1. The Z-Score Regression Method and You Tom Pagano tom.pagano@por.usda.gov 503-414-3010

2. Why do we need something new? What is a z-score? How does the regression work? How good are the results? How to stay out of trouble?

3. Why do we need something new or different? Challenges forecasters face: Data-rich mixed with data-poor stations Missing realtime data High cross-correlation of variables (“co-linearity”)

4. Mt. Rose Apr 1 Snowpack (1910-2006) Uneven record lengths Some stations have many years

5. Mt. Rose Apr 1 Snowpack (1910-2006) Mt. Rose Water Year Precipitation (1981-2005) Uneven record lengths Some stations have many years Others have fewer Typical regression requires completeness Overlapping record

6. Mt. Rose Apr 1 Snowpack (1910-2006) Mt. Rose Water Year Precipitation (1981-2005) Uneven record lengths Some stations have many years Others have fewer Typical regression requires completeness The choice in this situation has been: Use fewer stations or use fewer years Overlapping record

7. Why this is a problem To use new, younger stations, older information has to be “forgotten”. Otherwise, a station must exist for a long time before becoming useable.

8. Why this is a problem To use new, younger stations, older information has to be “forgotten”. Otherwise, a station must exist for a long time before becoming useable. If one piece of data is missing in realtime then no forecast at all is available, even if 95% of the “information” is there.

9. What does z-score regression do? 1. Combines predictors into weighted indices, emphasizing good stations, minimizing bad ones.

10. What does z-score regression do? 1. Combines predictors into weighted indices, emphasizing good stations, minimizing bad ones. 2. Compensates for missing data with remaining data.

11. What does z-score regression do? 1. Combines predictors into weighted indices, emphasizing good stations, minimizing bad ones. 2. Compensates for missing data with remaining data. 3. Regresses index against target predictand

12. What is a z-score? A z-score is a “normalized anomaly”: Z = value - average standard deviation

13. What is a z-score? A z-score is a “normalized anomaly”: Z = value - average standard deviation

14. What is a z-score? A z-score is a “normalized anomaly”: Z = value - average standard deviation 60 135 avgstdev 30 15

15. What is a z-score? A z-score is a “normalized anomaly”: Z = value - average standard deviation 60 135 avgstdev 30 15 Z = (90 – 60)/15 = +2

16. Z-scores wetter drier Stations are now on an “even footing” 0 avg stdev 1 What is a z-score? +2

17. Z-scores wetter drier If one station is partially missing, the other station hints at what it might have been. 0 avg stdev 1 What is a z-score?

18. 1. Normalize input time series (x – x )/σ April 1st inches swe x How does z-score regression work?

19. Standardized Anomalies (“z-scores”) 1. Normalize input time series (x – x )/σ x How does z-score regression work?

20. 2. Correlate each index with target (flow) to get weights Standardized Anomalies (“z-scores”) r^2 with Apr-Jul flow 0.48 0.52 0.61 How does z-score regression work?

21. 3. Develop weighted average of available sites Standardized Anomalies (“z-scores”) r^2 with Apr-Jul flow 0.48 0.52 0.61 Relative weightings e.g. A*x1 + B*x2 A + B How does z-score regression work?

22. 3. Develop weighted average of available sites Standardized Anomalies (“z-scores”) Relative weightings e.g. A*x1 + B*x2 A + B r^2 with Apr-Jul flow 0.48 0.52 0.61 How does z-score regression work? Weighted average

23. Multi-station z-score index Observed 4. Regress multi-station weighted index against flow How does z-score regression work?

24. In the case of multiple signals, stations with a like signal (e.g. fall precipitation) are combined by the user into their own respective “group index”, weighted by their combination with flow. The use of “groups” (aka components)

25. In the case of multiple signals, stations with a like signal (e.g. fall precipitation) are combined by the user into their own respective “group index”, weighted by their combination with flow. All the group indices are then combined into a “master index”, weighted, again, by their correlation with flow. The master index is regressed against flow. The use of “groups” (aka components)

26. Steps to z-score regression

31. A realtime numerical example (1 group, 2 sites) Site Fry Lk Mary Group Snow Avg 4” 5” Stdev 1” 2” Realtime Data 2” 2.5” Z-Score = -2.00 = -1.25 Correlation^2 with flow 0.75 0.50 Group Snow -2*0.75 + -1.25*0.50 0.75+0.50 Group index = -1.7 (2-4)/1 (2.5-5)/2

32. A realtime numerical example (1 group, 2 sites) Site Fry Lk Mary Group Snow Avg 4” 5” Stdev 1” 2” Realtime Data 2” 2.5” Z-Score = -2.00 = -1.25 Correlation^2 with flow 0.75 0.50 Group Snow -2*0.75 + -1.25*0.50 0.75+0.50 Group index = -1.7 (2-4)/1 (2.5-5)/2

33. A realtime numerical example (1 group, 2 sites) Site Fry Lk Mary Group Snow Avg 4” 5” Stdev 1” 2” Realtime Data 2” 2.5” Z-Score = -2.00 = -1.25 Correlation^2 with flow 0.75 0.50 Group Snow -2*0.75 + -1.25*0.50 0.75+0.50 Group index = -1.7 (2-4)/1 (2.5-5)/2

34. A realtime numerical example (3 sites) Site Fry Lk Mary Newman Group Snow Avg 4” 5” 12” Stdev 1” 2” 4” Realtime Data 2” 2.5” 6” Z-Score = -2.00 = -1.25 = -1.50 Correlation^2 with flow 0.75 0.50 0.65 Group Snow -2*0.75 + -1.25*0.50 + -1.5*0.65 0.75+0.50+0.65 Group index = -1.63 (2-4)/1 (2.5-5)/2 (6-12)/4

35. A realtime numerical example (3 sites, 1 missing) Site Fry Lk Mary Newman Group Snow Avg 4” 5” 12” Stdev 1” 2” 4” Realtime Data 2” missing 6” Z-Score = -2.00 = missing = -1.50 Correlation^2 with flow 0.75 0.50 0.65 Group Snow -2*0.75 + -1.25*0.50 + -1.5*0.65 0.75+0.50+0.65 Group index = -1.77 (2-4)/1 (6-12)/4

36. A realtime numerical example (2 groups, 3 sites) Site Fry Lk Mary Fry Group Snow Precip Avg 4” 5” 6” Stdev 1” 2” Realtime Data 2” 2.5” 3” Z-Score = -2.00 = -1.25 = -1.50 Correlation^2 with flow 0.75 0.50 0.25 Group Snow Precip -2*0.75 + -1.25*0.50 0.75+0.50 -1.5 * 0.25 0.25 Group index = -1.7 = -1.5 Group Correlation^2 with flow 0.6 0.25 Master index -1.7*0.6 + -1.5*0.25 = -1.64 0.6+0.25 (2-4)/1 (2.5-5)/2 (3-6)/2

37. How good are the results Under conditions of serially compete data, and relatively “normal” conditions PCA and Z-Score are effectively indistinguishable* Skill and behavior is similar to the official published outlooks** *Viper technical note - 1 basin** Pagano dissertation – 29 basins

38. How good are the results Under conditions of serially compete data, and relatively “normal” conditions PCA and Z-Score are effectively indistinguishable* Skill and behavior is similar to the official published outlooks** However… Any tool is a weapon if you hold it right. (aka “A fool with a tool is still a tool”) *Viper technical note - 1 basin** Pagano dissertation – 29 basins

39. Abuse of the z-score method r 2 =0.95 r 2 =0.18 If the main driver of skill is absent from certain years, those years will have overconfident forecasts. The set as a whole will not be as skillful as it could be. Fcst Obs

40. Abuse of the z-score method r 2 =0.95 If the main driver of skill is absent from certain years, those years will have overconfident forecasts. The set as a whole will not be as skillful as it could be. Solutions: 1.Remove poor skill years from calibration set Fcst Obs

41. Abuse of the z-score method r 2 =0.95 If the main driver of skill is absent from certain years, those years will have overconfident forecasts. The set as a whole will not be as skillful as it could be. Solutions: 1.Remove poor skill years from calibration set 2.Remove poor skill station entirely x x Fcst Obs

42. Abuse of the z-score method If the main driver of skill is absent from certain years, those years will have overconfident forecasts. The set as a whole will not be as skillful as it could be. Solutions: 1.Remove poor skill years from calibration set 2.Remove poor skill station entirely 3.If data for high skill station not available in realtime, remove high skill station x Fcst Obs

43. More z-score method atrocities Stations’ period of records should be representative station1 station2

44. Stations’ period of records should be representative station1 station2 Blue station’s “wet” years are actually normal over longer term. More z-score method atrocities

45. Z-Score Rescaling Stations’ period of records should be representative Blue station’s “wet” years are actually normal over longer term. More z-score method atrocities

46. Z-Score Rescaling Stations’ period of records should be representative Solutions: 1.Use consistent years 2.Eliminate one station 3.Estimate missing data ahead of time Blue station’s “wet” years are actually normal over longer term. More z-score method atrocities

47. Z-score regression – A regression methodology that, within reason, can handle uneven record lengths and missing data. It groups stations into indices, emphasizing good stations, minimizing the effect of poor stations. Multiple signals can be managed (e.g. snow, fall precip, baseflow). Can be abused especially if the input data set is highly uneven. Summary