In this tutorial, You'll learn Logistic Regression. Here you'll know what exactly is Logistic Regression and you'll also see an Example with Python. Logistic Regression is an important topic of Machine Learning and I'll try to make it as simple as possible. In the early twentieth century, Logistic regression was mainly used in Biology after this, it was used in some social science applications. If you are curious, You may ask where we should use logistic regression? So we use Logistic Regression when our independent variable is categorical.

This particular podcast covers the material in Chapter 1 of my new (unpublished) book "Statistical Machine Learning: A unified framework". In this episode we discuss Chapter 1 of my new book, which shows how supervised, unsupervised, and reinforcement learning algorithms can be viewed as special cases of a general empirical risk minimization framework. This is useful because it provides a framework for not only understanding existing algorithms but also for suggesting new algorithms for specific applications. Welcome to the 79th podcast in the podcast series Learning Machines 101. In this series of podcasts my goal is to discuss important concepts of artificial intelligence and machine learning in hopefully an entertaining and educational manner. This particular podcast is actually the second episode in a new special series of episodes designed to provide commentary on a new book that I am in the process of writing. The book's title is "Statistical Machine Learning: A unified framework" and it will be published by CRC Press in their "Texts in Statistical Science" series sometime in early 2021.

Many objects of interest can be expressed as a linear, mean square continuous functional of a least squares projection (regression). Often the regression may be high dimensional, depending on many variables. This paper gives minimal conditions for root-n consistent and efficient estimation of such objects when the regression and the Riesz representer of the functional are approximately sparse and the sum of the absolute value of the coefficients is bounded. The approximately sparse functions we consider are those where an approximation by some $t$ regressors has root mean square error less than or equal to $Ct^{-\xi}$ for $C,$ $\xi>0.$ We show that a necessary condition for efficient estimation is that the sparse approximation rate $\xi_{1}$ for the regression and the rate $\xi_{2}$ for the Riesz representer satisfy $\max\{\xi_{1} ,\xi_{2}\}>1/2.$ This condition is stronger than the corresponding condition $\xi_{1}+\xi_{2}>1/2$ for Holder classes of functions. We also show that Lasso based, cross-fit, debiased machine learning estimators are asymptotically efficient under these conditions. In addition we show efficiency of an estimator without cross-fitting when the functional depends on the regressors and the regression sparse approximation rate satisfies $\xi_{1}>1/2$.

In the 70s a novel branch of statistics emerged focusing its effort in selecting a function in the pattern recognition problem, which fulfils a definite relationship between the quality of the approximation and its complexity. These data-driven approaches are mainly devoted to problems of estimating dependencies with limited sample sizes and comprise all the empirical out-of sample generalization approaches, e.g. cross validation (CV) approaches. Although the latter are \emph{not designed for testing competing hypothesis or comparing different models} in neuroimaging, there are a number of theoretical developments within this theory which could be employed to derive a Statistical Agnostic (non-parametric) Mapping (SAM) at voxel or multi-voxel level. Moreover, SAMs could relieve i) the problem of instability in limited sample sizes when estimating the actual risk via the CV approaches, e.g. large error bars, and provide ii) an alternative way of Family-wise-error (FWE) corrected p-value maps in inferential statistics for hypothesis testing. In this sense, we propose a novel framework in neuroimaging based on concentration inequalities, which results in (i) a rigorous development for model validation with a small sample/dimension ratio, and (ii) a less-conservative procedure than FWE p-value correction, to determine the brain significance maps from the inferences made using small upper bounds of the actual risk.

In many graphs such as social networks, nodes have associated attributes representing their behavior. Predicting node attributes in such graphs is an important problem with applications in many domains like recommendation systems, privacy preservation, and targeted advertisement. Attributes values can be predicted by analyzing patterns and correlations among attributes and employing classification/regression algorithms. However, these approaches do not utilize readily available network topology information. In this regard, interconnections between different attributes of nodes can be exploited to improve the prediction accuracy. In this paper, we propose an approach to represent a node by a feature map with respect to an attribute $a_i$ (which is used as input for machine learning algorithms) using all attributes of neighbors to predict attributes values for $a_i$. We perform extensive experimentation on ten real-world datasets and show that the proposed feature map significantly improves the prediction accuracy as compared to baseline approaches on these datasets.

The development of algorithms for unsupervised pattern recognition by nonlinear clustering is a notable problem in data science. Markov clustering (MCL) is a renowned algorithm that simulates stochastic flows on a network of sample similarities to detect the structural organization of clusters in the data, but it has never been generalized to deal with data nonlinearity. Minimum Curvilinearity (MC) is a principle that approximates nonlinear sample distances in the high-dimensional feature space by curvilinear distances, which are computed as transversal paths over their minimum spanning tree, and then stored in a kernel. Here we propose MC-MCL, which is the first nonlinear kernel extension of MCL and exploits Minimum Curvilinearity to enhance the performance of MCL in real and synthetic data with underlying nonlinear patterns. MC-MCL is compared with baseline clustering methods, including DBSCAN, K-means and affinity propagation. We find that Minimum Curvilinearity provides a valuable framework to estimate nonlinear distances also when its kernel is applied in combination with MCL. Indeed, MC-MCL overcomes classical MCL and even baseline clustering algorithms in different nonlinear datasets.

Distance correlation has gained much recent attention in the statistics and machine learning community: the sample statistic is straightforward to compute, works for any metric or kernel choice, and equals 0 asymptotically if and only if independence. One major bottleneck is the testing process: the null distribution of distance correlation depends on the metric choice and marginal distributions, which cannot be readily estimated. To compute a p-value, the standard approach is to estimate the null distribution via permutation, which generally requires O(rn^2) time complexity for n samples and r permutations and too costly for big data applications. In this paper, we propose a chi-square distribution to approximate the null distribution of the unbiased distance correlation. We prove that the chi-square distribution either equals or well-approximates the null distribution, and always upper tail dominates the null distribution. The resulting distance correlation chi-square test does not require any permutation nor parameter estimation, is simple and fast to implement, works with any strong negative type metric or characteristic kernel, is valid and universally consistent for independence testing, and enjoys a similar finite-sample testing power as the standard permutation test. When testing one-dimensional data using Euclidean distance, the unbiased distance correlation testing runs in O(nlog(n)), rendering it comparable in speed to the Pearson correlation t-test. The results are supported and demonstrated via simulations.

Patients suffering from obstructive sleep apnea are mainly treated with continuous positive airway pressure (CPAP). Good compliance with this therapy is broadly accepted as more than 4h of CPAP average use nightly. Although it is a highly effective treatment, compliance with this therapy is problematic to achieve with serious consequences for the patients' health. Previous works already reported factors significantly related to compliance with the therapy. However, further research is still required to support clinicians to early anticipate patients' therapy compliance. This work intends to take a further step in this direction by building compliance classifiers with CPAP therapy at three different moments of the patient follow-up (i.e. before the therapy starts and at months 1 and 3 after the baseline). Results of the clinical trial confirmed that month 3 was the time-point with the most accurate classifier reaching an f1-score of 87% and 84% in cross-validation and test. At month 1, performances were almost as high as in month 3 with 82% and 84% of f1-score. At baseline, where no information about patients' CPAP use was given yet, the best classifier achieved 73% and 76% of f1-score in cross-validation and test set respectively. Subsequent analyses carried out with the best classifiers of each time point revealed that certain baseline factors (i.e. headaches, psychological symptoms, arterial hypertension and EuroQol visual analogue scale) were closely related to the prediction of compliance independently of the time-point. In addition, among the variables taken only during the follow-up of the patients, Epworth and the average nighttime hours were the most important to predict compliance with CPAP.

Many machine learning methods (classification, clustering, etc.) start with a known kernel that provides similarity or distance measure between two objects. Recent work has extended this to situations where the information about objects is limited to comparisons of distances between three objects (triplets). Humans find the comparison task much easier than the estimation of absolute similarities, so this kind of data can be easily obtained using crowd-sourcing. In this work, we give an efficient method of augmenting the triplets data, by utilizing additional implicit information inferred from the existing data. Triplets augmentation improves the quality of kernel-based and kernel-free data analytics tasks. Secondly, we also propose a novel set of algorithms for common supervised and unsupervised machine learning tasks based on triplets. These methods work directly with triplets, avoiding kernel evaluations. Experimental evaluation on real and synthetic datasets shows that our methods are more accurate than the current best-known techniques.

We propose a projection pursuit (PP) algorithm based on Gaussian mixture models (GMMs). The negentropy obtained from a multivariate density estimated by GMMs is adopted as the PP index to be maximised. For a fixed dimension of the projection subspace, the GMM-based density estimation is projected onto that subspace, where an approximation of the negentropy for Gaussian mixtures is computed. Then, Genetic Algorithms (GAs) are used to find the optimal, orthogonal projection basis by maximising the former approximation. We show that this semi-parametric approach to PP is flexible and allows highly informative structures to be detected, by projecting multivariate datasets onto a subspace, where the data can be feasibly visualised. The performance of the proposed approach is shown on both artificial and real datasets.