1. Inverse Problems: What are they, how can we approach them, and what can we learn from them?

2. Consider the Following: Johnny works two jobs. This past April he made $2100 at the first, and an additional $1300 at the second. How much did Johnny make in total for the month of April?

3. Consider the Following: Johnny works two jobs. This past April he made $2100 at the first, and an additional $1300 at the second. How much did Johnny make in total for the month of April? Not surprisingly, Johnny has made $3400 for April.

4. But How About This: Susie also works two jobs. If Susie made $4500 for herself during the month of April, how much did she earn at each job individually?

5. But How About This: Susie also works two jobs. If Susie made $4500 for herself during the month of April, how much did she earn at each job individually? There isn't a single answer for this question!

6. Here's a general diagram of the situations we just had.

7. In Johnny's case, we combined A & B to get C. Job #1 Salary Job #2 Salary Total Income

8. But in Susie's case, we only had C to try finding A & B. Job #1 Salary Job #2 Salary Total Income

9. This is an example of an inverse problem; how can we attempt to solve it? Job #1 Salary Job #2 Salary Total Income ? ?

10. One way is by utilizing contours, like on this map of Hawaii.

11. Here's a zoomed-in look at the Big Island.

12. Note that the contours represent lines of constant elevation.

13. We'll use that same idea, but applied in a different way.

14. For example, here's the combination A+B, where we require that C = 75.

15. All points on the blue contour represent A+B = 75.

16. We could also show many contours at once.

17. Here, the line in red represents the requirement that A+B = 110.

18. We could combine A & B any way we want actually; here are contours for AB.

19. So, in the regular problem, we find a unique answer C from our combined parameters A & B. However, in the inverse problem, we find a contour of solutions for A & B from a specified value for C. Mathematically speaking, there are infinitely-many solutions!

20. Things start to get even more complex when we take uncertainties into account.

21. This is a Normal Curve, also called a Bell Curve. Data can often be represented by this type of graph.

22. Notice how the graph has both a center (or mean) & a spread (or standard deviation.) Center Spread

23. The numbers at the bottom are a measure of how far from the center a value is: greater means farther away, and thus more unlikely to occur.

24. We can use this idea in our requirement for C...

25. ...with the Chi-square Statistic (pronounced “K-eye.”)

26. We can look at the value of Chi-square for C at many locations on our contour plot.

27. The higher the value, the less likely that combination of A & B is (given our requirements on C.)

28. Let's Try it Out: Suppose we are interested in the combination: A + B = C Say we also know that C = 110, with an uncertainty of 10. We might guess that we'd get a simple contour of possible A & B combos, like before. Is that the case?...

29. ...No! Now we find possible regions for A & B.

30. The darker regions are more likely combos for A & B, while lighter regions are more unlikely.

31. See how the uncertainty in C has spread out the combinations of A & B from a single contour?

32. As we can see, there are a LOT of reasonable possibilities for A & B in this situation.

33. Consider, Though: Say we were also interested in the combination: A – B = D Suppose we know that D = 30, with an uncertainty of 5. We realize now that we'll get a swath, just like with C from before. But what will it look like?...

34. ...Hey! This region runs in a different direction.

35. By itself, this isn't really any more interesting. If we put both regions together, though...

36. ...Presto! We've shrunk the possible A & B combinations dramatically.

37. In such situations, this Chi-square analysis has the potential to be quite powerful.

38. In Summary: Relationship: Parameters {A,B} & Constraint C {A,B} → C: Unique (Normal Problem) C → {A,B}: Contour (Inverse Problem) Uncertainties on C: Contour morphs into Region Compile C with D: Region can shrink smaller Apply this idea to tackle real problems!