a Dept of Computer Science, Michigan State University b Dept of - - PowerPoint PPT Presentation

CONTOUR REGRESSION: A distribution-regularized regression framework for climate modeling

Zubin Abrahama, Pang-Ning Tana, Julie A. Winklerb, Perdinanb, Shiyuan Zhongb, Malgorzata Liszewskac

aDept of Computer Science, Michigan State University bDept of Geography, Michigan State University cCentre for Math and Comp Modeling, Univ of Warsaw

Climate Change Modeling

• There are growing concerns about climate change and how it could impact

natural resources and various sectors of economy and society

• Agriculture,
• Health,
• Hydrology,
• Population migration and conflict, etc.
• Climate change impact assessment studies require long-term projections
• f future climate scenarios. [1]

Introduction : Climate Modeling

2

[1] Julie Winkler et. al. Climate Scenario Development and Applications for Local/Regional Climate Change Impact Assessments: An Overview for the Non-Climate Scientist: Part I: Scenario Development Using Dow nscaling Methods Climate scenario development and applications I- In proceeding of Geography Compass’11

Long Term Projection of Future Climate

• Introduction : Climate Modeling

3

Histogram of daily maximum temperature at a weather station in Michigan Warm bias Cold bias

Example: Multiple Linear Regression

• Introduction : Multiple Linear Regression

4

The cumulative distribution function (CDF) of observation variable Regression-based methods that minimize prediction error tend to have large distribution bias

Example: Quantile Mapping (QM)

• Quantile mapping is a bias correction method.

Where, ‘x’ is the RCM/GCM output and ‘y’ the observed response variable.

Introduction : Quantile Mapping

5

CDF of x and y Bias correction methods minimize bias but have large prediction error

• Eq. (1)

Comparison between QM and MLR

Introduction : Climate Modeling

6

• Fig. Histogram of daily maximum temperature at a weather station in Michigan

Contributions

Present framework (Contour Regression) that maximizes prediction

accuracy while minimizing bias in the distribution. [3,4]

We also present a linear, a non-linear and a quantile regression based

variations of contour regression

The framework can incorporate predictor variables from heterogeneous

data sources (semi-supervised) [4]

Contour Regression

7 [3] Zubin Abraham et al. Distribution regularized regression framework for climate modeling –SDM’13 [4] Zubin Abraham et al. Contour regression: A distribution-regularized regression framework for climate modeling –In proceeding of Statistical Analysis and Data Mining’ 14

Contour Regression (CR)

• General framework for contour regression

Contour Regression: Introduction

8

Where,

(2)

Minimize residual errors Minimize errors in CDF

y

Regression line

x

Multiple Linear Contour Regression (MLCR)

• Contour Regression: MLCR

9

Where,

• Eq. (6)
• Eq. (5)

• Kernel Contour regression (KCR).

Where,

• Ridge regression applied to CR.

Kernel Contour Regression (KCR)

Contour Regression: Non-Linear Setting

10

• Eq. (8)
• Eq. (9)
• Eq. (7)

Quantile Regression (QR)

Contour Regression: Conditional Quantiles

11

(Y)

Regression line

(X)

W1=7.0, W0=-0.4

Zubin Abraham et al. Extreme Value Prediction for Zero Inflated DataL-PAKDD’12

Quantile Regression (QR)

Contour Regression: Conditional Quantiles

12

u = Residual (Observation-prediction)

(10)

Where,

Quantile Contour Regression (QCR)

Contour Regression: Conditional Quantiles

13 Where

Linear programming is used to solve the above loss function.

(11) (12)

The preceding optimization problem can be converted to the following form Contour regression that uses a QR based loss function take the following form

Contour Regression for Multi-Source Data

Contour Regression: Multi-Source Data

14

Geometric Quantile Mapping:

Geometric quantile is the multi-dimensional equivalent of a univariate quantile mapping function.

Predictor variables:

Response variable:

• Eq. (13)
• Eq. (14)

Zubin Abraham et al. Position Preserving Multi-Output Prediction – ECML-PKDD’13

• J. I. Marden. Positions and qq plots. Statistical Science’ 04

EXPERIMENTAL EVALUATION

Experimental Setup

• Predicting surface precipitation, maximum temperature and min temperature at

a location using the following predictor variables obtained from regional climate models:

Contour Regression (CR) : Experimental Evaluation

16 Julie Winkler et. al. - Climate Scenario Development using Hybrid Downscaling: An Application to NARCCAP and ENSEMBLES simulations- In proceeding of AAG’12*

Data Sources

Data : RCM

17

Predictor variables are obtained from NCEP-driven regional climate models.

WRFG CRCM RCM3

Observation data obtained from 14 climate stations in Michigan. Daily data from 1980-1999

Training: 1980-1989 Testing: 1990-199

Experimental Evaluation

Performance of CR when using least square loss function.

Comparing residual error and distribution bias of MLR, QM, and MLCR

Performance of CR when using QR based loss function.

QCR versus QR

Performance of CR when using predictor variables from multiple data sources.

MLCR (in a semi-supervised setting)

Experimental Evaluation: Introduction

18

MLCR Results (Accuracy)

Multiple Linear Contour Regression (MLCR) : Experimental Results

19

A bar plot of maximum temperature RMSE of the 14 station belonging to the WRFG dataset

MLCR Results (Distribution Bias)

Multiple Linear Contour Regression (MLCR) : Experimental Results

20

The CDF plots of maximum temperature and precipitation of a station belonging to WRFG dataset

Summary for MLCR Results

Multiple Linear Contour Regression (MLCR) : Experimental Results

21

Relative performance gain of MLCR over baseline approaches.. MLCR had lower distribution bias in 14/14 stations for each dataset

Summary for MLCR Results

Multiple Linear Contour Regression (MLCR) : Experimental Results

22

Relative performance gain of MLCR over baseline approaches..

Summary of QCR Results

Quantile Contour Regression (QCR) : Experimental Results

23

QCR had lower distribution bias than QR in 14/14 stations for each dataset The CDF plots of minimum temperature and precipitation of a station belonging to WRFG dataset

Summary of QCR Results

Quantile Contour Regression (QCR) : Experimental Results

24

Percentage of stations that QCR outperformed QR QCR had better accuracy than QR in 14/14 stations for each dataset

MLCR Results (Heterogeneous Data)

Multiple Linear Contour Regression (MLCR) : Experimental Results

25

The CDF plots of maximum temperature and precipitation of a station belonging to WRFG dataset

Summary

• We presented a framework for contour regression, that maximizes prediction accuracy while

minimizing bias in the distribution.

• We show that the framework can be adapted to modeling non linear relationships and conditional

quantiles.

• We empirically showed that the framework outperformed or was at least on par with baseline

approaches on real world climate data.

• The framework can incorporate predictor variables from heterogeneous data sources

Contour Regression (CR) : References

26

References

• [1] Julie Winkler et. al. Climate Scenario Development and Applications for Local/Regional Climate Change Impact

Assessments: An Overview for the Non-Climate Scientist: Part I: Scenario Development Using Downscaling Methods Climate scenario development and applications I- In proceeding of Geography Compass’11

• [2] Themeßl, Jakob, M., Gobiet, A. and Leuprecht, A. (2011), Empirical-statistical downscaling and error correction of daily

precipitation from regional climate models. International Journal of Climatology, 31: 1530–1544.

• [3] Zubin Abraham et al. Distribution regularized regression framework for climate modeling –SDM’13
• [4] Zubin Abraham et al. Contour regression: A distribution-regularized regression framework for climate modeling –In

proceeding of Statistical Analysis and Data Mining’ 14

• [5] Zubin Abraham et al. Position Preserving Multi-Output Prediction – ECML-PKDD’13
• [6] Zubin Abraham et al. Extreme Value Prediction for Zero Inflated DataL-PAKDD’12
• [7] Zubin Abraham et al. An Integrated Framework for Simultaneous Classification and Regression of Time-Series data.

SDM’10

• [8] Julie Winkler et. al. - Climate Scenario Development using Hybrid Downscaling: An Application to NARCCAP and

ENSEMBLES simulations- In proceeding of AAG’12*

• [9] J. I. Marden. Positions and qq plots. Statistical Science’ 04
• [10] X. He, Y. Yang, and J. Zhang. Bivariate downscaling with asynchronous measurements. Journal of agricultural,

biological, and environmental statistics’ 12.

Contour Regression (CR) : Summary

27

THANK YOU!

This work is partially supported by NSF grant III-0712987 and subcontract for NASA award NNX09AL60G. This work is also partly supported by National Science Foundation Dynamics of Coupled Natural and Human Systems competition Program (CNH Award No.-0909378).