Missing Data Toolbox for Air Quality Datasets

Transcription

1 EnviroInfo 2002 (Wien) Environmental Communication in the Information Society - Proceedings of the 16th Conference Missing Data Toolbox for Air Quality Datasets Mikko Kolehmainen 1, Heikki Junninen 1,4, Harri Niska 1, Toni Patama 1, Anna Ruuskanen 2, Kari Tuppurainen 3 and Juhani Ruuskanen 1 Abstract The objective of the study was to find a useful missing data imputing method for air quality forecasting applications. The univariate methods studied were the linear interpolation, spline and nearest neighbour (univariate) interpolation. Multivariate methods studied were multivariate nearest neighbour (NN), Self-Organising Map (SOM) and Multi-Layer Perceptron (MLP). Additionally, a new approach was developed where univariate methods were combined with multivariate methods in order to utilise the best properties of both approaches. The results in general showed that the best overall performance can be achieved by combining univariate and multivariate methods and that the way of combining is dependent on the variable inspected. Based on these results a Missing Data Toolbox (MDT) with a Graphical User Interface (GUI) in Matlab environment was created. The MDT encapsulates the different algorithms and enables the treatment of missing data in a coherent way. The MDT and GUI were tested on Windows and Linux environments. 1. General Environmental databases and their effective use are often needed to be uninterrupted by missing data values. In many cases, however, the time-series recorded have discontinuities due to insufficient sampling, errors in measurement or faulty data acquisition and storing. The methods used for processing the data usually require the time-series to be uninterrupted. A commonly suggested way of handling this is to fill in the missing values by setting them to the mean value or some other statistical parameter. This leads, however, to corruption of the time-series properties of the data and is not usually as accurate as other methods. If the continuity of the time-series is not strictly required by the algorithm, which is often the case with multivariate algorithms, a common Departments of Environmental Sciences 1, Applied Physics 2 and Chemistry 3, University of Kuopio, P.O.Box 1627, FIN Kuopio, Finland Institute for Environment and Sustainability 4, EC Joint Research Centre, I-21020, Ispra (VA), Italy

2 446 procedure is to reject data lines having missing values and to continue with less data. In that case one must bear in mind that information in the rejected lines is lost. Therefore, it would be advantageous to be able to fill in the missing values using the best of the information available. 2. Imputing methods 2.1. Univariate methods The univariate methods studied were the linear (LI), spline (SPL) and univariate nearest neighbour (UNN) interpolations. Linear interpolation fits a straight line between the endpoints of a gap formed by missing values, which can be calculated straightforwardly employing the line equation. Spline interpolation is based on the polynomials of different degrees. Univariate nearest neighbour is likely the simplest imputation scheme: the endpoints of a gap are used as estimates for all missing values Multivariate methods Multivariate methods studied were multivariate nearest neighbour (NN), Self- Organising Map (SOM) and Multi-Layer Perceptron (MLP). The NN imputation handles a row of N variables as a co-ordinate in an N-dimensional space and takes the missing values from the nearest neighbour (row) in that space where they are available, weighting at the same time the distances proportionally to the number of missing values in each row. The goal of the SOM is to find vectors, which can represent the input data set with prototypes and at the same time realize a continuous mapping from input space to a lattice (Kohonen, 1997). The missing values can then be recovered using the lattice. Similarly with the NN also the SOM is using a whole data set so that the information in incomplete rows is utilised. During the training process of the SOM missing elements are ignored and only values available in a vector (data row) are used for updating the weights of the map. The idea of the MLP network is to learn a black box like mapping from input variables to one or several output variables. The usage of MLP in this study is actually a combination of several MLP networks so that for each missing data pattern a network of its own was trained. For more thorough description of the application of the SOM and MLP in air quality forecasting see Gardner and Dorling (1998) and Kolehmainen et al. (2001) Hybrid methods In our study, new approach was developed where univariate methods are combined

3 447 with multivariate methods in order to utilise the best properties of both approaches. In this approach short gaps are filled with univariate interpolation which gives the best performance in the given situation and longer gaps are filled with an advanced multivariate method which has no direct dependence on the gap length in time space. In this context, the shortness depends on the variable under study Measuring the goodness of the imputation In order to measure the degree in which the imputation is error free, the index-ofagreement (Willmott, 1982) was employed in this work: d 1 n i 1 i n i 1 ( P O P O ave i ave O i 2 O Where n = number of imputations, P i = predicted value, O i = observed value, and O ave = average of observed values. ave ) 2 (1) 2.5. Execution of the tests The datasets used for tests were extracted from the APPETISE (Air pollution Episodes: Modelling Tools for Improved Smog Management) database). The locations and years used were Cambridge 1996 and Belfast 1996 due to that they contain only little missing values (6.3 and 2.2 percent, respectively). The datasets from Cambridge consisted of NO x, NO 2, O 3 and CO concentrations, which had a time-scale of one per hour, together with seven meteorological parameters: wind speed, wind direction, temperature, relative humidity, precipitation, solar radiation and net radiation. Correspondently, the datasets from Belfast comprised of NO x, NO 2, O 3, PM 10, SO 2 and CO concentrations together with nine meteorological parameters: wind direction, wind speed, cloud base height, visibility, mean sea level pressure, cloud base level, temperature, dew point and wet bulb. These datasets were first imputed with the Nearest Neighbour interpolation (Dixon, 1979), which is known to be a safe method because it does not introduce new values into the data. Subsequently, missing data patterns (continuous sequences of the multivariate dataset having values missing at least in one of the variables) were extracted from the Cambridge 1995, Cambridge 1997, Belfast 1994, Belfast 1995 and Belfast 1998 datasets. Each year was mixed in random in order to construct seven different sets. These sets were then applied to the test sets of the same location in order to produce final test sets with known gaps. All the test datasets (14 for Cambridge and 21 for Belfast) were imputed separately with the methods described. The results of the tests were then combined into

4 448 tables of statistics giving the index-of-agreement value for them. 3. Results of the comparison 3.1. Univariate methods The results of comparing univariate methods are given in Table 1. One can conclude that the LI is slightly better than the UNN and that both of these are considerably better than the SPL method. Thus, the univariate method selected for further evaluation combined with the multivariate method was the LI. The results also showed that the performance of the univariate method is dependent on the length of gap in time space and that the performance depends also on a variable under study. Therefore, the length of gaps that can be replaced with the LI should be estimated separately for each variable before the replacement. Table 1. Comparison of univariate methods with the index-of-agreement. The values were calculated using all of the variables described in the chapter 2.5. LI = Linear Interpolation, UNN = Univariate Nearest Neighbour and SPL = SPLine. Method Mean Min Max Std LI UNN SPL Multivariate methods The results of the multivariate methods showed that both the SOM and MLP methods perform slightly better than multivariate NN method. The advantage of SOM compared with other methods is that it is less dependent on the actual location of the missing values. However, if computational speed is placed at first, multivariate NN is then recommended. It has also the advantage of not generating completely new values to the data. As with univariate methods, the differences in performance between variables were large. Table 2. Comparison of multivariate methods with the index-of-agreement. The values were calculated using all of the variables described in the chapter 2.5. NN = Nearest Neighbour, SOM = Self-Organizing Map and MLP = Multi-Layer Perceptron. Method Mean Min Max Std NN SOM MLP

5 Hybrid methods The results of comparing the hybrid methods are shown in Table 3. It can be seen that there is no big difference between the methods considering the mean value of the index-of-agreement and also the minimum and maximum values obtained. Therefore, the same conclusions given for the multivariate methods (3.2) also hold here. Table 3. Comparison of hybrid methods with the index-of-agreement. The values were calculated using all of the variables described in the chapter 2.5. HNN = Hybrid NN, HSOM = Hybrid SOM and HMLP = Hybrid MLP. Method Mean Min Max Std HNN HSOM HMLP Summary of the results The methods evaluated are summarised in Fig 1. The results in general showed that the best overall performance can be achieved by combining univariate and multivariate methods and that the way of combining is dependent on the variable inspected. The results in more detail have been reported in Junninen et al. (2002) Index-of-agreement LI UNN SPL NN SOM MLP HNN HSOM HMLP Fig. 1: Summary comparison of the methods evaluated in the study. The whiskers give the min and max values for the method described by the corresponding bar. LI = Linear Interpolation, UNN = Univariate Nearest Neighbour, SPL = SPLine, NN = Nearest Neighbour, SOM = Self-Organizing Map, MLP = Multi-Layer Perceptron, HNN = Hybrid NN, HSOM = Hybrid SOM and HMLP = Hybrid MLP.

6 Description of the MDT Based on these results a toolbox in Matlab environment was generated, which encapsulates the different algorithms and enables the treatment of missing data in a coherent way. The toolbox was further enhanced with a graphical user interface (GUI). The GUI has algorithms for loading and saving datasets in different file formats (Fig 2), calculating different statistics for missing data and illustrating missing data conditions with graphs (Fig 3 and Fig 5). Furthermore, end-users can choose a method for the replacement from various linear and non-linear methods such as interpolation, nearest neighbour, SOM and MLP (Fig 4). Combinations of univariate and multivariate methods are available as hybrid methods, too. 5. Dissemination of the MDT The MDT and GUI were tested on Windows and Linux environments. It was found to be usable in both environments but there are some restrictions on the Matlab version to be used with the package. The GUI was decided to be the best approach in order to bring the MDT for the end-users instead of an Internet server service. The MDT for Matlab can be downloaded from the web-address The usage is limited to academic purposes only. Currently, work is underway in order to construct a standalone MS Windows application with a GUI. However, it is likely to include some restrictions regarding the methods used (most notably the MLP). This is due to current versions of Matlab c- code generators not being able to translate Matlab code having object definitions, which is the case with the Neural Network Toolbox. The dissemination of the standalone software has not been decided yet but it is likely to be commercial at least for non-academic purposes. Bibliography Dixon, J.K. (1979): Pattern Recognition with Partly Missing Data, IEEE Transactions on Systems, Man, and Cybernetics, 10 (SMC-9), Gardner, M.W., Dorling, S.R. (1998): Artificial neural networks (the multiplayer perceptron) a review of applications in the atmospheric sciences, Atmospheric Environment, 32, Junninen, H (et al.) (2002): The Performance of Different Imputation Methods for Air Quality Data with Missing Values, Submitted to Atmospheric Environment Kohonen, T. (1997): Self-Organizing Maps, Springer, Berlin Kolehmainen, M. (et al.) (2001): Neural networks and periodic components used in air quality forecasting Atmospheric Environment, 35, Willmott, C.J. (et al.) (1985): Statistics for the evaluation and comparison of models, J. Geophys. Res., 90 (C5),

7 451 Fig 2: Main interface of the MDT Fig 3: Plot interface of the MDT Fig 4. Input method selection of the MDT Fig 5. Validation interface of the MDT