Adaptive Activation Functions for Predictive Modeling with Sparse Experimental Data

Neural networks have made impressive strides in predictive modeling tasks due to their ability to learn nested or composite functions [1, 2]. Therefore, the resulting expressive power to discern complex input-output relationships makes them suitable for a wide range of scientific and engineering disciplines, including biomedical engineering [3, 4, 5], structural engineering [6, 7], mechanical engineering [8, 9, 10, 11], and additive manufacturing [12, 13]. Neural networks are made up of neurons that are interconnected units that process and transmit information. They do this by performing two pivotal operations: computing a weighted sum of inputs and then applying a nonlinear activation function. This interplay among units equips neural networks to learn hierarchical features and multilayered representations to navigate the inherent complexities of diverse problems. In particular, neural networks excel in tasks that require simultaneous feature extraction and predictive modeling within high-dimensional feature spaces.

Among the various hyperparameters that users must set in advance, the selection of activation functions is a key element because of their role in encapsulating nonlinear patterns in the data. Typically, activation functions are predetermined and remain constant throughout the training process in the majority of scenarios. Classical examples of fixed activation functions include Sigmoid and hyperbolic tangent (tanh) functions [14], which squash input values into a specific range. However, fixed activation functions may suffer from problems such as the vanishing gradient problem [15], where gradients become extremely small during backpropagation, hindering the learning process. Therefore, a wide range of activation functions with varying forms of nonlinearity have been introduced in recent years, including Rectified Linear Unit (ReLU) [16], Exponential Linear Unit (ELU) [17], Softplus [18], and Swish [19], to name a few. For instance, the widely-used deep learning library Keras [20], as of version 2.14.0, offers a collection of 17 built-in activation functions.

Although the rapid increase in the number of fixed-shape activation functions offers potential benefits, a significant challenge arises from the intensive computational demands of conducting an exhaustive search to identify optimal functions for specific tasks. This challenge becomes more pronounced in dynamic and diverse scientific problems, necessitating frequent restarts of the search process to maintain the relevance of chosen activation functions. Consequently, the inflexibility inherent in fixed activation functions poses a hindrance to the streamlined development of neural network models within scientific and engineering disciplines.

To address this problem, one promising area of research is the use of adaptive or trainable activation functions [21, 22, 23]. Unlike traditional fixed activation functions such as ReLU or Sigmoid, which maintain a static form throughout training, adaptive functions evolve in response to the data distribution and the model’s learning progress. These functions contain training parameters themselves, allowing us to optimize and tailor the shape or behavior of activation functions during the learning process alongside the parameters of the neural network. Therefore, this adaptability has the potential to enhance the ability to learn complex representations and patterns, while reducing the reliance on costly search procedures and detailed domain expertise.

Although the effectiveness of adaptive activation functions has been extensively studied in fields with abundant annotated data, such as image classification tasks highlighted in Tables 2 and 3 of a recent survey [24], a significant gap exists in understanding their applicability in domains characterized by sparse labeled data sets. A key concern in such scenarios, where data availability is limited, is the potential adverse impact on performance due to the increase in trainable parameters. Furthermore, current research has primarily relied on the standard classification accuracy score to evaluate the success of adaptive activation functions. This score, defined as the ratio of correct predictions to the total number of test samples, poses limitations when applied to scientific and engineering problems with a small number of samples. This metric offers limited assurance regarding the confidence and stability in neural network predictions. Hence, it is imperative to move towards metrics that quantify the uncertainty in the predictions of neural networks to provide a more nuanced understanding of the utility of adaptive activation functions in scientific problems.

Therefore, this paper aims to address the shortcomings mentioned above by systematically investigating the use of adaptive activation functions in three distinct additive manufacturing problems, each limited to fewer than 100 training samples. Our goal is to provide new insights into the possible application of adaptive activation functions, which introduce additional training parameters into data-austere environments. This study involves a comprehensive evaluation of the selection and adaptability of activation functions, coupled with an analysis of their impact on the predictive accuracy and confidence. In particular, we summarize our main contributions as follows.

1. 1.

   We study the effectiveness of adaptive activation functions compared to their counterparts using fixed-shape activation functions, namely ELU, Softplus and Swish. In our investigation, we explore the common practice of sharing activation functions within a hidden layer and the less explored terrain of using activation functions with individual training parameters. Although individual parameter allocation increases the total number of training parameters, which may be a concern in small data settings, we exhibit that it improves the adaptability of neural networks and their predictive performance.

2. 2.

   To the best of our knowledge, we are exploring for the first time the effectiveness of adaptive activation functions in applications containing small data sets with fewer than 100 training samples. To conduct a comprehensive investigation, we consider three additive manufacturing problems, with different characteristics such as the number and type of features, as representative of data-austere problems. These problems include the selection of filament materials and 3D printers, as well as the prediction of printability in a complex additive manufacturing problem. We demonstrate that the use of adaptive activation functions is beneficial and justified for these different problems, eliminating the need for predetermined functions.

3. 3.

   Another distinctive feature of our research is the exploration of the effectiveness of adaptive activation functions through the generation of prediction sets using conformal inference [25, 26], as opposed to relying solely on point predictions. This approach enables us to assess how adaptive activation functions influence the predictive uncertainty of neural network models. In pursuit of this objective, we utilize two metrics: empirical coverage and the average size of the prediction set. In conformal inference, empirical coverage refers to the proportion of actual observed outcomes that fall within the constructed prediction set and the average size gives an indication of the typical range of uncertainty associated with the predictions. A smaller average size suggests more precise and narrow prediction sets, whereas a larger average size indicates a wider range of uncertainty that can be a major consideration in the deployment stage. Hence, this work involves a comprehensive assessment of the reliability of neural networks equipped with adaptive activation functions. This evaluation is particularly crucial in addressing scientific challenges characterized by limited data availability.

4. 4.

   We provide source code for the implementation of adaptive activation functions with shared and individual trainable parameters in a hidden layer. The source code, available on GitHub https://github.com/farhad-pourkamali/AdaptiveActivation, contains a Keras-compatible implementation of three trainable activation functions derived from ELU, Softplus, and Swish. The code provided and our insights will enable practitioners to use adaptive activation functions in a wider range of data-limited problems while eliminating the need to manually set them ahead of time.

The remainder of this paper is organized as follows. In Section 2, we discuss some mathematical notations and foundations of neural networks together with popular fixed-shape activation functions. Section 3 provides an overview of the existing work on the use of adaptive activation functions. In Section 4, we present the systematic approach we use to rigorously evaluate the predictive performance of adaptive activation functions with shared and individual parameters in small data settings. This evaluation is carried out in three testbeds, each characterized by sparse experimental data originating from additive manufacturing problems. The last section of this paper provides concluding remarks and outlines potential areas of future study.

An extensively used neural network type for structured or tabular data problems is the fully connected network, commonly called a multilayer perceptron (MLP) [27]. This architecture consists of densely interconnected layers organized in a sequential manner, presenting an elegant way to learn nested or composite functions crucial for capturing nonlinear input-output relationships. To be formal, imagine a network with $L$ hidden layers, located between the input and output layers. The $l$-th hidden layer consists of $N_{l}$ neurons or units. Also, assume that this network takes an input vector $x\in\mathbb{R}^{D}$, where $D$ is the number of given attributes or features. Furthermore, the weight matrix and the bias vector can be written as $W^{(l)}\in\mathbb{R}^{N_{l}\times N_{l-1}}$ and $b^{(l)}\in\mathbb{R}^{N_{l}}$, for each layer indexed by $l=1,\ldots,L+1$. The predicted output $f(x)$ is then defined from the input $x$ according to the following equations:

Therefore, the prediction model $f(x)$ takes on a composite or nested form. The number of units in the output layer $N_{L+1}$ and the corresponding activation function $g^{(L+1)}$ depend on the problem at hand. For example, in binary classification problems, a single neuron is typically placed, i.e., $N_{L+1}=1$, and the Sigmoid activation function is used to find the probability that the data point $x$ belongs to each class. For a given input $z$, the Sigmoid activation function takes the form of: $\text{Sigmoid}(z)=1/(1+e^{-z})$. However, when treating classification problems of more than two classes, the last layer contains one neuron for each class. In this case, we employ the Softmax function to find the categorical distribution for all classes. That is, if we have $C$ classes, then the Softmax function accepts a set of $C$ real-valued numbers $z_{1},\ldots,z_{C}$ and converts them into a valid probability distribution by returning $e^{z_{c}}/\sum_{c^{\prime}}e^{z_{c^{\prime}}}$, for $c=1,\ldots,C$ [28].

On the other hand, we have greater flexibility when selecting activation functions for intermediate or hidden layers because they provide “latent” representations. Note that in (1), the activation function is applied in the element-wise form, so the standard implementation of dense layers in popular deep learning libraries, such as Keras, follows the same activation function for all neurons in a particular layer. In other words, all neurons placed in a hidden layer apply the same nonlinear function to transfer information to the next layer. Consequently, significant efforts are required to conduct a comprehensive search to select appropriate levels of nonlinearity in order to maintain input-output relationships.

Initial efforts to train neural networks mainly focused on the hyperbolic tangent (tanh) function to activate hidden layers through $g^{(l)}$, $l=1,\ldots,L$. In the remainder of this paper, we omit the $(l)$ superscript to simplify the notation. The tanh activation function has the following form:

This function has a range of $(-1,1)$, which has an advantage due to the zero-centered structure. However, because this function has two different horizontal asymptotes, the derivatives of this function become very small as we move further away from the origin, which is problematic for gradient optimization techniques. Rectified Linear Unit or ReLU has been widely used in recent years to address the vanishing gradient problem [15]. ReLU is defined as $g(z)=\max(z,0)$. Hence, when $z\geq 0$, we have $g(z)=z$ and its derivative is always $1$. On the other hand, we get $g(z)=0$ for negative input values $z$, therefore, the derivative is equal to $0$. Despite the simplicity of ReLU, it exhibits a saturating region, which can be problematic for gradient descent optimization. Specifically, ReLU discards negative values, leading to the known problem of dying ReLU [29]. Consequently, several activation functions have been devised to address these concerns while maintaining the fundamental structure of ReLU.

Exponential Linear Unit (ELU) shares a structure similar to ReLU for nonnegative inputs $z$, yet it facilitates the flow of information to some degree for negative values of $z$ through the expression $(e^{z}-1)$. Additionally, Softplus is another popular activation function that provides a smooth approximation of ReLU in the form of $g(z)=\log(e^{z}+1)$ for all values of $z$. Consequently, for sufficiently large positive $z$ values, this function mimics a linear behavior, i.e., $g(z)\approx z$, owing to the inverse relationship between the logarithmic and exponential functions. Another way of approximating the ReLU activation function was proposed in [19], named the Swish activation function, which takes the form of $g(z)=z\cdot\text{Sigmoid}(z)=z/(1+e^{-z})$. Similar to ReLU, Swish acts as a linear function for large positive $z$ values because the denominator becomes very close to $1$. However, what sets Swish apart is its departure from the widely-used monotonicity property. Building on the empirical success of Swish in various computer vision problems, several Swish variants have been proposed in recent years, as highlighted in [30].

In recent advancements in neural networks, there has been a growing interest in exploring trainable or adaptive activation functions to enhance their flexibility and adaptability. Traditional activation functions that we discussed in the previous section have fixed shapes, and their performance may be suboptimal for certain tasks. Thus, users must make significant efforts to choose an appropriate activation function from the collection of existing built-in functions. By introducing adaptive activation functions, neural networks can progressively learn the optimal form of input-output relationship for each unit or layer and accelerate the training process. To elucidate this concept, consider the standard empirical risk minimization problem for model fitting with fixed activation functions:

where $\mathcal{D}_{\text{train}}=\{(x_{n},y_{n})\}_{n=1}^{N_{\text{train}}}$ represents the training data set comprising feature and response pairs. The loss function serves as a metric to evaluate the accuracy of predictions, employing specific formulations like the quadratic loss function in regression problems and cross-entropy in classification tasks. In the optimization process, the variable $\theta$ serves as a container for weight matrices $W^{(l)}$ and bias vectors $b^{(l)}$, $l=1,\ldots,L+1$. These values are obtained through an initialization step, followed by an iterative refinement of their values. The refinement entails computing the gradient of the loss function with respect to the elements of $\theta$ and applying gradient descent to minimize the loss function over iterations.

The core concept of adaptive activation functions lies in parameterizing activation functions to learn the optimal form of nonlinearity introduced in (1). To this end, let $g(z;\alpha)$ be a parameterized activation function and assume that we utilize $N_{h}$ distinct activation functions in a neural network model. In this case, we can modify the optimization variable in (3) by adding the new parameters, i.e., $\theta=\{W^{(1)},b^{(1)},\ldots,W^{(L+1)},b^{(L+1)},\alpha_{1},\ldots,\alpha_{N_{h}}\}$. As a result, the total number of parameters to be learned through gradient descent increases, prompting the key question of whether the set of enhanced parameters can improve predictive power compared to fixed-shape activation functions characterized by predefined values of $\alpha$.

In this work, we implement and examine some popular parameterized activation functions, starting with the trainable Exponential Linear Unit (ELU) function, which has the following form:

In many deep learning libraries, the default value for $\alpha$ is set to $1$. The derivative of the parameterized ELU is given by $\alpha e^{z}$ for negative values of $z$, making the selection of $\alpha$ a crucial consideration during the training process. For instance, when dealing with negative values of $z$, where $e^{z}$ tends to be very small, opting for a larger value of $\alpha$ becomes essential to prevent the derivative from approaching zero.

Similarly, the parameterized version of Softplus can be written as $\text{Softplus}(z;\alpha)=\log(e^{z}+\alpha^{2})$. While the default value of $1$ results in a smooth approximation of ReLU, choosing very small values for $\alpha$ renders this activation function nearly linear. This attribute plays a crucial role in finely regulating the complexity of the mapping accomplished by every hidden layer. Lastly, we can introduce an additional tuning parameter for the Swish activation function in the form of $\text{Swish}(z;\alpha)=z\cdot\text{Sigmoid}(\alpha z)=z/(1+e^{-\alpha z})$. Notably, when $\alpha=0$, this activation function behaves as a scaled linear function, and gradually converging to ReLU as $\alpha$ increases. Furthermore, we can show that the derivative of the parameterized Swish activation function has the following form:

Thus, setting $\alpha$ to 0 results in a constant derivative of 0.5. Conversely, when $\alpha$ takes nonzero values, the derivative of Swish around the origin becomes significantly dependent on the chosen $\alpha$.

Figure 1 depicts the three parameterized activation functions, showcasing their distinctions across a range of $\alpha$ values to provide a visual understanding of their differences. Setting $\alpha=0$ in ELU yields the well-known ReLU activation function. However, by increasing $\alpha$, ELU enables the information flow for negative values of $z$. It should be noted that for nonnegative input values $z$, the parameter $\alpha$ does not alter the structure of ELU. In contrast, $\alpha$ plays a pivotal role in shaping the overall characteristics of Softplus and Swish for all values of $z$.

Recent research dedicated to investigating the effectiveness of adaptive activation functions has predominantly focused on the analysis of image data sets within the domain of computer vision [31]. A comprehensive survey by Apicella et al. [24] synthesized findings from around 20 studies, extensively utilizing established benchmarks such as MNIST, CIFAR10, CIFAR100, and ImageNet. For example, on the CIFAR10 data set, the median classification accuracy scores using the fixed ReLU and ELU activation functions were approximately $0.91$ and $0.93$, respectively. However, employing adaptive ELU and Swish resulted in slightly higher accuracy scores of $0.94$ and $0.95$, respectively. Additionally, recent papers by Dubey et al. [23] and Emanuel et al. [32] delved into the exploration of adaptive activation functions, specifically for tasks related to language translation, speech recognition, and text classification.

Expanding the domain, Wang et al. [33] extended the scope by exploring the application of adaptive activation functions across six benchmark data sets in the social and e-commerce domains. Moreover, Klopries et al. [34] investigated adaptive activation functions in the context of unsupervised learning for training autoencoders. In addition, Jagtap et al. [35] and another recent work [36] contributed to the literature by studying adaptive activation functions in the context of solving various forward and inverse problems within the framework of physics-informed neural networks, where the goal is to combine the expressive power of neural networks with the physical constraints provided by partial differential equations.

Despite the abundance of research in the area of adaptive activation functions, there is a notable gap in understanding the effectiveness of adaptive activation functions in addressing sparse data challenges, particularly those arising from intricate experimental scenarios within scientific and engineering domains. To bridge this gap, this paper concentrates on three distinct experimental data sets derived from various additive manufacturing problems. Improving the predictive performance of neural networks and reducing the need for predetermined activation functions are indispensable to accelerate the design and discovery of novel materials with enhanced flexibility and efficiency.

In our study, we specifically evaluate the efficacy of adaptive activation functions in scenarios with fewer than $100$ training samples. This focus provides a unique perspective compared to existing research, which has predominantly focused on data sets with significantly larger sizes spanning multiple orders of magnitude. To illustrate this point, consider the CIFAR10 data set—a collection of 60,000 32x32 color images distributed across 10 classes, with 6,000 images per class. In contrast, in many experimental setups, the average sample size per class is on the order of a few tens of samples. It is worth noting that the expense of gathering a few tens of samples in the world of material experiments may significantly surpass the cost of collecting and labeling images with sizes several orders of magnitudes larger due to machine, maintenance, material, and labor costs. As a result, there is a compelling need for new research to understand the trade-off between performance gains and the increase in trainable parameters associated with the utilization of adaptive activation functions.

In this section, we undertake a comprehensive evaluation of adaptive activation functions using three distinct experimental data sets. Additionally, we explore three activation functions—ELU, Softplus, and Swish—utilizing both fixed and trainable parameters, denoted as $\alpha$. As mentioned earlier, fixed activation functions correspond to the specific choice of $\alpha=1$, while adaptive activation functions dynamically learn and optimize the optimal value of $\alpha$ through gradient descent optimization during the training process. To ensure a fair comparison, we initialize the optimization process with a value of $1$ for $\alpha$ and then report the optimized value of $\alpha$ after completing the training stage. Furthermore, we configured the number of epochs to be 100, employing the Adam optimization algorithm with a learning rate of $0.05$.

In terms of the neural network architecture, our focus in this paper is on MLPs consisting of a single hidden layer, a deliberate choice driven by the constraint of a small sample size. This choice of $L=1$ also enables us to explore a critical aspect of adaptive activation functions that has been overlooked in previous research. To elucidate, we examine the common scenario of identical activation functions, where all units in a layer share the same $\alpha$ parameter. However, we go a step further and investigate the impact of allocating individual parameters for units within a hidden layer. This exploration aims to understand the trade-off between the number of trainable parameters and the adaptability of neural networks, particularly in small data settings.

Figure 2 depicts our systematic investigation, where the model M1 represents networks with fixed activation functions, M2 corresponds to networks that share the same trainable parameter $\alpha$, and M3 allows for individual trainable parameters for each activation function. In our investigation, we set the number of units in the hidden layer to be $N_{h}=2$ due to the limited availability of data. However, for completeness, we will explore the impact of $N_{h}$ on the performance gap between neural networks with adaptive and fixed activation functions at the end of this section.

To gauge the influence of employing trainable activation functions on the confidence of prediction models, we propose a shift from the conventional evaluation method, which relies on singular point predictions, to the creation and assessment of prediction sets. Formally, consider a classification problem with $C$ classes, denoting the output space as $\mathcal{Y}={1,\ldots,C}$. Adopting the conformal inference framework [25, 37], our objective is to construct a prediction set for a new or unseen input vector $x_{\text{test}}$, expressed as $\mathcal{I}(x_{\text{test}})\subseteq\mathcal{Y}$, adhering to the following probabilistic rationale:

where the desired coverage level $1-\delta$ is set by users. In this paper, we opt for the conventional selection of $\delta=0.1$ to establish the desired coverage level at a reasonable level $0.9$.

Next, we discuss an efficient algorithm to find the prediction set in our numerical experiments. During the training stage, we define the score function as one minus the softmax score of the true class. That is, we have $s(x_{n},y_{n})=1-f(x_{n})_{y_{n}}$, where $f(x_{n})_{y_{n}}$ represents the softmax score of the true class indicated by the ground-truth label $y_{n}$. Then, we find the empirical quantile of $\{s(x_{n},y_{n})\}_{n=1}^{N_{\text{train}}}$ at the adjusted level $\lceil(1-\delta)(N_{\text{train}}+1)/N_{\text{train}}\rceil$ [37]. Once we obtain the quantile $\hat{q}$, the prediction set for the test vector $x_{\text{test}}$ can be found as follows:

This implies that the prediction set encompasses all classes with softmax scores that exceed the threshold of $1-\hat{q}$.

Given these prediction sets, we evaluate the impact of adaptive activation functions using two metrics: empirical coverage and uncertainty score. Let $\mathcal{D}_{\text{test}}$ represent a distinct testing set with $N_{\text{test}}$ input-output pairs for evaluation. Empirical coverage denotes the proportion of data points with true outputs within the corresponding prediction sets. Similar to classification accuracy, higher values are preferable, and we aim for a minimum coverage level of $0.9$. On the other hand, the uncertainty score in this context is the average size of prediction sets. A perfect score is $1$ for confident and accurate classifiers that predict a single class while adhering to the probabilistic argument. However, higher scores are less desirable, indicating a classifier with less confidence in its predictions.

In the following sections, we partition $30\%$ of the data at random to construct the testing set $\mathcal{D}_{\text{test}}$, while the remaining portion is allocated for forming the training data set $\mathcal{D}_{\text{train}}$. Recognizing the sensitivity of neural networks to data splits, we conduct $20$ independent splits and visualize the distribution of evaluation metrics across these experiments. Our chosen visualization method is the violin plot, which combines key elements from box plots and kernel density plots. The width of this plot signifies the density of data points at various values, and the vertical axis represents the probability density. Additionally, the tick in a violin plot denotes the median of each evaluation metric.

As a final note before delving into our three testbeds, it is important to highlight that the version of conformal prediction discussed here differs slightly from the split conformal prediction methods. The key distinction lies in the fact that split conformal prediction typically employs a separate calibration data set for quantile determination. However, in this paper, we opt not to divide the training set for two reasons. First, across all testbeds, our training samples are fewer than 100, and the objective is to utilize as many data points as possible for training neural networks, including optimizing trainable parameters for each activation function. Second, utilizing the training data set enables us to indirectly assess the risk of overfitting—a significant concern in predictive modeling with neural networks. The discussed approach allows us to account for artificially small score functions due to overfitting, negatively impacting empirical coverage on the test set, a factor considered in our comprehensive analysis.

In this study, we investigated the adaptability of neural network models in scenarios with limited data, leveraging parameterized activation functions. Employing three real-world testbeds derived from additive manufacturing problems, we demonstrated that neural network models equipped with individual trainable parameters—going beyond the conventional practice of employing identical activation functions for a given hidden layer—resulted in improved prediction models. We evaluated these enhancements through the lens of point predictions, gauged by the standard classification accuracy score, and further delved into the realm of prediction sets through conformal inference. Specifically, we noted the following key observations.

1. 1.

   When dealing with sparse scientific data sets, opting for ELU and Softplus activation functions with individual trainable parameters proved advantageous over fixed and parameterized activation functions shared across all units in a hidden layer. The adoption of individual trainable activation functions in M3 demonstrated remarkable flexibility, allowing each unit to discern the optimal degree of nonlinearity introduced when conveying information to the next layer.

2. 2.

   Leveraging conformal inference emerged as a versatile and crucial approach to assess the confidence in the predictions of neural network models with trainable activation functions. Due to the increase in the number of parameters that must be inferred during the training process, measuring the empirical coverage score and the average prediction set size were informative measures to quantify predictive uncertainty.

3. 3.

   The performance trade-offs between fixed and adaptive activation functions were heavily contingent on the neural network architecture. Consequently, in scenarios with limited data, striking the right balance between the number of hidden units and their adaptability becomes crucial.

4. 4.

   As automated machine learning (AutoML) methods gain popularity to democratize the utilization of prediction models among practitioners and engineers [53], incorporating adaptive activation functions can yield substantial improvements. This approach mitigates the dependence on predetermined and fixed-shape activation functions, which can significantly impact predictive performance.

In future work, we plan to explore several extensions of our study. In particular, we aim to investigate the performance of adaptive activation functions in more complex neural network models, such as convolutional neural networks (CNNs). This exploration may find application in the development of adaptive CNNs for constructing accurate prediction models for in situ monitoring of manufacturing technologies with minimal reliance on human supervision. Another extension will revolve around the incorporation of ensemble activation functions. Here, the objective is to leverage richer parameterized activation functions, offering greater flexibility compared to the activation functions employed in this study.

Research was sponsored by DEVCOM Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-19-2-0100. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of DEVCOM Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.