.1.2 The LCA Model as a Black Box
		14.1.3 From Deterministic LCA to Probabilistic LCA
	14.2 Towards a Guide for Including Uncertainty and Sensitivity
		14.2.1 Elements of a Guide
		14.2.2 Why a True Guide on Uncertainty and Sensitivity  Is Not Feasible
		14.2.3 A Guidance Document
	14.3 Overall Guidance
	References
15 Guidance for Standard LCA
	15.1 Uncertainty Analysis
		15.1.1 Input Data and Choices in the Framework of LCA
		15.1.2 Uncertain Data
		15.1.3 Uncertainty Propagation
		15.1.4 Non-comparative LCA
		15.1.5 Comparative LCA
	15.2 Sensitivity Analysis
		15.2.1 Local Sensitivity Analysis, One-at-a-Time
		15.2.2 Global Sensitivity Analysis, One-at-a-Time
		15.2.3 Discrete Sensitivity Analysis
		15.2.4 Global Sensitivity Analysis, All-at-a-Time
		15.2.5 Uncertainty Apportioning
	References
16 Guidance for Special Types of LCA
	16.1 Input–Output-Based LCA and Hybrid LCA
	16.2 Attributional and Consequential LCA
	16.3 Parametrized and Non-linear LCA
	16.4 Agent-Based LCA
	16.5 Fleet-Based LCA
	16.6 Life Cycle Optimization
	16.7 Other Pillars of Sustainability
		16.7.1 Life Cycle Costing
		16.7.2 Social Life Cycle Assessment
		16.7.3 Environmental and Economic Aspects
		16.7.4 All Three Pillars
	References
Appendix A Symbols
A.1 Abbreviations
A.2 General Symbols and Conventions
Appendix B Matrix Topics
B.1 General Conventions and Symbols
B.2 Abstract Vectors and Euclidean Vectors
B.3 Transposition and Diagonalization
B.4 Multiplication of Vectors and Matrices
B.5 Matrix Inversion and the Determinant
B.6 Systems of Linear Equations
B.7 Vector and Matrix Norms
B.8 Partitioned Matrices and the Vec-Operator
Appendix C Special Functions
C.1 The Exponential, Logarithmic and Logit Functions and the Fisher Transformation
C.2 The Gamma, Beta and Error Functions
C.3 The Binomial Coefficient
C.4 The Floor, Ceiling, Sign, Heaviside and Indicator Functions and the Kronecker Delta
	References
Subject Index