Estimation and Uncertainty

1. Estimation and Uncertainty Scott Matthews 12-706/73-359 Lecture 6 - Sept 16, 2002

2. Problem of Unknown Numbers • If we need a piece of data, we can: • Look it up in a reference source • Collect number through survey/investigation • Guess it ourselves • Get experts to help you guess it • Often only ‘ballpark’, ‘back of the envelope’ or ‘order of magnitude needed • Situations when actual number is unavailable or where rough estimates are good enough • E.g. 100s, 1000s, … (102, 103, etc.) • Source: Mosteller handout

3. Notes about Reference Sources • Some obvious: Statistical Abstract of US • Always check sources and secondary sources of data • Usually found in footnotes – also tells you about assumptions/conditions for using • Sometimes the summarized data is wrong! • Look in multiple sources • Different answers implies something about the data and method – and uncertainty

4. Estimation in the Course • We will encounter estimation problems in sections on demand, cost and risks. • We will encounter estimation problems in several case studies. • Projects will likely have estimation problems. • Need to make quick, “back-of-the-envelope” estimates in many cases. • Don’t be afraid to do so!

5. Estimation gets no respect • The 2 extremes - and the respect thing • Aristotle: • “It is the mark of an instructed mind to rest satisfied with the degree of precision which the nature of the subject permits and not to seek an exactness where only an approximation of the truth is possible.” • Archbishop Ussher of Ireland, 1658 AD: • “God created the world in 4028 BC on the 9th of September at nine o’clock in the morning.” • We consider it somewhere in between

6. In the absence of “Real Data” • Are there similar or related values that we know or can guess? (proxies) • Mosteller: registered voters and population • Are there ‘rules of thumb’ in the area? • E.g. ‘Rule of 72’ for compound interest • r*t = 72: investment at 6% doubles in 12 yrs • MEANS construction manual • Set up a ‘model’ to estimate the unknown • Linear, product, etc functional forms • Divide and conquer

7. Methods • Similarity – do we have data that can be made applicable to our problem? • Stratification – segment the population into subgroups, estimate each group • Triangulation – create models with different approaches and compare results • Convolution – use probability or weightings (see Selvidge’s table, Mosteller p. 181) • Note – example of a ‘secondary source’!!

8. Notes on Estimation • Move from abstract to concrete, identifying assumptions • Draw from experience and basic data sources • Use statistical techniques/surveys if needed • Be creative, BUT • Be logical and able to justify • Find answer, then learn from it. • Apply a reasonableness test

9. Attributes of Good Assumptions • Need to document assumptions in course • Have some basis in known facts or experience • Are unbiased towards the answer • Example: what is inflation rate next year? • Is past inflation a good predictor? • Can I find current inflation? • Should I assume change from current conditions? • We typically use history to guide us

10. How many TV sets in the US? • Can this be calculated? • Estimation approach #1: Survey/similarity • How many TV sets owned by class? • Scale up by number of people in the US • Should we consider the class a representative sample? Why not?

11. TV Sets in US – another way • Estimation approach # 2 (segmenting): • work from # households and # tvs per household - may survey for one input • Assume x households in US • Assume z segments of ownership (i.e. what % owns 0, owns 1, etc) • Then estimated number of television sets in US = x*(4z5+3z4+2z3+1z2+0z1)

12. TV Sets in US – sample • Estimation approach # 2 (segmenting): • work from # households and # tvs per household - may survey for one input • Assume 50,000,000 households in US • Assume 19% have 4, 30% 3, 35% 2, 15% 1, 1% 0 television sets • Then 50,000,000*(4*.19+3*.3+2*.35+.15) = 125.5 M television sets

13. TV Sets in US – still another way • Estimation approach #3 – published data • Source: Statistical Abstract of US • Gives many basic statistics such as population, areas, etc. • Done by accountants/economists - hard to find ‘mass of construction materials’ or ‘tons of lead production’. • How close are we?

15. How well did we do? • Most recent data = 1997 • But ‘recently’ increasing < 3% per year • TV/HH - 125.5 tvs, StatAb – 229M tvs, • % error: (229M – 125.5M)/125.5M ~ 82% • What assumptions are crucial in determining our answer? Were we right? • What other data on this table validate our models?

16. Significant Figures • We estimated 125,500,000 tvs in US • How accurate is this - nearest 50,000, the nearest 500,000, the nearest 5,000,000 or the nearest 50,000,000? • Should only report estimates to your confidence - perhaps 1 or 2 “significant figures” could be reported here. • Figures are only carried along to document calculations or avoid rounding errors.

17. Some handy/often used data • Population of US btw 275-300 million • Number of households ~ 100 million • Average personal income ~$30,000

18. Estimate Annual Vehicle Miles Travelled (VMT) in the US • Estimate “How many miles per year are passenger automobiles driven in the US?” • Types of models • Similar to TVs: Guess number of cars, segment population into miles driven per year • Find fuel consumption data, guess at fuel economy ratio for passenger vehicles • Other ideas? Let’s try it on the board.

19. Estimate VMT in the US • Table 1033 of Stat. Abstract suggests 1995 VMT was 2.068 trillion miles (yes - twice as much as 1972 implied in the Mosteller handout)! • 230 billion ‘passenger car trips’ per year • About 200 million cars • Avg VMT 21,000 mi.

20. More clever: Cobblers in the US • Cobblers repair shoes • On average, assume 20 min/task • Thus 20 jobs / day ~ 5000/yr • How many jobs are needed overall for US? • I get shoes fixed once every 5 years • About 280M people in US • Thus 280M/4 = 56 M shoes fixed/year • 56M/5000 ~ 11,000 => 10^4 cobblers in US • Actual: Census dept says 5,120 in US

21. An Energy Example • Energy measured in SI units = Watts (as opposed to BTUs, etc) • In practice, we usually talk about kilowatts or kilowatt-hours of energy • Rule: 1 Watt of energy used for one hour is One watt-hour(compound unit) = 1Wh • 1000 Watts used for one hour = 1kWh • ‘How much energy used by lighting in US residences?’

22. ‘How much energy used by lighting in US residences?’ • Assume 50 light fixtures per house • Assume each in use avg 2 hours per day • Assume average fixture is 50W • Thus each fixture uses 100Wh/day • Each house uses 5000Wh/day (5kWh/day) • 100 million households would use 500 million kWh/day • 182,500 million kWh/yr

23. ‘How much energy used by lighting in US residences?’ • Our guess: 182,500 million kWh/yr • DOE: “lighting is 5-10% of household elec” • http://www.eren.doe.gov/erec/factsheets/eelight.html • 2000 US residential Demand ~ 1.2 million million kWh (source below) • 10% is 120,000 million kWh • 5% is 60,000 million kWh • 2000 demand source: http://www.eia.doe.gov/cneaf/electricity/epm/ epmt44p1.html

24. A Random Example • Select a random panel of data from the Statistical Abstract of the U.S. • Can you formulate an ‘estimation question’? • Can you estimate the answer? • How close were you to the ‘actual answer’? • Let’s try this ourselves

25. Uncertainty • Investment planning and benefit/cost analysis is fraught with uncertainties • forecasts of future are highly uncertain • applications often made to preliminary designs • data is often unavailable • Statistics has confidence intervals – economists need them, too.