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Business Analytics and Optimization: A Technical Introduction 
Oleksandr Romanko, Ph.D. 
Senior Research Analyst, Risk Analytics – Business Analytics, IBM 
Adjunct Professor, University of Toronto 
Toronto SMAC Meetup 
September 18, 2014
© 2014 IBM Corporation 
2 
Making the world work better – pioneering the science 
2008 
1973 
1969 
1981
© 2014 IBM Corporation 
3 
IBM Centennial: 100 Years of Innovation
© 2014 IBM Corporation 
 Analytics Jobs
Created by: Dennis Buttera
Business Analytics and Optimization Introduction
© 2014 IBM Corporation 
Data science 
7
© 2014 IBM Corporation 
 Business Analytics
© 2014 IBM Corporation 
Predictive Analytics 
What will happen? 
Descriptive Analytics 
What has happened? 
Prescriptive Analytics What should we do? 
What is analytics? 
Data 
Insight 
Action 
Decide 
Analyze 
Business Value 
9 
Analytics is the scientific process of deriving insights from data in order to make decisions
Business analytics and optimization (video)
© 2014 IBM Corporation 
11 
IBM Business Analytics portfolio 
IBM Business Analytics 
Financial 
Services Public Sector Distribution Industrial Communications 
Customer Risk 
Industry 
Solutions 
Finance Operations 
Risk 
Analytics 
Business 
Intelligence 
Software 
Categories 
Predictive 
Analytics 
Performance 
Management 
Functional 
Solutions 
Core 
Capabilities 
REPORT MODEL COLLABORATE PREDICT 
Budgeting & 
Forecasting 
Financial 
Consolidation 
Disclosure 
Management 
Risk 
Identification 
Risk & Control 
Assessment 
Resource 
Optimization 
Social Media 
Analytics 
Profitability 
Modeling & 
Optimization 
Production 
Planning 
Asset 
Management 
Customer 
Acquisition 
Customer 
Lifetime 
Value 
Customer 
Loyalty 
& Retention 
Risk Mitigation 
Planning 
Risk Aware 
Decisioning 
Sales Performance 
Management 
ANALYZE PLAN 
Visualize Discover 
Forecast Mine 
Govern 
Score Decide 
Simulate Contribute 
Survey 
Decision 
Management
© 2014 IBM Corporation 
12 
Operations research 
Operations Research (O.R.) is the discipline of applying advanced analytical methods to help make better decisions 
Analytical techniques: 
Simulation – giving you the ability to try out approaches and test ideas for improvement 
Optimization – narrowing your choices to the very best when there are virtually innumerable feasible options and comparing them is difficult 
Probability and Statistics – helping you measure risk, mine data to find valuable connections and insights, test conclusions, and make reliable forecasts 
Mathematical Modeling – algorithms and software
© 2014 IBM Corporation 
13 
Our planet is a complex, dynamic, highly interconnected $54 Trillion system-of-systems (OECD-based analysis) 
Communication 
$ 3.96 Tn 
Transportation 
$ 6.95 Tn 
Leisure / Recreation / Clothing 
$ 7.80 Tn 
Healthcare 
$ 4.27 Tn 
Food $ 4.89 Tn 
Infrastructure 
$ 12.54 Tn 
Govt. & Safety $ 5.21 Tn 
Finance $ 4.58 Tn 
Electricity $ 2.94 Tn 
Education 
$ 1.36 Tn 
Water 
$ 0.13 Tn 
Global system-of-systems $54 Trillion (100% of WW 2008 GDP) 
Same Industry Business Support IT Systems Energy Resources Machinery Materials Trade 
Legend for system inputs 
Note: 1. Size of bubbles represents systems’ economic values 2. Arrows represent the strength of systems’ interaction 
Source: IBV analysis based on OECD 
This chart shows ‘systems‘ (not ‘industries‘) 
1 Tn
14 © 2014 IBM Corporation 
Economists estimate, that all systems carry inefficiencies of up 
to $15 Tn, of which $4 Tn could be eliminated 
Global economic value of 
System-of-systems 
$54 Trillion 
100% of WW 2008 GDP 
Inefficiencies $15 Trillion 
28% of WW 2008 GDP 
Improvement 
potential 
$4 Trillion 
7% of WW 2008 GDP 
How to read the chart: 
For example, the Healthcare system‘s 
value is $4,270B. It carries an estimated 
inefficiency of 42%. From that level of 42% 
inefficiency, economists estimate that 
~34% can be eliminated (= 34% x 42%). 
Source: IBM economists survey 2009; n= 480 
System inefficiency as % of total 
economic value 
Improvement potential as 
% of system inefficiency 
Education 
1,360 
Building & Transport 
Infrastructure 
12,540 
Healthcare 
4,270 
Government & Safety 
5,210 
Electricity 
2,940 
Financial 
4,580 
Food & Water 
4,890 
Transportation (Goods 
& Passenger) 
6,950 
Leisure / Recreation 
/ Clothing 
7,800 
Communication 
3,960 
Analysis of inefficiencies in the 
planet‘s system-of-systems 
Note: Size of the bubble indicate absolute 
value of the system in USD Billions 
42% 
34% 
This chart shows ‘systems‘ (not ‘industries‘) 
15% 
20% 
25% 
30% 
35% 
40% 
15% 20% 25% 30% 35% 40% 45%
© 2014 IBM Corporation 
15 
History of analytics
© 2014 IBM Corporation 
16 
History of business analytics
© 2014 IBM Corporation 
 Business Analytics Examples
© 2014 IBM Corporation 
Pit stop analytics 
7 
Calculations showed that time spent changing tires and refilling the tank was more 
than offset by the improved performance of the car on the track. 
1. Softer tires stuck to the track better during turns than their harder cousins, 
though they wore out more quickly. 
2. Less gas in the tank translated into a lighter, and therefore faster, car. 
Optimized F1 pit teams can change four tires in two seconds
© 2014 IBM Corporation 
19 
Movies
© 2014 IBM Corporation 
20 
Smarter Cities
© 2014 IBM Corporation 
21 
We can collect information from almost everything to make better decisions 
Camera phones in existence able to document accidents, damage, and crimes 
1 billion 
RFID tags embedded into our world and across entire ecosystems 
30 billion 
Of new automobiles will contain event data recorders collecting travel information 
85% 
Instrumented 
Interconnected 
Intelligent
© 2014 IBM Corporation 
22 
Big data 
Big data are datasets that grow so large that they become awkward to work with using on-hand database management tools. Difficulties include capture, storage, search, sharing, analytics, and visualizing. Source: Wikipedia
© 2014 IBM Corporation 
23 
Big social data
© 2014 IBM Corporation 
24 
Applications of big data analytics 
Homeland Security 
Finance 
Smarter Healthcare 
Multi-channel sales 
Telecom 
Manufacturing 
Traffic Control 
Trading Analytics 
Fraud and Risk 
Log Analysis 
Search Quality 
Retail: Churn, NBO
© 2014 IBM Corporation 
25 
Police use analytics to reduce crime (video)
© 2014 IBM Corporation 
26 
Marketing and supply chain analytics (video)
© 2014 IBM Corporation 
27 
Marketing analytics
28 © 2014 IBM Corporation 
Intelligent transport systems 
 Real time monitoring & forecasting of congestion in cities enables real time action to 
reduce traffic and emissions 
– Can charge drivers at point of use for access to city centers 
 Stockholm Congestion Tax Project 
– Involves 18 barrier-free control points 
– Allows differentiated pricing by time of day, congestion level, and potentially emissions level 
– Results: 
• Traffic reduced by 100,000 vehicle passages per day (25%) 
• Public transportation passengers increased by 40,000 / day 
• Congestion during peak hours and CO2 emissions were dramatically reduced
© 2014 IBM Corporation 
29 
Analytics for green vehicles and technology (video)
© 2014 IBM Corporation 
30 
Artificial intelligence 
Source: A Brief Overview and Thoughts for Healthcare Education and Performance Improvement by Watson Team
© 2014 IBM Corporation 
31 
Artificial intelligence 
Source: A Brief Overview and Thoughts for Healthcare Education and Performance Improvement by Watson Team 
On 27th May 1498, Vasco da Gama landed in Kappad Beach 
On 27th May 1498, Vasco da Gama landed in Kappad Beach 
celebrated 
May 1898 
400th anniversary 
arrival in 
In May 1898 Portugal celebrated the 400th anniversary of this explorer’s arrival in India. 
Portugal 
landed in 
27th May 1498 
Vasco da Gama 
Temporal Reasoning 
Statistical Paraphrasing 
GeoSpatial Reasoning 
explorer 
On 27th May 1498, Vasco da Gama landed in Kappad Beach 
On the 27th of May 1498, Vasco da Gama landed in Kappad Beach 
Kappad Beach 
Para- phrases 
Geo- KB 
Date 
Math 
India 
Search Far and Wide 
Explore many hypotheses 
Find Judge Evidence 
Many inference algorithms
© 2014 IBM Corporation 
32 
Artificial intelligence (video)
© 2014 IBM Corporation 
33 
Watson Analytics
© 2014 IBM Corporation 
34 
Watson Analytics
© 2014 IBM Corporation 
 Cloud
© 2014 IBM Corporation 
36 
Bluemix 
www.bluemix.net
© 2014 IBM Corporation 
37 
Bluemix
© 2014 IBM Corporation 
 Business Analytics Education
© 2014 IBM Corporation 
IBM Academic Initiative program 
Cognos SPSS ILOG
© 2014 IBM Corporation 
Master of Business Analytics programs – top 20 universities
© 2014 IBM Corporation 
Industry support for Master of Business Analytics programs
© 2014 IBM Corporation 
Business Analytics programs – curriculum 
Applied Statistics and Probability 
Fundamentals of Computational Mathematics 
Data Mining and Knowledge Discovery 
Simulation Modelling 
Optimization 
Financial Decision Making 
Computational Methods for Business Data Analysis 
Computational Finance and Risk Management 
Visual Analytics and Knowledge Representation 
Mathematical Modelling for Business 
Machine Learning, Cognitive Computing and Artificial Intelligence 
Marketing Analytics 
Strategies for Managing Innovations 
Analytics of Web, Social Networks and Business News
© 2014 IBM Corporation 
 Applied Statistics
© 2014 IBM Corporation 
What kind of data are we dealing with? 
Types of data 
•Quantitative 
•Categorical (ordered, unordered) 
Data collection 
•Independent observations (one observation per subject) 
•Dependent observations (repeated observation of the same subject, relationships within groups, relationships over time or space) 
Type of data drives the direction of your analysis 
•How to plot 
•How to summarize 
•How to draw inferences and conclusions 
•How to issue predictions 
44
© 2014 IBM Corporation 
Quantitative data 
Examples: temperature, age, income 
Quick check: “Does it makes sense to calculate an average?” 
Appropriate summary statistics: 
–Mean and Median 
–Standard Deviation 
–Percentiles 
More advanced predictive methods: Regression, Time Series Analysis, … 
Plot your data! 
45
© 2014 IBM Corporation 
Summarizing quantitative data 
One-number summaries 
–Mean Average, obtained by summing all observations and dividing by the number of obs. 
–Median The center value, below and above which you will find 50% of the observations. 
Summarizing your data with one number may not tell the whole story: 
46 
Median = 19.8 
Median = 19.8 
Median = 10.5
© 2014 IBM Corporation 
47 
Flaw of averages 
“Plans based on average assumptions are wrong on average” 
Average depth 3 ft
© 2014 IBM Corporation 
“Most observations fall within ±2 standard deviations of the mean.” 
Standard deviation 
 
48 
If the data is normally distributed 
95 % of observations 
Standard Deviation = 4.2 
~95% of observations between 11.4 and 28.2
© 2014 IBM Corporation 
Descriptive statistics - example 
Random sample of 5000 customers of a credit card company 
49 
Amount spent on primary card last month 
Debt to income ratio (x100) 
N 
Valid 
5000 
5000 
Missing 
0 
0 
Mean 
1683.7340 
9.9578 
Median 
1690.0670 
8.8000 
Std. Deviation 
210.26680 
6.42317 
Minimum 
.00 
.00 
Maximum 
2482.72 
43.10
© 2014 IBM Corporation 
Percentiles 
Generalizations of the median (50th percentile). 
The pth is the data point below which p percent of the observations fall. 
Often used to compare a single observation to a general population. 
Examples: 
–Standardized test scores If you scored in the 93th percentile, your score was higher than that of 93% of test takers. 
–Child growth percentiles 
50
© 2014 IBM Corporation 
Percentiles - example 
Percentiles can be another way of describing how spread out data values are. Example: 5-Number Summary Minimum – 25th percentile – Median – 50th percentile - Maximum 
51 
Amount spent on primary card last month 
Debt to income ratio (x100) 
Minimum 
.00 
.00 
Percentiles 
25 
1567.4658 
5.1250 
50 
1690.0670 
8.8000 
75 
1814.5430 
13.5000 
Maximum 
2482.72 
43.10
© 2014 IBM Corporation 
Distributions: Normal distribution 
52
© 2014 IBM Corporation 
Distributions 
53
© 2014 IBM Corporation 
54 
Distributions 
Estimate of the probability distribution of global mean temperature resulting from a doubling of CO2 relative to its pre-industrial value, made from 100000 simulations
© 2014 IBM Corporation 
55
© 2014 IBM Corporation 
56 
Questions?

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Business Analytics and Optimization Introduction

  • 1. Business Analytics and Optimization: A Technical Introduction Oleksandr Romanko, Ph.D. Senior Research Analyst, Risk Analytics – Business Analytics, IBM Adjunct Professor, University of Toronto Toronto SMAC Meetup September 18, 2014
  • 2. © 2014 IBM Corporation 2 Making the world work better – pioneering the science 2008 1973 1969 1981
  • 3. © 2014 IBM Corporation 3 IBM Centennial: 100 Years of Innovation
  • 4. © 2014 IBM Corporation  Analytics Jobs
  • 7. © 2014 IBM Corporation Data science 7
  • 8. © 2014 IBM Corporation  Business Analytics
  • 9. © 2014 IBM Corporation Predictive Analytics What will happen? Descriptive Analytics What has happened? Prescriptive Analytics What should we do? What is analytics? Data Insight Action Decide Analyze Business Value 9 Analytics is the scientific process of deriving insights from data in order to make decisions
  • 10. Business analytics and optimization (video)
  • 11. © 2014 IBM Corporation 11 IBM Business Analytics portfolio IBM Business Analytics Financial Services Public Sector Distribution Industrial Communications Customer Risk Industry Solutions Finance Operations Risk Analytics Business Intelligence Software Categories Predictive Analytics Performance Management Functional Solutions Core Capabilities REPORT MODEL COLLABORATE PREDICT Budgeting & Forecasting Financial Consolidation Disclosure Management Risk Identification Risk & Control Assessment Resource Optimization Social Media Analytics Profitability Modeling & Optimization Production Planning Asset Management Customer Acquisition Customer Lifetime Value Customer Loyalty & Retention Risk Mitigation Planning Risk Aware Decisioning Sales Performance Management ANALYZE PLAN Visualize Discover Forecast Mine Govern Score Decide Simulate Contribute Survey Decision Management
  • 12. © 2014 IBM Corporation 12 Operations research Operations Research (O.R.) is the discipline of applying advanced analytical methods to help make better decisions Analytical techniques: Simulation – giving you the ability to try out approaches and test ideas for improvement Optimization – narrowing your choices to the very best when there are virtually innumerable feasible options and comparing them is difficult Probability and Statistics – helping you measure risk, mine data to find valuable connections and insights, test conclusions, and make reliable forecasts Mathematical Modeling – algorithms and software
  • 13. © 2014 IBM Corporation 13 Our planet is a complex, dynamic, highly interconnected $54 Trillion system-of-systems (OECD-based analysis) Communication $ 3.96 Tn Transportation $ 6.95 Tn Leisure / Recreation / Clothing $ 7.80 Tn Healthcare $ 4.27 Tn Food $ 4.89 Tn Infrastructure $ 12.54 Tn Govt. & Safety $ 5.21 Tn Finance $ 4.58 Tn Electricity $ 2.94 Tn Education $ 1.36 Tn Water $ 0.13 Tn Global system-of-systems $54 Trillion (100% of WW 2008 GDP) Same Industry Business Support IT Systems Energy Resources Machinery Materials Trade Legend for system inputs Note: 1. Size of bubbles represents systems’ economic values 2. Arrows represent the strength of systems’ interaction Source: IBV analysis based on OECD This chart shows ‘systems‘ (not ‘industries‘) 1 Tn
  • 14. 14 © 2014 IBM Corporation Economists estimate, that all systems carry inefficiencies of up to $15 Tn, of which $4 Tn could be eliminated Global economic value of System-of-systems $54 Trillion 100% of WW 2008 GDP Inefficiencies $15 Trillion 28% of WW 2008 GDP Improvement potential $4 Trillion 7% of WW 2008 GDP How to read the chart: For example, the Healthcare system‘s value is $4,270B. It carries an estimated inefficiency of 42%. From that level of 42% inefficiency, economists estimate that ~34% can be eliminated (= 34% x 42%). Source: IBM economists survey 2009; n= 480 System inefficiency as % of total economic value Improvement potential as % of system inefficiency Education 1,360 Building & Transport Infrastructure 12,540 Healthcare 4,270 Government & Safety 5,210 Electricity 2,940 Financial 4,580 Food & Water 4,890 Transportation (Goods & Passenger) 6,950 Leisure / Recreation / Clothing 7,800 Communication 3,960 Analysis of inefficiencies in the planet‘s system-of-systems Note: Size of the bubble indicate absolute value of the system in USD Billions 42% 34% This chart shows ‘systems‘ (not ‘industries‘) 15% 20% 25% 30% 35% 40% 15% 20% 25% 30% 35% 40% 45%
  • 15. © 2014 IBM Corporation 15 History of analytics
  • 16. © 2014 IBM Corporation 16 History of business analytics
  • 17. © 2014 IBM Corporation  Business Analytics Examples
  • 18. © 2014 IBM Corporation Pit stop analytics 7 Calculations showed that time spent changing tires and refilling the tank was more than offset by the improved performance of the car on the track. 1. Softer tires stuck to the track better during turns than their harder cousins, though they wore out more quickly. 2. Less gas in the tank translated into a lighter, and therefore faster, car. Optimized F1 pit teams can change four tires in two seconds
  • 19. © 2014 IBM Corporation 19 Movies
  • 20. © 2014 IBM Corporation 20 Smarter Cities
  • 21. © 2014 IBM Corporation 21 We can collect information from almost everything to make better decisions Camera phones in existence able to document accidents, damage, and crimes 1 billion RFID tags embedded into our world and across entire ecosystems 30 billion Of new automobiles will contain event data recorders collecting travel information 85% Instrumented Interconnected Intelligent
  • 22. © 2014 IBM Corporation 22 Big data Big data are datasets that grow so large that they become awkward to work with using on-hand database management tools. Difficulties include capture, storage, search, sharing, analytics, and visualizing. Source: Wikipedia
  • 23. © 2014 IBM Corporation 23 Big social data
  • 24. © 2014 IBM Corporation 24 Applications of big data analytics Homeland Security Finance Smarter Healthcare Multi-channel sales Telecom Manufacturing Traffic Control Trading Analytics Fraud and Risk Log Analysis Search Quality Retail: Churn, NBO
  • 25. © 2014 IBM Corporation 25 Police use analytics to reduce crime (video)
  • 26. © 2014 IBM Corporation 26 Marketing and supply chain analytics (video)
  • 27. © 2014 IBM Corporation 27 Marketing analytics
  • 28. 28 © 2014 IBM Corporation Intelligent transport systems  Real time monitoring & forecasting of congestion in cities enables real time action to reduce traffic and emissions – Can charge drivers at point of use for access to city centers  Stockholm Congestion Tax Project – Involves 18 barrier-free control points – Allows differentiated pricing by time of day, congestion level, and potentially emissions level – Results: • Traffic reduced by 100,000 vehicle passages per day (25%) • Public transportation passengers increased by 40,000 / day • Congestion during peak hours and CO2 emissions were dramatically reduced
  • 29. © 2014 IBM Corporation 29 Analytics for green vehicles and technology (video)
  • 30. © 2014 IBM Corporation 30 Artificial intelligence Source: A Brief Overview and Thoughts for Healthcare Education and Performance Improvement by Watson Team
  • 31. © 2014 IBM Corporation 31 Artificial intelligence Source: A Brief Overview and Thoughts for Healthcare Education and Performance Improvement by Watson Team On 27th May 1498, Vasco da Gama landed in Kappad Beach On 27th May 1498, Vasco da Gama landed in Kappad Beach celebrated May 1898 400th anniversary arrival in In May 1898 Portugal celebrated the 400th anniversary of this explorer’s arrival in India. Portugal landed in 27th May 1498 Vasco da Gama Temporal Reasoning Statistical Paraphrasing GeoSpatial Reasoning explorer On 27th May 1498, Vasco da Gama landed in Kappad Beach On the 27th of May 1498, Vasco da Gama landed in Kappad Beach Kappad Beach Para- phrases Geo- KB Date Math India Search Far and Wide Explore many hypotheses Find Judge Evidence Many inference algorithms
  • 32. © 2014 IBM Corporation 32 Artificial intelligence (video)
  • 33. © 2014 IBM Corporation 33 Watson Analytics
  • 34. © 2014 IBM Corporation 34 Watson Analytics
  • 35. © 2014 IBM Corporation  Cloud
  • 36. © 2014 IBM Corporation 36 Bluemix www.bluemix.net
  • 37. © 2014 IBM Corporation 37 Bluemix
  • 38. © 2014 IBM Corporation  Business Analytics Education
  • 39. © 2014 IBM Corporation IBM Academic Initiative program Cognos SPSS ILOG
  • 40. © 2014 IBM Corporation Master of Business Analytics programs – top 20 universities
  • 41. © 2014 IBM Corporation Industry support for Master of Business Analytics programs
  • 42. © 2014 IBM Corporation Business Analytics programs – curriculum Applied Statistics and Probability Fundamentals of Computational Mathematics Data Mining and Knowledge Discovery Simulation Modelling Optimization Financial Decision Making Computational Methods for Business Data Analysis Computational Finance and Risk Management Visual Analytics and Knowledge Representation Mathematical Modelling for Business Machine Learning, Cognitive Computing and Artificial Intelligence Marketing Analytics Strategies for Managing Innovations Analytics of Web, Social Networks and Business News
  • 43. © 2014 IBM Corporation  Applied Statistics
  • 44. © 2014 IBM Corporation What kind of data are we dealing with? Types of data •Quantitative •Categorical (ordered, unordered) Data collection •Independent observations (one observation per subject) •Dependent observations (repeated observation of the same subject, relationships within groups, relationships over time or space) Type of data drives the direction of your analysis •How to plot •How to summarize •How to draw inferences and conclusions •How to issue predictions 44
  • 45. © 2014 IBM Corporation Quantitative data Examples: temperature, age, income Quick check: “Does it makes sense to calculate an average?” Appropriate summary statistics: –Mean and Median –Standard Deviation –Percentiles More advanced predictive methods: Regression, Time Series Analysis, … Plot your data! 45
  • 46. © 2014 IBM Corporation Summarizing quantitative data One-number summaries –Mean Average, obtained by summing all observations and dividing by the number of obs. –Median The center value, below and above which you will find 50% of the observations. Summarizing your data with one number may not tell the whole story: 46 Median = 19.8 Median = 19.8 Median = 10.5
  • 47. © 2014 IBM Corporation 47 Flaw of averages “Plans based on average assumptions are wrong on average” Average depth 3 ft
  • 48. © 2014 IBM Corporation “Most observations fall within ±2 standard deviations of the mean.” Standard deviation  48 If the data is normally distributed 95 % of observations Standard Deviation = 4.2 ~95% of observations between 11.4 and 28.2
  • 49. © 2014 IBM Corporation Descriptive statistics - example Random sample of 5000 customers of a credit card company 49 Amount spent on primary card last month Debt to income ratio (x100) N Valid 5000 5000 Missing 0 0 Mean 1683.7340 9.9578 Median 1690.0670 8.8000 Std. Deviation 210.26680 6.42317 Minimum .00 .00 Maximum 2482.72 43.10
  • 50. © 2014 IBM Corporation Percentiles Generalizations of the median (50th percentile). The pth is the data point below which p percent of the observations fall. Often used to compare a single observation to a general population. Examples: –Standardized test scores If you scored in the 93th percentile, your score was higher than that of 93% of test takers. –Child growth percentiles 50
  • 51. © 2014 IBM Corporation Percentiles - example Percentiles can be another way of describing how spread out data values are. Example: 5-Number Summary Minimum – 25th percentile – Median – 50th percentile - Maximum 51 Amount spent on primary card last month Debt to income ratio (x100) Minimum .00 .00 Percentiles 25 1567.4658 5.1250 50 1690.0670 8.8000 75 1814.5430 13.5000 Maximum 2482.72 43.10
  • 52. © 2014 IBM Corporation Distributions: Normal distribution 52
  • 53. © 2014 IBM Corporation Distributions 53
  • 54. © 2014 IBM Corporation 54 Distributions Estimate of the probability distribution of global mean temperature resulting from a doubling of CO2 relative to its pre-industrial value, made from 100000 simulations
  • 55. © 2014 IBM Corporation 55
  • 56. © 2014 IBM Corporation 56 Questions?