We work on dynamic problems with collected data $\{\mathsf{x}_i\}$ that distributed on a manifold $\mathcal{M}\subset\mathbb{R}^p$. Through the diffusion map, we first learn the reaction coordinates $\{\mathsf{y}_i\}\subset \mathcal{N}$ where $\mathcal{N}$ is a manifold isometrically embedded into an Euclidean space $\mathbb{R}^\ell$ for $\ell \ll p$. The reaction coordinates enable us to obtain an efficient approximation for the dynamics described by a Fokker-Planck equation on the manifold $\mathcal{N}$. By using the reaction coordinates, we propose an implementable, unconditionally stable, data-driven upwind scheme which automatically incorporates the manifold structure of $\mathcal{N}$. Furthermore, we provide a weighted $L^2$ convergence analysis of the upwind scheme to the Fokker-Planck equation. The proposed upwind scheme leads to a Markov chain with transition probability between the nearest neighbor points. We can benefit from such property to directly conduct manifold-related computations such as finding the optimal coarse-grained network and the minimal energy path that represents chemical reactions or conformational changes. To establish the Fokker-Planck equation, we need to acquire information about the equilibrium potential of the physical system on $\mathcal{N}$. Hence, we apply a Gaussian Process regression algorithm to generate equilibrium potential for a new physical system with new parameters. Combining with the proposed upwind scheme, we can calculate the trajectory of the Fokker-Planck equation on $\mathcal{N}$ based on the generated equilibrium potential. Finally, we develop an algorithm to pullback the trajectory to the original high dimensional space as a generative data for the new physical system.

Inverse probability weighted estimators are the oldest and potentially most commonly used class of procedures for the estimation of causal effects. By adjusting for selection biases via a weighting mechanism, these procedures estimate an effect of interest by constructing a pseudo-population in which selection biases are eliminated. Despite their ease of use, these estimators require the correct specification of a model for the weighting mechanism, are known to be inefficient, and suffer from the curse of dimensionality. We propose a class of nonparametric inverse probability weighted estimators in which the weighting mechanism is estimated via undersmoothing of the highly adaptive lasso, a nonparametric regression function proven to converge at $n^{-1/3}$-rate to the true weighting mechanism. We demonstrate that our estimators are asymptotically linear with variance converging to the nonparametric efficiency bound. Unlike doubly robust estimators, our procedures require neither derivation of the efficient influence function nor specification of the conditional outcome model. Our theoretical developments have broad implications for the construction of efficient inverse probability weighted estimators in large statistical models and a variety of problem settings. We assess the practical performance of our estimators in simulation studies and demonstrate use of our proposed methodology with data from a large-scale epidemiologic study.

We study supervised learning by extreme learning machines and regression for autonomous objects moving in a non-stationary spatial environment. In general, this results in non-stationary data in contrast to the i.i.d. sampling typically studied in learning theory. The stochastic model for the environment and data collection especially allows for algebraically decaying weak dependence and spatial heterogeneity, for example induced by interactions of the object with sources of randomness spread over the spatial domain. Both least squares and ridge learning as a computationally cheap regularization method is studied. Consistency and asymptotic normality of the least squares and ridge regression estimates is shown under weak conditions. The results also cover consistency in terms of bounds for the sample squared predicition error. Lastly, we discuss a resampling method to compute confidence regions.

You're looking for a complete Linear Regression course that teaches you everything you need to create a Linear Regression model in Python, right? You've found the right Linear Regression course! Identify the business problem which can be solved using linear regression technique of Machine Learning. Create a linear regression model in Python and analyze its result. A Verifiable Certificate of Completion is presented to all students who undertake this Machine learning basics course.

Data has become the new currency now and when the new norm of the life will be push us more towards adoption of digital products, data will play crucial role in determining consumer behaviour and personalising the digital solution. The demand for the digital products will grow day by day and the responsibility of a product manager will also increase, which will push them to learn new skills and technology. I will keep on sharing my experience and learning with fellow product professionals to solve consumers problem in a better way. Let us start our journey with a brief understanding of machine learning. Machine learning is an application of artificial intelligence (AI) that provides systems the ability to automatically learn and improve from experience.

We investigate meta-learning procedures in the setting of stochastic linear bandits tasks. The goal is to select a learning algorithm which works well on average over a class of bandits tasks, that are sampled from a task-distribution. Inspired by recent work on learning-to-learn linear regression, we consider a class of bandit algorithms that implement a regularized version of the well-known OFUL algorithm, where the regularization is a square euclidean distance to a bias vector. We first study the benefit of the biased OFUL algorithm in terms of regret minimization. We then propose two strategies to estimate the bias within the learning-to-learn setting. We show both theoretically and experimentally, that when the number of tasks grows and the variance of the task-distribution is small, our strategies have a significant advantage over learning the tasks in isolation.

Paradigmatic problems in supervised machine learning (ML) involve predicting an output response from an input, based on patterns extracted from a (training) dataset. In classification, the output response is (finitely) discrete and we need to classify input data into one of these discrete categories. In regression, the output is continuous, typically a real number or a vector. Owing to this important distinction in output response, the two tasks are typically treated differently. The differences in treatment manifest in two phases of modern ML: optimization (training), which consists of an algorithmic procedure to extract a predictor from the training data, typically by minimizing the training loss (also called empirical risk); and generalization (testing), which consists of an evaluation of the obtained predictor on a separate test, or validation, dataset. Traditionally, the choice of loss functions for both phases is starkly different across classification and regression tasks. The squared-loss function is typically used both for the training and the testing phases in regression. In contrast, the hinge or logistic (cross-entropy for multi-class problems) loss functions are typically used in the training phase of classification, while the very different 0-1 loss function is used for testing.

In this work we provide a review of basic ideas and novel developments about Conformal Prediction -- an innovative distribution-free, non-parametric forecasting method, based on minimal assumptions -- that is able to yield in a very straightforward way predictions sets that are valid in a statistical sense also in in the finite sample case. The in-depth discussion provided in the paper covers the theoretical underpinnings of Conformal Prediction, and then proceeds to list the more advanced developments and adaptations of the original idea.

We study the nested model averaging method on the solution path for a high-dimensional linear regression problem. In particular, we propose to combine model averaging with regularized estimators (e.g., lasso and SLOPE) on the solution path for high-dimensional linear regression. In simulation studies, we first conduct a systematic investigation on the impact of predictor ordering on the behavior of nested model averaging, then show that nested model averaging with lasso and SLOPE compares favorably with other competing methods, including the infeasible lasso and SLOPE with the tuning parameter optimally selected. A real data analysis on predicting the per capita violent crime in the United States shows an outstanding performance of the nested model averaging with lasso.

Convolutional neural networks (CNNs) are widely used for image recognition and text analysis, and have been suggested for application on one-dimensional data as a way to reduce the need for pre-processing steps. Pre-processing is an integral part of multivariate analysis, but determination of the optimal pre-processing methods can be time-consuming due to the large number of available methods. In this work, the performance of a CNN was investigated for classification and regression analysis of spectral data. The CNN was compared with various other chemometric methods, including support vector machines (SVMs) for classification and partial least squares regression (PLSR) for regression analysis. The comparisons were made both on raw data, and on data that had gone through pre-processing and/or feature selection methods. The models were used on spectral data acquired with methods based on near-infrared, mid-infrared, and Raman spectroscopy. For the classification datasets the models were evaluated based on the percentage of correctly classified observations, while for regression analysis the models were assessed based on the coefficient of determination (R$^2$). Our results show that CNNs can outperform standard chemometric methods, especially for classification tasks where no pre-processing is used. However, both CNN and the standard chemometric methods see improved performance when proper pre-processing and feature selection methods are used. These results demonstrate some of the capabilities and limitations of CNNs used on one-dimensional data.