Statistical tools in the analysis of pollutant concentrations measured with optoelectronic instruments

Transcription

1 Proceedings of the th WSEAS International Conference on Sustainability in Science Engineering Statistical tools in the analysis of pollutant concentrations measured with optoelectronic instruments IOAA IOEL a, SABI IOEL b, IOA LIE b a Department of Thermal Machines and Transport, b Department of Applied Electronics "Politehnica" University of Timişoara Bv. V. Pârvan, nr., 33 Timişoara ROMAIA ioana.ionel@mec.upt.ro, sabin.ionel@etc.upt.ro, ioan.lie@etc.upt.ro Abstract: - This paper is related to a measuring campaign of pollutant concentrations, using several optoelectronic instruments. Due to their random character, pollutant concentrations signals can be analysed using statistical processing methods. In this frame, statistical tools like correlation coefficient, covariance and correlation functions play a central role. Possible interpretations and facilities offered by several statistical concepts are discussed. Computer experiments with real pollutant concentration signals lead to some practical recommendations concerning the acquisition parameters for future measuring campaigns. Keywords: - Optoelectronic instruments, Pollutant concentration, Correlation coefficient, Correlation functions. Introduction In the frame of international and national partnership research projects, several air quality measuring campaigns were realized {*** ROSE (-4), []; *** TOP (6-8), []} by means of classic and optoelectronic devices. Some of the conclusions of these projects are already reported {Bisorca D., Ionel Ioana, Hanna L., Cooke K., (3), [3]; Bisorca D., Ionel Ioana, Popescu Fr., Ionel S., Ungureanu C., (3), [4]; Ionel Ioana, Ionel S., Bisorca D., (4), [5]; Ionel Ioana, Ionel S., (4), [6]; Ionel Ioana, Ionel S., Doina icolae, (7), [7],[8]}. This paper presents some statistical tools useful in a comparative analysis of pollutant concentrations signals. The analyzed signals, representing O, O 3, SO and HC concentrations, were measured with several optoelectronic instruments. The main statistical tools taken into account are histograms, correlation coefficients, correlation and covariance functions. Section refers to the pollutant signals measured in a recent experimental campaign. Possible utilization of histograms and associated parameters is presented in section 3. Part 4 of the paper is dedicated to Pearson correlation coefficient, revealed as a useful analysis tool in a comparative study of pollutant concentrations measured with different instruments. Section 5 presents possible utilizations of correlation and covariance functions. General conclusions of this study are presented in section 6.. The pollutant concentrations measured with optoelectronic instruments The data are related to an experimental campaign from 7, accomplished by several partners in the research project TOP (6-8) []. As result of the different positioning of the optoelectronic instruments according to the necessities of being electrically connected and having a free optical open path, the measured species and the owner, the experimental results are of huge amount. A typical plot of pollutant concentration signals is presented in Fig.. After a visual analysis of the raff database, one may come to the conclusion that not all are correct registered or that noise or dysfunctions interfere essentially. Different signal conditioning methods like high-pass and low-pass filtering, interpolation etc. can be used in order to eliminate the artefacts [7] /5/ 7 :4 /5/ 7 4:48 /5/ 7 7: /5/ 7 9:36 HC.5. 7 /5/ 7 : /5/ 7 4:4 /5/ 7 6:48 /5/ 7 9: /5/ 7 :36 FTIR FF-UB POIT UPT POIT ICIM POIT3 IOE /5/ 7 : Fig.. Sample results of the measuring campaign episode May 7, accomplished in partnership by IOE Bucuresti, Universitatea Politehnica Timisoara, ICIM Bucuresti and Universitatea Bucuresti. The measured species in three different points were O (using FTIR of Universitatea Bucuresti and point ISS: ISB:

2 Proceedings of the th WSEAS International Conference on Sustainability in Science Engineering measurement of ICIM working in chemiluminescence s according SR E 4:5), HC (DOAS of IOE, point measurement accomplished by Universitatea Politehnica Timisoara, using SR E 466-3: 5 applying FDI and FTIR of Universitatea Bucuresti), O 3 (in UV according SR E 465:5 of Universitatea Politehnica Timisoara and DOAS of IOE ), SO (according SR E 4:5 applying fluorescence principle by Universitatea Politehnica Timisoara and DOAS of IOE ). 3. Histograms and related parameters The histogram is a representation of the signal amplitudes according to their values, regardless of the time variable: on the pollutant concentration axis a number of equally spaced containers are defined and the histogram returns the number of signal samples in each container [9], [], []. This function can be easily implemented using the MATLAB software [], [3]. Point-ICIM DOAS-IOE Point-ICIM DOAS-IOE The Signals O -5 in [ppb] Mean Value=3.885; Deviation= Mean Value=4.465; Deviation= Sample umber Fig.. Two O signals measured simultaneously Histograms for O-5 Mean Value=3.885; Deviation= Mean Value=4.465; Deviation= O concentrations in [ppb] Fig.3. Histograms of two O signals The histogram can be a useful tool for a preliminary characterisation of the pollutant concentration signals, complementary to there usual temporal representation. For example, Fig. presents two O concentration signals, each containing 83 measured samples. The corresponding histograms of the same signals can be seen in Fig. 3. The realisations represented in Fig. are containing the full information about the measured concentrations. Particularly, one can observe the similarity of there evolution. The upper signal has two artefacts which must be eliminated during the pre-processing step. However, the soft limitation of the upper signal in Fig. appears with greater clarity in the corresponding histogram from Fig.3. At the same time, the quasi normal (Gaussian) character of the lower signal is more evident in the corresponding histogram then in the temporal representation. The good continuity of both histograms suggests a sufficient size of the measured signals. The normalized histogram (with area equal to one) is an approximate representation for the probability density function of the analysed signal. Some statistical parameters related to the histogram (through the probability density function) can be utilized for a quantitative or qualitative characterisation of the pollutant concentration signals. Thus, the mean value [9], [], [], x = x i () i= and the standard deviation [9], [], [], σ = x i x () i= ( ) usually reported together, as in Fig. and Fig.3, offer a quantitative description for the location of the pollutant concentration values. Two other statistical parameters can be utilized for a rather qualitative characterisation of the data set in comparison with a normal distribution. amely, the skewness (the third central moment divided by the cube of the standard deviation) 3 = ( x ) i i x s = (3) 3 σ describes the lack of symmetry of the histogram. One can mention that for any symmetrical distribution (e.g. the normal distribution), the skewness is near zero. The kurtosis (the fourth central moment divided by fourth power of the standard deviation) 4 = ( x ) i i x k = (4) 4 σ offers a characterization of whether the histogram is peaked or flat relative to a Gaussian distribution. The kurtosis of a normal distribution equals three. Pollutant concentration signals with high kurtosis have a sharp peak near the mean value while signals with low kurtosis have a rather flat top near the mean value. Obviously, in (), (), (3) and (4), x i is the current sample from the samples of the pollutant concentration signal. 4. The Pearson correlation coefficient Taking into account the usual lack of stationarity of pollutant concentration signals, the correlation coefficient ISS: ISB:

3 Proceedings of the th WSEAS International Conference on Sustainability in Science Engineering is, probably, the most useful statistical tool revealing possible influences between such data sets. Particularly, the Pearson correlation coefficient [9], is defined by the following relation: [ x ] i i yi x i i y = = i= i r = (5) [ x ] [ ] σ xσ y i =,,..., y i where i and, are two simultaneously measured pollutant concentration signals, with standard deviations σ and σ, respectively. One x can prove that r. The variability of the two data sets can be interpreted according to the range including the particular value of the correlation coefficient. Thus, r.33 denotes negative correlation, while.33 r signifies that the two signals are positive correlated. If.33 r.33, the two pollutant concentration signals are said to be uncorrelated (independent). In this interpretation of the estimated value of r, one must take care of the confidence interval of this parameter [7], [4]. Practically, not only the value of the correlation coefficient but also his confidence interval must be in one of the three ranges in order to conclude that the analyzed signals are positive correlated, negative correlated or independent. Point UPT - Point3 IOE Point UPT - DOAS IOE Correlation Coefficients for Moving Segments of Data O 3-5; Mean CC= Segment umber SO -5; Mean CC= Segment umber Fig.4. Correlation coefficients for moving segments of data, each containing 3 samples. Two acquisition parameters of the pollutant concentration signals are important for a reliable determination of the correlation coefficient: the sampling frequency and the number of samples in the measured data sets. In order to establish some practical rules for the acquisition phase of future pollutant concentration measuring campaigns, several signal processing experiments were carried out. Thus, Fig.4 presents the evolution of the correlation coefficient calculated for a moving segment of data, containing only 3 samples measured with a sampling period of five minutes. The huge variation of the correlation coefficients shows that the size of the data is totally insufficient, at least for this sampling frequency. y Fig.5 shows the dependence of the correlation coefficient on size of the measured pollutant concentration signals. The correlation coefficient between PointUPT and Point3 IOE O 3 signals has a transient behaviour. The final value of the correlation coefficient proves that the two signals are strongly positive correlated. Just 6 measured samples of the signals could be sufficient for a good determination of the correlation coefficient for these signals, achieved at a sampling period of five minutes. Fig.5 shows also the oscillatory behaviour of the correlation coefficient between Point UPT and DOAS IOE SO signals measured with a sampling period of five minutes. The low values of the correlation coefficient, proves that the two pollutant concentration signals are practically independent. Taking the confidence interval of the analysed data sets into account, the relative high values of r for segment length between and 4 are not high enough to draw the conclusion that the two signals are positive correlated. The final decay of the correlation coefficient for segment length greater than 4 definitively proves the independence of the two SO concentration signals. Point UPT - Point3 IOE Point UPT - DOAS IOE Correlation Coefficients for Increasing Length Segments of Data O 3-5; Mean CC=.5689; Final CC= Segment Length SO -5; Mean CC=.538; Final CC= Segment Length Fig.5. The dependence of the correlation coefficient on the length of data segments. A practical conclusion can be drawn from this experiment: the calculation of a correlation coefficient with transient behaviour is practically completed as soon as the value of r reaches the quasi-steady state; if the evolution of the correlation coefficient is characterized by small amplitude variations, an optimal segment length is difficult to establish; however, since the duration of a pollutant concentrations measuring campaign is measured in days while the calculation of the correlation coefficient takes a few seconds, the interactive signal processing, during the measuring campaign is highly recommended. The dependence of the correlation coefficient on the sampling period is illustrated in Fig.6. The correlation coefficients are presented for two signal pairs achieved at a sampling period of 5,, 5,, 5 and 3 minutes, respectively. Corresponding to the increasing sampling period, the number of samples utilized in the calculation of the correlation coefficient was 7, 86, 57, 4, 34 and 8, respectively, so that the temporal signal length is maintained approximately constant (about 85 minutes). Comparing to the whole range [- to +] of possible values ISS: ISB:

4 Proceedings of the th WSEAS International Conference on Sustainability in Science Engineering for the correlation coefficient, the variations of the results shown in Fig.6 are small. This proves a high positive correlation between Point UPT and Point3 IOE O 3 signals (correlation coefficient about.73) and statistical independence between Point UPT and DOAS IOE SO signals (correlation coefficient about.). Point UPT - Point3 IOE Point UPT - DOAS IOE Correlation Coefficients for Different Sampling Periods Signals: O 3 ; Mean CC= Sampling Period in [Minutes] Signals: SO ; Mean CC= Sampling Period in [Minutes] Fig.6. Correlation coefficients calculated for different sampling periods. The practical conclusion of this experiment is that a reduced number of samples in the data sets can be compensated by a corresponding higher sampling period. This conclusion differs from the usual statement found in books on statistics, where the confidence interval of the correlation coefficient is determined by the number of sample in the analyzed signals. Our interpretation of this fact is that pollutant concentration signals are not artificial random signals generated by a certain mathematical rule, but physical signals where the generating mechanism can change in time. Consequently, one must give the signal enough time to show his features, and this can be done assuring a minimal temporal signal length, i.e. a minimal value of the product (Segment Size) (Sampling Period). Certainly, the condition of a minimal number of measured samples (about 3 samples) must be also fulfilled. Table shows the global correlation coefficients for different pairs of signals representing the measured concentrations of pollutants. In parenthesis one mentioned the number of samples (rows) of each signal and the number of signals (columns). For example, O_5 (83,) means a matrix with 83 rows and columns, containing the samples of the pollutant O, measured with 5 minutes sampling period. The values of the corresponding correlation coefficients must be interpreted taking the whole arrangement of the experimental area into account. Optoelectronic pollutant concentrations measuring instruments, working on different principles can be compared using the correlation coefficient. Specific preprocessing procedures like ideal filtering for the rejection of measurement noise and artifacts or interpolation in order to increase the number of samples of the pollution level signals can be used [7]. The correlation coefficient proves to be an useful tool in analyzing the dependencies between pollutant concentration signals, also relevant for the influence of the meteorological factors on the pollution level [7]. One ca use such analysis to decide which optoelectronic measuring instrument is best suited for a certain application. Table : Global correlation coefficients for pairs of measured pollutant signals. Concentration - Signals Correlation Coefficients O_5 (83,) Point ICIM and DOAS IOE.749 O_ (9,3) FTIR FF-UB and Point ICIM FTIR FF-UB and DOAS IOE Point ICIM and DOAS IOE HC_3 (3,4) FTIR FF-UB and Point UPT FTIR FF-UB and Point ICIM FTIR FF-UB and DOAS IOE Point UPT and Point ICIM Point UPT and DOAS IOE Point ICIM and DOAS IOE O3_5 (7,3) Point UPT and Point3 IOE Point UPT and DOAS IOE Point3 IOE and DOAS IOE SO_5 (7,) Point UPT and DOAS IOE Correlation and covariance functions The correlation and covariance functions can put into evidence power and temporal relations between pollutant concentration signals. The sample crosscorrelation function is defined by the formula [9] l xi+ l yi ; for l < l i= Rxy[ l] = (6) l < < xi y ; for l i+ l l i= Autocorrelation of the signal O 3-5 Point UPT Power= Autocorrelation of the signal O 3-5 Point3 IOE Power= Delay Fig.7. Typical autocorrelation functions ISS: ISB:

5 Proceedings of the th WSEAS International Conference on Sustainability in Science Engineering can be calculated if the measured pollutant concentration signals are stationary, but this is rarely the case. However, for [ yi ] [ xi ] we obtain the autocorrelation R xx [l] of the sequence [xi ]. Fig.7 shows the autocorrelation functions for PointUPT and Point3 IOE O 3 signals. The value of the autocorrelation al lag zero, R xx [], gives the power of the corresponding signal: approximately 4 ppb for PointUPT and 78 ppb for Point3 IOE O 3 signals. After some experience, one can appreciate the spectral content of the signal from the shape of the autocorrelation function. The sample autocovariance, C xx [l], is the sample autocorrelation of the centred sequence [ x i ] (the sequence after removing the mean) while the sample crosscovariance, C xy [l], is the crosscorrelation of the centred sequences [ x i ] and [ y i ]. Thus, Fig.8 presents the autocovariance functions for the same signals, namely PointUPT and Point3 IOE O 3 pollutant concentrations. The autocovariance at zero lag, gives the power of the variable part of the analysed signal: approximately 74 ppb for PointUPT and 3 for Point3 IOE O 3 signals. The shapes of the autocovariance functions show that PointUPT O 3 signal has a highly random character while in the Point3 IOE O 3 signal, the spectral components are concentrated in a small range near.6mhz. 8 6 Autocovariance of the signal O 3-5 Point UPT Power= were achieved with sampling periods of 5 to 3 minutes. The consequence is that the peak of the autocorrelation function appears in the origin (zero lag) or they have a flat maximum around zero, as shown in Fig Value at Zero Delay= Crosscovariance of the signals O 3-5 Point UPT and Point3 IOE Cosscovariance of the signals O 3-5 Point UPT and DOAS IOE - Value at Zero Delay= Delay Fig.9. Experimental crosscovariance functions Summarizing, the autocorrelation and autocovariance functions are useful tools in establishing power relations between pollutant concentration signals measured with optoelectronic instruments. For reliable results, a sufficient temporal length of measured data sets must be assured, so that the signals can manifest there features. Temporal relations i.e. delays between certain signals measured simultaneously can be revealed using the crosscovariance function. But, for this purpose a supplementary condition must be fulfilled: a small sampling period must assure a good resolution on the time axis Conclusions Autocovariance of the signal O 3-5 Point3 IOE Power= Delay Fig.8. Typical autocovariance functions The crosscovariance functions offer an interesting possibility to determine delays between pollutant concentrations measured simultaneously. The position of the peak in the covariance function around zero gives the temporal delay between two signals. Depending on the wind direction and intensity the pollutants can be transported from one place to the other in the experimental area. However such temporal relations can be put into evidence only if the resolution on the time axis is sharp enough to allow the proper localization of the crosscovariance peak. This condition was not fulfilled during the related measuring campaign: the sampling period should be in the range of seconds while the signals Several pollutant concentrations were measured using different optoelectronic instruments. Due to random character of the pollutant concentration signals, statistical processing methods are recommended. Histograms, correlation coefficients, correlation and covariance functions as well as statistical parameters like mean, standard deviation, skewness and kurtosis can be useful tools in analyzing such signals. However, according to the purpose of the measuring campaign, the experiment must be carefully designed in order to obtain reliable results. The most important practical rules are as follow: assure the temporal length of the measured signal, assure the necessary resolution on the time axis, and make interactive verifications of the acquisition parameters during the measuring campaign. These conclusions will be applied during future measuring campaigns in the idea of optimizing parameters like sampling period and data size according to the purpose of each experimental research. Acknowledgement One addresses thanks for the financial support of the Romanian Ministry of Research and Education. ISS: ISB:

6 Proceedings of the th WSEAS International Conference on Sustainability in Science Engineering References [] *** ROSE EU Project G6RD/CT/ / 434, (-4). [] *** TOP-Optical Teledetection of pollutants, Excellence Parnership Contract 964 /.7. 6, 6-8, (7-9). [3] D. Bisorca, Ioana Ionel, L. Hanna, K. Cooke, Dispersion modelling in a canyon street with application to the Romanian city of Timişoara, In Transport et pollution de l'air, Avignon, Vol. II, pp. 7-3 (3). [4] D. Bisorca, Ioana Ionel, Fr. Popescu, S. Ionel, C. Ungureanu, Air quality investigation by means of remote sensing, with application to CO thermodynamic measurements in the city of Timisoara, 3-th international conference on thermal engineering and thermo-grammetry, Budapest, pp , http: // (3). [5] Ioana Ionel, S. Ionel, D. Bisorca, A correlative comparison of measured traffic induced CO concentrations, In Transport and Air Pollution, Boulder, Colorado, USA, pp (4). [6] Ioana Ionel, S. Ionel, Beurteilung von Luftqualität mittels optischen Fernmessystemen in Vergleich zu der D-IR Methode, In OPTAM, VDI, Düsseldorf, Germany, pp.8-85 (4). [7] Ioana Ionel, S. Ionel, Doina icolae, Correlative comparison of two optoelectronic carbon monoxide measuring instruments, JOAM, nr - Journal for Optoelectronics and Advanced Materials, Vol 9, pp (7). [8] Ioana Ionel, S. Ionel, Doina icolae, Correlative Comparison of two optoelectronic carbon monoxide measuring instruments, OTEM 7, st International Workshop on Optoelectronic Techniques for Environmental Monitoring and Risk Assessment, May 7, Bucharest, Romania, pp (7). [9] A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, Inc., 3 rd edition ew York (99). [] C.W. Therrien, Discrete Random Signals and Statistical Signal Processing, Prentice-Hall International, Inc., Englewood Cliffs (99) [] R.G. Brown, P.Y.C. Hwang, Introduction to Random Signals and Applied Kalman Filtering, John Wiley & Sons, Inc., nd edition, ew York (99). [] *** MATLAB User s Guide, MathWorks () [3] J. Hoffmann, MATLAB und SIMULIK in der Signalverarbeitung und Kommunikationstechnik, Addison-Wesley, München (999). [4] D. Shen, Z. Lu, Computation of Correlation Coefficient and Its Confidence Interval in SAS, (6) ISS: ISB: