The Z-Score Regression Method and You

Tom Paganotom.pagano@por.usda.gov

503-414-3010

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?

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”)

Mt. Rose Apr 1 Snowpack (1910-2006) Uneven record lengths

Some stations have many years

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

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

Why this is a problem

To use new, younger stations, older information has to be “forgotten”.

Otherwise, a station must existfor a long time before becoming useable.

Why this is a problem

To use new, younger stations, older information has to be “forgotten”.

Otherwise, a station must existfor a long time before becoming useable.

If one piece of data is missing in realtimethen no forecast at all is available,

even if 95% of the “information” is there.

What does z-score regression do?

1. Combines predictors into weighted indices,emphasizing good stations, minimizing bad ones.

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.

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

What is a z-score?

A z-score is a “normalized anomaly”:Z = value - average

standard deviation

What is a z-score?

A z-score is a “normalized anomaly”:Z = value - average

standard deviation

What is a z-score?

A z-score is a “normalized anomaly”:Z = value - average

standard deviation

60

135

avg stdev

30

15

What is a z-score?

A z-score is a “normalized anomaly”:Z = value - average

standard deviation

60

135

avg stdev

30

15

Z = (90 – 60)/15 = +2

Z-s

core

s

wetter

drier

Stations are now on an “even footing”

0

avg stdev

1

What is a z-score?

+2

Z-s

core

s

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?

1. Normalize input time series (x – x )/σ

April 1stinches swe

x

How does z-score regression work?

StandardizedAnomalies(“z-scores”)

1. Normalize input time series (x – x )/σx

How does z-score regression work?

2. Correlate each index with target (flow) to get weights

StandardizedAnomalies(“z-scores”)

r^2 with Apr-Jul

flow

0.480.520.61

How does z-score regression work?

3. Develop weighted average of available sites

StandardizedAnomalies(“z-scores”)

r^2 with Apr-Jul

flow0.480.520.61

Relativeweightings

e.g.A*x1 + B*x2

A + B

How does z-score regression work?

3. Develop weighted average of available sites

StandardizedAnomalies(“z-scores”)

Relativeweightings

e.g.A*x1 + B*x2

A + B

r^2 with Apr-Jul

flow0.480.520.61

How does z-score regression work?

Weighted average

Multi-station z-score index

Observed

4. Regress multi-station

weighted index against flow

How does z-score regression work?

In the case of multiple signals, stations with a like signal (e.g. fall precipitation)

are combined by the userinto their own respective “group index”, weighted by their combination with flow.

The use of “groups” (aka components)

In the case of multiple signals, stations with a like signal (e.g. fall precipitation)

are combined by the userinto 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)

Steps to z-score regression

Steps to z-score regression

Steps to z-score regression

Steps to z-score regression

Steps to z-score regression

A realtime numerical example (1 group, 2 sites)

Site

FryLk Mary

Group

SnowSnow

Avg

4”5”

Stdev

1”2”

RealtimeData

2”2.5”

Z-Score

= -2.00= -1.25

Correlation^2with flow

0.750.50

Group

Snow-2*0.75 + -1.25*0.50

0.75+0.50

Group index

= -1.7

(2-4)/1(2.5-5)/2

A realtime numerical example (1 group, 2 sites)

Site

FryLk Mary

Group

SnowSnow

Avg

4”5”

Stdev

1”2”

RealtimeData

2”2.5”

Z-Score

= -2.00= -1.25

Correlation^2with flow

0.750.50

Group

Snow-2*0.75 + -1.25*0.50

0.75+0.50

Group index

= -1.7

(2-4)/1(2.5-5)/2

A realtime numerical example (1 group, 2 sites)

Site

FryLk Mary

Group

SnowSnow

Avg

4”5”

Stdev

1”2”

RealtimeData

2”2.5”

Z-Score

= -2.00= -1.25

Correlation^2with flow

0.750.50

Group

Snow-2*0.75 + -1.25*0.50

0.75+0.50

Group index

= -1.7

(2-4)/1(2.5-5)/2

A realtime numerical example (3 sites)

Site

FryLk MaryNewman

Group

SnowSnowSnow

Avg

4”5”

12”

Stdev

1”2”4”

RealtimeData

2”2.5”6”

Z-Score

= -2.00= -1.25= -1.50

Correlation^2with flow

0.750.500.65

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

A realtime numerical example (3 sites, 1 missing)

Site

FryLk MaryNewman

Group

SnowSnowSnow

Avg

4”5”

12”

Stdev

1”2”4”

RealtimeData

2”missing

6”

Z-Score

= -2.00= missing

= -1.50

Correlation^2with flow

0.750.500.65

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

A realtime numerical example (2 groups, 3 sites)

Site

FryLk Mary

Fry

Group

SnowSnow

Precip

Avg

4”5”

6”

Stdev

1”2”

2”

RealtimeData

2”2.5”

3”

Z-Score

= -2.00= -1.25

= -1.50

Correlation^2with flow

0.750.50

0.25

Group

Snow

Precip

-2*0.75 + -1.25*0.50 0.75+0.50

-1.5 * 0.250.25

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

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

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

Abuse of the z-score method

r2=0.95

r2=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.

FcstObs

Abuse of the z-score method

r2=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

FcstObs

Abuse of the z-score method

r2=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 set2.Remove poor skill station entirelyx

x

FcstObs

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 set2.Remove poor skill station entirely

3.If data for high skill station not available in realtime, remove high skill

station

x

FcstObs

More z-score method atrocities

Stations’ period of recordsshould be representative

station1

station2

Stations’ period of recordsshould be representative

station1

station2

Blue station’s “wet” years are actually

normal over longer term.

More z-score method atrocities

Z-Score Rescaling

Stations’ period of recordsshould be representative

Blue station’s “wet” years are actually

normal over longer term.

More z-score method atrocities

Z-Score Rescaling

Stations’ period of recordsshould be representative

Solutions:1.Use consistent years2.Eliminate one station3.Estimate missing data

ahead of time

Blue station’s “wet” years are actually

normal over longer term.

More z-score method atrocities

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 signalscan be managed (e.g. snow, fall precip, baseflow).

Can be abused especially if the input data set is highly uneven.

Summary