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Expectation-Maximization Model for Substitution of Missing Values Characterizing Greenness of Organic Solvents

Gabriela Łuczyńska
Francisco Pena-Pereira
Marek Tobiszewski
4 and
Jacek Namieśnik
Division of Applied Mathematics and Probability, Institute of Mathematics, Faculty of Mathematics, University of Gdansk, 8 J. Bażyńskiego St., 80-309 Gdańsk, Poland
Department of Nonlinear Analysis and Statistics, Faculty of Applied Mathematics, Gdańsk University of Technology (GUT), 11/12 G. Narutowicza St., 80-233 Gdańsk, Poland
Department of Analytical and Food Chemistry, Faculty of Chemistry, University of Vigo, Campus As Lagoas-Marcosende s/n, 36310 Vigo, Spain
Department of Analytical Chemistry, Chemical Faculty, Gdańsk University of Technology (GUT), 11/12 G. Narutowicza St., 80-233 Gdańsk, Poland
Author to whom correspondence should be addressed.
Molecules 2018, 23(6), 1292;
Submission received: 13 April 2018 / Revised: 21 May 2018 / Accepted: 25 May 2018 / Published: 28 May 2018
(This article belongs to the Special Issue Green Analytical Chemistry)


Organic solvents are ubiquitous in chemical laboratories and the Green Chemistry trend forces their detailed assessments in terms of greenness. Unfortunately, some of them are not fully characterized, especially in terms of toxicological endpoints that are time consuming and expensive to be determined. Missing values in the datasets are serious obstacles, as they prevent the full greenness characterization of chemicals. A featured method to deal with this problem is the application of Expectation-Maximization algorithm. In this study, the dataset consists of 155 solvents that are characterized by 13 variables is treated with Expectation-Maximization algorithm to predict missing data for toxicological endpoints, bioavailability, and biodegradability data. The approach may be particularly useful for substitution of missing values of environmental, health, and safety parameters of new solvents. The presented approach has high potential to deal with missing values, while assessing environmental, health, and safety parameters of other chemicals.

1. Introduction

Green Chemistry has been defined as the “design of chemical products and processes to reduce or eliminate the use and generation of hazardous substances” [1]. With this aim, Anastas and Warner, introduced, in 1998, the twelve principles of Green Chemistry that charted a path towards sustainability in chemical processes [1]. Several principles of Green Chemistry point out the need to eliminate or replace solvents by less harmful alternatives [2]. Particularly, the 5th principle of Green Chemistry specifically recommends the use of innocuous solvents when avoiding the use of solvents is not possible. The employment of harmful solvents is also indirectly discouraged, as can be deduced from additional principles, such as waste prevention (1st principle) and the prevention or minimization of potential chemical accidents (12th principle) that are associated to their use. Furthermore, advances toward the design of safer chemicals (4th principle) that at the end of their function can be transformed into innocuous non-persistent products (10th principle), and importantly, renewable feedstocks, rather than depleting non-renewable resources (7th principle), are highly recommended [3,4].
Organic solvents are increasingly used in scientific and technological activities, with an estimated worldwide consumption of roughly 30 million metric tons per year [5]. Apart from the steady increase of solvent consumption in the last years, especially worrisome is the fact that certain solvents of very high concern are still being widely used. Thus, the implementation of solventless process is strongly advisable from the point of view of Green Chemistry. While remarkable efforts have been made in certain areas toward the implementation of solventless processes (e.g., solventless sample preparation approaches [6] and greener reactions under solvent free conditions [7]), these strategies are, in general terms, still far from reaching the desired level of implementation. Alternatively, the minimization of solvent consumption and/or the replacement of hazardous solvents by cleaner alternatives are highly recommended strategies to reduce the risks that are associated to solvent usage [8,9]. In this vein, a number of solvent selection guides have been reported in the literature for a convenient selection of alternatives to harmful solvents [10,11,12,13,14,15,16,17,18]. However, the lack of relevant data, such as physicochemical properties and environmental impact, might hinder their implementation in scientific and technological processes [19]. The problem of missing data in solvents assessments is managed by default, substitution with value for nearest neighbor (homologue), and substitution with mean value for the entire chemical class [20].
Expectation-Maximization (E-M) algorithm [21] was developed in the 1970‘s and it is widely applied in different branches of sciences as the tool for the substitution of missing values [22]. It is applied to deal with missing values for the characterization of patients to predict breast cancer recurrence [23]. The algorithm is used to predict missing values in genetic arrays [24]. Application in chemistry include the prediction of missing data in environmental monitoring [25] or to construct regression models in the case of missing data in the raw dataset [26]. Other applications of E-M algorithm for predictions in the chemistry related field include the prediction of biomarkers essentiality [27] or the prediction of peptide bounding [28].
The aim of the study is to substitute missing values in the dataset characterizing organic solvents with the application of E-M algorithm, and to find relations between the characteristics from the estimated distribution. Toxicity, biodegradability, and bioavailability parameters are predicted with E-M algorithm.

2. Materials and Methods

2.1. Dataset

The dataset consists of 155 solvents that are described by 13 variables. The values of variables are extracted mainly from the Handbook of Physical-Chemical Properties and Environmental Fate for Organic Chemicals [29] and from material safety data sheets of solvents. Physicochemical properties include melting point, boiling point, vapor pressure, density, water solubility, Henry’s law constant, logarithms of octanol-water partitioning coefficient, and logarithm of octanol-air partitioning coefficient. Also, toxicity towards rodents when being administered orally (Oral LD50), toxicity towards rodents via inhalation exposure pathway (Inhalation LC50), toxicity towards fish (Fish LC50), half-life time needed for biodegradation, and logarithm of bioconcentration factors were taken as variables for analysis.
Solvents included in the dataset are compounds of different chemical classes–from hydrocarbons, terpenes, chlorinated solvents to alcohols, ketones, ethers, and esters and carboxylic acids. 85 out of the 155 solvents were fully characterized in terms of abovementioned variables, whereas the dataset contained at least one gap in case of the remaining solvents.

2.2. E-M Model

To complete the data we use E-M algorithm. This algorithm consists of two steps: an Expectation step or the E-step and a Maximization step or the M-step.
We observe a data y = ( y 1 , , y n ) , where y i are realizations of a random vector Y , i = 1 , , n . Let Y has the probability distribution function depending on a vector of unknown parameters Ψ .
Let X be a k dimensional random vector corresponding to a complete-data x = ( x 1 , , x n ) , where x i are realizations of X , i = 1 , , n . We consider the case where the vector X has multivariate normal distribution, which means that Ψ = ( μ , Σ ) , where μ is a vector of means and Σ is a covariance matrix.
Suppose that there are G groups with distinct missing patterns. Then, the observed-data log likelihood can be expressed as
log L ( Ψ ) = Σ g = 1 G log L g ( Ψ ) .
The likelihood function for g th group formed from the observed data y g = ( y 1 g , , y n g g ) is, discarding a proportionality constant, given by
L g ( Ψ ) = | Σ g | n g 2 exp { 1 2 Σ i = 1 n g ( y i g μ g ) T Σ g 1 ( y i g μ g ) }
An estimate Ψ ^ of Ψ can be obtained by solving the log likelihood equation
log L ( Ψ ) Ψ = 0
The E-M algorithm approaches the problem of solving the incomplete-data log likelihood Equation (3) indirectly by proceeding iteratively in terms of complete-data log likelihood function log L c ( Ψ ) , where
L c ( Ψ ) = | ( 2 π ) k Σ | n 2 exp { 1 2 Σ i = 1 n ( x i μ ) T Σ 1 ( x i μ ) }
The E-step calculates the conditional expectation of the complete-data log likelihood, E Ψ ( 0 ) ( log L c ( Ψ ) | y ) , given the observed data and the parameter estimates. Then, the M-step finds the parameter estimates to maximize the complete-data log likelihood from the E-step.
The steps are carried out until the value of
L ( Ψ ( k + 1 ) ) L ( Ψ ( k ) )
is smaller than arbitrarily amount in case of convergence of the sequence of likelihood values ( L ( Ψ ( k ) ) ) k . The extended description of this procedure can be found in Supplementary Material. More details for normal distribution can be found in [30].

2.3. Dataset Preparation

We consider 155 solvents that are described by 13 attributes, which are listed in Table 1. Some chemical compounds do not have the value for the parameter “Inhalation LC50”, because they are not volatile, so there is no possibility to be intoxicated via inhalation. We decided to give them values 5001. This is the expert judgment caused by the threshold of danger to the environment. According to Globally Harmonized System of Classification and Labelling of Chemicals, the values above 5000 ppm are not characterized as “harmful if inhaled” [31].

3. Results and Discussion

3.1. Basic Statistics

Table 1 shows the basic statistics of investigated dataset, including the number of missing data (N missing). For boiling point, density, water solubility, vapor pressure, and log KOW, all of the values are available. The biggest problems with data availability are in case of Oral LD50, inhalation LC50, fish LC50, and BOD t1/2 (biodegradation half-life), as they are characterized by a big fraction of missing values.

3.2. Predictions with Bayesian Model

Application of E-M Algorithm

To complete the data set, we use the programming language SAS 4GL. There is a proper procedure in SAS, called PROC MI, which performs the E-M algorithm by function EM. The extensive description of this procedure can be found in [32].
We assume that the random vector X corresponding to a complete-data vector x has the multivariate normal distribution, X ~N(µ,Σ) where µ is a vector of means and Σ is a covariance matrix. To approach that assumption, we use a logarithmic transformation of the properties Henry’s law constant, Oral LD50, Inhalation LC50, fish LC50, and BOD t1/2.
In this case, we have to find a parameter Ψ = (µ, Σ) where µ is a vector of means and Σ is a covariance matrix of the unknown distribution. The initial estimates Ψ(0) are the means and the standard deviations from available cases. The correlations are set to zero.
On the prepared set, we carry out the E-M algorithm in SAS. To satisfy the convergence requirement the difference (5) has to be smaller than 0.0001. The method converges after 69 iterations. It means that
L(Ψ(69)) − L(Ψ(68)) < 0.0001
due to (5). Thereby, we received the completion of the data set. The most important thing is that we received full information about the set by finding the distribution of the data, N(µ,Σ). The E-M algorithm evaluated the parameters µ and Σ. Despite the Mardia’s kurtosis test has not shown that the data has a multivariate normal distribution we think that such a model gives a good approximation of relations. We will find these relations using principal component analysis (PCA) [33]. Table 2 shows the obtained results of first three principal components.
The first three components explain 65.9% of variability of the raw dataset. Fourth principal component explained 8.4% of variability, and it was decided not to include it in the assessment result. The first component consists of melting point, boiling point, vapor pressure, and log KOA, and it explains 26.2% of the initial variability. This component can be identified as responsible for the characterization of solvents in terms of their basic physicochemical properties, especially volatility. The second component is loaded with Henry’s law constant, log Kow, and log BCF, and it explains 24.4% of variability. This relation can be explained by the polarity of solvents. All three variables are related to interphase transfer and the ability to be transferred out of water. Apart from these three variables, weaker negative loading (−0.3774) is observed for water solubility, which additionally supports the polarity related origin of this group. The third component is formed by inhalation and oral toxicities and the negative loading of density. It carries 15.3% of initial dataset variability. It can be defined as toxicity relation and the presence of density in this group is due to the fact that more toxic solvents are usually more dense (i.e., chlorinated solvents).
The prediction of missing values are presented in Table 3—shaded are modeled with the E-M algorithm, whereas non-shaded are input data. Algorithm allows to substitute missing values for alkyl glycerol esters (numbers 15–29 in the Table 3), bio-based solvents originating from biodiesel production. Glycerol is formed as a byproduct during biodiesel production, and it is a platform molecule for the synthesis of its alkylated ester derivatives [34]. Because they originate from renewable resource, undergo biodegradation, and are low cost, they are potentially attractive from a Green Chemistry point of view. However, they are not fully characterized in terms of their toxicology or environmental fate related properties. The means of completed values are of 2017 mg kg−1 LD50 by oral administration (mean for solvents with available data—3667 mg kg−1), 13945 ppm of LC50 by inhalation (mean for solvents with available data—4658 ppm), 158 mg dm−3 of LC50 towards fish (mean for solvents with available data—970 mg dm−3), 1.96 days of biodegradability half-lifes (mean for solvents with available data—55 days), and 0.46 of logarithm of bioconcentration factors (mean for solvents with available data—1.15). According to Globally Harmonized System of Classification and Labelling of Chemicals, oral toxicity of >2000 mg kg−1 and inhalation toxicity of >5000 ppm indicate that they are chemicals of low acute toxicity. The predicted results show that alkyl glycerol esters may be toxic (especially towards fish) and they should be characterized in this manner to confirm their green status. Other predicted that missing values show that alkyl glycerol esters are biodegradable and they do not undergo bioaccumulation.
Esters (mainly methyl or isopropyl esters of fatty acids) are characterized by low completed inhalation toxicity, rather low oral toxicity, but they some have low values (at the level of single mg dm−3) of toxicity towards fish, which suggest they are potential threats to aquatic life. Their completed values suggest that they are biodegradable, with half-lifes at the level of few to 20 days. Gamma valerolactone is considered to be green solvent, and recently it gained much attention [35]. E-M algorithm showed that LC50 by inhalation is 1186 ppm and LC50 towards fish is 756.6 mg dm−3 and biodegradability half-life is 7.8 days. These values suggest that this compound is not very toxic and it undergoes biodegradation.
Chloropropane, chlorobutane, and chloropentane are characterized by mean predicted missing value of inhalation LC50 = 12572 ppm, which is in accordance with available values for cis-1,2-dichloroethene (13,700 ppm) and 1,1-dichloroethane (13,000 ppm). The completed values show that they are slightly less toxic than the mean of the entire dataset. Mean value of LC50 towards fish is 82 mg dm−3, what shows they can be toxic to fish as the mean value of this parameters equals to 970 mg dm−3. Biodegradability half-lifes are predicted to be 18.2 and 10.5 days for 1-chlorobutane and 1-chloropentane, respectively. This value for 1-chloropropane of 30 days is available in the original dataset. What is more, the completed value for 1,1,2,2-tetrachloroethane is equal to 134 days. Chlorinated solvents are not concerned as green solvents and predicted missing values confirm this statement.
To obtain the information about the standard errors bootstrap analysis is performed [36]. The results of analysis are presented in Table 4. Standard errors are small for almost all of the variables. Only those variables, which do not contain data gaps and are not transformed, have the high standard errors.

4. Conclusions

E-M algorithm is useful in predicting of organic solvents missing parameters. Algorithm allows for completing 35 values of LD50 by oral administration to rodents, 46 values of LC50 by inhalation, 57 values of LC50 towards fish, 57 values of biodegradability, and 62 values of bioconcentration factors. Some of the solvents that are considered in this study are promising from the Green Chemistry point of view, even though they have not been fully characterized yet.
E-M algorithm can be useful in characterization of other novel, potential green alternatives of other chemicals. It is important for the characterization of chemicals for their rapid screening.

Supplementary Materials

The following are available online. E-M algorithm description.

Author Contributions

G.Ł. performed the calculations with PCA and E-M algorithm, and wrote substantial part of paper, F.P.-P. prepared the dataset and wrote substantial part of paper, M.T. prepared the dataset, interpreted the results and wrote substantial part of paper, J.N. was guiding the research.


This research received no external funding.


F.P.-P. thanks Xunta de Galicia for financial support as a postdoctoral researcher of the I2C program. Authors would like to express their gratitude to Karol Dziedziul for pointing to E-M algorithm and for his precious remarks and suggestions.

Conflicts of Interest

Authors declare no conflict of interests


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Sample Availability: Samples of the compounds are not available from the authors.
Table 1. Basic statistics of the dataset.
Table 1. Basic statistics of the dataset.
VariableMeanStd. Dev.VarianceMinimumMaximumNN Missing
Melting point (°C)−43.90148.7282374.453−14049.521523
Boiling point (°C)142.38568.6264709.488203231550
Density (g cm−3)0.9520.2140.0460.621.681550
Water solubility (mg dm−3)116,796.63244,339.685.97 × 10100.0009271,000,0001550
Vapor pressure (Pa)11,901.62628,631.7818.2 × 1080241,9001550
Henry law constant (Pa m3 mol−1)60,736.714267,946.027.18 ×1 0108.03×10−62,219,0171532
log KOW2.2292.3525.531−2.328.731550
log KOA4.4341.9993.9951.45112.1011523
Oral LD50 (mg kg−1)3667.3834658.4821,701,436531,50012035
Inhalation LC50 (ppm)10,532.28418,252.9573.33 × 10834123,00010946
Fish LC50 (mg dm−3)970.0962813.0937,913,4900.116,7009857
BOD t1/2 [days]55.360127.19216,17818009362
log BCF1.1541.0161.032−1.634.71514
LD50: lethal dose administered orally to rodents that kills half of population; LC50: toxicity towards rodents via inhalation exposure pathway; Std. Dev.: Standard Deviation; Kow: octanol-water partitioning coefficient; KOA: octanol-air partitioning coefficient; BOD: biodegradation half-life; BCF: bioconcentration factor.
Table 2. The results of data treatment with principal component analysis. Dark red is for very negative values, yellow is for neutral values and green stands for positive values.
Table 2. The results of data treatment with principal component analysis. Dark red is for very negative values, yellow is for neutral values and green stands for positive values.
Comp. 1Comp. 2Comp. 3
Melting point−0.4448−0.1465−0.0451
Boiling point−0.4963−0.08830.0598
Water solubility0.1150−0.37440.0708
Vapor pressure0.34780.0295−0.0888
log Henry law const0.12470.51030.0289
log KOW−0.24620.43400.1635
log KOA−0.4352−0.17950.1165
log Oral LD50−0.05280.09330.5678
log Inhalation LC500.22870.06000.4662
log fish LC500.1673−0.26420.2638
log BOD t1/20.08480.2915−0.3085
log BCF−0.24920.42020.0174
Table 3. Input and completed values (shaded) for solvents that were not fully characterized. Green color indicate predicted values.
Table 3. Input and completed values (shaded) for solvents that were not fully characterized. Green color indicate predicted values.
SolventCAS NumberOral LD50 (mg kg−1)Inhalation LC50 (ppm)Fish LC50 (mg dm−3)BOD t1/2 (days)log BCF
14oleic alcohol143-28-2960413,04946.79.02.623
161,3-dimethoxypropan-2-ol 13933794104.72.50.5
171,3-di-n-butoxy-2-propanol 113088569.42.10.603
181-ethoxy-3-iso-propoxy-2-propanol 12561889377.74.30.5
191-methoxy-3-(propan-2-yloxy)propan-2-ol 14982945160.82.10.5
201-n-butoxy-3-ethoxy-2-propanol 22202347232.21.80.5
211-n-butoxy-3-iso-propoxy-2-propanol 30474273188.01.50.168
221-n-butoxy-3-methoxy-2-propanol 18832582197.62.10.5
231-tert-butoxy-3-ethoxy-2-propanol 2568460197.11.30.5
241-tert-butoxy-3-methoxy-2-propanol 1477330535.31.40.5
253-butoxypropane-1,2-diol 38752818203.41.50.5
263-ethoxypropane-1,2-diol 25382663186.21.90.5
273-methoxypropane-1,2-diol 20811985272.42.60.5
283-n-butoxy-1-tert-butoxy-2-propanol 5660516751.71.40.517
29Isopropylidene glycerol100-79-87000167,19716,7001.30.125
31Benzyl ethyl ether539-30-02428262538.66.61.374
321,2,3-trimethoxypropane 13052815135.85.30.5
331,2,3-tri-n-butoxypropane 43905001261.24.52.276
342-methylfuran 1965935294.316.00.725
352-methyltetrahydrofuran 450024,083319.66.40.343
363-n-butoxy-1-tert-butoxy-2-methoxypropane 2392165695.42.91.094
37Isosorbide dimethyl ether 154518,269213.84.90.5
41Dihydrolevoglucosenone 2021291659.34.40.5
44Neryl acetate141-12-84550500141.78.72.365
45Propionic acid79-19-4350054225110
46Ethyl formate 18509800276.6150.5
47Butyl levulinate2052-15-55000500126.33.30.278
48Ethyl levulinate539-88-850004735121.33.30.5
49Glycerol triacetate102-76-13000500172.52.20.5
50Methyl caprylate111-11-510,8009987957.01.856
51Methyl lactate27871-49-450001350828.611.80.5
52Methyl levulinate624-45-32051288892.73.40.5
53Methyl linoleate112-63-0397750014.520.43.051
54Isopropyl myristate110-27-0834811,2078.410.73.07
55Methyl oleate112-62-9200050016.118.92.694
56Methyl palmitate112-39-0478650011.89.42.789
57Isopropyl palmitate142-91-617,78145,41450.313.01.725
58Methyl stearate112-61-8523750012.810.41.46
59Tributyl 2-acetylcitrate77-90-731,500226,17460141.6
60Benzyl benzoate120-51-417006656.25.32.357
67Dimethyl sulphide75-18-3535515687.110.70.561
68Dimethyl sulfoxide67-68-52758429136.91.60.349
Table 4. Standard errors of predictions calculated with bootstrap.
Table 4. Standard errors of predictions calculated with bootstrap.
VariableMeanMean Error
Melting point (°C)−43.1378−0.0056
Boiling point (°C)142.3852−0.4245
Density (g cm−3)0.9521−0.0005
Water solubility (mg dm−3)116,796.6328−279.7044
vapor pressure (Pa)11,901.6258403.5914
Henry law constant (Pa m3 mol−1)2.46770.0955
log KOW2.22850.0125
log KOA4.4644−0.0269
Oral LD50 (mg kg−1)7.6122−0.0093
Inhalation LC50 (ppm)8.27340.0014
Fish LC50 (mg dm−3)4.2248−0.0069
BOD t1/2 (days)2.24310.0065
log BCF1.14500.0076

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Łuczyńska, G.; Pena-Pereira, F.; Tobiszewski, M.; Namieśnik, J. Expectation-Maximization Model for Substitution of Missing Values Characterizing Greenness of Organic Solvents. Molecules 2018, 23, 1292.

AMA Style

Łuczyńska G, Pena-Pereira F, Tobiszewski M, Namieśnik J. Expectation-Maximization Model for Substitution of Missing Values Characterizing Greenness of Organic Solvents. Molecules. 2018; 23(6):1292.

Chicago/Turabian Style

Łuczyńska, Gabriela, Francisco Pena-Pereira, Marek Tobiszewski, and Jacek Namieśnik. 2018. "Expectation-Maximization Model for Substitution of Missing Values Characterizing Greenness of Organic Solvents" Molecules 23, no. 6: 1292.

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