Use of robust methods in the analysis of suspended particulate air pollution: A case study in...

Use of Robust Methods in the Analysis of Suspended Particulate Air Pollution:

A Case Study in Malaysia

Mokhtar Abdullahl and Sham Sani2

ABSTRACT

This paper describes several analyses in which a robust method, namely the least median of squares (LMS) has been used in the development of a mathematical model to relate suspended particulate air pollution concentration to meteorological factors. The robust procedure provides a formal method that not only pays due attention to anomalies or outliers in the data but also reduces the influence of the outliers. The study showed that the robust procedure yielded more reliable results than did the classical least squares (LS) approach.

1. INTRODUCTION

When pollutants are released into the atmosphere their subsequent fate and concentration will depend on three major factors, namely, the wind direction and speed in a horizontal plane, turbulence, and the stability of the air based on vertical temperature. As the last two factors are sufficiently related, for practical purposes only wind velocities and stability are consid- ered. Another meteorological aspect which is directly contributory to air quality as a sink is precipitation scavenging. Although its effectiveness as a cleansing agent varies greatly depending on the sizes and surface properties of the collector and of the collected particles, precipitation however does remove a certain amount of pollutants from the atmosphere either through washout or rainout processes (Shaw and Munn 1971).

'i2University Kebangsaan Malaysia, Bangi, Selangor, Malaysia.

11 80-4009/91/020201-15$0?.50 @John Wiley & Sons, Ltd.

Received 17 October 1990 Revised 30 Julv 1991

202 M. ABDULLAH AND S. SANI

The present paper analyses total suspended particulate (TSP) and meteorological data as recorded at the meteorological station, University Ke- bangsaan Malaysia (UKM), Bangi, Malaysia. The aims of this paper are two-fold. First, to examine if recognizable trend can be observed using time series analysis. Second, to examine the extent of relationships that exists between TSP and selected meteorological factors under wet tropical conditions. Results of preliminary analysis of the available data using simple statistical techniques were published earlier by one of the authors (Sham 1987) but no meaningful significance was found in the relationships. In the present analysis the data have been subjected to a more rigorous technique using the robust regression method called the Least Median of Squares (LMS).

2. DATA

Total suspended particulate (TSP) levels were measured using a high- volume sampler (HVS) and were expressed in pg/m3. All laboratory analyses were carried out in the Department of Geography, UKM, Bangi.

The HVS was located at the University meteorological station some two metres above ground level and sufficiently removed from the nearest road (about 50 m awa,y) (Figure 1). The site characteristics conform closely to specifications for a ‘neighbourhood station’ of the USEPA (1977). Generally, the station site is well exposed with only some low vegetation in the vicinity. To the northwest of the station however is Bandar Baru Bangi, a developing new town with some light industries bordering the Kuala Lumpur - Seremban Highway and the residential section of the town. The latest estimate shows that the new town contains some 30,000 people but is expected to grow at a faster rate in the future (Selangor State Government 1985). The UKM campus is situated to the south of the station while much of the rest of the area is still under vegetation.

Apart from TSP monitoring, the meteorological station at UKM also keeps records of temperature, rainfall, wind speed and direction, sunshine, global radiation, and evaporation. For most of the parameters, the data are available since 1979.

In the present study the period data coverage was between 20 May 1985 to 18 July 1986 involving 52 daily observations.

3. FACTORS AFFECTING THE OBSERVED TOTAL SUSPENDED PARTICULATE (TSP)

Our interest in studying the TSP data is to see what relationship, if any, could be found between the observed levels of TSP and meteorological factors. Earlier studies have shown that wind speed, temperature and

ANALYSIS OF SUSPENDED PARTICULATE 203

Figure 1. Location of the sampling station and the local geography showing Ban- dar Baru Bangi and UKM Campus.


some other meteorological variables such as wind direction and rainfall are important factors that may greatly influence the concentration of suspended particulates (see Owens and Tapper 1977; Bringfelt 1971; Turner 1961; Az- man et af. 1989; and, Sham 1987).

Sham (1987) examined the influence of some meteorological factors on suspended particulate concentrations using simple correlation analysis. The study revealed a positive relationship between total suspended particle concentrations and average daily wind speeds. It was found to be contrary to earlier findings by Owens and Tapper (1977) and Turner (1961) who obtained a negative relationship between pollution concentrations and wind speeds. It is not clear what factors have contributed to the conflicting finding by Sham (1987). However, the presence of outlying observations, the use of statistical methods that are sensitive to anomalous observations, and inadequate model specification might be some of the possible reasons behind such a finding.

In this paper the robust procedure, i.e, the least median of squares which is insensitive to the presence of outliers is used to formulate a reasonable relationship between the pollution concentrations and the meteorological factors.

One major difficulty in assessing the effects of meteorological factors on the pollution concentrations is that concentrations were measured on a daily basis whereas the measurements of wind speed and temperature were made on an hourly basis. It was observed that the wind speeds for example, were consistently high between late morning and early evening. It was not clear how best to condense such hourly observations into a single descriptive statistic which would accurately represent the wind condition for the day and have the most influence on pollution concentration. Because of the high afternoon wind speed, the normally simple daily average speed might not have that much influence on pollution concentration. In the present study, and after examining the data set it was decided that the hourly readings over 12-hours prior to the collection time at 8.00 am was most appropriate and appeared to have the most bearing on air pollution concentration for the day.

For the temperature, the variations in the hourly readings were rela- tively small. Therefore, the daily temperature can simply be represented by the average of the hourly readings over the 24-hour period.

Apart from the meteorological factors, the previous values of the pollution concentrations may also have an influence on the present level. Natu- rally, the largest impact comes from the most recent one. The data on the concentrations, daily wind speed, and temperature are presented in Table 1.


OBS TSP Wind Speed Temp. (OF) t c1 Wl (m.p.h.) TI

1 2 3 4 5 6 7 8 9

10

11 12 13 14 15

16 17 18 19 20

21 22 23 24 25

26 27 28 29 30 31 32 33 34 35

36 37 38 39 40

41 42 43 44 45

46 47 48 49 50

51 52

24.38 29.68 29.33 25.44 40.99 53.00 40.28 25.79 26.50 34.98

40.99

45.23 30.74 40.28

37.46 30.03 64.66 41.34 40.28

44.12 49.12 57.24 56.54 10.95 59.72 36.04 64.31 74.56 77.38 57.24 56.18 54.77 50.18 55.12

61.13 59.36 55.48 55.48 70.67

31.10 55.83 55.83 50.18 49.12

43.82 46.64 40.99 55.48 54.06

53.36 65.72

38.52

2.78 3.25 1.39 3.26 1.64

1.14 2.29 1.69 1.40 1.54

1.44 1.55 1.69 2.14 1.37

2.85 1.69 2.49 1.37 1.79

2.01 3.72 1.63 1.70 1.66 1.73 1.82 1.78 1.88 1.77 1.62 2.14 1.58 1.47 1.19

1.34 1.92 1.62 1 .so 1.43

1.55 1.51 2.16 1.45 1.83

1.42 1.72 1.47 1.59 1.53

1.82 1.84

78.7 75.8 79.4 79.4 77.4 78.3 78.2 75.9 77.2 78.7

79.1 79.3 79.4 75.6 79.4

79.4 79.6 79.3 78.6 78.8

80.7 79.3 81.8 79.3 79.5

78.7 79.8 79.6 79.1 79.4

80.6 81.1 80.7 77.7 80.5

80.1 81.3 80.3 79.9 78.9

79.9 79.4 79.7 79.9 78.0

78.3 79.3 76.9 78.9 79.0

79.0 78.5

Table 1. TSP levels, wind speed and temperature at UKM, Bangi, Malaysia.


3.1 THE MODEL

is An appropriate model that relates the TSP to the meteorological factors

C ~ = P O + P I C ~ - I t P z W t + P 3 T t + ~ t , t = l , . . . , T (3.1) were Ct is the total daily concentration at time period t , Ct-l the concentration at time period t - 1, Wt the average wind speed, T the average temperature, and ~t the error term at time period t , respectively.

Model (3.1) which includes lagged values of the response variable Ct-1 as regressors is also known as an ‘autoregressive linear model’. In this model ‘contemporaneous’ correlation exists between the regressor Ct-l and the error ~ t . It is also possible that the error terms are not pairwise independent, i.e., the errors exhibit serial correlation or they are said to be ‘autocorre- lated’. In this case we assume that the errors are generated by a first-order autoregressive process; that is

where p is the autocorrelation coefficient, and vt is a normally distributed random variable with zero mean and variance 62.

For the autoregressive linear model in (3.1), ordinary least squares estimators of the parameters Pj, j = 0,1,2,3 are consistent if the errors are uncorrelated. However, the least squares estimators are biased, inconsistent and asymptotically inefficient if the error terms are correlated (for further discussion of models of this type, see Goldberger 1964). Furthermore, if the model contains lagged variables, the Durbin-Watson test for testing autocorrelation is no longer appropriate. In this case Durbin’s h-test provides a means of testing the presence of autocorrelation (see Durbin 1970).

For the model in (3.1), consistent and asymptotically efficient estimators of pj and p can be obtained as follows;

Rewrite model (3.1) in the form

Estimate /?, and p simultaneously by minimizing

T

where


This is a nonlinear least squares problem which is computationally cumber- some. As an alternative, direct search procedures can be used to minimize S in which a value of p is selected and then linear least squares applied to estimate @,. This process would be repeated using the specified values of p until a minimum value of S is found. An example of this approach which is usually known as the ‘conditional’ least squares is given by Chatterjee and Price (1977).

However, the conditional least squares procedure may still be sensitive to outliers. In such a situation a robust procedure may provide a better alternative for estimating the parameters of the model.

4. ROBUST AUTOREGRESSIVE LINEAR MODEL

In the classical approach to regression problems the least squares method is known to be sensitive to the presence of bad data points or outliers. Accordingly, robust regression methods have been created to modify the least squares procedure so that the outliers have much less influence on the regression parameter estimates.

Generally speaking, a robust estimator is one whose performance re- mains quite good when the true distribution of the data deviates from the assumed distribution.

Tukey (1960), Huber (1964) and Hampel (1971) provide more precise notions of robustness, namely ‘efficiency’ robustness, ‘min-ma’ robustness and ‘qualitative’ robustness, respectively. Hampel (1974) also introduced the ‘influence function’ and the ‘breakdown point’, both useful concepts in robustness.

The influence function measures how an estimator reacts to a small fraction of outlicrs, i.e, the effect of infinitesimal perturbations on the estimator. The breakdown point is, roughly speaking, the maximum fraction of outliers an estimator can cope with, that is, without taking on arbitrary values. Hampel argues that a highly robust estimator should have:

(i) The highest possible breakdown point. The breakdown point E* = 50% is the best we can hope to accomplish, because for larger amounts of contamination no estimator can distinguish between the ‘good’ and the ‘bad’ parts of the sample, The least squares estimator has a breakdown point E* = 096, which reflects the extreme sensitivity of the estimator to outliers.

(ii) A bounded and continuous influence function. The non-robustness of the least squares estimator is due to the fact that its influence function is unbounded.

208 M. ABDULLAH A N D s. S A N I

Hampel (1975), Rousseeuw (1984) and Rousseeuw and Leroy (1987) pointed out that the so-called least median of squares (LMS) estimator has a maximal breakdown point, i.e. E+ = 50%.

For the autoregressive linear model in (3.2) the least median of squares (LMS) are defined as the values which minimize

med r: t

where rt, t = 1,2,. . . , T are given by (3.3). Because of its high breakdown properties, the LMS procedure is a very

valuable diagnostic tool in exploratory data analysis. Unfortunately, the LMS performs poorly when the errors are really normally distributed. To overcome this problem, one can apply a ‘weighted least squares’ defined by minimizing

t

where

and

This means simply that observation t will be retained in the weighted least squares if its absolute (standardized) residual is reasonably small or moderate, but omitted if it is an outlier. The criterion may be interpreted as ‘hard’ rejection of outliers in which only ‘good’ observations are retained in the data set.

5 . RESULTS AND DISCUSSION

Table 2 presents the summary statistics for the model in (3.1) from the least squares analysis of the data in Table 1. The results in Table 2 show that none of the variables included in model (3.1) contributes significantly (at 5% level) to the model. The presence of potential outliers in the data might contribute to the insignificance of all the variables in the model.

Since we are dealing with a linear autoregressive model, the least squares estimates obtained are biased, and consequently the results shown above are


Variable Coeff. Est. Stand. error t-stat. pvalue

C 0.268 0.139 1.931 0.059 W -1 -2.615 3.458 -0.756 0.453

T 2.886 1.480 1.950 0.057 Intercept - 189.195 115.706 -1.635 0.108

R2 = 0.213

Table 2. Summary statistics based on LS.

unreliable. Bccause the data are time series, we suspect that autocorrelation may be present.

We now use Durbin’s h-test for testing

Ho:p=O against H 1 : p f O . From the simple linear regression of Ct on Ct-l , we obtained the estimate of the autocorrelation coefficient and the h-statistic {which is distributed as

j? = -0.484 and h = -3.72

respectively. We reject HO and conclude that the errors are negatively correlated.

In order to obtain consistent and asymptotically efficient estimates of the model in (3.1), we use the conditional least squares on the transformed model in (3.2) as described in the earlier section. That is, define

“A 1))

and

i.e., the transformed model to be estimated is

Table 3 presents the summary statistics from the conditional least squares analysis based on the transformed model. From the results in Table 3 we

210 M. ABDULLAH AND s. SANI

Variable Coeff. Est. Stand. error t-stat. pvalue I ~ ~ ~ ~~~

C' 0.548 0.119 4.623 0.000 -3.355 3.106 - 1.080 0.285

T' 2.978 1.264 2.356 0.023 Intercept -308.754 143.428 -2.153 0.036

R2 = 0.559

W'-1

Table 3. Summary statistics based on Conditional LS.

see that only the variable W is not significant (at 5% level). For the other variables not only are they significant but also their efficiencies increase as exhibited from the lower standard errors. We also see that R2 increases al- most three times with the transformed model. The Durbin h-test also shows that the residuals based on the transformed model are no longer correlated. Therefore, no additional analysis using the search procedure is required.

However, the residual plot (see Figure 2) between the fitted values c; and the residuals reveals that at least one observation (observation 25, Czs = 10.95) is an extreme outlier. This is indicated by its standardized residual which exceeds the lower cutoff value of -2.5. The fact that the variable W' (wind speed) was found to be insignificant is rather surprising. It is suspected that this might have been due to the presence of some other potential outliers that failed to be highlighted by the conditional least squares method.

In order to have a clearer picture about the effects of the potential outliers, we fitted the transformed model in (3.2) using the least median of squares (LMS) procedure in (4.1)-(4.2). The estimated coefficients, their standard errors, t-statistics, and pvalues appear in Table 4. It can be observed that all the variables in the model are highly significant (at 5% level).

The most notable changes occur in the variable W' which was insignificant with the least squares procedure but now became highly significant with the robust procedure. This seems to agree with the findings made by Owens and Tapper (1977) and Turner (1961). The value of R2 increased by 36% with the robust method. Table 5 presents percent differences between the least squares (LS) and least median of squares (LMS) in the values of coefficient estimates, and their standard errors. The percent difference is given by

LS - LMS

LS % difference = x 100% .

The results show that the LMS has improved the efficiencies of the estimates as depicted from the considerable reductions in their standard


1 . 5 1

Variable Coeff. Est. Stand. error t-stat. pvalue I

C' 0.577 0.092 6.289 0.000

T' 2.836 0.883 3.211 0.002 Intercept -289.454 99.665 -2.904 0.006

R2 = 0.75

w+-* -5.128 2.131 -2.406 0.020

Table 4. Summary statistics based on robust LMS.

errors. The robust residual plot shown in Figure 3 illustrates that the LMS not only identifies observation 25 as an extreme outlier but also four other similar observations, i.e., observations 18,29,30 and 41. Their standardized residuals exceeded the cutoff values of either -2.5 or 2.5. The elimination of these extreme data points through assigning zero weights by the LMS as shown in Table 6, seems to have contributed to the high significances of all


I Variable Coeff. Est. Stand. Error I

C' 5.4 53.6

T' 4.0 Intercept 6.0

w*-1 25.0 31.5 30.1 30.5

Table 5. % differences between LS and LMS.

EST. W E N T R .

Figure 3. LMS residual plot.

the variables in the model, substantial reductions in their standard errors and also a marked increase in the value of R2.

6. CONCLUDING REMARKS

In the analysis of the suspended particulate air pollution data, we have formulated a mathematical model which includes some of the important


LS LMS LMS Obll Standardized Standardid weight

Rerid. Raid

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

- 1.40 .32

-.51 -1.10

.51 1.16 -.12 - .95 - .74 - -25 -.lo - .60 -.l6 -.28 -.19 -.31 -1.29 1.80 -.15 -.87 -.41 .62 .39 .23

-3.47 1.01 - .34 .97 2.36 1.88 -.41 - .60 -.31

.02

.06

.40

.05 -.35 -.12 1 .50

-1.86 .03 .88 - .33 .M

-.15 -.12 -.04 .92 .58 .28 1 .so

-1.78 .w - .60

-1.24 .96

1.60 -.17 -1.42 -1.16 - .37 -.17 - .93 - .23 - .37 - .30 - 20 -1.75 2.89 -27 - 1.35 -.SO 1.41 .80 .30

-5.19 1.59 - .47 1.47 3.47 2.68 -.77 - .86 -.47 -.11 - .08

.43

.04 -.55 - .28 2 .08

-2.92 -.01 1.36 -.54

.04

- .33 - 2 2 -.16 1.29 .77 .35 2.20

1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 0 1 .o 1 .o 1.0 1 .o 1 .o 1.0 0 1 .o 1 .o 1.0 0 0 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o 0 1 .o 1 .o 1 .o 1 .o 1.0 1 .o 1 .o 1 .o 1 .o 1 .o 1 .o

Table 6. Residuals for the LS and LMS fits.

214 M. ABDULLAH AND s. SANI

variables that contribute significantly to the variations in the pollution concentrations. The application of the robust least median of squares method has led to the automatic identification of observations which are extreme outliers. Ignoring these extreme outliers temporarily, the robust least median of squares method has also produced a fit that can best describe the data set that is much superior to that of simple correlation analysis. In many ways, a robust method such as the LMS can thus be viewed as an effective procedure for isolating unusually influential data points so that they may be further examined.

Of course, in pollution studies such as this, the relationship between dependent and independent variables may not be at all linear. In the previous study using a similar data set but a simpler correlation analysis, Sham (1987) did indicate that the relationship between particulate pollution and wind speeds at the sampling site could be far more complex than that sug- gested by the regression line. It might be possible, for example, that above a certain critical value, the relationship could be just the reverse. Such a complex relationship between smoke concentrations and wind speed ( w ) was reported by Tyson (1963) in Durban, South Africa. In this case smoke concentration decreased rapidly for 0 < zu < 5 , decreased slowly for 5 < w < 11 increased for 11 < w < 15 and decreased for w > 15 (wind speeds in m.p.h.). In the case of the present study, pollution brought about by 'strong' winds might be a significant feature.

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