3.1 Random forest regressor algorithm
3.1随机森林回归器算法

As the aim here is to predict a continuous numerical value for sea-surface iodide, a regression approach is taken. As discussed in the introduction, previous approaches have been made to parameterise sea-surface iodide, and the most commonly used relationships employ sea-surface temperature as the predictor variable. Here we take a different multivariate and non-parametric approach, using the computationally cheap and interpretable random forest regressor (RFR) algorithm (Breiman, 2001; Pedregosa et al., 2011).
由于此处的目的是预测海面碘化物的连续数值，因此采用回归方法。正如引言中所讨论的，之前已经采用了参数化海面碘化物的方法，最常用的关系式采用海面温度作为预测变量。在这里，我们采用不同的多变量和非参数方法，使用计算成本低且可解释的随机森林回归（RFR）算法（ Breiman ， 2001 ； Pedregosa 等人， 2011 ） 。

Random forest regression is based on finding a number of decisions trees, which predict the dependent variable. All of the trees contribute to the prediction, and they are collectively referred to as a “forest”. These trees can be explained as a record of the way the algorithm has linearly traversed a subset of the training data, splitting the data into two parts at each decision point or “node” in a way that minimised the internal differences of the parts. The best split is chosen between the available variables based on an error metric (e.g. mean square error), and this process is continued until a criterion of purity is reached or a minimum number of data points are left from a split. This is essentially a classification problem. The prediction of the forest is the mean value of the prediction of all of the different decision trees, which attempts to make the results more of a regression problem. More details of this approach can be found in Friedman et al. (2009).
随机森林回归基于查找多个决策树，这些决策树预测因变量。所有树木都对预测做出贡献，它们统称为“森林”。这些树可以解释为算法线性遍历训练数据子集的方式的记录，在每个决策点或“节点”将数据分成两部分，以最小化各部分内部差异的方式。基于误差度量（例如均方误差）在可用变量之间选择最佳分割，并且继续该过程直到达到纯度标准或分割留下最小数量的数据点。这本质上是一个分类问题。森林的预测是所有不同决策树的预测的平均值，它试图使结果更像是一个回归问题。有关此方法的更多详细信息，请参见Friedman 等人。 （ 2009 ） 。

This approach differs from previous approaches which have individually tested proposed relationships and selected the best-performing model(s) as a parameterisation (e.g. Table A1). Here, an algorithm uses the data it is provided to build a model that gives a prediction, and therefore it is the data themselves that define the model that is used to predict new values. A key difference of this approach is also that only a subset, the training set, is used to build the model, and the rest (or withheld set) is then used to test the performance of the model. Here we use 80  % of the data for the training set and use the remaining 20 % as the withheld set (also commonly referred to as the “testing set”).
该方法不同于先前的方法，先前的方法单独测试了所提出的关系并选择性能最佳的模型作为参数化（例如表A1 ）。在这里，算法使用所提供的数据来构建给出预测的模型，因此数据本身定义了用于预测新值的模型。这种方法的一个关键区别还在于，仅使用一个子集（训练集）来构建模型，然后使用其余部分（或保留集）来测试模型的性能。这里我们使用 80% 的数据作为训练集，使用剩余的 20% 作为保留集（通常也称为“测试集”）。

To ensure that the models built are generalisable and mitigate overfitting, the random forest approach used here artificially increases the randomness within the forest (Pedregosa et al., 2011). This is done by randomly combining single decision trees by an approach referred to as “bootstrap aggregation” or “bagging” (Breiman, 2001; Tong et al., 2003). This additional bagging approach randomly samples observations within the training dataset and so mitigates overfitting of the trees to the dataset (Friedman et al., 2009). Furthermore, a stratified sampling approach was taken. Specifically, the overall dataset was split into quartiles according to iodide concentration value, and training data were randomly selected from each quartile. This approach was used to maintain the same statistical distribution in the training/testing data as the overall dataset.
为了确保建立的模型具有普适性并减轻过度拟合，这里使用的随机森林方法人为地增加了森林内的随机性（ Pedregosa et al. , 2011 ） 。这是通过称为“引导聚合”或“装袋”的方法随机组合单个决策树来完成的（ Breiman ， 2001 ； Tong 等人， 2003 ） 。这种额外的装袋方法对训练数据集中的观察结果进行随机采样，从而减轻了树与数据集的过度拟合（ Friedman 等人， 2009 ） 。此外，还采取了分层抽样的方法。具体来说，根据碘化物浓度值将整个数据集分成四分位数，并从每个四分位数中随机选择训练数据。这种方法用于保持训练/测试数据与整个数据集相同的统计分布。

Machine learning algorithms can generally be tuned to increase performance using settings called hyperparameters. However, random forests are known to generally perform well without tuning. The default hyperparameters were therefore used here (Pedregosa et al., 2011), except for increasing the number of trees (“n_estimators”) from 10 to 500. Mean square error (MSE) was used as the criterion for evaluating each split (also referred to as a “node”). The maximum number of “features” (the ancillary variables provided to the algorithm, such as temperature or nitrate concentration) considered when looking for the best split is set to the number provided to the algorithm. The number of splits a tree is allowed to make (“max_depth”) is not restricted, and further nodes are made until leaves contain less than two samples (“min_samples_split”) and a minimum of one (“min_samples_leaf”). All the random forest models are built using bootstrapping.
通常可以使用称为超参数的设置来调整机器学习算法以提高性能。然而，众所周知，随机森林通常无需调整即可表现良好。因此，这里使用默认的超参数（ Pedregosa et al. ， 2011 ） ，除了将树的数量（“n_estimators”）从 10 增加到 500 之外。均方误差（MSE）被用作评估每个分割的标准（也称为“节点”）。寻找最佳分割时考虑的“特征”（提供给算法的辅助变量，例如温度或硝酸盐浓度）的最大数量设置为提供给算法的数量。允许树进行分割的数量（“max_depth”）不受限制，并且会生成更多节点，直到叶子包含少于两个样本（“min_samples_split”）且至少包含一个样本（“min_samples_leaf”）。所有随机森林模型都是使用引导构建的。

3.2 Error and uncertainty calculations
3.2误差和不确定性计算

Understanding the errors and uncertainties in the global iodide distribution is important due to any sensitivities to this value within the modelled Earth system. We consider three sources of error in our predictions: the “dataset selection” error due to the splitting of the dataset into training and withheld parts, the “model selection error” due to the choice of dependent variables, and the “observational error” on the iodide measurements.
由于模拟地球系统内对该值的敏感性，了解全球碘化物分布的误差和不确定性非常重要。我们在预测中考虑三个误差来源：由于将数据集分为训练部分和保留部分而导致的“数据集选择”误差，由于因变量的选择而导致的“模型选择误差”，以及由于变量的选择而导致的“观察误差”碘化物测量。

To quantify the range of the dataset selection error, we construct models from 20 pseudo-random splits of the dataset into training and withheld parts. The hyperparameters and input ancillary variables are kept the same for the generation of the 20 models, so that the only difference between the models is the training dataset. These 20 models are then used to predict the withheld data. Performance metrics (e.g. root mean square error (RMSE) and average absolute prediction) can then be calculated for each model. This gives a range of 20 values, which can then be converted to a percentage range as the error. This is done by dividing the largest range in predicted values for a model by the minimum predicted value, to give a maximum value for the range of error, and then taking the smallest range in the predicted values and dividing this by the maximum value, to give a minimum value for the error range. Significant differences between the model's performance metrics would suggest important sensitivity to the training/withheld dataset splits.
为了量化数据集选择误差的范围，我们将数据集的 20 个伪随机分割构建为训练部分和保留部分。生成 20 个模型时，超参数和输入辅助变量保持相同，因此模型之间的唯一区别是训练数据集。然后使用这 20 个模型来预测保留的数据。然后可以计算每个模型的性能指标（例如均方根误差（RMSE）和平均绝对预测）。这给出了 20 个值的范围，然后可以将其转换为百分比范围作为误差。这是通过将模型预测值的最大范围除以最小预测值来完成的，以给出误差范围的最大值，然后取预测值的最小范围并将其除以最大值，得到给出误差范围的最小值。模型性能指标之间的显着差异表明对训练/保留数据集分割的重要敏感性。

We define the “model selection” error as the uncertainty resulting from the choice of input ancillary variables. A number of combinations of input variables are possible in generating the models, and each will generate a different prediction. We quantify this error as the difference in performance against the withheld dataset and prediction value (e.g. average global value). Similarly to our calculation of the dataset selection error, this can be converted to percentage error by considering the range in these values and dividing them by minimum and maximum values.
我们将“模型选择”误差定义为由于选择输入辅助变量而产生的不确定性。在生成模型时可以使用多种输入变量组合，并且每种组合都会生成不同的预测。我们将此误差量化为与保留数据集和预测值（例如平均全局值）的性能差异。与我们计算数据集选择误差类似，可以通过考虑这些值的范围并将它们除以最小值和最大值来将其转换为百分比误差。

For the observational error we refer to Chance et al. (2019a), who provide individual error estimates for each of the iodide observations in the data compilation. Over half (51 %) of the data points have an error of 5 % or less, and a further ∼25 % have an uncertainty in the range of 5 %–10 %. We therefore use a value of 10 % as a conservative estimate of the observational error.
对于观察误差，我们参考Chance 等人。 ( 2019 a ) ，他们为数据汇编中的每个碘化物观测值提供了单独的误差估计。超过一半 (51%) 的数据点的误差为 5% 或更少，另外约 25 % 的数据点的不确定性在 5%–10% 范围内。因此，我们使用 10% 的值作为观测误差的保守估计。

3.3 Outlier identification and removal
3.3异常值识别和去除

Our dataset consists of values for ancillary variables and iodide concentration for all of the 1342 measurement locations in the observational dataset (Sect. 2). As discussed in Sect. 3.1, we split this dataset into two parts: (i) a training set for use in building and optimising models, and (ii) a withheld set to evaluate the models built. Particular care was taken to ensure the withheld and training datasets were representative of the entire dataset in the way the models are built, therefore improving performance and generalisability to unseen data (see Sect. 3.1).
我们的数据集由观测数据集中所有 1342 个测量位置的辅助变量值和碘化物浓度组成（第2节）。正如第 3 节中所讨论的那样。 3.1中，我们将该数据集分为两部分：（i）用于构建和优化模型的训练集，以及（ii）用于评估构建的模型的保留集。我们特别注意确保保留数据集和训练数据集以模型构建的方式代表整个数据集，从而提高性能和对未见数据的通用性（参见第3.1节）。

We take a random forest regressor model built with variables that were intuitively assumed to give a reasonable ability to differentiate the observations (using depth, temperature, and salinity as the independent variables – abbreviated to “RFR(DEPTH+TEMP+SAL)” following Table 1). The RFR(DEPTH+TEMP+SAL) model was then used to explore the variation of error in the predictions using the dataset selection error approach described in Sect. 3.2. This builds multiple versions of the same model with different splits of training and test data and yields a distribution of root mean square error in the predicted iodide for withheld data as summarised in the final column of Table 2 and shown graphically in Appendix Fig. A1.
我们采用随机森林回归模型，该模型使用变量构建，直观地假设这些变量具有区分观测结果的合理能力（使用深度、温度和盐度作为自变量 - 缩写为“RFR(DEPTH+TEMP+SAL)”，如下表1 ）。然后使用 RFR(DEPTH+TEMP+SAL) 模型来探索预测中误差的变化，使用第 1 节中描述的数据集选择误差方法。 3.2 .这使用不同的训练和测试数据分割构建了同一模型的多个版本，并产生了保留数据的预测碘化物的均方根误差分布，如表2最后一列中总结并​​在附录图A1中以图形方式显示。

We define outliers here as values greater than the third quartile plus 1.5 times the interquartile range (Frigge et al., 1989). Removing these 49 values categorised as outliers (>309.5 nM) leads to a vast improvement in the RMSE error in the ensemble prediction from 95.1 to 37.6 nM (Table 2). This is shown graphically in Appendix Fig. A1, with the other subsets of the data explored (Table 2). This demonstrates that the high values are not well enough represented by the dataset to be able to be captured by the RFR approach. The removal of these high values from the dataset can also be justified as the driver for these concentrations is not yet well understood (Chance et al., 2014, 2019c; Cutter et al., 2018).
我们在这里将异常值定义为大于第三个四分位数加上 1.5 倍四分位距的值（ Frigge 等， 1989 ） 。删除这 49 个被分类为异常值 ( >309.5 nM) 的结果是，集合预测中的 RMSE 误差从 95.1 nM 大幅改善至 37.6 nM（表2 ）。附录图A1中以图形方式显示了这一点，并探索了其他数据子集（表2 ）。这表明数据集不足以很好地表示高值，无法通过 RFR 方法捕获。从数据集中删除这些高值也是合理的，因为这些浓度的驱动因素尚未得到很好的理解（ Chance 等人， 2014，2019 c ； Cutter 等人， 2018 ） 。

Removing these outliers reduces RMSE in the prediction with the 20 independent model builds from 48.2 to 2.3 nM (third quartile–first quartile). Once these outliers are excluded, more modest changes in average RMSE are then seen if models are built only using coastal or non-coastal data. Figure A1 also shows this is seen when removing lower salinity data (“Salinity ≥30 PSU and no outliers”), which is indicative of estuarine water. This highlights the strength in this approach's ability to predict iodide in different biogeochemical regions (i.e. not just coastal or non-coastal locations).
删除这些异常值可将 20 个独立模型构建的预测中的 RMSE 从 48.2 降低到 2.3 nM（第三个四分位数 - 第一个四分位数）。一旦排除这些异常值，如果仅使用沿海或非沿海数据构建模型，则平均 RMSE 会出现更温和的变化。图A1还显示了在删除较低盐度数据（“盐度≥30 PSU 并且没有异常值”）时看到的情况，这表明是河口水。这凸显了该方法预测不同生物地球化学区域（即不仅仅是沿海或非沿海地区）碘化物的能力。

An additional removal of a single dataset of 19 observations from the Skagerrak strait (Truesdale et al., 2003) was made due to it exerting a disproportionate influence on iodide prediction in high northern latitudes (≥65∘$\ge \mathrm{65}{}^{\circ }$ N), an area that is almost entirely unconstrained by local observations. We note that the Skagerrak is relatively unusual oceanographically, being an estuarine location with high ship traffic, and is considered unlikely to be an analogue for iodine speciation in the Arctic. This is decision is discussed further in the Appendix (Sect. A1), and the predictions made including this dataset are also provided in the shared output (Sect. 5).
额外删除了斯卡格拉克海峡的 19 个观测值的单个数据集（ Truesdale 等， 2003 ），因为它对北高纬度地区的碘化物预测产生了不成比例的影响（ ≥65∘$\ge \mathrm{65}{}^{\circ }$ N），一个几乎完全不受当地观测限制的区域。我们注意到，斯卡格拉克海峡在海洋学上相对不寻常，是一个船舶交通频繁的河口位置，并且被认为不太可能与北极的碘形态类似。这一决定将在附录（ A1节）中进一步讨论，并且包括该数据集在内的预测也在共享输出中提供（第5节）。

From here, only the 1293 observational points excluding outliers and the data from the Skagerrak strait (Truesdale et al., 2003) are used.
从这里开始，仅使用排除异常值的 1293 个观测点和斯卡格拉克海峡的数据（ Truesdale 等， 2003 ） 。

3.4 Selection of ancillary variables and building an ensemble prediction
3.4辅助变量的选择和建立集合预测

To decide which ancillary variables (temperature, salinity, etc.; see Table 1 and Sect. 2) should be used to predict sea-surface iodide concentration, RFR models were built and evaluated with different combinations of variables. Thirty eight combinations were considered (see first column of Appendix Table A2).
为了确定应使用哪些辅助变量（温度、盐度等；参见表1和第2节）来预测海面碘化物浓度，建立了 RFR 模型并使用不同的变量组合进行评估。考虑了 38 种组合（参见附录表A2第一列）。

The top 20 performing models, based on their root mean square error against the withheld data, are plotted in Fig. 2, alongside existing parameterisations. The standard deviation for all predicted values is also shown to illustrate variation in the predictions. A complete list of the performance and of all models built here and their performance is given in the appendix (Table A2).
根据保留数据的均方根误差，表现最好的 20 个模型与现有参数一起绘制在图2中。还显示所有预测值的标准差以说明预测的变化。附录（表A2 ）中给出了此处构建的所有模型及其性能的完整列表。

Figure 2Random forest regression (RFR) model performance (root mean square error (RMSE), blue) against the withheld data for the top 20 models on the left-hand y axis, along with values from the parameterisations from Chance et al. (2014) and MacDonald et al. (2014). The right-hand y axis is standard deviation of the prediction for the withheld data (orange). The top 10 performing models and the two exiting parameterisations considered here (Chance et al., 2014; MacDonald et al., 2014) are shown in bold. Parameterisations are ordered by their RMSE. Abbreviations are given in Table 1.
图 2随机森林回归 (RFR) 模型性能（均方根误差 (RMSE)，蓝色）与左侧y轴上前 20 个模型的保留数据，以及来自Chance 等人的参数化的值。 （ 2014 ）和麦克唐纳等人。 （ 2014 ） 。右侧y轴是保留数据（橙色）的预测的标准偏差。这里考虑的前 10 个性能模型和两个现有参数化（ Chance 等人， 2014 年； MacDonald 等人， 2014 年）以粗体显示。参数化按其 RMSE 排序。表1中给出了缩写。

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The RMSE values in Fig. 2 show the increased skill present in the new predictions compared to the existing parameterisations. The RMSE improves from the 75.3 and 50.2 nM found for the Chance et al. (2014) and MacDonald et al. (2014) parameterisations, respectively, to 33.2–37.4 nM for the top 10 models created here. Only modest gains are seen in RMSE between models with three variables or more.
图2中的 RMSE 值显示，与现有参数化相比，新预测中的技巧有所提高。 RMSE 较Chance 等人发现的 75.3 和 50.2 nM 有所提高。 （ 2014 ）和麦克唐纳等人。 ( 2014 )将此处创建的前 10 个模型的参数分别设置为 33.2–37.4 nM。具有三个或更多变量的模型之间的 RMSE 仅出现适度的增长。

The best-performing model in the list is only marginally better than the 10th best-performing one; therefore there is not an obvious “best” performing set of ancillary variables. Thus going forward we use an ensemble prediction approach based on the mean value from an ensemble of the 10 top-performing models.
列表中性能最好的模型仅比性能排名第 10 的模型稍好一些；因此，不存在明显的“最佳”表现辅助变量集。因此，接下来我们将使用基于 10 个表现最好的模型的集合平均值的集合预测方法。

3.5 Model descriptions
3.5型号说明

Unlike many machine learning approaches, the random forest regressor algorithm is interpretable. The decision trees can be visualised to explain the main features driving the splits. Figure 3 shows schematically the whole regression approach taken here. Panel (a) shows single trees, of which 500 are built with the same input variables and then combined into forest (b). Then this forest is combined with the nine other top-performing models (made from different combinations of ancillary variables) to make an ensemble (c). The 10 predictions of (c) are then arithmetically averaged into a single prediction, which thus includes the predictions of 5000 trees with 10 different combinations of input variables. In Fig. 3a, the colour of a limb or “branch” following a node is given by the variable driving that split within the training dataset. For Fig. 3b and c it shows the percent of times that a variable drives that node within the forest. The value of the ancillary variable that sets the split is shown inside the circle (a, b, c). The thickness of the branch scales to the throughput of training dataset samples contained within that split. The trees are shown to a depth of five nodes for aesthetic reasons and due to increased divergence of the trees within a forest the deeper you go. However the trees themselves are unlimited in the depth they can reach.
与许多机器学习方法不同，随机森林回归算法是可解释的。决策树可以可视化来解释驱动分裂的主要特征。图3示意性地显示了此处采用的整个回归方法。面板 (a) 显示单棵树，其中 500 棵使用相同的输入变量构建，然后组合成森林 (b)。然后将该森林与其他九个表现最好的模型（由辅助变量的不同组合组成）组合起来形成一个集合（c）。然后，(c) 的 10 个预测被算术平均为单个预测，从而包括具有 10 种不同输入变量组合的 5000 棵树的预测。在图3a中，节点后面的肢体或“分支”的颜色由训练数据集中分割的变量驱动给出。对于图3 b 和 c，它显示了变量驱动森林中该节点的次数百分比。设置分割的辅助变量的值显示在圆圈内（a、b、c）。分支的厚度与该分割中包含的训练数据集样本的吞吐量成比例。出于美观原因，树木显示为五个节点的深度，而且随着深度的加深，森林内树木的发散性也会增加。然而，树木本身的深度是无限的。

Figure 3Schematic illustration of how (a) multiple decision trees are combined into (b) a forest and then combined into (c) a further 10-member ensemble. (a) shows individual trees in a forest. (b) represent a forest of 500 trees as a single figurative tree. (c) shows the 10 forests of 500 trees combined into a single prediction. The branches in plots (a)–(c) are coloured by the percentage of the decisions at a given node that are driven by a given variable. That value within the circle gives the value of the main ancillary variable driving a split. The thickness of branches gives the throughput of the dataset through a given node for single trees (a) or the average for plots of forests (b, c). The 10 forests shown as thumbnails in panel (c) are also shown in larger form in Appendix Fig. A5. Variable names are coloured as per the following coloured text: temperature (blue, ^∘C), depth (orange, metres), chlorophyll a (green, mg m⁻³), salinity (pink, PSU), nitrate (brown, µg m⁻³), mixed layer depth (MLD; purple, metres), phosphate (red, µg m⁻³), and shortwave radiation (grey, W m⁻²).
图 3示意图显示(a)多个决策树如何组合成(b)森林，然后组合成(c)进一步的 10 成员集成。 (a)显示了森林中的单棵树。 (b)将 500 棵树组成的森林表示为一棵象征性的树。 (c)显示将 10 个森林的 500 棵树组合成一个预测。图中(a) – (c)中的分支根据给定节点上由给定变量驱动的决策的百分比进行着色。圆圈内的值给出了驱动分裂的主要辅助变量的值。树枝的粗细给出了单棵树(a)的给定节点的数据集吞吐量或森林图块的平均值(b, c) 。图(c)中以缩略图显示的 10 个森林也在附录图A5中以放大形式显示。变量名称按照以下彩色文本着色：温度（蓝色， ^∘ C）、深度（橙色，米）、叶绿素a （绿色，mg m ⁻³ ）、盐度（粉红色，PSU）、硝酸盐（棕色， μg m ^-3 ）、混合层深度（MLD；紫色，米）、磷酸盐（红色， μg m ^-3 ）和短波辐射（灰色，W m ^-2 ）。

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The first and larger splits in the data at decision nodes in the models can be simply read, which can provide understanding of the main variables driving the initial and largest splits in the prediction. For all models in the ensemble, the initial split is driven by temperature, with a split occurring at around 21.1 ^∘C (with a standard deviation of 1.2 ^∘C). The data are then split by two further nodes from this, a left- and right-hand split (e.g. Fig. 3b). If depth or temperature is present as a variable, then they drive the majority of the next splits. If depth is not present as a variable, then either nitrate or mixed layer depth (MLD) is the most common variable to dictate the split in the data at the next node in the tree. Thus a qualitative way of interpreting the initial splits of the dataset would be to say that the model is primarily differentiating between warmer and shallower locations.
可以简单地读取模型中决策节点处的数据中的第一个和较大的分割，这可以提供对驱动预测中的初始和最大分割的主要变量的理解。对于系综中的所有模型，初始分裂由温度驱动，分裂发生在 21.1 ^∘ C 左右（标准差为 1.2 ^∘ C）。然后，数据被另外两个节点分割，即左手分割和右手分割（例如图3 b）。如果深度或温度作为变量存在，那么它们会驱动接下来的大部分分裂。如果深度不作为变量存在，则硝酸盐或混合层深度 (MLD) 是最常见的变量，用于指示树中下一个节点处的数据分割。因此，解释数据集初始分割的定性方法是说该模型主要区分较温暖和较浅的位置。

4.1 Prediction of iodide at observational locations
4.1观测地点碘化物预测

Figure 4 shows a point-by-point comparison between parameterised and observed iodide for the entire dataset, the withheld dataset, and the withheld coastal dataset and withheld non-coastal dataset. Predictions are shown for the ensemble random forest regressor approach described here and for both the Chance et al. (2014) and MacDonald et al. (2014) parameterisations. The root mean square errors of observed and predicted values are given in Fig. 4 and in Table 3.
图4显示了整个数据集、保留数据集、保留沿海数据集和保留非沿海数据集的参数化碘化物和观测到的碘化物之间的逐点比较。显示了此处描述的集成随机森林回归器方法以及Chance 等人的预测。 （ 2014 ）和麦克唐纳等人。 （ 2014 ）参数化。观测值和预测值的均方根误差如图4和表3所示。

Figure 4Regression plots showing comparisons between predicted values and observations in the entire (blue, N=1293) and withheld data (orange, N=259), withheld data classed as coastal (green, N=157), and the withheld data classed as non-coastal (pink, N=102). Solid lines give the orthogonal distance regression line of best fit. The dashed grey line gives the 1:1 line. Root mean square error for each line is annotated by subplot in nanomolar (nM).
图 4回归图显示了整个数据（蓝色， N = 1293 ）和保留数据（橙色， N = 259 ）、保留数据归类为沿海（绿色， N = 157 ）以及保留数据分类的预测值和观测值之间的比较作为非沿海地区（粉红色， N = 102 ）。实线给出最佳拟合的正交距离回归线。灰色虚线表示1:1线。每条线的均方根误差由纳摩尔 (nM) 的子图注释。

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Table 3Statistics for observations and predictions by the ensemble prediction (RFR(ensemble)) and existing parameterisations against the entire dataset of observations. The root mean square error is shown against the entire and the withheld data.
表 3集合预测 (RFR(ensemble)) 的观测值和预测的统计数据以及针对整个观测数据集的现有参数化。显示针对整个数据和保留数据的均方根误差。

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The new ensemble prediction is the best performing, with a lower RMSE (35 nM) compared to the existing parameterisations (75 and 50 nM for the MacDonald et al., 2014, and Chance et al., 2014, respectively) for the withheld data. Both the new parameterisation and Chance et al. (2014) parameterisation are relatively unbiased (both have best-fit-line slopes of 0.84, against the withheld data), but the new parameterisation shows less noise than Chance et al. (2014). MacDonald et al. (2014) shows a significant low bias and significant noise. The improved skill from the RFR ensemble is consistent for both coastal and non-coastal observations.
新的集合预测性能最佳，与保留数据的现有参数化（ MacDonald 等人， 2014 年和Chance 等人， 2014 年分别为 75 和 50 nM）相比，RMSE 更低（35 nM） 。新的参数化和Chance 等人。 ( 2014 )参数化相对无偏差（相对于保留的数据，两者的最佳拟合线斜率为 0.84），但新的参数化显示出比Chance 等人更少的噪声。 （ 2014 ） 。麦克唐纳等人。 ( 2014 )显示出显着的低偏差和显着的噪声。 RFR 集合改进的技能对于沿海和非沿海观测都是一致的。

Figure 5 shows comparisons between the probability distribution functions (PDFs) of the observed iodide and the predictions, together with the PDFs of the biases for the entire, coastal, and non-coastal withheld datasets. The PDF of the new parameterisation shows the greatest similarity to the observations. The PDF from Chance et al. (2014) shows a similar range to the observations and structure to the observations, whereas the PDF from MacDonald et al. (2014) shows again a significant underestimate. The bias plots show the new predictions are generally clustered around zero with a relatively narrow peak. Chance et al. (2014) is again roughly clustered around zero but shows a wider peak. The largest biases are found from MacDonald et al. (2014), which systematically underestimates observed iodide concentrations.
图5显示了观测到的碘化物的概率分布函数 (PDF) 与预测之间的比较，以及整个沿海和非沿海保留数据集的偏差 PDF。新参数化的概率密度函数显示出与观测结果的最大相似性。来自Chance 等人的 PDF。 ( 2014 )显示了与观察结果相似的范围和结构，而MacDonald 等人的 PDF。 （ 2014 ）再次显示出明显的低估。偏差图显示，新的预测通常聚集在零附近，峰值相对较窄。机会等人。 ( 2014 )再次大致聚集在零附近，但显示出更宽的峰值。最大的偏见来自麦克唐纳等人。 ( 2014 ) ，系统地低估了观测到的碘化物浓度。

Figure 5Probability density function (bars) and Gaussian kernel density (lines) estimate of observations and predicted concentrations (left), and bias (right, model minus observations) in entire withheld dataset (upper, N=259), the withheld coastal dataset (middle, N=157), and the withheld non-coastal dataset (lower, N=102).
图 5整个保留数据集（上， N = 259 ）（保留沿海数据集）中观测值和预测浓度（左）的概率密度函数（条形图）和高斯核密度（线）估计值和预测浓度（左）以及偏差（右，模型减去观测值） （中， N = 157 ），以及保留的非沿海数据集（下， N = 102 ）。

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The dataset selection error range, which shows the influence of the choice of how the dataset is split into training and withheld data on model prediction, is described in Sect. 3.2. Within the 20-member ensemble of different testing/withdrawn choices, the average variation in RMSE was 8.4 nM (5.9–11.02 nM), and in the range of average predicted values it was 6.1 nM (5.4–6.6 nM). This translates to a percentage error range of 13.9 %–36.3 % on the RMSE and 5.4 %–7.3 % on the average predicted value.
数据集选择误差范围，显示了数据集如何分割为训练数据和保留数据的选择对模型预测的影响，在第 1 节中进行了描述。 3.2 .在不同测试/撤回选择的 20 名成员中，RMSE 的平均变异为 8.4 nM (5.9–11.02 nM)，在平均预测值范围内为 6.1 nM (5.4–6.6 nM)。这意味着 RMSE 的百分比误差范围为 13.9%–36.3%，平均预测值的百分比误差范围为 5.4%–7.3%。

The model selection error, which is the influence of the different independent variables used, is described in Sect. 3.2. The difference in the average prediction of the 10 members of the ensemble is 1.8 nM (with a range of average prediction from 96.0 to 97.8 nM), and the range of the difference in model performance is 3.9 nM (33.2–37.2 nM). As a percentage this model selection translates to a percentage uncertainty on the RMSE of 10.6 %–11.9 % and on the average of 1.8 %–1.9 %.
模型选择误差是所使用的不同自变量的影响，在第 4 节中进行了描述。 3.2 .集成的 10 个成员的平均预测差异为 1.8 nM（平均预测范围为 96.0 至 97.8 nM），模型性能差异范围为 3.9 nM（33.2-37.2 nM）。作为百分比，该模型选择转化为 RMSE 的百分比不确定性为 10.6%–11.9%，平均为 1.8%–1.9%。

The dataset selection and model selection compare to an error on the observations of ∼10 %. Uncertainty from dataset selection has a far greater effect on the prediction error than model selection. This is expected due to the small dataset size. The combined error in the prediction (dataset selection + model selection error) is either comparable to (7.2 %–9.2 % in terms of average prediction) or greater (24 %–48 % in terms of RMSE) than the observational error.
数据集选择和模型选择与观测值的误差相比约为10 %。数据集选择的不确定性对预测误差的影响比模型选择的影响大得多。由于数据集较小，这是预期的。预测中的组合误差（数据集选择 + 模型选择误差）与观测误差相当（就平均预测而言为 7.2%–9.2%），或者大于观测误差（就 RMSE 而言为 24%–48%）。

From this analysis we have shown that the new ensemble RFR model performs significantly better than those currently in the literature. We now turn to explore the predicted global distribution of sea-surface iodide using our ensemble model.
通过此分析，我们表明新的集成 RFR 模型的性能明显优于现有文献中的模型。我们现在转而使用我们的集合模型来探索海面碘化物的预测全球分布。

4.2 Global sea-surface iodide distribution
4.2全球海面碘化物分布

From the ensemble prediction system we calculate monthly global grids (0.125∘×0.125∘$\mathrm{0.125}{}^{\circ }\times \mathrm{0.125}{}^{\circ }$, ∼12.5km×12.5km$\sim \mathrm{12.5}\mathrm{km}\times \mathrm{12.5}\mathrm{km}$) of sea-surface iodide using the gridded ancillary data (Sect. 2). The annual average spatial predictions are shown in Fig. 6 with the observations overlaid in circles. Similar to previous work, annual average maximum concentrations of 220 nM are found in tropical and coastal regions (e.g. Oceania and in the Caribbean/Gulf of Mexico), with the lowest concentrations in mid-latitude waters (22.4 nM). Seasonal variability is also seen within the monthly prediction (Appendix Fig. A3). However, this spatial and temporal variability is not well constrained by observations. For example, some of the highest concentrations are predicted for the South China Sea, a region without any observations (Fig. 6). Some features visible in the concentration field appear to be associated with deep bathymetric features (e.g. the higher concentrations over the Mid-Atlantic Ridge – Fig. 6), even though a physical explanation for such a link seems unlikely.
从集合预测系统中，我们计算每月的全球网格（ 0.125∘×0.125∘$\mathrm{0.125}{}^{\circ }\times \mathrm{0.125}{}^{\circ }$ , ∼12.5km×12.5km$\sim \mathrm{12.5}\mathrm{km}\times \mathrm{12.5}\mathrm{km}$ ）使用网格辅助数据计算海面碘化物（第2节）。年平均空间预测如图6所示，观测结果以圆圈重叠。与之前的工作类似，热带和沿海地区（例如大洋洲和加勒比海/墨西哥湾）的年平均最大浓度为220 nM，中纬度水域的浓度最低（22.4 nM）。在每月预测中也可以看到季节性变化（附录图A3 ）。然而，这种空间和时间的变化并不能很好地受到观测的限制。例如，一些最高浓度预计出现在南海，该地区没有任何观测结果（图6 ）。浓度场中可见的一些特征似乎与深水深特征相关（例如大西洋中脊上方的较高浓度 - 图6 ），尽管对这种联系的物理解释似乎不太可能。

Figure 6Annual average predicted sea-surface iodide by the 10-member ensemble of models (RFR(Ensemble)), overlaid with iodide observations from Chance et al. (2019a) without outliers. Outliers are defined here as values greater than the third quartile plus 1.5 times the interquartile range (Frigge et al., 1989). Only locations that are entirely water are included in the spatial average. Earth raster and vector map data used are freely available from Natural Earth (http://www.naturalearthdata.com/, last access: 23 July 2019).
图 6由 10 人模型集合 (RFR(Ensemble)) 预测的年平均海面碘化物，与Chance 等人的碘化物观测结果重叠。 ( 2019 a )没有异常值。异常值在此定义为大于第三个四分位数加上 1.5 倍四分位距的值（ Frigge 等， 1989 ） 。只有完全由水组成的位置才包含在空间平均值中。使用的地球栅格和矢量地图数据可从 Natural Earth 免费获取（ http://www.naturalearthdata.com/ ，最后访问日期：2019 年 7 月 23 日）。

Summary statistics on the global predictions are shown in Table 4. These show that, as for comparisons at the observed locations (Sect. 4.1), the ensemble prediction is broadly in between the two existing parameters. The new ensemble model predicts a mean value of 106 nM (with members ranging from 102.3 to 108.8 nM), with predicted values from existing parameterisations ranging from 58.9 (MacDonald et al., 2014) to 128.1 nM (Chance et al., 2014).
全球预测的汇总统计数据如表4所示。这些表明，对于观测位置的比较（第4.1节），集合预测大致介于两个现有参数之间。新的集成模型预测平均值为 106 nM（成员范围为 102.3 至 108.8 nM），现有参数化的预测值范围为 58.9 （ MacDonald 等人， 2014 ）至 128.1 nM （ Chance 等人， 2014 ） 。

Table 4Statistics on predicted global annual sea-surface values from new and existing parameters at a horizontal resolution of 0.125∘×0.125∘$\mathrm{0.125}{}^{\circ }\times \mathrm{0.125}{}^{\circ }$. Existing parameterisations (Chance et al., 2014; MacDonald et al., 2014) and the ensemble prediction are shown in bold.
表 4根据新参数和现有参数预测的全球年海面值的统计数据，水平分辨率为 0.125∘×0.125∘$\mathrm{0.125}{}^{\circ }\times \mathrm{0.125}{}^{\circ }$ 。现有的参数化（ Chance 等人， 2014 年； MacDonald 等人， 2014 年）和集合预测以粗体显示。

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The annual latitudinal average of these fields, together with predictions from Chance et al. (2014) and MacDonald et al. (2014), and the observations are shown in Fig. 7. Far greater structure is seen compared to the two existing parameterisations (Fig. 7) due to the multivariate and non-parametric ensemble approach used here. All parameterisations capture the broad observed feature of decreasing iodide from lower to higher latitude. The new predicted values lie between Chance et al. (2014) and MacDonald et al. (2014) in the tropics; however, within the polar regions, the new prediction is significantly higher than both of the previous parameterisations. The lower concentrations in the predicted values from MacDonald et al. (2014) for most of the global sea surface are clear.
这些油田的年纬度平均值以及Chance 等人的预测。 （ 2014 ）和麦克唐纳等人。 ( 2014 ) ，观察结果如图7所示。由于此处使用的多元和非参数集成方法，与两个现有参数化（图7 ）相比，可以看到更大的结构。所有参数化都捕获了从低纬度到高纬度碘化物减少的广泛观察到的特征。新的预测值介于Chance 等人之间。 （ 2014 ）和麦克唐纳等人。 （ 2014 ）在热带地区；然而，在极地地区，新的预测明显高于之前的参数化。 MacDonald 等人的预测值中浓度较低。 （ 2014 ）全球大部分海面都是晴朗的。

Figure 7Predicted annual average sea-surface iodide plotted against latitude (lines), overlaid with observed concentrations (diamonds). Solid lines give mean values and shaded regions give (±) the average standard deviation. The standard deviation is the monthly standard deviation across a latitude between all 10 ensemble members (RFR(Ensemble)) or within a single prediction for existing parameterisations (Chance et al., 2014; MacDonald et al., 2014). Filled diamonds show non-coastal observations and non-filled ones show coastal values. Extent of x axis is shown for grid boxes that are entirely water.
图 7根据纬度绘制的预测年平均海面碘化物（线），与观测到的浓度（菱形）重叠。实线给出平均值，阴影区域给出 ( ± ) 平均标准偏差。标准差是跨纬度所有 10 个集合成员 (RFR(Ensemble)) 或现有参数化的单个预测内的月标准差( Chance 等人， 2014 年； MacDonald 等人， 2014 年) 。填充的菱形显示非沿海观测值，未填充的菱形显示沿海值。显示完全由水组成的网格框的x轴范围。

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The range of dataset error is found for the 20 models with different training data splits, as described in Sect. 3.2. This gives an uncertainty in the form of a average range in predicted global mean surface iodide for all of the multiple builds of ensemble members of 4.0 nM (2.8–5.0) compared to a annual mean prediction of 106 nM. This maximum and minimum of this range in predicted values can then be divided by the minimum and maximum predicted global mean surface iodide values (98 and 109.3 nM, respectively) to give percent range of 2.5 % to 5.1 %. This is lower than that calculated for the individual locations of observations (Sect. 4.1) due to large global areas being similar in chemical and physical regimes compared to the subset of sampled locations within the observations.
找到具有不同训练数据分割的 20 个模型的数据集误差范围，如第 2 节所述。 3.2 .这给出了所有多重构建的集合成员的预测全球平均表面碘化物平均范围为 4.0 nM (2.8–5.0) 的不确定性，而年平均预测为 106 nM。然后可以将该预测值范围的最大值和最小值除以预测的全球平均表面碘化物值的最小值和最大值（分别为 98 和 109.3 nM），得到 2.5% 至 5.1% 的百分比范围。这低于针对单个观测位置计算的值（第4.1节），因为与观测中采样位置的子集相比，大范围的全球区域在化学和物理状况方面相似。

The model selection error due to variability within ensembles' 10 members, generated with different independent variables, gives a global average surface concentration between 102.3 and 108.8 nM. This range in prediction gives a model selection error of 6.45 nM, which equates to 6.0 %–6.3 %. Like with the global uncertainty from dataset selection, the global value would be expected to be lower than the uncertainty at the specific locations of the observations (Sect. 4.1) due to the more homogeneous nature of the predicted areas. However, a greater variation is seen from different model predictions than within predictions for the observation locations. This highlights the importance of the different ancillary variables considered here and also therefore the strength gained from the ensemble approach taken here.
由于由不同自变量生成的集合 10 个成员内的变异性导致的模型选择误差给出了 102.3 至 108.8 nM 之间的全局平均表面浓度。此预测范围给出的模型选择误差为 6.45 nM，相当于 6.0%–6.3%。与数据集选择的全局不确定性一样，由于预测区域的更均匀性质，预计全局值将低于观测特定位置的不确定性（第4.1节）。然而，与观测位置的预测相比，不同模型的预测存在更大的差异。这凸显了此处考虑的不同辅助变量的重要性，以及从此处采用的集成方法中获得的力量。

Within members of the ensemble, variation is modest except for two ensemble members which diverge north of ≥65∘$\ge \mathrm{65}{}^{\circ }$ N (Appendix Fig. A2). As noted earlier (Sect. 2), the values in this region are very poorly constrained by the observational dataset (Fig. 6).
在该群的成员中，除了两个群成员在北面发散之外，变化不大。 ≥65∘$\ge \mathrm{65}{}^{\circ }$ N（附录图A2 ）。如前所述（第2节），该区域的值受观测数据集的约束非常弱（图6 ）。

In addition to the three errors described above, we also attempt to gain an understanding of the spatial uncertainty in the 10-model ensemble prediction. We do this by calculating the differences in the predicted spatial fields from the 10 ensemble members. Figure 8 shows the monthly average of the standard deviation of the 10 model ensemble as a percentage of the annual mean of the ensemble prediction. This is also shown in absolute terms in the Appendix (Fig. A4). Relative uncertainties are largest at the poles where predicted concentrations are lowest and where (at least in the Northern Hemisphere) very few observations are available to constrain the system. The southern oceans show an distinct pattern, where values close to coastal Antarctica appear well constrained but values further north appear poorly constrained.
除了上述三个错误之外，我们还尝试了解 10 模型集成预测中的空间不确定性。我们通过计算 10 个集合成员的预测空间场的差异来实现这一点。图8显示了 10 个模型集合的标准差的月平均值占集合预测年平均值的百分比。这也以绝对值的形式显示在附录中（图A4 ）。两极的相对不确定性最大，那里的预测浓度最低，并且（至少在北半球）可用于约束系统的观测数据非常少。南部海洋显示出一种独特的模式，其中靠近南极洲沿海的值似乎受到很好的约束，但更北的值似乎受到很弱的约束。

Figure 8Annual average spatial percent uncertainty in predicted sea-surface iodide for the ensemble of models. Percent spatial uncertainty was calculated as the standard deviation in monthly average values for all models, divided by the annual mean. Values are limited to 25 % for contrast, but the maximum plotted value is 77 % in the northern high latitudes. Only locations that are entirely water are included in the spatial average. Earth raster and vector map data used are freely available from Natural Earth (http://www.naturalearthdata.com/, last access: 23 July 2019).
图 8模型集合预测的海面碘化物的年平均空间百分比不确定性。空间不确定性百分比计算为所有模型月平均值的标准差除以年平均值。对比度值限制为 25%，但在北部高纬度地区绘制的最大值为 77%。只有完全由水组成的位置才包含在空间平均值中。使用的地球栅格和矢量地图数据可从 Natural Earth 免费获取（ http://www.naturalearthdata.com/ ，最后访问日期：2019 年 7 月 23 日）。