GEOS-Chem is a global 3D chemical transport model (CTM) driven by assimilated meteorological observations from the Goddard Earth Observing System (GEOS) of the NASA Global Modeling and Assimilation Office (GMAO). Several research groups develop and use this model, which contains numerous state-of-the-art modules treating emissions and various chemical and aerosol processes e.g., for solving a variety of atmospheric composition research problems. The ISORROPIA II scheme is used to calculate the thermodynamic equilibrium of inorganic aerosols. Secondary organic aerosol formation and aging are based on the mechanisms developed by and . MEGAN v2.1 (Model of Emissions of Gases and Aerosols from Nature; ) implements biogenic emissions and GFED4 (Global Fire Emissions Database; ) implements biomass burning emissions in GEOS-Chem.

The present study uses GEOS-Chem version 10-01 with the implementation of the Advanced Particle Microphysics (APM) package , henceforth referred to as GCAPM. The APM model has the following features of relevance towards accurate simulation of CCN number concentrations: (1) 40 bins to represent secondary particles with a high size resolution for the size range important for growth of nucleated particles to CCN sizes ; (2) a state-of-the-art ternary ion-mediated nucleation (TIMN) mechanism and temperature-dependent organic nucleation parameterization ; (3) calculation of H2SO4 condensation and the successive oxidation aging of secondary organic gases (SOGs) and explicit kinetic condensation of low-volatility SOGs onto particles ; (4) contributions of nitrate and ammonium via equilibrium uptake and semi-volatile organics through partitioning to particle growth considered . CCN number concentrations simulated by GCAPM have previously been shown to agree well with measurements .

The horizontal resolution of GCAPM in this study is 2∘ × 2.5∘, with 47 vertical layers (14 layers from the surface to 2 km above the surface). The period of global simulation is 30 years from 1989 to 2018. For 47 sites spread across the globe, co-located GCAPM data are output at the half-hourly time step for all model layers in the troposphere. In the present application, we use those at six selected vertical heights: surface, ≈ 1, ≈ 2, ≈ 4, ≈ 6, and ≈ 8 km.

In the present study, we choose to use the random forest (RF) technique from the large suite of machine learning techniques, for the following reasons: (a) our objective of predicting (regressing) values of [CCN0.4], (b) the ease of physical interpretability of RF models, (c) ease of implementation, and (d) the ability to tune this supervised machine learning, which is learning by example.

A random forest is an ensemble of decision trees. A decision tree is a supervised machine learning algorithm that recursively splits the data into subsets based on the input variables that best split the data into homogeneous sets. This is a top-down “greedy” approach called recursive binary splitting. Decision trees are easy to visualize, are not influenced by missing data or outliers, and are nonparametric. They can, however, overfit on the data. Random forest modeling is an ensemble technique of growing numerous decision trees from subsets (bags) of the training data and then using all the decision trees to make an aggregated (typically mean) prediction. This approach corrects for the overfitting of single decision trees. Additionally, the bootstrap aggregating bagging; allows for model validation during training, by evaluating each component tree of the random forest with the out-of-bag training examples (training data that were not subsetted in growing the decision tree). Random forest models are advantageous due to the component decision trees being able to resolve complex nonlinear relationships between predictor variables regardless of their interdependencies or cross correlations and the outcome to be predicted. Further, they are relatively easier to visualize and interpret as compared to black-box neural network or deep-learning methods. For the purpose of predictions, random forest models are one of the most accurate machine learning models with the ability to be trained fast due to the parallelizability of the growth of decision trees. For these reasons, random forest is our chosen machine learning tool.

We utilize a fast implementation of random forest models in R trained on the GCAPM modeled [CCN0.4] detailed in Sect. . Further details of model development and applications are in Sect. .

In this study, we use the Kendall rank correlation coefficient (τ) and mean fractional bias (MFB) as statistical estimators of correlation and deviation, respectively. These statistical parameters are more robust (as discussed later in this section) than the conventionally used Pearson product-moment correlation coefficient (r) and mean normalized error (MNE) (or similar parameters).

Pearson product-moment correlation coefficient (r) is as follows: 1r=∑i=1nCim-Cm‾Cio-Co‾∑i=1nCim-Cm‾2∑i=1nCio-Co‾2. And the Kendall rank correlation coefficient (τ) is as follows: 2τ=∑i=2nsignCim-Ci-1msignCio-Ci-1on2-12∑i=1ntimtim-1n2-12∑i=1ntiotio-1, where n is the sample size, C is the value, t is the number of ties in the ith group of ties, and []m denotes modeled and []o denotes observed values.

As noted in :

In the use of Pearson's r are the following assumptions: (1) continuous measurements with pairwise complete observations for the two samples being compared (2) absence of outliers (3) Gaussian distribution of values (4) linearity between the two distributions, with minimal and homogenous variation about the linear fit (homoscedasticity).

Kendall's τ is a nonparametric rank correlation coefficient that is not constrained by the assumptions in the use of Pearson's r. This parameter is also intuitive and simpler to interpret due to (a) the maximum possible value of +1 indicative of complete concordance and the minimum possible value of -1 indicative of complete discordance and (b) the ratio of concordance to discordance being (1+τ)/(1-τ) .

MNE and MFB are defined as 3MNE=1n∑i=1nCim-CioCioandMFB=1n∑i=1nCim-CioCim+Cio2, where n is the sample size, C is the value, []m denotes modeled and []o denotes observed values, and i is the ith of their n pairs.

As noted in :

In the use of MNE, it is assumed that observed values are true values and not just estimates. MNE can easily blow up to ∞ when observed values are very small. Further, positive bias is weighted more than negative bias. Related parameters such as Normalised Mean Bias and Error (NMB and NME) also suffer from these deficiencies. Mean Fractional Bias (MFB) is not limited by the issues in the use of MNE. As a measure of deviation, Mean Fractional Bias (MFB), ranging from [-2,+2], is symmetric about 0 and also not skewed by extreme differences in the compared values.

The following quantitative ranges are provided (arbitrarily) to qualitatively describe the degree of correlation: (1) poor agreement – τ≤0.2; (2) fair agreement – 0.2<τ≤0.4; (3) moderate agreement – 0.4<τ≤0.6; (4) good agreement – 0.6<τ≤0.8); and (5) excellent agreement – 0.8<τ≤1.0. Additionally, it is defined (arbitrarily; factor of 1.86 deviation) that there is good agreement between derived and expected values when the MFB is within [-0.6,+0.6].

Statistical analyses are performed using R: a freely available language and environment for statistical computing and graphics and with the aid of the “Kendall” and “pcaPP” packages.

For validation of the developed RFRM, we use in situ measurements of atmospheric state and composition as inputs to the RFRM and compare the output [CCN0.4] with its measurements. The U.S. Department of Energy's (DOE) ARM Southern Great Plains (SGP) Central Facility located in Lamont, Oklahoma (36∘36′18′′ N, 97∘29′6′′ W; 318 m; Fig. ) was established with the mission statement of “provid[ing] the climate research community with strategically located in situ and remote-sensing observatories designed to improve the understanding and representation, in climate and earth system models, of clouds and aerosols as well as their interactions and coupling with the Earth's surface”. We use data from this facility, which has the longest record of [CCN0.4] at the hourly resolution.

The GCAPM output for 47 sites across the globe, for six selected vertical levels from surface to ≈ 8 km and for 30 years (1989–2018) at the half-hour time step (≈ 150 million rows, i.e., sets of predictors and [CCN0.4]) is considered in training the RFRM. The predictors of importance in controlling [CCN0.4] are listed in Table . The RFRM is trained on a subset of this data. First, the ARM SGP site is ignored; this is to establish a completely independent analysis with available observational data in Sect. . The remaining GCAPM data for 46 sites and six vertical levels each are partitioned into training (≈ 101 million rows) and testing sets (≈ 44 million rows) in a 7:3 ratio. Due to the large number of training and testing examples, these sets are reduced to a 1 % random subset; it is ensured to be representative, with almost identical statistical properties, of the training datasets (Fig. ).

Once the data have been selected to train the RFRM, we tune the hyperparameters, which govern the training of the machine learning model. The default implementation of comes with reasonable (balancing speed and accuracy) choices for these. Based on literature review and references therein as well as our preliminary examination of RFRMs with varying hyperparameters, we identify the following as most important to optimize: numtrees – number of trees in the forest; mtry – the minimum number of variables to consider for each split; and min.node.size – the minimum node size, i.e., the minimum size of homogeneous data to prevent overfitting. By setting the minimum number of training examples in the terminal nodes of the component trees of the RF, the individual tree depth is controlled, which further mitigates the overfitting associated with decision tree algorithms (discussed in Sect. ). The default hyperparameter values in are as follows: numtrees = 500, mtry = rounded-down square root of the number of variables, and min.node.size = 5. We verify whether these hyperparameter choices are optimal by performing a grid search of the hyperparameters and training multiple random forest models and not just examining their performance with the training set but also additionally with the test set. By evaluating the RFRMs with the test set (data that the machine learning algorithm was not exposed to during its training), additional mitigation of possible overfitting is achieved. Figures  and show the results of this exercise.

Figure  has nine panels, as follows from top to bottom:

the top three are for varying number of trees in the random forest: numtrees from 1 (a single decision tree) to 1800, through the default choice of 500 (marked with vertical black dotted line);

From left to right, the panels are as follows:

the left three (green) show the overall out-of-bag error, i.e., the mean square error for the entire random forest computed using the complement of the bootstrapped data used to train each tree;

A single decision tree (left-most point in each of the top three panels of Fig. ) is able to explain the variance (R2≈0.81) in [CCN0.4] through the predictors' variabilities and has an interquartile range in the MFB (which has a median of ≈0.004: not pictured as the symmetry of MFB means the median → 0) of 0.41, which corresponds to a deviation of ≈±20%. However, this is drastically improved when moving beyond a simple decision tree to even a small ensemble of 30 trees (R2≈ 0.91; IQR(MFB) ≈ 0.27), which plateaus (within a range of ± 0.0005) after ≈ 500 trees. The out-of-bag (OOB) error shows a similar trend. Growing a random forest of 500 trees with a min.node.size of 5, we see the effect of varying mtry in the middle three panels. Instead, keeping the mtry fixed at the default value of 4; the rounded down square root of the 19 predictor variables and the random forest performance metrics with varying min.node.size are shown in the bottom three panels. Figure  thus shows the possibility of improving the random forest derivation of [CCN0.4] by changing the default choices of for this specific work. It must be kept in mind that a reasonable cutoff, beyond which there is imperceptible gain in performance at increased computational cost, should be considered.

The results shown in Fig.  motivate a zoomed-in hyperparameter grid search to choose the optimal (accurate and fast) RFRM. Figure  shows this for the best-performing random forest models with numtrees ranging from 600 to 1400, mtry from 6 to 18, and min.node.size from 3 to 6. While there is variability, it must be noted that the y axes range over 2 %, 0.2 %, and 2 % of the values of OOB error, R2, and IQR(MFB), respectively. While, indeed, considering a larger (numtrees) forest is beneficial, considering the cost-to-benefit ratio, the hyperparameters we choose are a maximum number of 800 trees in the forest, 12 (mtry) variables randomly chosen at each split, a minimum node size of 3 as the only control on tree depth, and the splitting rule as the minimization of variance. With these hyperparameters, the random forest has IQR(MFB), R2, and OOB error of ≈100.34%, ≈99.93%, and ≈100.56% of the best-performing model for each hyperparameter, respectively.

We train the optimized RFRM using the 19 predictors listed in Table  as predictors of [CCN0.4]; these are as follows: eight fractions of PM2.5 (ammonium, sulfate, nitrate, secondary organic aerosol (SOA), black carbon (BC), primary organic carbon (POC), dust, and salt), seven gaseous species (nitrogen oxides (NOx), ammonia (NH3), ozone (O3), sulfur dioxide (SO2), hydroxyl radical (OH), isoprene, and monoterpene), and four meteorological variables (temperature, relative humidity (RH), precipitation, and solar radiation).

Figure  shows the importance of each predictor in determining the CCN0.4 number concentration in the above-trained RFRM. This importance measure is obtained by randomly permuting values of each predictor to break the association with CCN0.4. Also, the model is fed a pseudo-predictor of randomly generated white noise, labeled “Random” in Fig. . Most important are component mass fractions of PM2.5, especially its inorganic fraction (ammonium, sulfate, and nitrate), [SO2], the other PM2.5 fractions excluding the salt and dust fractions, [NOx], and [NH3]. The Random predictor is least important, contributing imperceptibly to the [CCN0.4] prediction.

To quantify the importance of each of the 19 predictors, we do the following: (1) RFRM-Blind – train RFRMs without considering one variable at a time; (2) RFRM-Random – randomize each predictor variable and input into the RFRM. These exercises will provide a more robust, as well as more intuitive measure of the importance of each predictor by analyzing and comparing the deviation in predicted [CCN0.4] compared to the baseline optimized RFRM (hereafter, RFRM-AllVars). Table  shows the median ± median absolute deviation of MFB for each such trained RFRM-Blind and RFRM-Random evaluated with the test dataset, with the values for the baseline RFRM-AllVars at the bottom. For the MFB, its median absolute deviation (and even IQR(MFB)) is a stronger indicator of the performance than the median, due to the symmetry of MFB. Examining the median absolute deviation of the MFB, the “Blind” approach shows that the ammonium (PM2.5NH4), sulfate (PM2.5SO4), and secondary organic (PM2.5SOA) fractions of PM2.5 are most important (in increasing order) in determining [CCN0.4]. The Blind approach may, however, underestimate the importance of a predictor due to possible correlations with other predictors, as seen in Fig. . These cross correlations could mean the implicit participation of a predictor despite its absence in training the RFRM. To overcome this limitation, in the Random approach, the trained RFRM-AllVars is then input randomized predictors (one at a time) from the testing dataset. This breaks the association of each predictor with the outcome ([CCN0.4]) as well as with other predictors. The resulting change (if any) in the RFRM-AllVars would show the importance of the specific predictor. RFRM-Random for each predictor shows that all predictors (except solar radiation) are important, with the most important being PM2.5SOA, PM2.5SO4, and PM2.5NH4.

Table  thus shows the importance of each predictor towards determining [CCN0.4] in decreasing order, which complements the results in Fig.  that shows the out-of-bag increase in mean square error upon the permutation of a specific predictor. We modify the approach in (a) to leverage the advantages of the MFB; (b) to account for any implicit correlations between the data used to train (bagged) and evaluate (out-of-bag) the RFRM by using the unseen testing dataset; and (c) to dissociate the effects of cross correlations between predictors. The results of this modified evaluation of the RFRM are in Table . These exercises to probe into the working of the RFRM show that all of the predictors were deemed necessary to capture [CCN0.4] magnitude and variability. The most important predictors are the PM2.5 speciated components, gases including [SO2], [O3], and [NOx], and temperature and relative humidity.

There are a number of limitations in the application of the present study to augment empirical measurements of [CCN]. These are as follows:

The RFRM is trained on GCAPM, with the assumption that the physical and chemical processes that relate the “predictor” variables to the [CCN0.4] outcome are accurate. Previous studies show that GCAPM performs reasonably when compared to observations, but uncertainties in both model and observation may contribute to uncertainties in the RFRM derivation.

Despite the caveats associated with this work, this proof of concept shows promise for wide-ranging development and deployment. This machine learning approach can provide improved representation of cloud condensation nuclei numbers:

in locations where their direct measurements are limited, but measurements of other atmospheric state and composition variables are available. Typically, since measurements of PM2.5 speciation, trace gases, and meteorology are easier than those of [CCN0.4], there are longer and more widely (spatially) distributed in situ measurement records, especially through air quality monitoring networks. The developed RFRM can derive [CCN0.4] from these ubiquitous measurements to complement [CCN0.4] measurements when available and fill in the gaps in their absence.

This work is an initial step towards fast and accurate derivation of [CCN0.4], in the absence of its measurements, constrained by empirical data for other measurements of atmospheric state and composition. This work demonstrates the possible applications of machine learning tools in handling the complex, nonlinear, ordinal, and large amounts of data in the atmospheric sciences.