2.3.1 SMA Adaptation for Animation and Visualisation

This adaptation is designed for carrying out animation and graphical data visualisation of selected variables, to reflect the multi-dimensional impact of COVID–19 in a two–dimensional space. Its specific applications will vary and must typically be guided by the framework in Figure 2. For example, the choice of attributes to be animated and/or visualised will depend on the intended purpose of the study. Which variables to display and which cut-off points to trigger which alarms are decisions that require underlying domain knowledge and not a purely data science problem. Algorithm 1 represents a simple variant of the SMA algorithm. It is designed to display multiple variables in a two–dimensional space, comparing relevant parameters and firing a message on meeting pre-specified criteria. Its mechanics are illustrated below, using the notation in Table 1, collectively featuring a super set of data sources Γ.

The subset ø ⊆ Γ contains variables of interest, based on which the algorithm iteratively displays multi-dimensional data in a two-dimensional space, triggering alarms in accordance with pre-set conditions. For example, as the unemployment rate in a particular borough in England reaches a specific level, e.g. ξ ≥ 3.5% while death rates are above 1000 per day, at time τ = t^*, the Chancellor of the Exchequer may need to consider taking action on the furlough scheme, say. Presenting structured data in both visual static and animated forms, provides clear insights to stakeholders in addressing societal challenges such as COVID–19. The main focus is on both Γ and ø which will always need to be adapted to handle new cases. Practical illustrations of the algorithm’s mechanics are given in Section 3.1.

2.3.2 SMA Adaptation for Convolutional Neural Networks

Figure 3 provides a graphical illustration of our adaptation of the SMA algorithm in addressing data randomness via multiple model training, testing and assessing. The data repository is a large data source from which multiple training and testing samples are drawn, with or without replacement. Its data contents can be either structured or unstructured.

At the preparatory level, the investigator examines the overall behaviour of the data through visualisation, animation or other methods of inspection, such as outlier detection and missing values, in order to ascertain its validity for applying the adopted modelling technique. A machine learning model trained and tested on different training samples will typically yield different outcomes. Performance assessment is made on the basis of specific metrics generated and assessed via the two algorithms, as described in Section 2.3. Given a dataset with class labels, y, the SMA algorithm applies a learning model which, without loss of generality, we can define as in Equation 1

where $\mathcal{D}$M12 \documentclass[10pt]{article} \usepackage{wasysym} \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \[ {\cal D} \] \end{document} is the underlying distribution and P[ø(x) ≠ y] is the probability of disparity between the predicted and actual values. By repeatedly sampling from the provided data source, modelling and carrying out a comparative assessment of the results, the SMA algorithm provides a unifying environment with the potential to yield consistent results across samples. For classification problems, it proceeds by training and validating the model in Equation 1 on random samples, keeping the samples stateless across all iterations. Thus, multiple machine learning models are fitted, compared and updated over several iterations, finally selecting the best performing model based on the probability

where 𝔼[Δ] is the estimated difference between the population error ψ_D,POP and the validation error ψ_B,POP. Adaptation of the SMA algorithm is illustrated via Convolutional Neural Network (CNN)–a machine learning technique, typically used for classifying image data such as the X–Ray data, in this case. Its original ideas derive from the work of a Japanese Scientist, Kunihiko Fukushima (Fukushima 1980), on neocognitron–a basic image recognition neural network and developed through the work of LeCun, Jackel, Boser, Denker, Graf, Guyon, Henderson, Howard & Hubbard (1989), LeCun, Boser, Denker, Henderson, Howard, Hubbard & Jackel (1989) into the modern day CNN via the ImageNet data challenge Krizhevsky et al. (2012).

A CNN model performs classification based on image inputs and a target variable of known classes of the images. It is typically composed of multiple layers of artificial neurons, imitating biological neurons, as graphically illustrated in Figure 4. It processes the convolution computing for the input multichannel extracting features on its plane.

Each convolutional kernel is convolved across the width and height of 2D input volumes from the previous layer, computing the dot product between the kernel and the input. If we let X be an n × n data matrix and W a k × k matrix of weights, which is a 2-dimensional filter with k ≤ n (see Figure 5), then

where X_k(i,j) is the k × k submatrix of X and 1 ≤ i, j ≤ n – k + 1. Now, given a k × k matrix Λ ∈ ℝ^k × ^k

We can then express the 2-dimensional convolution of X and W using the sums of the element-wise products as

such that ${\displaystyle \sum \left(X_k\left(i,j\right)\odot W\right)}={\displaystyle {\sum }_{\alpha =1}^k{\displaystyle {\sum }_{\beta =1}^k{x}_{i+\alpha -1,j+\beta -1}\hspace{0.17em}.\hspace{0.17em}w}}$M9 \documentclass[10pt]{article} \usepackage{wasysym} \usepackage[substack]{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage[mathscr]{eucal} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \[\sum {\left({{{\bf{X}}_k}\left({i,j} \right) \odot {\bf{W}}} \right)} = \sum\nolimits_{\alpha = 1}^k {\sum\nolimits_{\beta = 1}^k {{x_{i + \alpha - 1,j + \beta - 1}}\,.\,w} } \] \end{document} , for i, j = 1,2,3, … n – k + 1. It can be shown that the convolution of X ∈ ℝ^k × ^k is an (n – k + 1) × (n – k + 1) matrix (Zaki & Mera 2020). A CNN is driven by mathematical functions that calculate the weighted sum of multiple inputs to generate an output based on an activation value function.

We can envision a CNN output, y_i,j,k as denoting the neuron output in the i^th row and the j^th column of feature map k of the l^th convolutional layer. To get the convolutional values, we slide the kernel over the input data, multiplying the corresponding values and summing up and fill the matrix of the same dimension as the kernel, as shown in Figure 5. Note that the filter has reduced the input matrix to a smaller dimension of its size.

It is also important to consider the vertical and horizontal strides as they impinge on the model’s capability of feature capturing. The pooling layer reduces the dimensionality of the rectified feature map, using different filters to identify different parts of the image – like edges, corners, curves etc. Flattening converts the 2-D arrays from the pooled layer into a one-dimensional vector. The Fully Connected layer then receives this as input, for classifying the image. A 2 × 2 pooling layer, filtering with a sliding of 2 downsamples at every depth of the input discards 75% of the activations. The Rectified Linear Unit (ReLU) applies the activation function in Equation 6.

which basically replaces negative values from the activation map by zero and adopting the actual values otherwise. Other activation functions like the hyperbolic tangent and the sigmoid, in Equation 7, are also commonly used.

Central to the performance of the CNN is its architecture. Figure 4 exhibits a typical structure of a CNN model, consisting of the input, convolutional, pooling and the fully connected layers. Each input is weighted in a similar way as are the coefficients in a linear regression model. CNN models are trained using an optimization process, driven by a loss function that calculates the classification error. The maximum likelihood methods is a framework that describes the loss function choice. Other common methods include the cross-entropy and mean squared error. Typically, a CNN model is trained using the stochastic gradient descent optimization algorithm, via which the weights are updated by back-propagating the error. That is, the model with specific weights performs predictions and the resulting allocation error is calculated. At every epoch, the weights are changed to improve performance at the next stage. Equation 8 shows how each weight expresses the rate of change in the total loss (L), as the weight (w) changes by one unit.

As with all models that learn rules from data, performance of the CNN is associated with variations due to data randomness (Mwitondi et al. 2013, Mwitondi & Said 2013). Hence, our CNN implementation will draw multiple samples from the data sources in Section 2.2 in order to attain a generalised performance and attain model robustness.

A number of factors are known to affect the accuracy of CNN–they include the network’s number of layers, number of neurons and the learning rate–see, for instance, Géron (2019), Rawat et al. (2020), Wang et al. (2019). These parameters impinge on the model’s accuracy and loss–two crucial parameters to the performance of CNN–accuracy and loss. The former is the number of correctly predicted data points as a proportion of the total number of predictions. Loss is the quantitative measure of deviation or difference between the predicted and actual values–it measures the mistakes the CNN makes in applying Equation 1 to any given dataset. Due to the random nature of the sampled data, the accuracies and mistakes the model makes will vary. We shall be seeking to stabilise these variations across samples, using our adaptation of the SMA algorithm as shown below. Thus, the second adaptation of the SMA algorithm, described below, conditions these random samples to the foregoing parameters and varying proportions of training, validation and testing samples for CNN classification. Different strategies to reduce the learning rate during training are known, including those outlined in Table 2.

Algorithm 2 experiments with as many learning rates as possible in search of an optimal model and it only stops once it is evident that the changes have no much impact on the parameters δ_t and δ_t, i.e., changes in the training and validation errors respectively. To arrive at the best model at step #36, the CNN model runs with a check point that monitors both training and validation accuracy, saving the best weights and reporting each time performance improves. In Python checkpointer = ModelCheckpoint(filepath=”best_weights, monitor = ‘accuracy’, save_best_only=True) is called by the best CNN fit as an argument alongside other training and validation parameters and number of epochs.

While the implementations in Section 3 were carried out by a combination of R and Python libraries, the two algorithms are amenable to any data analytics tool. Our animation and visualisation in Section 3.1 were driven by the gapminder package in R, hence their explicit inclusion in Algorithm 1 but, again, these steps are transferable to other packages and libraries. The same applies to the implementation in Section 3.2, which was carried out in Python’s Keras.

3.2.1 Training and Validation

Implementation of the adapted Algorithm 2 was driven mainly by Keras–a deep learning Application Programming Interface (API) running on top of TensorFlow–an open–source machine learning platform (Géron 2019, Grattarola & Alippi 2020). The open–source models were chosen in consideration of the interdisciplinary nature of the proposed methods, as they provide easy access to a wide range of stakeholders–beginners and experts alike (Zhang et al. 2021). Further discussions on the relevance of interdisciplinarity to modelling mechanics of various machine learning models are in Section 3.3.2. As already noted, learning rules from data is inevitably associated with variations due to data randomness (Mwitondi & Said 2013, Mwitondi et al. 2013), which can negatively affect model performance. For generalisation and robustness, we adopted the SMA algorithm (Mwitondi et al. 2018a, b, 2020), taking multiple samples and conditioning each on the training and validation proportions as in Table 3.

Training and validation data for the first run was split into 80%–20% respectively, running 50 epochs on two classes with 744 training and 186 validation images. Other samples were of 70% (training) and 30% (validation), corresponding to 651 images and 279 images respectively, 60%–40% (558 training and 372 validation) and 50% for training and validation on 2 classes. All but the first sample started and converged at high training and validation accuracy.

The panels in Figure 10 correspond to model accuracy (left–hand side) and model loss (right–hand panel). They are based on the training–validation split of 80%–20% respectively. In this case, both start with very high accuracy and quickly converge, after only ten epochs. The panel to the right shows the model loss–a quantitative measure of deviation between the predicted and actual values. This is the measure of the mistakes the CNN model makes in predicting the output. As the loss is approximately equal to validation loss, the model is perfectly fitting on both training and validation data, as can be seen on the left–hand side panel.

The two panels in Figure 11 are based on the training–validation split of 70%–30% respectively. In this case, while they start with high accuracy, it isn’t until about 25 epochs that the training accuracy converges while the validation accuracy continues to oscillate around 97%. In the panel to the right, the training loss exceeds validation loss up until 30 epochs, an indication of underfitting, but the model fits well at higher epochs, except for the two spikes.

The two panels at the top of Figure 12 correspond to the 60%–40% split, while the bottom panels represent the 50%–50% split. In both cases, the training and loss rates are stable above 25 epochs, but the validation rates appear to be consistently oscillating, an indication of huge variations attributed to randomness in unseen data. In all plots in Figures 10 through 12 attention is on the model’s consistency, i.e., whether the model is predicting the classes well. The panels to the right show the model loss–a measure of the mistakes the CNN model makes in predicting the output.

When training loss exceeds validation loss, we have the case of underfitting, a rarity. The most common scenario is that of over-fitting–i.e., when training loss is significantly less than validation loss, which implies that the model is adapting so well to the training data that it considers random noise as meaningful data. In other words, the model fails to generalize well to previously unseen data. The ideal scenario is when training loss is approximately equal to validation loss, as that would mean that the model is perfectly fitting on both training and validation data. It is important to note that while these technical issues are fundamental, interpreting data visualisation and modelling findings must always be considered in problem–specific and interdisciplinary context.

3.2.2 Testing and Assessment

Ensuring that the CNN model performs well after training is crucial before its deployment on previously unseen data. The model was tested on new data and yielded convincingly high accuracy and consistent loss patterns. We did this by repeatedly running the CNN model with a “check point” via Algorithm 2, monitoring both training and validation accuracy, saving the best weights and reporting each time performance improves. The saved best weights (the model) were then used to predict the class of any previously unseen X–ray images as illustrated in Figure 13.

We assessed model performance based on the metrics inside Algorithm 2, measuring loss as the distance between the predicted and true values. Minimising this loss means making fewer errors on the data. In our binary classification, application we had access to probabilities of class membership and we computed the loss as the sum of the difference between the predicted probability of the real class of the test image and 1. Parameter tuning is necessary to achieve optimal results and different applications may require different tunings. However, this can generally be monitored at the model refinement stage in Figure 3. Adapting Equation 2 to a loss function informs how the model is performing. In a binary classification, anything above 0.5 will allocate to one class and to another class, otherwise. We used the loss function to evaluating how well the CNN model functioned through the algorithm in modelling our dataset. Figure 12 exhibits a very low loss output, which indicates a good performance of the algorithm.

3.3.1 Potential Extensions of Algorithm 1 Applications

Table 4 provides selected examples of the role of interdisciplinarity (Figure 2) in addressing SDG. The complex interactions of SDG present an ideal case for the mechanics of Algorithm 1. Given established interactions, the algorithm can be applied to monitor SDG at all levels–national, regional or global. Algorithm 1 provides scope for interventions based on automated multi-dimensional animation for a wide range of applications. More specifically, describing “what is interesting” (the basis for problem identification), is based on Figure 2–i.e., underlying domain knowledge, problem space and modelling expertise. In Algorithm 1, this amounts to identifying Γ and ø ⊆ Γ.

The impact of COVID-19 on SDG has recently been widely studied, particularly during the first 18 months of the pandemic. In one recent publication, the pandemic is reported to have led to an unprecedented rise in poverty, in a generation, in parts of the world. For example, the Government of Bangladesh is said to have struggled to provide social safety net packages for marginalised groups, leading to a huge socio-economic inequality and exclusion (SDG #10) (IISD 2021). The three examples in Table 4 underline not only the SDG overlaps but also the need for interdisciplinary consensus in adapting and executing Algorithm 1.

3.3.2 Potential Extensions of Algorithm 2 Applications

For Algorithm 2, variability derives from data, deployed models and model parameters. Like in all other applications, validity of the results is hugely influenced by Ω and model-specific parameters, which implies that interdisciplinarity plays a crucial role in identifying “what is interesting”. Table 5 highlights two examples for the rationale of the framework in Figure 2 in searching for optimal machine learning models–unsupervised or supervised.

For all learning models, the choice and/or tuning of parameters is inherently interdisciplinary. For instance, pre-specifying the number of clusters in the data hugely impinges on the performance of the K-Means algorithm, implying that this decision has to be made based on some level of prior knowledge of the phenomenon. In applying Algorithm 2 for K-Means clustering, these considerations must be made. For example, instead of using a single predefined number of centroids, multiple sets might be considered. For the second example, in Table 5, the convergence of the back propagation network in neural computing is a function of factors such as initial weights, learning rate, updating rule as well as the quality and size of training and validation data. The multi-parameter dependence yields different results, the interpretations of which determines whether the underlying problem is addressed or not.

Attaining model optimisation through training and validation is crucial for the performance of all learning algorithms, yet data randomness remains a major challenge to researchers. The plots in Figures 10 through 12 exhibit one common challenge in predictive modelling–attaining generalisation for which we need to avoid both underfitting and overfitting. They reflect the challenges of model optimisation and while they provide guidelines in selecting the best performing model, we can attain unified understanding of the concepts and work towards scientific consensus if we work collaboratively across regions and disciplines, openly sharing resources. Some of the general commonalities for addressing global challenges in problem–specific and interdisciplinary contexts are summarised in Table 6.

For our sustainability and that of species around us, we are required to make right decisions at the right time. Co-ordinated initiatives are required in responding to global challenges that defy geographical boundaries and national or regional legislations. While the foregoing geo-political variations may not disappear overnight, the scientific community is duty bound to engage in co-ordinated studies for addressing the current and potential future global challenges.