The common approach to compositional data analysis is to transform the data by means of logratios. Logratios between pairs of compositional parts (pairwise logratios) are the easiest to interpret in many research problems. When the number of parts is large, some form of logratio selection is a must, for instance by means of an unsupervised learning method based on a stepwise selection of the pairwise logratios that explain the largest percentage of the logratio variance in the compositional dataset. In this article we present three alternative stepwise supervised learning methods to select the pairwise logratios that best explain a dependent variable in a generalized linear model, each geared for a specific problem. The first method features unrestricted search, where any pairwise logratio can be selected. This method has a complex interpretation if some pairs of parts in the logratios overlap, but it leads to the most accurate predictions. The second method restricts parts to occur only once, which makes the corresponding logratios intuitively interpretable. The third method uses additive logratios, so that $K-1$ selected logratios involve exactly $K$ parts. This method in fact searches for the subcomposition with the highest explanatory power. Once the subcomposition is identified, the researcher's favourite logratio representation may be used in subsequent analyses, not only pairwise logratios. Our methodology allows logratios or non-compositional covariates to be forced into the models based on theoretical knowledge, and various stopping criteria are available based on information measures or statistical significance with the Bonferroni correction. We present an illustration of the three approaches on a dataset from a study predicting Crohn's disease. The first method excels in terms of predictive power, and the other two in interpretability.

This work aims to provide an engagement decision support tool for Beyond Visual Range (BVR) air combat in the context of Defensive Counter Air (DCA) missions. In BVR air combat, engagement decision refers to the choice of the moment the pilot engages a target by assuming an offensive stance and executing corresponding maneuvers. To model this decision, we use the Brazilian Air Force's Aerospace Simulation Environment (Ambiente de Simula\c{c}\~ao Aeroespacial - ASA in Portuguese), which generated 3,729 constructive simulations lasting 12 minutes each and a total of 10,316 engagements. We analyzed all samples by an operational metric called the DCA index, which represents, based on the experience of subject matter experts, the degree of success in this type of mission. This metric considers the distances of the aircraft of the same team and the opposite team, the point of Combat Air Patrol, and the number of missiles used. By defining the engagement status right before it starts and the average of the DCA index throughout the engagement, we create a supervised learning model to determine the quality of a new engagement. An algorithm based on decision trees, working with the XGBoost library, provides a regression model to predict the DCA index with a coefficient of determination close to 0.8 and a Root Mean Square Error of 0.05 that can furnish parameters to the BVR pilot to decide whether or not to engage. Thus, using data obtained through simulations, this work contributes by building a decision support system based on machine learning for BVR air combat.

If you've ever searched for evaluation metrics to assess model accuracy, chances are that you found many different options to choose from. Accuracy is in some sense the holy grail of prediction so it's not at all surprising that the machine learning community spends a lot time thinking about it. In a world where more and more high-stake decisions are being automated, model accuracy is in fact a very valid concern. But does this recipe for model evaluation seem like a sound and complete approach to automated decision-making? Some would argue that we need to pay more attention to model uncertainty. No matter how many times you have cross-validated your model, the loss metric that it is being optimized against as well as its parameters and predictions remain inherently random variables.

Let's say I am given an Excel sheet with data about various fruits and I have to tell which look like Apples. What I will do is ask a question "Which fruits are red and round?" and divide all fruits which answer yes and no to the question. Now, All Red and Round fruits might not be apples and all apples won't be red and round. So I will ask a question "Which fruits have red or yellow color hints on them? " on red and round fruits and will ask "Which fruits are green and round?" on not red and round fruits. Based on these questions I can tell with considerable accuracy which are apples. This cascade of questions is what a decision tree is. However, this is a decision tree based on my intuition.

Current seismic design codes primarily rely on the strength and displacement capacity of structural members and do not account for the influence of the ground motion duration or the hysteretic behavior characteristics. The energy-based approach serves as a supplemental index to response quantities and includes the effect of repeated loads in seismic performance. The design philosophy suggests that the seismic demands are met by the energy dissipation capacity of the structural members. Therefore, the energy dissipation behavior of the structural members should be well understood to achieve an effective energy-based design approach. This study focuses on the energy dissipation capacity of reinforced concrete (RC) shear walls that are widely used in high seismic regions as they provide significant stiffness and strength to resist lateral forces. A machine learning (Gaussian Process Regression (GPR))-based predictive model for energy dissipation capacity of shear walls is developed as a function of wall design parameters. Eighteen design parameters are shown to influence energy dissipation, whereas the most important ones are determined by applying sequential backward elimination and by using feature selection methods to reduce the complexity of the predictive model. The ability of the proposed model to make robust and accurate predictions is validated based on novel data with a prediction accuracy (the ratio of predicted/actual values) of around 1.00 and a coefficient of determination (R2) of 0.93. The outcomes of this study are believed to contribute to the energy-based approach by (i) defining the most influential wall properties on the seismic energy dissipation capacity of shear walls and (ii) providing predictive models that can enable comparisons of different wall design configurations to achieve higher energy dissipation capacity.

An individualized decision rule (IDR) is a decision function that assigns each individual a given treatment based on his/her observed characteristics. Most of the existing works in the literature consider settings with binary or finitely many treatment options. In this paper, we focus on the continuous treatment setting and propose a jump interval-learning to develop an individualized interval-valued decision rule (I2DR) that maximizes the expected outcome. Unlike IDRs that recommend a single treatment, the proposed I2DR yields an interval of treatment options for each individual, making it more flexible to implement in practice. To derive an optimal I2DR, our jump interval-learning method estimates the conditional mean of the outcome given the treatment and the covariates via jump penalized regression, and derives the corresponding optimal I2DR based on the estimated outcome regression function. The regressor is allowed to be either linear for clear interpretation or deep neural network to model complex treatment-covariates interactions. To implement jump interval-learning, we develop a searching algorithm based on dynamic programming that efficiently computes the outcome regression function. Statistical properties of the resulting I2DR are established when the outcome regression function is either a piecewise or continuous function over the treatment space. We further develop a procedure to infer the mean outcome under the (estimated) optimal policy. Extensive simulations and a real data application to a warfarin study are conducted to demonstrate the empirical validity of the proposed I2DR.

Linear regression is a technique frequently used in statistical and data analysis. The task for standard linear regression is to fit a linear relationship among variables in a data set. Often, the goal is to find the most parsimonious model that can describe the majority of the data. In this work, we consider the situation where only a small portion of the data can be accurately modeled using linear regression. More generally, in many kinds of real-world data, portions of the data of significant size can be predicted significantly more accurately than by the best linear model for the overall data distribution: Rosenfeld et al. (2015) showed that there are attributes that are significant risk factors for gastrointestinal cancer in certain subpopulations, but not in the overall population. Hainline et al. (2019) demonstrated that a variety of standard (real-world) regression benchmarks have portions that are fit significantly better by a different linear model than the best model for the overall data set; Calderon et al. (2020) presented further, similar findings. We will consider cases where linear regression fits well when the data set is conditioned on a simple condition, which is unknown to us. We study the task of finding such a linear model, together with a formula on the data attributes describing the condition, i.e., the portion of the data for which the linear model is accurate. This problem was introduced by Juba (2017), who gave an algorithm for conditional sparse linear regression, using the maximum residual as the objective.

as in the previous article, I have given you an introduction to polynomial regression now I will tell you how to make a basic polynomial regression model in this article with some lines of codes. the…

In this paper we study the statistical properties of Principal Components Regression with Laplacian Eigenmaps (PCR-LE), a method for nonparametric regression based on Laplacian Eigenmaps (LE). PCR-LE works by projecting a vector of observed responses ${\bf Y} = (Y_1,\ldots,Y_n)$ onto a subspace spanned by certain eigenvectors of a neighborhood graph Laplacian. We show that PCR-LE achieves minimax rates of convergence for random design regression over Sobolev spaces. Under sufficient smoothness conditions on the design density $p$, PCR-LE achieves the optimal rates for both estimation (where the optimal rate in squared $L^2$ norm is known to be $n^{-2s/(2s + d)}$) and goodness-of-fit testing ($n^{-4s/(4s + d)}$). We also show that PCR-LE is \emph{manifold adaptive}: that is, we consider the situation where the design is supported on a manifold of small intrinsic dimension $m$, and give upper bounds establishing that PCR-LE achieves the faster minimax estimation ($n^{-2s/(2s + m)}$) and testing ($n^{-4s/(4s + m)}$) rates of convergence. Interestingly, these rates are almost always much faster than the known rates of convergence of graph Laplacian eigenvectors to their population-level limits; in other words, for this problem regression with estimated features appears to be much easier, statistically speaking, than estimating the features itself. We support these theoretical results with empirical evidence.

Machine learning (ML) is increasingly often used to inform high-stakes decisions. As complex ML models (e.g., deep neural networks) are often considered black boxes, a wealth of procedures has been developed to shed light on their inner workings and the ways in which their predictions come about, defining the field of 'explainable AI' (XAI). Saliency methods rank input features according to some measure of 'importance'. Such methods are difficult to validate since a formal definition of feature importance is, thus far, lacking. It has been demonstrated that some saliency methods can highlight features that have no statistical association with the prediction target (suppressor variables). To avoid misinterpretations due to such behavior, we propose the actual presence of such an association as a necessary condition and objective preliminary definition for feature importance. We carefully crafted a ground-truth dataset in which all statistical dependencies are well-defined and linear, serving as a benchmark to study the problem of suppressor variables. We evaluate common explanation methods including LRP, DTD, PatternNet, PatternAttribution, LIME, Anchors, SHAP, and permutation-based methods with respect to our objective definition. We show that most of these methods are unable to distinguish important features from suppressors in this setting.