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**Atmospheric Chemistry and Physics**
An interactive open-access journal of the European Geosciences Union

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**Research article**
03 Sep 2018

**Research article** | 03 Sep 2018

Exploring non-linear associations between atmospheric new-particle formation and ambient variables: a mutual information approach

^{1}Institute for Atmospheric and Earth System Research/Physics, Helsinki University, 00560 Helsinki, Finland^{2}Aalto Science Institute, School of Science, Aalto University, 00076 Espoo, Finland^{3}Department of Applied Physics, Aalto University, 00076 Espoo, Finland^{4}Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kongens Lyngby, Denmark^{5}Institute of Physics, University of Tartu, Ülikooli 18, 50090 Tartu, Estonia^{6}Aerosol and Haze Laboratory, Beijing University of Chemical Technology, 100096 Beijing, China^{7}WPI Nano Life Science Institute (WPI-NanoLSI), Kanazawa University, Kakuma-machi, Kanazawa 920-1192, Japan^{8}Graduate School Materials Science in Mainz, Staudinger Weg 9, 55128 Mainz, Germany

^{1}Institute for Atmospheric and Earth System Research/Physics, Helsinki University, 00560 Helsinki, Finland^{2}Aalto Science Institute, School of Science, Aalto University, 00076 Espoo, Finland^{3}Department of Applied Physics, Aalto University, 00076 Espoo, Finland^{4}Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kongens Lyngby, Denmark^{5}Institute of Physics, University of Tartu, Ülikooli 18, 50090 Tartu, Estonia^{6}Aerosol and Haze Laboratory, Beijing University of Chemical Technology, 100096 Beijing, China^{7}WPI Nano Life Science Institute (WPI-NanoLSI), Kanazawa University, Kakuma-machi, Kanazawa 920-1192, Japan^{8}Graduate School Materials Science in Mainz, Staudinger Weg 9, 55128 Mainz, Germany

**Correspondence**: Martha A. Zaidan (martha.zaidan@helsinki.fi)

**Correspondence**: Martha A. Zaidan (martha.zaidan@helsinki.fi)

Abstract

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Atmospheric new-particle formation (NPF) is a very non-linear process that includes atmospheric chemistry of precursors and clustering physics as well as subsequent growth before NPF can be observed. Thanks to ongoing efforts, now there exists a tremendous amount of atmospheric data, obtained through continuous measurements directly from the atmosphere. This fact makes the analysis by human brains difficult but, on the other hand, enables the usage of modern data science techniques. Here, we calculate and explore the mutual information (MI) between observed NPF events (measured at Hyytiälä, Finland) and a wide variety of simultaneously monitored ambient variables: trace gas and aerosol particle concentrations, meteorology, radiation and a few derived quantities. The purpose of the investigations is to identify key factors contributing to the NPF. The applied mutual information method finds that the formation events are strongly linked to sulfuric acid concentration and water content, ultraviolet radiation, condensation sink (CS) and temperature. Previously, these quantities have been well-established to be important players in the phenomenon via dedicated field, laboratory and theoretical research. The novelty of this work is to demonstrate that the same results are now obtained by a data analysis method which operates without supervision and without the need of understanding the physics deeply. This suggests that the method is suitable to be implemented widely in the atmospheric field to discover other interesting phenomena and their relevant variables.

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Zaidan, M. A., Haapasilta, V., Relan, R., Paasonen, P., Kerminen, V.-M., Junninen, H., Kulmala, M., and Foster, A. S.: Exploring non-linear associations between atmospheric new-particle formation and ambient variables: a mutual information approach, Atmos. Chem. Phys., 18, 12699–12714, https://doi.org/10.5194/acp-18-12699-2018, 2018.

1 Introduction

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New-particle formation (NPF) is an important source of aerosol particles and cloud condensation nuclei (CCN) and in a vast number of atmospheric environments ranging from remote continental areas to heavily polluted urban centres (Kulmala et al., 2004; Dunne et al., 2016; Wang et al., 2017). The occurrence and strength of NPF and its influence on the CCN budget in different atmospheric environments depends on a delicate balance between the factors that favour NPF and subsequent particle growth and the factors that suppress these processes (Kerminen and Kulmala, 2002; Pierce and Adams, 2007; Westervelt et al., 2014; Kulmala et al., 2017). As a result, researchers have not managed to find a general framework, or formulae, on how to relate atmospheric NPF to the concentrations of various trace gases, meteorological quantities and radiation parameters.

Based on data from field measurements, several studies investigated the relations between NPF and meteorological conditions (Nilsson et al., 2001) and various chemical compounds (Bonn and Moortgat, 2003; Kulmala et al., 2004; Almeida et al., 2013; Nieminen et al., 2014). Such studies have found the ideal conditions for NPF events to consist of low atmospheric water content, low preexisting particle concentration and high solar radiation (Boy and Kulmala, 2002). In addition, sulfuric acid is believed to be the single most important compound to participate in the atmospheric NPF (Kerminen et al., 2010; Sipilä et al., 2010; Petäjä et al., 2011; Nieminen et al., 2014).

Due to the practical limitations, the measurement campaigns typically last from weeks to months and they often have a dedicated focus. On the one hand, such an approach enables a very detailed inspection for a somewhat narrower scope, but, on the other hand, there is a risk of overlooking important processes falling outside the chosen, predetermined scope. One way to circumvent this issue is to have long-term continuous measurements of a wide variety of atmospheric variables. Nowadays there is more and more focus on continuous observations as described by Kulmala (2018). However, such enterprises then open a new set problems: how to analyse all the collected data? It is clear that techniques offered by the modern data science, such as data mining and machine learning, should be consulted.

Previously, Mikkonen et al. (2006) studied the effects of gas and
meteorological parameters as well as aerosol size distribution to nucleation
events. The used data were measured in the Po Valley, Italy, for about 3 years
(2002–2005). In this case, they used a discriminant analysis method where
relative humidity (RH), ozone and radiation are found to give the best
classification performance. Next, similar atmospheric variables were also
included in their further study (Mikkonen et al., 2011). The used
data were measured from three polluted sites, which are the Po Valley,
Italy; and Melpitz and Hohenpeissenberg, Germany. In this study, they applied a
multivariate non-linear mixed effects model to examine the variables
affecting the number concentration of Aitken particles (50 nm). They also
found that relative humidity and ozone give the best predictor variables. In
addition, the model indicated that the temperature, condensation sink (CS), and
concentrations of sulfuric dioxide (SO_{2}) and nitrogen dioxide
(NO_{2}) influence NPF as well as the number concentration of Aitken
mode particles.

In order to understand the effects of atmospheric variables to NPF in
Hyytiälä, Finland, a comprehensive study was done by
Hyvönen et al. (2005). They utilized two main types of data mining methods
on 8 years of continuous measurements of 80 variables. Their first
method was based on unsupervised *K*-means clustering. The first method
demonstrated that the relative humidity, global radiation and sensible heat
have data separation power and correlate with NPF. In addition to those,
their results indicated that ozone (O_{3}) and carbon dioxide
(CO_{2}) concentrations might also correlate with NPF. The second method
was based on a supervised learning classification. Several machine learning
models (such as linear discriminant analysis, support vector machine and logistic regression) were set up to perform a classification task for each
day as an event or a non-event day. The goal was not to separate event days
from non-event days, but to understand which atmospheric variables should be
used to clearly separate the two groups. In this case, the mean and standard
deviation of atmospheric variables were calculated as the input, whereas the
aerosol particle formation event and non-event days database was used as the
output. Due to the initial model's random parameters, the models were run
1000 times using different training and test sets to ensure the result
stability. The selected models used a pair and triplet combination of
atmospheric variables. The models were ranked based on the classification
performance and the best model was used to evaluate all pair and triplet
combinations of the atmospheric variables. In this case, the supervised
classification models found that the best pair of atmospheric variables to
classify events–non-events is condensation sink and relative humidity. The
latter was also found through the clustering method. The results of
Hyvönen et al. (2005) support some earlier conclusions from
Boy and Kulmala (2002) stating that NPF events are largely explained by
three parameters: temperature, the atmospheric water content and radiation.
However, they did not find significant correlations between NPF and radiation
variables as suggested by the aforementioned studies.

The previously used data mining approaches are mostly based on classification methods. Although these methods seem to be suitable tools for finding correlation between variables in complex systems, the used implementation may not be always effective for this case. The first reason concerns the used features, such as mean and standard deviations. This practice compresses the measurement data into a single quantity for each day, which may potentially lead to information loss in the data. Secondly, the implementation procedure is computationally expensive. This requires the exploration of all possible models and variable combinations to find the best pairs. The models also need to be run multiple times to ensure their stability.

To overcome the above-mentioned issues, we propose here an alternative method – based on information theory – to be used in atmospheric data analysis. Mutual information (MI), one of the many information quantities, measures the amount of information that can be obtained about one random variable by observing another one. In this paper, MI is first introduced and then used to find the maximal amount of shared information between atmospheric variables and NPF. In other words, the goal is to find the most relevant atmospheric variables in relation to NPF events using a data-driven information theoretic method based on the data set measured at Hyytiälä, Finland.

2 Atmospheric database measured at the SMEAR II station in Hyytiälä, Finland

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In this study, we utilize the data measured during the years 1996–2014 at the Station for Measuring Forest Ecosystem-Atmosphere Relations (SMEAR) II station in Hyytiälä, Finland, operated by Helsinki University (SMEAR website, 2017).

The SMEAR II station is located in Hyytiälä forestry field station in
southern Finland (61^{∘}51^{′} N, 24^{∘}17^{′} E; 181 m above sea
level), about 220 km northwest of Helsinki. It also lies between two large
cities, Tampere and Jyväskylä, that are about 60 and 90 km from the
measurement site, respectively. Homogeneous 55-year-old (in 2017)
scots-pine-dominated forests surround the station. SMEAR II is classified as
a rural background site considering the levels of air pollutants, shown by
for example submicron aerosol number size distributions
(Asmi et al., 2011a; Nieminen et al., 2014).

The SMEAR II station has been established for multidisciplinary research, including atmospheric sciences, soil chemistry and forest ecology. The station consists of a measurement building, a 72 m high mast, a 15 m tall tower and two mini-watersheds. It is equipped with extensive research facilities for measurement of various gases' concentration, various fluxes, meteorological parameters (e.g. temperature, wind speed and direction, relative humidity), solar and terrestrial radiation (e.g. ultraviolet rays), and atmospheric aerosols (e.g. particle size distribution). The measurements for forest ecophysiology and productivity, such as photochemical reflectance, and the measurements for soil and water balance also take place there. A detailed description of the continuous measurements performed at this station can be found in Kulmala et al. (2001a), Hari and Kulmala (2005) and SMEAR website (2017).

In this study, we used four types of continuous measurement data: gas
concentrations, meteorological conditions, radiation variables and aerosol
particle concentrations. The gases include nitrogen monoxide (NO) and
other oxides (NO_{x}), ozone (O_{3}), sulfur dioxide
(SO_{2}), water (H_{2}O), carbon dioxide (CO_{2}) and carbon
monoxide (CO). Meteorological data include the temperature, humidity,
pressure, and wind speed and direction, among others. The gas concentrations
and meteorological data measurements are performed at the heights of 4.2,
8.4, 16.8, 33.6, 50.4 and 67.2 m. The radiation variables include
UV-A, UV-B, PAR, global, net, reflected global and reflected PAR. These
measurements are mostly performed at a radiation tower (18 m). The measured
aerosol particle number size distribution ranges were between 3 and
500 nm until December 2004, and after that it has been extended to cover the
size range from 3 to 1000 nm. The sampling height was at 2 m until
2015 when the instrument was moved to the tower at 35 m.

Table 1 collects all the atmospheric variables used in this study, including the adapted shorthand notation used throughout the current paper together with few details on the measurements. The raw data can be accessed free of charge via the SMEAR website (2017), which also contains more information on the measurements. It should be noted that not all the measured atmospheric variables are included in the current analysis.

In addition to directly measured variables, there are few derived variables included in this study. The aerosol particle condensation sink determines how rapidly molecules and small particles condense onto preexisting aerosol particles and it is strongly related the shape of the size distribution (Pirjola et al., 1999; Kulmala et al., 2001b). CS is formulated as

$$\begin{array}{}\text{(1)}& {\displaystyle}\mathrm{CS}{\displaystyle}=\mathrm{4}\mathit{\pi}D\sum _{i}{\mathit{\beta}}_{{M}_{i}}{r}_{i}{N}_{i},\end{array}$$

where *r*_{i} is the radius of a particle for size class *i*, *N*_{i} is the
particle concentration in the respective class *i*, *D* is the diffusion
coefficient of the condensing vapour and *β*_{M} is the transitional
correction factor, defined in Fuks and Sutugin (1970).

Sulfuric acid (H_{2}SO_{4}) concentration is included in the study since
it is believed to be one of the key factors in atmospheric aerosol particle
formation (Nieminen et al., 2014). Unfortunately, there are no continuous
long-term measurements of sulfuric acid concentrations at SMEAR II in
Hyytiälä. In order to gauge sulfuric acid, we need to calculate its proxy
concentration based on the measured gas concentrations, solar radiation and
the measured aerosol particle size distributions acting as CS
(Kulmala et al., 2001b). Petäjä et al. (2009) proposed two proxies
by using CS and solar radiation in the UV-B range as well as global radiation
(Glob). The proxy formulations are given by

$$\begin{array}{}\text{(2)}& {\displaystyle}{p}_{\mathrm{2}}& {\displaystyle}={k}_{\mathrm{2}}\cdot {\displaystyle \frac{\left[{\mathrm{SO}}_{\mathrm{2}}\right]\cdot \text{UV-B}}{\text{CS}}},\text{(3)}& {\displaystyle}{p}_{\mathrm{3}}& {\displaystyle}={k}_{\mathrm{3}}\cdot {\displaystyle \frac{\left[{\mathrm{SO}}_{\mathrm{2}}\right]\cdot \text{Glob}}{\text{CS}}},\end{array}$$

where *k*_{2} and *k*_{3} are median values for the scaling factors, which are
$\mathrm{9.9}\times {\mathrm{10}}^{-\mathrm{7}}$ and $\mathrm{2.3}\times {\mathrm{10}}^{-\mathrm{9}}$ m^{2} W^{−1} s, respectively. Here, we include the proxies 2
and 3 (*p*_{2} and *p*_{3}) calculated for the years 1996–2014 in our analysis.

Finally, it is essential to have a database of aerosol particle formation
days – without such database the correlation analysis between NPF and
atmospheric variables cannot be performed. We used a database of the years
1996–2014, generated by the atmospheric scientists at Helsinki University.
The database has been created by visual inspection of the continuously
measured aerosol size distributions over a size range of 3–1000 nm at
the SMEAR II Hyytiälä forest (Dal Maso et al., 2005). The method
classifies days into three main groups: event, non-event and undefined days.
An event day occurs when there is a growing new mode in the nucleation size
range prevailing over several hours, whilst a non-event day takes place when
the day is clear of all traces of particle formation. Finally, an undefined
day is assumed when it cannot be unambiguously classified as either an event
or non-event day. In order to prevent bias in the data, we did not consider
the undefined days because this group cannot be unambiguously classified as
either an event or non-event day. Undefined days may belong to event or
non-event days if further investigation is made. Therefore, the undefined
day's group was excluded from our database. Figure 1 shows two
examples of the day when NPF and growth (event day) and the day when no
particle formation is observed (non-event day) on April 2005 at
Hyytiälä station. The *x* axis displays the 24 h time period whilst
*y* axis denotes the range of particle diameters (from 3 to 1000 nm). The
colour indicates the particle concentration level (cm^{−3}).

3 Computational methods: concept and their application

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Before the raw data can be fed into an analysis model, they need to be preprocessed first and these steps will be outlined below. After that, the mutual information method will be introduced.

The (raw) data used in this paper range from 1 January 1996 to 31 December 2014, totalling 18 years. The first step in preprocessing is to exclude the undefined days, as the focus is to find the correlation between aerosol particle formation days and atmospheric variables. In order to reduce the amount of irrelevant data, we then eliminate nighttime data points in all atmospheric variables. When the atmospheric photochemistry is most intense (during the daytime), the strongest and long-lasting events of the atmospheric NPF are typically observed (Kulmala and Kerminen, 2008; Nieminen et al., 2014). Due to significant variation in daytime and nighttime in the Hyytiälä forest during a year, it is necessary to use accurate sunrise and sunset times (Duffett-Smith and Zwart, 2011; National Oceanic and Atmospheric Administration, 2017). Since bivariate analysis is performed, between NPF and an atmospheric variable, the time resolution varies for every variable. If a variable is measured every 10 min, it means 10 min time resolution is used.

Information theory is a mathematical representation of the conditions and parameters affecting the transmission and processing of information (Stone, 2015). It was proposed firstly by Claude E. Shannon in 1948 (Shannon, 1948). Information theory has been applied to a wide range of applications, such as communication (Xie and Kumar, 2004), cryptography (Bruen and Forcinito, 2011) and seismic exploration (Mukerji et al., 2001). Although, information theory has not been used yet to analyse NPF phenomena, this theory has been used in the field of atmospheric sciences, such as acquisition of aerosol size distributions (Preining, 1972), aerosol remote sensing (Li et al., 2009, 2012) and land-precipitation analysis (Brunsell and Young, 2008).

This subsection introduces briefly the basic concepts of information quantities, as well as the definitions and notations of probabilities that will be used throughout the paper. In-depth explanation concerning the principles of information theory can be found for example in MacKay (2003), Cover and Thomas (2012) and Stone (2015).

Entropy is a key measure in information theory. It quantifies the amount of
uncertainty involved in the value of a random variable. If 𝕏 is
the set of all data points $\mathit{\{}{x}_{\mathrm{1}},\mathrm{\cdots},{x}_{N}\mathit{\}}$ that *X* could take, and
*p*(*x*) is the probability of some *x*∈𝕏, then the entropy of
*X*, *H*(*X*), is defined as

$$\begin{array}{}\text{(4)}& H\left(X\right)\equiv -\sum _{x\in \mathbb{X}}p\left(x\right)\phantom{\rule{0.125em}{0ex}}\mathrm{log}p\left(x\right).\end{array}$$

Using the concept of information entropy *H*(*X*), one can further define two related and useful quantities: the joint and conditional entropies.

*Joint entropy* measures the amount of uncertainty in two random
variables *X* and *Y* taken together, and it is defined by

$$\begin{array}{}\text{(5)}& H(X,Y)\equiv -\sum _{x\in \mathbb{X},y\in \mathbb{Y}}p(x,y)\mathrm{log}p(x,y),\end{array}$$

where the random variable *Y* can take values from the set of points
𝕐 = $\mathit{\{}{y}_{\mathrm{1}},\mathrm{\cdots},{y}_{N}\mathit{\}}$ and *p*(*x*,*y*) is the joint probability
of *x* and *y*.

*Conditional entropy* quantifies the amount of uncertainty remaining
in the random variable *Y* when the value of the random variable *X* is
known. This can be defined mathematically by

$$\begin{array}{ll}{\displaystyle}H\left(Y\right|X)& {\displaystyle}\equiv -\sum _{x\in \mathbb{X}}p\left(x\right)\sum _{y\in \mathbb{Y}}p\left(y\right|x\left)\mathrm{log}p\right(y\left|x\right)\\ \text{(6)}& {\displaystyle}& {\displaystyle}=-\sum _{x\in \mathbb{X},y\in \mathbb{Y}}p(x,y)\mathrm{log}{\displaystyle \frac{p(x,y)}{p\left(x\right)}},\end{array}$$

where *p*(*y*|*x*) is the conditional probability of *y* given *x* satisfying the
chain rule of probability: $p(x,y)=p\left(y\right|x\left)p\right(x)$. It follows directly from
the definition (6) that conditional entropy fulfils the property

$$\begin{array}{}\text{(7)}& H\left(Y\right|X)=H(X,Y)-H(X),\end{array}$$

which relates the two-variable conditional and joint entropies with the single-variable information entropy.

The mutual information (MI) of two random variables is a measure of the
mutual dependence between these two variables. MI is thus a method for
measuring the degree of relatedness between data sets. MI and its relation to
joint and conditional entropies is illustrated visually in
Fig. 2 with the help of correlated variables *X* and *Y*. The
left disk (red and orange surface area) shows the entropy *H*(*X*), while the
right disk (yellow and orange surface area) shows the entropy *H*(*Y*). The
total surface area covered by the two disks is the joint entropy *H*(*X*,*Y*).
The conditional entropy *H*(*X*|*Y*) is the red surface on the left, while the
conditional entropy *Y* given *X*, *H*(*Y*|*X*), is the yellow surface area on
the right. The intersection of the red and yellow disks, the orange surface
area in the middle, is the mutual information *I*(*X*;*Y*) between *X* and *Y*.

More formally the mutual information of *X* relative to *Y* is given as

$$\begin{array}{ll}{\displaystyle}I(X;Y)& {\displaystyle}\equiv H(X,Y)-H\left(X\right|Y)-H(Y\left|X\right)\\ \text{(8)}& {\displaystyle}& {\displaystyle}=H\left(X\right)+H\left(Y\right)-H(X,Y).\end{array}$$

From the Eq. (8) it is clear that MI is symmetric with
respect to the variables *X* and *Y*. In terms of probabilities, MI is given
by

$$\begin{array}{}\text{(9)}& I(X;Y)\equiv \sum _{x\in \mathbb{X},y\in \mathbb{Y}}p(x,y)\mathrm{log}{\displaystyle \frac{p(x,y)}{p\left(x\right)\phantom{\rule{0.125em}{0ex}}p\left(y\right)}}.\end{array}$$

From the definition (9), one can see that for completely independent and uncorrelated variables, $p(x,y)=p\left(x\right)p\left(y\right)$, the MI vanishes, as expected. It can be also seen that, in the other extreme where the variables are the same, MI reduces into the corresponding information entropy.

MI has found its use in modern science and technology, for example in search
engines (Su et al., 2006), in bioinformatics (Lachmann et al., 2016), in
medical imaging (Cassidy et al., 2015) and in feature selection
(Peng et al., 2005). Probably at least a part of the MI method's appeal comes
from its capability to effectively measure non-linear correlation between
data sets (Steuer et al., 2002; Chen et al., 2010). In this aspect MI
is superior to the standard Pearson correlation coefficient (PCC)
(Pearson, 1895), which is only suitable for measuring linear
correlation (Wang et al., 2015). To illustrate this,
Fig. 3 shows a comparison between PCC (commonly represented
by *ρ*), the Spearman correlation coefficient (represented by *Sp*)
(Spearman, 1904) and MI using a standard test set of linearly and
non-linearly correlated data that is publicly available. The upper row shows
six linear data sets, whereas the bottom row plots six non-linear data sets;
both rows also contain one uncorrelated data set (the middle one). All
methods estimate similar correlation for the linear data sets and correctly
detect the uncorrelated data. In the case of the non-linear data, the PCC and Spearman
correlation coefficient method simply fail, whereas the MI method is able to
measure the correlation in the data.

The MI implementation is straightforward for discrete distributions because the required probabilities for calculating MI can be computed precisely based on counting. However, the MI implementation for continuous distributions may be tricky because the probability distribution function is often unknown. A binning method can be implemented for calculating MI involving continuous distribution. This method makes the data completely discrete by grouping the data points into bins in the continuous variables. Nevertheless, the choice of binning size (i.e. the number of data points per bin) is a non-trivial task, since this choice often leads to different MI result. The binning method does not allow MI calculation between two data sets that have different resolution – this would be a major obstacle in this study. Therefore, in the current investigation we will use the so-called nearest-neighbour method (Kraskov et al., 2004; Ross, 2014). It has been shown to be accurate, insensitive to the choice of model parameter and also computationally relatively fast.

This subsection explains the nearest-neighbour MI method adopted from
Ross (2014). Suppose *x* is a discrete variable and *y* is a
continuous variable. The method computes a number *I*_{i} for each data point
*i*, based on its nearest neighbours in the continuous variable *y*. First,
using Euclidean distance (or other types of distance metrics), we find the
*k*th closest neighbour to point *i* among ${N}_{{x}_{i}}$, where ${N}_{{x}_{i}}$ is the
data point whose value of the discrete variable equals *x*_{i}. This results
in *d*, that is the distance to this *k*th neighbour. Next, we count the
number of neighbours *m*_{i} in the full data set that lie within distance *d*
to point *i* (including the *k*th neighbour itself). Based on ${N}_{{x}_{i}}$ and
*m*_{i}, MI for every data point *i* can be computed using

$$\begin{array}{}\text{(10)}& {I}_{i}=\mathit{\psi}\left(N\right)-\mathit{\psi}\left({N}_{{x}_{i}}\right)+\mathit{\psi}\left(k\right)-\mathit{\psi}\left({m}_{i}\right),\end{array}$$

where *N* is the number of full data points and *k* is the user choice for
the number of nearest neighbours. The symbol *ψ*(.) is the digamma function,
defined as the logarithmic derivative of the gamma function. This can be
expressed as

$$\begin{array}{}\text{(11)}& \mathit{\psi}\left(z\right)={\displaystyle \frac{\mathrm{d}}{\mathrm{d}z}}\text{ln}\left(\mathrm{\Gamma}\right(z\left)\right)={\displaystyle \frac{{\mathrm{\Gamma}}^{\prime}\left(z\right)}{\mathrm{\Gamma}\left(z\right)}},\end{array}$$

where Γ(.) is a gamma function. The detailed explanation about gamma and digamma functions can be found in Abramowitz and Stegun (2012).

After obtaining MI for every point *i*, in order to estimate the MI from our
data set, we average *I*_{i} over all data points, symbolized by
〈.〉, to give

$$\begin{array}{}\text{(12)}& {\displaystyle}I(X;Y)& {\displaystyle}=\langle {I}_{i}\rangle \text{(13)}& {\displaystyle}& {\displaystyle}=\mathit{\psi}\left(N\right)-\langle \mathit{\psi}\left({N}_{x}\right)\rangle +\mathit{\psi}\left(k\right)-\langle \mathit{\psi}\left(m\right)\rangle ,\end{array}$$

where *k* is determined by a user. In order to bound the MI estimates within
the interval $(-\mathrm{1},\mathrm{1})$ and make it comparable with the Pearson correlation
coefficient (Pearson, 1895), the proposed scaling factor from
Numata et al. (2008) is used to give

$$\begin{array}{}\text{(14)}& \widehat{I}(X;Y)=\text{sign}\left[I(X;Y)\right]\sqrt{\mathrm{1}-\mathrm{exp}(-\mathrm{2}|I(X;Y)\left|\right)},\end{array}$$

where sign is a signum function and $|.|$ is the absolute value. In this case, the negative values of $\widehat{I}(X;Y)$ should not be interpreted as anti-correlations.

Figure 4 illustrates the concept of the nearest-neighbour MI method.
This MI implementation is capable of analysing two data sets with different
time resolutions. This motivates the adoption of the method in this study,
where the time resolution between the measured atmospheric variables and the
classification of aerosol particle formation days is not uniform. Hence, the
calculation of time-domain features, such as the mean and standard deviation,
is not required here. These features naturally compress the data and
typically lead to information loss. Panel (a) illustrates the time-series
measurement of an atmospheric variable for each day. Every single day can be
associated with two classes that are event (E) or non-event days (NE). It can
be seen that there are multiple measurements in a day, whereas there are only
single event–non-event data available for each day. The distances between the
measurement vectors themselves are then calculated as illustrated in panel (b). Here, we take the example of day index 100 (*D*_{100}).
Here, the distance between the measurement vectors at *D*_{100} from
the same class is calculated. In panel (c), the distance vector of
*D*_{100} calculated from the same class (event days) is then ranked
in ascending order, shown on the top line. In this particular case, the user
choice parameter *k*th closest neighbour is selected to be 3. So the
distance threshold is found at *D*_{98}. The distance vector from the same
class (the red sign) is then projected on the bottom line. The bottom line
contains the distance vector of *D*_{100} calculated from all classes. The
dashed line, representing the threshold from point distance *D*_{100} out to
the third neighbour, is drawn until the bottom line. After that, it is found
that the number of distance points, which is the third closest neighbour to
*D*_{100} on the top lines, is the seventh closest neighbour on the bottom line
(*m*=7). The above processes point out that the parameter *m* becomes a
crucial factor in the MI estimator, shown in the Eq. (13). This
parameter is obtained through the above processes involving the distances
calculation between different data resolution. This is advantageous in
computing MI between event classification data and atmospheric variable data,
which typically vary in different time resolution. In summary, besides its
effectiveness in estimating non-linear correlation, the nearest-neighbour MI
is also advantageous for the current problem because (1) it is a
non-parametric method making no assumptions about the functional form
(Gaussian or non-Gaussian) of the statistical distribution underlying the
data, (2) there is no need for computationally costly binning to generate
histograms, (3) it is computationally fairly light and (4) the model contains
only one free model parameter (*k*) and it is easy to tune.

Prior to demonstrating the result of MI application on the atmospheric data in Sect. 4, the following subsection discusses first how MI is capable of estimating a non-linear relationship, tested on a simulated physics equation.

MI capability in detecting a non-linear relationship between two variables on an artificial benchmark data set is already illustrated in Fig. 3. Before applying nearest-neighbour MI to real atmospheric data, this subsection shows another, more physical case study demonstrating how well MI is able to detect a non-linear relationship between two correlated variables.

We consider the intensity of blackbody radiation. The monochromatic emissive
power of a blackbody *F*_{B}(*λ*) (W m^{−2} µm^{−1}) is
related to temperature *T* and wavelength *λ* by
(Seinfeld and Pandis, 2016)

$$\begin{array}{}\text{(15)}& {F}_{\mathrm{B}}\left(\mathit{\lambda}\right)={\displaystyle \frac{\mathrm{2}\mathit{\pi}{c}^{\mathrm{2}}h{\mathit{\lambda}}^{-\mathrm{5}}}{{e}^{ch/k\mathit{\lambda}T}-\mathrm{1}}},\end{array}$$

where *k* is the Boltzmann constant ($k=\mathrm{1.381}\times {\mathrm{10}}^{-\mathrm{23}}$ J K^{−1}), *h*
is the Planck constant ($\mathrm{6.626}\times {\mathrm{10}}^{-\mathrm{34}}$ Js) and *c* is the speed of
light in vacuum ($c=\mathrm{2.9979}\times {\mathrm{10}}^{\mathrm{8}}$ m s^{−1}). The solar spectral
irradiance at the top of the Earth's atmosphere at 5777 K is shown in
Fig. 5a. If the temperature is varied (randomly between 10 and
10 000 K in this case), the solar spectral irradiance for the same range of
wavelengths looks quite different, as is shown in Fig. 5b. The
correlation level for both scenarios using PCC (again symbolized by *ρ*)
and the nearest-neighbour MI is also shown. It can be seen that, when the
temperature is fixed, the Pearson correlation is still able to detect the
correlation between wavelength and solar spectral irradiance, but fails in
detecting the relationship between these variables when the data are more
messy due to the variation in the temperature. On the other hand, MI is able
to detect the correlation between *λ* and *F*_{B}(*λ*) in both cases.

4 Results and discussion

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The results section is divided into two subsections. The first part presents the result of MI correlation analysis between atmospheric variables and NPF. The second part then discusses the scatter plot of several relevant atmospheric variables to NPF.

In this study, the atmospheric variables are continuous values while the aerosol formation days classification is discrete. Hence, we implemented the MI based on the nearest-neighbour method for finding the correlation between these two data sets, explained earlier in Sect. 3.3. MI attempts to find the best atmospheric factors/variables which differentiate between event and non-event days. In general, there is no specific level for MI or threshold that indicates a correlation between different variables, which is also similar to the Pearson correlation, where this correlation value gives an only indication of the variables relationship. The value of MI depends on the distribution and the amount data. Unless MI gives a very high value (very close to one) or a very low number (very close to zero), scientists need to make their own judgement about the variable correlation. In this case, similar variables are grouped based on their measurement types (traced gases, radiation, etc.), and their correlation level is ranked. The variables that have the highest MI level indicate that they are more favourable to the NPF process compared to other variables.

Figure 6 presents the correlation results in the form of bar charts, including gases and aerosols (top), meteorology (middle) and radiation (bottom). Several atmospheric variables are measured at different heights, such as gas concentrations and meteorological parameters. In this case, the mean and standard deviation of their MI correlation level were calculated. For those variables, the rectangular bar represents the mean of the MI correlation level, whereas the whisker is its two standard deviations. For the variables which are measured only at one particular height or location, their MI correlation is only represented as the rectangular bar without any whisker.

The top subplot in Fig. 6 shows the MI correlation level between
NPF and gas concentrations as well as aerosol (CS). It can be seen that the
water concentration (H_{2}O) has the highest correlation among others.
This finding is in agreement with those presented by
Boy and Kulmala (2002) and Hyvönen et al. (2005). The reason for the
high MI correlation between NPF occurrence and H_{2}O concentration has
so far not been explained. Whether this relation is truly causal or appears
because of correlations in diurnal or annual cycles of air masses related to
other NPF-related variables remains to be assessed in future studies. The
second highest correlation variable in this group is condensation sink. The
high correlation with CS can be expected, since CS describes the main sink
for vapours participating in NPF and it is also an effective sink for freshly
formed new particles. Previous studies have shown that the average value of
CS is typically lower on NPF days compared with non-event days
(Dal Maso et al., 2007; Asmi et al., 2011b; Dada et al., 2017). Furthermore, this subplot
shows that sulfuric acid (H_{2}SO_{4}), evaluated using two proxies,
correlates well with NPF. It is known that H_{2}SO_{4} is one of the
key vapours participating in NPF (Kulmala et al., 2013). The correlation
between NPF and H_{2}SO_{4} has been proven through analysis on the
data obtained from a number of measurement sites
(Kuang et al., 2008; Nieminen et al., 2009; Paasonen et al., 2010; Wang et al., 2011)
as well as in laboratory experiments (Almeida et al., 2013).

The MI found that ozone (O_{3}) and carbon dioxide (CO_{2})
might be related to the NPF process. The correlations of these variables were
also indicated by Hyvönen et al. (2005) via a *K*-means clustering method.
The correlation with O_{3} is probably related to the formation of
extremely low volatile organic compounds (ELVOCs), which can be initiated by
the ozonolysis of monoterpenes (Ehn et al., 2014). ELVOCs are presumed to
participate in NPF. The correlation with CO_{2}, on the other hand,
might be related to the coupling between photosynthesis and emission of
monoterpenes, as suggested by Kulmala et al. (2014).

On the other hand, the result suggests that sulfur dioxide (SO_{2}) and
nitrogen oxides (NO_{x}) do not correlate strongly with NPF.
The SO_{2} observation is inconclusive: its concentration has been found
to be higher for NPF event days in some studies
(Boy et al., 2008; Young et al., 2013) and lower in others
(Wu et al., 2007; Dai et al., 2017). Previously, Boy and Kulmala (2002)
already stated that, in the cases of SO_{2} and NO_{x} at
this measurement site, there are no significant differences found between
event and non-event days.

The middle subplot presents the MI correlation level for all measured
meteorological variables. Some variables with the subscript “ave” are
averages of meteorological variables measured at different heights. As the
top subplot, we calculated the mean and standard deviation of their MI
correlation level and display them as a rectangular bar with a whisker. The
middle subplot shows that there is a very strong correlation between NPF and
relative humidity (RHURAS_{ave} as well as RHTd). A
similar result was also reported in Hyvönen et al. (2005). On NPF event
days, the average ambient RH is typically lower than non-event days in both
clean and polluted environments
(Vehkamäki et al., 2004; Hamed et al., 2007; Jun et al., 2014; Qi et al., 2015; Dada et al., 2017). High values of RH tend to have a
negative influence on the solar radiation intensity, photochemical reactions
and atmospheric lifetime of aerosol precursor vapours (Hamed et al., 2011).
Our result points out that the temperature (*T*_{ave} and
*T*_{d}) correlates with NPF, as also observed by
Boy and Kulmala (2002) and Hyvönen et al. (2005) for this site. The
relationship between NPF and temperature may take place due to indirect
influences from other factors. For instance, NPF often takes place during the
sunny days, when the radiation level and temperature are relatively high. The
temperature connection may also occur due to its influence in some chemical
reactions leading to NPF. One example might be related emissions of
monoterpenes (Tunved et al., 2006; Kiendler-Scharr et al., 2009), which is known as a
strong function of temperature (Guenther et al., 1995). However, the
temperature is associated with so many atmospheric variables (e.g. boundary
layer height, turbulence, radiation, RH and the volatility of the vapours)
that the correlation might be caused by several different variables.

In contrast, wind speed (WS_{ave} and
WSU_{ave}) and wind direction (WD_{ave} and
WDU_{ave}) have little correlation with NPF. Similar results
were also reported by Boy and Kulmala (2002). They stated that the small
correlation persists due to pollution from the west–southwest (station
building and city of Tampere). The correlations between NPF and rain
indicator (SWS) as well as the atmospheric pressure (Pamb0)
at Hyytiälä were also found to be weak. Several other meteorological
variables (not displayed) were excluded from the analysis due to the data
scarcity. It is also important to note that on both subplots (top and middle)
the whiskers for most bar variables are very short. This means that the MI
correlation level for the same variables measured at various heights is
similar. The whisker for wind speed (WSU_{ave}) is slightly
longer because the measured wind speed varies moderately at different
heights.

The bottom subplot shows the MI level of several radiation variables. It can
be seen that most radiation variables have a strong relation with NPF. This
fact was discussed earlier by Boy and Kulmala (2002), especially on the
variable ultraviolet A (UV_{A}). The high level of correlation
in the global radiation (Glob) was also found by
Hyvönen et al. (2005). In all measurement sites, the average solar
radiation intensity tends to be higher on NPF event days compared with
non-event days (Birmili and Wiedensohler, 2000; Vehkamäki et al., 2004; Hamed et al., 2007; Kristensson et al., 2008; Pierce et al., 2014; Qi et al., 2015; Wonaschütz et al., 2015). Radiation is known as
the driving force for atmospheric chemistry, producing low-volatility vapours
(e.g. sulfuric acid, ELVOCs) that participate in NPF.

The correlation between concentrations of particles with different sizes from
3 to 1000 nm and NPF is illustrated as a coloured panel in
Fig. 7. There are four columns in the *x* axis. The first three
columns represent three periods between years 1996 and 2014, where each
period comprises the correlation level for 6 years. The last column is the
total correlation level for 18 years. The period division observes if the
correlation level for all periods is similar and consistent. The *y* axis
shows the aerosol particle sizes. There are 51 ranges of particles size in
the *x* axis of the coloured panel, but we downsample the 51 particles
size ranges to be only 11 sizes for simplification. The colour bar
represents the MI correlation level between the specified aerosol particles
and NPF. It can be seen that NPF correlates very well with particles in the
nucleation mode size range (3–25 nm). This can be expected, since in a
relatively clean environment, such as Hyytiälä, NPF is the main
source of nucleation mode particles. Clear correlations between particle
concentrations and the NPF event occurrence are also detected in the size
range from 150 to 550 nm. In this size range, the correlation can be
expected, since it is the concentration of these particles that has the
largest impact on the condensation sink. Thus, the high concentrations of
150–550 nm particles disfavour NPF and the correlation can be presumed to
be negative (see the explanation related to CS in the top panel of
Fig. 6).

In order to understand in depth the results from the aforementioned MI
analysis, a scatter plot matrix was generated, as shown in
Fig. 8. The plot involves some of the most important
atmospheric variables in the NPF process, according to via the MI analysis
made in the previous subsection, including the sulfuric acid concentration
(H_{2}SO_{4}), average temperature (*T*_{ave}), relative humidity
(RHTd), global radiation (Glob) and condensation sink. The logarithm was
applied to the variables H_{2}SO_{4} and CS to ease the scatter plot
visualization. The red and blue dots in the plot represent event and
non-event days, respectively. Along the diagonal are histogram plots of each
column of *x*. Here, the same data as the above study were used (e.g.
18 years SMEAR II data sets). The undefined days were excluded. Next, the
daily mean of all measurements during the daytime was computed and then
normalized (between 0 and 1). Finally, for MI comparison, we performed a
linear correlation to analyse the relationship between atmospheric variables
and event–non-event days. Since the latter is a dichotomous variable (i.e.
it contains two categories or discrete), we used the point-biserial
correlation coefficient (*r*_{pb}), which is mathematically equivalent
to PCC (Howell, 2012). This correlation coefficient is
displayed on each histogram.

First, we focus on the histogram plots located on the sub-axes along the
diagonal. It can be seen that the event and non-event days are well separated
in the cases of Glob and RHTd. These histogram plots demonstrate very well
that NPF has a positive (*r*_{pb}=0.639) and negative
(${r}_{\mathrm{pb}}=-\mathrm{0.707}$) correlation with the variables Glob and RHTd,
respectively. When the value of global radiation is high, NPF days are likely
to occur. On the other hand, non-event days tend to take place when RH is
high. The correlations between NPF and these variables were found earlier by
MI in the previous subsection. This fact supports the view that the MI method
is an effective tool to provide early correlation detection between
atmospheric variables and NPF.

The next focus is on the variables H_{2}SO_{4}, CS and T_{ave}.
The variable H_{2}SO_{4} can still be detected through the linear
correlation method (*r*_{pb}=0.403). The histogram plot of
H_{2}SO_{4} shows that NPF event days do not take place when the
concentration of H_{2}SO_{4} is very low, whereas the event days usually
occur when it is high. However, both event and non-event days may take place
if the H_{2}SO_{4} concentration level is medium (i.e. see the
intersection between the red and blue histograms). Nevertheless, the scatter
plots between Glob, RHTd and H_{2}SO_{4} indicate that these variables
are connected in the process of NPF. It is known that the formation of 3 nm
particles occurs on the days with strong solar radiation. In other words, to
form H_{2}SO_{4} in the atmosphere, high solar radiation is typically
required. Likewise, high H_{2}SO_{4} concentration in the atmosphere
increases cluster formation and growth rate and hence favours the occurrence
of an NPF event (Almeida et al., 2013; Kulmala et al., 2013). On the other
hand, when RHTd value is high, the radiation is typically low and therefore
the H_{2}SO_{4} concentration also tends to be low.

The above conclusion would be very challenging to make by using a linear
correlation analysis for variables *T*_{ave} (*r*_{pb}=0.134) and
CS (*r*_{pb}=0.007). These correlation coefficients do not reveal that
the variables are related to NPF, which we know from previous literature
results. Likewise, by observing the histogram plots, both event and non-event
days may take place on any values of *T*_{ave}, except in very low or
very high temperature regimes. This situation is also similar for the case of
CS for which event and non-event days are not separable. Since the histogram
plots of CS and *T*_{ave} present the complication in understanding
their connection with NPF, their scatter plots should also be analysed. For
instance, the event and non-event days seem to be separated on the scatter
plots between *T*_{ave} and Glob as well as RHTd, where the last two
variables are known to be correlated with NPF. This may explain how they are
connected, but their correlation may be non-linear, since the separation
takes place in the middle of the plot. Likewise, the scatter plots between CS
and the variables Glob, RHTd and H_{2}SO_{4} show a separation between
the event and non-event days. Even though the separation is not perfect, this
may still clarify how they are connected.

This subsection demonstrates the analysis complexity by observing the
histogram and scatter plots for some variables, such as H_{2}SO_{4},
*T*_{ave} and CS. The intricacy might occur because the relationship
among some atmospheric variables and NPF may be complex, non-linear or
indirect, in addition to which there might be other variables influencing the
process of NPF. CS and temperature are known to impact NPF directly or
indirectly, as discussed in Sect. 4.1. A sole investigation through
linear correlation analysis, histogram and scatter plots for finding the
relationship among atmospheric variables sometimes poses a challenge. This
problem explains why MI should be used in the first place for finding early
correlation detection between atmospheric variables and their phenomena.

5 Conclusions

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This paper extends and complements the analysis of a previous data mining
study on atmospheric data, conducted by Hyvönen et al. (2005). Both papers
exploit the strengths of data-driven methods, but there are two notable
distinctions between this study and the previous investigation. First, our
work utilizes 18 years (1996–2014) of atmospheric measurements from the
SMEAR II station in Hyytiälä, Finland. This means that the current
work deals with 10 more years of data. The utilization of a larger data set
is expected to provide more reliable results and thus a more accurate
conclusion. Second, instead of using data mining methods based on clustering
and classification, this paper promotes the use of MI for identifying the key
variables in atmospheric aerosol particle formation. The applied
nearest-neighbour MI method is a powerful and computationally light tool
capable of finding both linear and non-linear relationships between the
measured atmospheric variables and observed NPF events. The method also
contains only one free parameter (the number of nearest neighbours, *k*) and
its value does not affect the results significantly (Ross, 2014).
Furthermore, the method operates directly on the data and does not require
the calculation of characterizing compressed features (i.e. mean, standard
deviation) which might potentially lead to a partial information loss.

The MI method reports very similar findings with the previous atmospheric studies. The water content and sulfuric acid concentration are found to be strongly correlated with NPF. Furthermore, the results also suggest that NPF is influenced by temperature, relative humidity, CS and radiation. According to the results from the MI analysis, the measurements taken at different heights have similar correlation with NPF.

As shown in the previous subsection, this method is more powerful than a linear correlation analysis. Therefore, this method should be used in the first place before performing a deeper data analysis method, such as through histogram and scatter plots. This method could act as an early correlation detection for any atmospheric variables.

This work uses the longest available data sets of NPF observations with simultaneously measured ambient variables. As future works, we will seek to investigate the use of the method on different atmospheric data sets. For instance, robust correlation analysis is required for understanding other variables influencing atmospheric process, such as volatile organic compounds (VOCs) and aerosol particles at sizes below 3 nm.

In order to enrich the analysis, the database from other SMEAR stations as well as previous research campaigns should be included. The data may contain more variation because they are measured in different locations. One anticipated obstacle is the scarceness of NPF days classification databases. Although an automatic classification algorithm to create such a database has been called for (Kulmala et al., 2012), currently the event–non-event days are labouriously classified using a manual visualization method (Dal Maso et al., 2005). There has been an attempt to use machine learning for automating aerosol database classification, but the performance has not been completely satisfactory yet (Zaidan et al., 2017). One possibility to enhance the performance of the machine learning classification is to use the correlated atmospheric variables found in this study as additional inputs for such models. Similar concept can also be applied in developing any atmospheric process or proxy. Proxy-dependent variables can be selected by finding the most correlated variables to the interested proxy via MI.

Data availability

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Data availability.

Data measured at the SMEAR II station are available on the following web page: https://avaa.tdata.fi/web/smart (last access: 5 August 2018).

Author contributions

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Author contributions.

MAZ, VH, RR, HJ and ASF designed the study. MAZ and RR developed the methodology. MAZ performed data and statistical analysis. VH, PP, VMK and MK contributed to the interpretation of the data and the results. VH and HJ suggested the use of experimental data from the SMEAR II station. All authors contributed to writing the manuscript.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

This work is supported by the European Research Council (ERC) via ATM-GTP (grant
number 742206) and the Academy of Finland Centre of Excellence in Atmospheric
Sciences (project number 307331).

Edited by:
Fangqun Yu

Reviewed by: two anonymous referees

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Short summary

This article promotes the use of the mutual information method for finding any non-linear associations among atmospheric variables. We demonstrate that the same results from previous studies are obtained by this method, which operates without supervision and without the need of understanding the physics deeply. This suggests that the method is suitable to be implemented widely in the atmospheric field to discover other interesting phenomena and their relevant variables.

This article promotes the use of the mutual information method for finding any non-linear...

Atmospheric Chemistry and Physics

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