Large-scale models require substantial computational resources for analysis and studying treatment conditions. Specifically, estimating treatment effects using simulations may require a lot of infeasible resources to allocate at every treatment condition. Therefore, it is essential to develop efficient methods to allocate computational resources for estimating treatment effects. Agent-based simulation allows us to generate highly realistic simulation samples. FRED (A Framework for Reconstructing Epidemiological Dynamics) is an agent-based modeling system with a geospatial perspective using a synthetic population constructed based on the U.S. census data. Given its synthetic population, FRED simulations present a baseline for comparable results from different treatment conditions and treatment conditions. In this paper, we show three other methods for estimating treatment effects. In the first method, we resort to brute-force allocation, where all treatment conditions have an equal number of samples with a relatively large number of simulation runs. In the second method, we try to reduce the number of simulation runs by customizing individual samples required for each treatment effect based on the width of confidence intervals around the mean estimates. In the third method, we use a regression model, which allows us to learn across the treatment conditions such that simulation samples allocated for a treatment condition will help better estimate treatment effects in other conditions. We show that the regression-based methods result in a comparable estimate of treatment effects with less computational resources. The reduced variability and faster convergence of model-based estimates come at the cost of increased bias, and the bias-variance trade-off can be controlled by adjusting the number of model parameters (e.g., including higher-order interaction terms in the regression model).

This paper introduces a new data analysis method for big data using a newly defined regression model named multiple model linear regression(MMLR), which separates input datasets into subsets and construct local linear regression models of them. The proposed data analysis method is shown to be more efficient and flexible than other regression based methods. This paper also proposes an approximate algorithm to construct MMLR models based on $(\epsilon,\delta)$-estimator, and gives mathematical proofs of the correctness and efficiency of MMLR algorithm, of which the time complexity is linear with respect to the size of input datasets. This paper also empirically implements the method on both synthetic and real-world datasets, the algorithm shows to have comparable performance to existing regression methods in many cases, while it takes almost the shortest time to provide a high prediction accuracy.

We explore the minimax optimal error associated with a demographic parity-constrained regression problem within the context of a linear model. Our proposed model encompasses a broader range of discriminatory bias sources compared to the model presented by Chzhen and Schreuder (2022). Our analysis reveals that the minimax optimal error for the demographic parity-constrained regression problem under our model is characterized by $\Theta(\frac{dM}{n})$, where $n$ denotes the sample size, $d$ represents the dimensionality, and $M$ signifies the number of demographic groups arising from sensitive attributes. Moreover, we demonstrate that the minimax error increases in conjunction with a larger bias present in the model.

This research enhances linear regression models by integrating a Kalman filter and analysing curve areas to minimize loss. The goal is to develop an optimal linear regression equation using stochastic gradient descent (SGD) for weight updating. Our approach involves a stepwise process, starting with user-defined parameters. The linear regression model is trained using SGD, tracking weights and loss separately and zipping them finally. A Kalman filter is then trained based on weight and loss arrays to predict the next consolidated weights. Predictions result from multiplying input averages with weights, evaluated for loss to form a weight-versus-loss curve. The curve's equation is derived using the two-point formula, and area under the curve is calculated via integration. The linear regression equation with minimum area becomes the optimal curve for prediction. Benefits include avoiding constant weight updates via gradient descent and working with partial datasets, unlike methods needing the entire set. However, computational complexity should be considered. The Kalman filter's accuracy might diminish beyond a certain prediction range.

Accurate demand forecasting in the retail industry is a critical determinant of financial performance and supply chain efficiency. As global markets become increasingly interconnected, businesses are turning towards advanced prediction models to gain a competitive edge. However, existing literature mostly focuses on historical sales data and ignores the vital influence of macroeconomic conditions on consumer spending behavior. In this study, we bridge this gap by enriching time series data of customer demand with macroeconomic variables, such as the Consumer Price Index (CPI), Index of Consumer Sentiment (ICS), and unemployment rates. Leveraging this comprehensive dataset, we develop and compare various regression and machine learning models to predict retail demand accurately.

A basic task in explainable AI (XAI) is to identify the most important features behind a prediction made by a black box function $f$. The insertion and deletion tests of Petsiuk et al. (2018) can be used to judge the quality of algorithms that rank pixels from most to least important for a classification. Motivated by regression problems we establish a formula for their area under the curve (AUC) criteria in terms of certain main effects and interactions in an anchored decomposition of $f$. We find an expression for the expected value of the AUC under a random ordering of inputs to $f$ and propose an alternative area above a straight line for the regression setting. We use this criterion to compare feature importances computed by integrated gradients (IG) to those computed by Kernel SHAP (KS) as well as LIME, DeepLIFT, vanilla gradient and input$\times$gradient methods. KS has the best overall performance in two datasets we consider but it is very expensive to compute. We find that IG is nearly as good as KS while being much faster. Our comparison problems include some binary inputs that pose a challenge to IG because it must use values between the possible variable levels and so we consider ways to handle binary variables in IG. We show that sorting variables by their Shapley value does not necessarily give the optimal ordering for an insertion-deletion test. It will however do that for monotone functions of additive models, such as logistic regression.

We study a fundamental transfer learning process from source to target linear regression tasks, including overparameterized settings where there are more learned parameters than data samples. The target task learning is addressed by using its training data together with the parameters previously computed for the source task. We define a transfer learning approach to the target task as a linear regression optimization with a regularization on the distance between the to-be-learned target parameters and the already-learned source parameters. We analytically characterize the generalization performance of our transfer learning approach and demonstrate its ability to resolve the peak in generalization errors in double descent phenomena of the minimum L2-norm solution to linear regression. Moreover, we show that for sufficiently related tasks, the optimally tuned transfer learning approach can outperform the optimally tuned ridge regression method, even when the true parameter vector conforms to an isotropic Gaussian prior distribution. Namely, we demonstrate that transfer learning can beat the minimum mean square error (MMSE) solution of the independent target task. Our results emphasize the ability of transfer learning to extend the solution space to the target task and, by that, to have an improved MMSE solution. We formulate the linear MMSE solution to our transfer learning setting and point out its key differences from the common design philosophy to transfer learning.

Regression analysis is a key area of interest in the field of data analysis and machine learning which is devoted to exploring the dependencies between variables, often using vectors. The emergence of high dimensional data in technologies such as neuroimaging, computer vision, climatology and social networks, has brought challenges to traditional data representation methods. Tensors, as high dimensional extensions of vectors, are considered as natural representations of high dimensional data. In this book, the authors provide a systematic study and analysis of tensor-based regression models and their applications in recent years. It groups and illustrates the existing tensor-based regression methods and covers the basics, core ideas, and theoretical characteristics of most tensor-based regression methods. In addition, readers can learn how to use existing tensor-based regression methods to solve specific regression tasks with multiway data, what datasets can be selected, and what software packages are available to start related work as soon as possible. Tensor Regression is the first thorough overview of the fundamentals, motivations, popular algorithms, strategies for efficient implementation, related applications, available datasets, and software resources for tensor-based regression analysis. It is essential reading for all students, researchers and practitioners of working on high dimensional data.

The regression of a functional response on a set of scalar predictors can be a challenging task, especially if there is a large number of predictors, or the relationship between those predictors and the response is nonlinear. In this work, we propose a solution to this problem: a feed-forward neural network (NN) designed to predict a functional response using scalar inputs. First, we transform the functional response to a finite-dimensional representation and construct an NN that outputs this representation. Then, we propose to modify the output of an NN via the objective function and introduce different objective functions for network training. The proposed models are suited for both regularly and irregularly spaced data, and a roughness penalty can be further applied to control the smoothness of the predicted curve. The difficulty in implementing both those features lies in the definition of objective functions that can be back-propagated. In our experiments, we demonstrate that our model outperforms the conventional function-on-scalar regression model in multiple scenarios while computationally scaling better with the dimension of the predictors.

Climate projections using data driven machine learning models acting as emulators, is one of the prevailing areas of research to enable policy makers make informed decisions. Use of machine learning emulators as surrogates for computationally heavy GCM simulators reduces time and carbon footprints. In this direction, ClimateBench [1] is a recently curated benchmarking dataset for evaluating the performance of machine learning emulators designed for climate data. Recent studies have reported that despite being considered fundamental, regression models offer several advantages pertaining to climate emulations. In particular, by leveraging the kernel trick, regression models can capture complex relationships and improve their predictive capabilities. This study focuses on evaluating non-linear regression models using the aforementioned dataset. Specifically, we compare the emulation capabilities of three non-linear regression models. Among them, Gaussian Process Regressor demonstrates the best-in-class performance against standard evaluation metrics used for climate field emulation studies. However, Gaussian Process Regression suffers from being computational resource hungry in terms of space and time complexity. Alternatively, Support Vector and Kernel Ridge models also deliver competitive results and but there are certain trade-offs to be addressed. Additionally, we are actively investigating the performance of composite kernels and techniques such as variational inference to further enhance the performance of the regression models and effectively model complex non-linear patterns, including phenomena like precipitation.