a study of the air quality of some major cities in china · a study of the air quality of some...

A Study of the AirQuality of SomeMajor Cities in

Emma Simpson

Supervisor: YeLiu, JBA

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Extreme Value Theory

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

A Study of the Air Quality of SomeMajor Cities in China

Emma Simpson

Supervisor: Ye Liu, JBA

6 September 2013

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

What is Air Pollution?

Beijing during periods of low and high air pollution

I Air pollution is composed of sulphur oxides, nitrogenoxides, carbon monoxide and particulates.

I Particulates are small particles of solid or liquid materialin the air.

I PM2.5 and PM10 are particulates that are smaller than2.5 and 10 micrometres respectively.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Measuring Air Pollution

I The US Embassy and Chinese Government releasehourly PM2.5 readings for Beijing.

I Some people believe that there is a discrepancy betweenthe two sources of data.

I Both use the same formula to calculate the PM2.5 indexI from the concentration C :

I =Ihigh − IlowChigh − Clow

(C − Clow ) + Ilow , (1)

but the breakpoints are different for the US AQI andChinese API.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Measuring Air Pollution

I The US Embassy and Chinese Government releasehourly PM2.5 readings for Beijing.

I Some people believe that there is a discrepancy betweenthe two sources of data.

I Both use the same formula to calculate the PM2.5 indexI from the concentration C :

(C − Clow ) + Ilow , (1)

but the breakpoints are different for the US AQI andChinese API.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Air Quality Index & Air Pollution Index

US breakpoints China breakpoints

Clow Chigh Ilow Ihigh Clow Chigh Ilow Ihigh

0 12 0 50 0 35 0 50

12.1 35.4 51 100 35.1 75 51 100

35.5 55.4 101 150 75.1 115 101 150

55.5 150.4 151 200 115.1 150 151 200

150.5 250.4 201 300 150.1 250 201 300

250.5 350.4 301 400 250.1 350 301 400

350.5 500 401 500 350.1 500 401 500

Table: PM2.5 breakpoints for the US AQI and Chinese API.

(C − Clow ) + Ilow

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Air Quality Index & Air Pollution Index

0 100 200 300 400 500

Plot of AQI/API vs Concentration

AQI/API

USChina

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Air Pollution Data

Our data consists of:

I six months of hourly PM2.5 readings for Beijing fromthe US and Chinese sources;

I twelve years of daily PM10 readings for Beijing, Tianjin,Shanghai and Suzhou from the Chinese Government.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

I There are two methods for deciding which data pointsare extreme:

0 50 100 150 200

Block Maxima

0 50 100 150 200

Threshold Exceedances

1. Separate the data into blocks and take the maximumvalue in each block;

2. Choose a suitable threshold above which points areconsidered extreme.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

0 50 100 150 200

Block Maxima

0 50 100 150 200

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

0 50 100 150 200

Block Maxima

0 50 100 150 200

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Generalised Pareto Distribution (GPD)

I For the hourly PM2.5 data, we first took the dailymaxima and then applied a threshold to determine theextremes.

I A GPD distribution could then be fitted to the data.

I The GPD has distribution functions of the form:

H(y) =

(1 + ξy

)−1/ξif ξ 6= 0

1− exp(− yσ ) if ξ = 0

for y > 0, and subject to the constraint (1 + ξyσ ) > 0.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Generalised Pareto Distribution (GPD)

I For the hourly PM2.5 data, we first took the dailymaxima and then applied a threshold to determine theextremes.

I A GPD distribution could then be fitted to the data.

I The GPD has distribution functions of the form:

H(y) =

(1 + ξy

)−1/ξif ξ 6= 0

1− exp(− yσ ) if ξ = 0

for y > 0, and subject to the constraint (1 + ξyσ ) > 0.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Testing the Reliability of the US/Chinese Data

We want to test whether there is a difference between theUS AQI and Chinese API data.

I Since the US and Chinese data are measured ondifferent scales, it cannot be compared directly.

I Instead, we fit GPD models to the US and Chinese datasets separately and compared the threshold exceedanceprobabilities.

I Then we used a bootstrapping technique to test fordifferences.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Bootstrapping

Suppose we have data x1,...,xn, and a model fitted to thisdata with parameters θ. Bootstrapping works as follows:

1. Resample (with replacement) from these nobservations, obtaining another sample also of length n.

2. Fit the model to the resampled data to get a new set ofparameters θ1.

3. Repeat the process of resampling and fitting the modelN times, obtaining new parameters θi each time, fori = 1, ...,N.

4. These θ1,...,θN , then allow us to make inferences aboutthe parameter θ.

5. Block bootstrapping involves taking blocks of theoriginal data when resampling rather than individualdata points.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Result of the Bootstrapping Test

I The block bootstrapping procedure was applied to theprobabilities that the PM2.5 concentrations exceed the500 threshold, with:

I blocks of seven days;I 1000 iterations.

I 95% confidence intervals were found for US andChinese bootstrapped probabilities.

I If the confidence intervals overlap, there is nosignificant difference between the sets of data.

I The confidence intervals were:I US: (0.00530, 0.05989)I China: (0.00540, 0.06373).

I The confidence intervals overlap, suggesting there is nosignificant difference in the two data sets.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

I The boxplots of the bootstrapped probabilities are alsovery similar.

Figure: Boxplot of the bootstrapped probabilities

I This reiterates that there is no significant differencebetween the data from the US and China.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Asymptotic Dependence

I It is interesting to investigate whether high API/AQIlevels in one city correlate with high readings elsewhere.

I Two sets of data, X1 and X2, are:I asymptotically dependent if

limu→∞

Pr(X1 > u|X2 > u) = α > 0;

I asymptotically independent if

limu→∞

Pr(X1 > u|X2 > u) = 0.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Asymptotic Dependence

I It is interesting to investigate whether high API/AQIlevels in one city correlate with high readings elsewhere.

I Two sets of data, X1 and X2, are:I asymptotically dependent if

limu→∞

Pr(X1 > u|X2 > u) = α > 0;

I asymptotically independent if

limu→∞

Pr(X1 > u|X2 > u) = 0.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Modelling Bivariate Extremes

I The data, X1 and X2, first needs to be transformed tounit Frechet random variables, Y1 and Y2, using aProbability Integral Transform.

I Then the model is as follows:

Pr(Y1 > y ,Y2 > y) ∼ c(y)y−1/η, for y ≥ u, (2)

where u is the threshold of interest, c is a slowly varyingfunction of y, and η ∈ (0, 1].

I The parameter η can be used as a measure ofasymptotic dependence:

I If η = 1, there is asymptotic dependence;I if 0 < η < 1, there is asymptotic independence.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Comparison Between Beijing and Shanghai

Initially, the asymptotic dependence of the PM10 levels inBeijing and Shanghai was tested.

I The η value was 0.619804, which relates to asymptoticindependence.

I Applying block bootstrapping gave a 95% confidenceinterval of (0.4573141, 0.6360939) for the η values.

I This confidence interval does not contain 1, suggestingthat the PM10 levels in Beijing and Shanghai areasymptotically independent.

It is possible that the distance between Beijing and Shanghaiis causing the asymptotic independence.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Comparison Between Beijing and Shanghai

Initially, the asymptotic dependence of the PM10 levels inBeijing and Shanghai was tested.

I The η value was 0.619804, which relates to asymptoticindependence.

I Applying block bootstrapping gave a 95% confidenceinterval of (0.4573141, 0.6360939) for the η values.

I This confidence interval does not contain 1, suggestingthat the PM10 levels in Beijing and Shanghai areasymptotically independent.

It is possible that the distance between Beijing and Shanghaiis causing the asymptotic independence.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Time Series for Shanghai and Suzhou

Time Series Plot of Shanghai API

0 1000 2000 3000 4000

Time Series Plot of Suzhou API

0 1000 2000 3000 4000

PM10 levels are known to vary between seasons, so we focuson just the summer data for Shanghai and Suzhou.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Asymptotic Dependence: Shanghai & Suzhou

50 100 150

Plot of Suzhou and Shanghai Summer APIs

Suzhou

The correlation between all the data is approximately 0.82.There is some positive linear correlation between the PM10

levels in Shanghai and Suzhou.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

50 100 150

Suzhou

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

50 100 150

Suzhou

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

50 100 150

Suzhou

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

The results for the bootstrapping of the η values were asfollows:

Bootstrapped Eta ValuesE

I The 95% confidence interval for the η values was(0.4147481, 0.6823547).

I This suggests there is asymptotic independence betweenthe air pollution levels in Shanghai and Suzhou.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Conclusion

I The correlation coefficient of 0.82 shows that overall,there is a positive linear relationship between the PM10

data from Shanghai and Suzhou.

I The bootstrapping test revealed that there is noasymptotic dependence between the two sets of data.

I We can conclude that there are underlying factors thataffect the pollution levels of cities in the same region,but that different factors contribute to the extreme airpollution levels in individual cities.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Conclusion

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Conclusion

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Coles, S. (2001)An Introduction to Statistical Modelling of ExtremeValues, Springer, 2001.

Ledford, A.W. and Tawn, J.A. (1996)Modelling Dependence within Joint Tail Regions,Journal of the Royal Statistical Society, 1996.

Hill, B.M. (1975)A Simple General Approach to Inference About the Tailof a Distribution The Annals of Statistics, 1975.

Emma Simpson

Introduction

Air Pollution

AQI & API

Air Pollution Data

Methodology 1

Testing Reliability

Bootstrapping

Results 1

Bootstrapping Test

Methodology 2

Testing Dependence

Results 2

Beijing & Shanghai

Shanghai & Suzhou

Conclusion

References

Any Questions?

a study of the air quality of some major cities in china · a study of the air quality of some...

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