olga ivina phd thesis presentation short

Conformal prediction of air pollution concentrations forthe Barcelona Metropolitan Region

PhD Thesis summary

Olga Ivina

University of GironaGRECS research group

CIBER de Epidemiologıa y la Salud Publica

November 22, 2012

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Outline

IntroductionAir pollution and its effectsAir pollution exposure assessmentConformal predictors for air pollution problem

ObjectivesMethods and data

KrigingConformal predictorsComputingData

ResultsOrdinary kriging and RRCM models in default settingKernelisation: a Gaussian kernelKernelisation: other kernelsComparison of models

DiscussionConclusion

Conformal predictors and geostatisticsFuture research

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Air pollution and its effectsIntroduction

Air pollutant is a problem of growing concern all over the world.There exists great body of scientific evidence of hazardous effect of airpollution on people’s health and well-being, as well as on generalecological condition of our planet.

In people: association with adverse health outcomes - both in adults andin children. Children are specially susceptible to pollution. They getaffected from the very first stages of their lives and on. Linked outcomes(to name a few):

- preterm birth and low birth weight

- asthma aggravation, cough and bronchitis

- allergies: hay fever, rhinitis, ...

- excess risk of mortality

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Air pollution and its effects - 2Introduction

Adults are influenced by pollution as well. In them, pollution is linked toboth long-term and short-term health effects (to name a few):

- respiratory: COPD, asthma, chronic bronchitis

- lung cancer

- cardiovascular morbidity

- mortality: cancer, all-cause, cardiopulmonary, non-accidental,...

Special factors of impact: SES and geographical location of a person.

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Global air pollution map produced by Envisat’s SCIAMACHY.

Authors: S. Beirle, U. Platt and T. Wagner, University of Heidelberg’s Institute for Environmental Physics.

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The main contributor to air pollution in urban areas is traffic. Two -”criteria” - traffic-related air pollutants are taken up in this study:

- nitrogen dioxide (NO2)

- particulate matter PM10

NO2 effects:

short-term: respiratory effects and asthma aggravation

long-term: risk of coronary heart disease and fatal events

PM10 effects:

short-term: aggravation of respiratory and cardiovascular diseases,premature death, ...

long-term: development of heart and lung diseases, prematuredeath,...

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Air pollution exposure assessmentIntroduction

Problem: direct measurements of pollution not always available.

There exists a large number of models aimed t predict pollution at a givenspot. The main classes are:

- proximity models

- geostatistical models

- land use regression (LUR) models

- dispersion models

- integrated meteorological emission (IME) models

- hybrid models

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Conformal predictors for air pollution problemIntroduction

Problem: nowadays existing methods for air pollution exposureassessment may lack confidence in predictions.

In order to tackle this problem, this research suggests making use of anewly developed approach that is conformal predictors. A conformal

predictor is a “confidence predictor”, where the level of confidence forprediction is introduced ad hoc. This prediction is always valid - providedby definition of conformal predictor.

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Conformal predictors for air pollution problem - 2Introduction

A conformal predictor is defined by some nonconformity measure, and ithas two major desiderata:

- validity of predictions

- efficiency of preditions

Conformal predictors are flexible: they can be based upon almost anyunderlying statistical algorithm.

In air pollution modeling, if a regression-based algorithm is taken up, suchas LUR or kriging, regression residuals serve as a nonconformity measure.

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Objectives

This dissertation has two major objectives:

1 To demonstrate the capacity of conformal predictors as a method forspatial environmental modeling.

2 To provide valid estimates of nitrogen dioxide and fine particulatematter for Barcelona Metropolitan Region.

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KrigingMethods and data

Kriging is a spatial interpolation method. Provides a prediction of a factorof interest in an unobserved point on the basis of a set of observed points.Also provides an estimate of error variance (called “kriging variance”).

First introduced in 1951 by a South African engineer D.H. Krige in hismaster work devoted to estimation of a mineral ore body. The method hasbeen further developed: nowadays the notion “kriging” stands for asset ofmethods such as ordinary kriging, simple kriging, co-kriging, Bayesiankriging etc.

In its simples form, a kriging estimate of the data at an unobservedlocation is a linear combination of the observed data. The coefficients ofthe equation depend on spatial structure of the data and on the spatialcovariance.

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Kriging - 2Methods and data

The most common kriging is ordinary kriging. It is used when the meanof the second order stationary process is unknown. It is based on ageostatistical concept of variogram, and its approach - covariance function.Let there be n neighboring observed locations, x1, . . . , xn, and anunobserved location x0, on a spatial domain D. Let Z (x) : x ∈ D denotethe process, and let it have a variogram γ(h). Then the ordinary krigingestimate Z ∗OK (x0) at the unobserved point x0 will take the followinganalytical form:

Z ∗OK (x0) =n∑

α=1

ωαZ (xα), (1)

where ωα are the kriging weights. Ordinary kriging provides BLUEestimates of a random field, together with an error variance estimate(kriging variance.)

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New methods. Conformal predictorsMethods and data

How it works? Provided: pairs of observations of (xi , yi ) where xi is anobject and yi is a label. Then

Z := X× Y (2)

denotes the example space. Z is a measurable space. Given an incompletedata sequence (x1, y1), (x2, y2), . . . , (xn−1, yn−1) ∈ Z∗, the aim is to predicta label yn for an object xn. An operator:

D : Z∗ × X→ Y (3)

denotes then a simple predictor. (e.g., an ordinary kriging predictor).

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New methods. Conformal predictors - 2Methods and data

The prediction can be described as:

yn = D(x1, y1, x2, y2, . . . ; xn−1),Yn ∈ Y. (4)

Let us allow the predictor to output the prediction sets Yn large enough toprovide the confidence in prediction. This means, that the real value of ynwill fall in Yn with a given level of confidence, which is chosen andprovided to a predictor ad hoc.

A conformal predictor is a confidence predictor defined by somenonconformity measure. Given the measure, a conformal predictor outputsthe prediction set assuming that the new example conforms with theobserved ones.

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Ridge regression confidence machine (RRCM) is a regression-basedconformal predictor. It makes use of the ridge regression procedure (A. E.Hoerl, 1971) as an underlying algorithm.

Suppose Xn is the n × p matrix of objects (independent variables), and Yn

is the vector of labels (dependent variables). Then, a RRCM estimate ofparameters ω takes form:

ω = (X ′nXn + aIp)−1X ′nYn, (5)

where a is a ridge factor. a = 0 yields a standard least squares estimate.The nonconformity scores for this predictor are the regression residuals:|ei | := |yi − yi |.

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Based on a significance level for prediction introduced (roughly, aprobability of error not to exceed), a RRCM predictor outputs a set oflabels y for yn:

Si := {y : αi (y) ≥ αn(y)} = {y : |ai + biy | ≥ |an + bny |}, (6)

where ai and bi are the components of the vectors A and B.

RRCM outputs prediction sets instead of point predictions (what krigingdoes). These sets can be in form of a point, an interval, a ray, a union oftwo rays, the whole real line, or empty. Usually, it is an interval.

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When the number of parameters p is large, computation is hard. “Kerneltrick” is a method that helps deal with hight-dimensional data. It allows toconsider nonlinearity in RRCM.

A kernel is a similarity measure that operates in a feature space. Providedan input space X with a dot product, and an operator Φ that maps X to afeature space H:

Φ : X → H

x 7→ x := Φ(x)

a kernel will be defined as follows. For xα, xβ ∈ X :

k(xα, xβ) = 〈Φ(xα),Φ(xβ)〉 (7)

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Any conventional covariance function for kriging can be taken up asa kernel for RRCM. This research uses three (positive definite) kernels:

a dot product kernel (default)

a radial basis Gaussian kernel

an inhomogeneous polynomial kernel of a second degree

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ComputingMethods and data

All computational work made with R.

- Kriging: geoR package. Function krige.conv

- RRCM: PredictiveRegression package. Function iidpred.

- “Kernel trick”: self-developed (on the basis of the PredictiveRegressionpackage) functions for RRCM in “dual form” and for implementing thekernels.

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DataMethods and data

The data for this study has been kindly provided by XVPCA (Network forMonitoring and Forecasting of Air Pollution) of the Generalitat deCatalunya.

Mean annual concentrations of two criteria pollutants, NO2 and PM10, areprovided for the Barcelona Metropolitan Region, together with thegeographical coordinates of the monitoring stations(Mercator, UTM 31).

Time frames:

- NO2: 1998 - 2009, ex. 2003

- PM10: 2001 - 2009, ex.2003

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Data - 2Methods and data

49 monitoring stations over the area in total.

Barcelona Metropolitan Region has a territory of about 3200 km2 andaccommodates over 5 million inhabitants.

In BMR, there happen about 107 million displacements weekly, 54.1% ofthem - by means of motorized transport.

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Table: 1. Data on mean annual nitrogen dioxide concentrations

Available observations for each year1998 1999 2000 2001 2002 2004 2005 2006 2007 2008 2009

24 25 25 25 25 24 22 24 25 25 24

Table: 2. Data on mean annual particulate matter concentrations

Available observations for each year2001 2002 2004 2005 2006 2007 2008 2009

22 24 28 28 29 30 33 36

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Two major drawbacks, or limiting factors, of the data set:

Size: there was a small number of observations for each year andpollutant,

Distribution: the measurement spots are situated quite far apartfrom one another, and they are distributed, or placed, unevenly overthe geographic region.

Also, the data is the mean averages, and more frequent observations wereunavailable for this study.

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Ordinary kriging and RRCM modeling resultsResults

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Ordinary kriging and RRCM modeling results - 2Results

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Ordinary kriging and RRCM modeling results - 3Results

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Kernelisation: a Gaussian kernelResults

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Kernelisation: a Gaussian kernel - 2Results

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Kernelisation: a Gaussian kernel - 3Results

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Comparison of the RRCM modelsResults

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Comparison of the RRCM models - 2Results

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Table: Comparison of models for different ridge factors (µg/m3)

linear iid RBF polynomialridge 0.01 1 2 0.01 1 2 0.01 1 22001 64.46 64.44 67.13 71.08 63.11 66.06 71.95 74.63 77.242002 43.43 42.46 45.54 47.41 42.91 45.05 50.44 53.17 55.822004 47.26 39.17 34.59 51.48 39.29 35.19 34.66 37.00 39.512005 39.65 45.14 49.28 35.50 47.60 51.91 51.44 54.76 57.762006 47.68 45.40 48.63 55.51 46.09 48.86 52.48 55.27 57.862007 91.43 94.02 96.45 85.40 94.09 96.65 99.83 102.11 104.292008 49.48 50.90 52.58 45.42 55.27 58.21 55.60 57.26 58.912009 28.42 27.32 29.01 29.16 26.11 27.79 32.26 33.67 35.09

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Table: Comparison of models for different ridge factors (µg/m3)

linear iid RBF polynomialridge 0.01 1 2 0.01 1 2 0.01 1 21998 76.08 72.33 68.27 65.81 72.37 68.37 65.27 64.71 65.991999 66.31 60.11 61.44 67.68 60.57 60.39 65.32 68.20 70.872000 51.69 55.27 57.89 50.91 52.90 55.63 61.89 64.19 66.382001 36.25 41.30 44.90 35.32 38.65 42.36 49.54 52.34 54.952002 52.12 46.57 49.51 47.78 51.44 57.38 54.51 56.99 59.372004 53.65 59.11 62.46 53.89 56.95 60.41 67.06 69.36 71.602005 78.75 84.77 88.57 79.44 82.18 86.14 94.41 96.94 99.432006 61.79 66.39 69.78 61.24 63.82 67.38 74.90 77.36 79.762007 47.01 49.35 53.13 48.15 47.11 51.04 57.15 59.91 62.482008 46.96 50.15 53.58 47.45 48.04 51.55 57.63 60.21 62.632009 55.59 55.17 53.89 48.38 54.35 52.68 52.79 55.19 57.57

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Efficiency of predictionsDiscussion

Kriging predictions are smooth and vary little, also made for mean annualdata. Error estimates, however, are huge in case of nitrogen dioxide, andsmall in case of airborne particles - subject to properties of the substances:NO2 is known to have a generally larger variability than PM10.

Kriging intervals can be derived, assuming the Gaussianity of datadistribution. This assumption is common, but not always correct. RRCMmakes no assumption on data distribution, apart from being iid.

Two factors help boost the efficiency of RRCM prediction: kernels andridge factor. The least is chosen by the brute force method (or the methodof consecutive approximations).

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Conformal predictors and geostatisticsConclusion

Table: Comparison of OK and RRCM

OK RRCMpoint predictions prediction sets (usually intervals)

regression algorithm regression algorithm

Gaussianity assumption iid assumption

estimates error variance -

uses variogram and uses any appropriatecovariance function kernel

to approach it

- ridge factor

may lack confidence confidence level ischosen and guaranteed

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Future researchConclusion

Extend the existing data set for BMR

Provide additional validation for the methods

Test these models on the data for other cities

Develop conformal predictors on the basis of other popular airpollution exposure modeling algorithms (land use regression,dispersion models etc.)

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Selected references

V.Vovk, A.Gammerman, G.Shafer, Algorithmic learning in a randomworld, Springer (2005).

V.Vovk, I.Nouretdinov, A. Gammerman, On-line predictive linearregression, The Annals of Statistics (2009).

H. Wackernagel, Multivariate geostatistics: an introduction withapplications, Springer (2003).

B. Scholkopf, J. Smola, Learning with kernels: support vectormachines, regularization, optimization, and beyond, MIT Press(2002).

A. Lertxundi-Manterola, M. Saez, Modelling of nitrogen dioxide (NO2)and fine particulate matter (PM10) air pollution in the metropolitanareas of Barcelona and Bilbao, Spain, Environmetrics (2009).

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Selected references - 2

A. Hoerl, R. Kennard, Ridge regression: Biased estimation fornonorthogonal problems, Technometrics 12.1 (1970).

P. Diggle, P. Ribeiro Jr., Model-Based Geostatistics, Springer (2007).

P. Ribeiro Jr., P. Diggle, geoR: a package for geostatistical analysis,R-NEWS 1.2 (2001).

N. Cressie, Statistics for spatial data, Wiley (1993).

M. Jerrett et al., A review and evaluation of intraurban air pollutionexposure models, Journal of exposure analysis and environmentalepidemiology (2005).

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