P-values for coefficients of cylinders, horsepower and acceleration are all greater than 0.05. This means that the relationship between the dependent and these independent variables is not significant at the 95% certainty level. I'll drop 2 of these variables and try again. High p-values for these independent variables do not mean that they definitely should not be used in the model. It could be that some other variables are correlated with these variables and making these variables less useful for prediction (check Multicollinearity).

Even though standard regression seems to be performing much better, predictions for individual salary - regression versus Jackknife - are not far off, as illustrated in the top figure. Both for regression and Jackknife, only 8 different estimated values are generated, since we have just 8 codes. Note that if we boost correlations to the point that Correl(Python, R) 1, then the linear regression model will crash, while the Jackknife will perform nicely. Rudimentary, approximate methods such as Jackknife regression (not to be confused with Efron's bootstrap) are just nearly as good as so-called exact models such as traditional regression, for predictive modeling. The reason is because data is anything but exact, and statistical models are approximate representations of the reality: all models are wrong, some are not as wrong as others. Approximate solutions provide substantial advantages: easy to code (even in SQL) and understand, robust, and easy to interpret. In short, they are a good choice for inclusion in black-box, automated data science.

Many modern data sets are sampled with error from complex high-dimensional surfaces. Methods such as tensor product splines or Gaussian processes are effective/well suited for characterizing a surface in two or three dimensions but may suffer from difficulties when representing higher dimensional surfaces. Motivated by high throughput toxicity testing where observed dose-response curves are cross sections of a surface defined by a chemical's structural properties, a model is developed to characterize this surface to predict untested chemicals' dose-responses. This manuscript proposes a novel approach that models the multidimensional surface as a sum of learned basis functions formed as the tensor product of lower dimensional functions, which are themselves representable by a basis expansion learned from the data. The model is described, a Gibbs sampling algorithm proposed, and is investigated in a simulation study as well as data taken from the US EPA's ToxCast high throughput toxicity testing platform.

The former belongs to the realm of explanatory models, the latter to the realm of predictive analytics. Explanatory models, often involving linear regression, are concerned with explaining a given phenomenon and finding causal relationships between an output (dependent) variable, and a host, often very few, input (independent) variables. The objective is to find a good regression model that fits the data very well which meets the underlying assumption of linear regression. The emphasis here is on hypothesis testing, p-values, confidence intervals,…Once a good model is found, one can use it for estimating the value of the output variable for given values of the input variables. It is OK to estimate an output value based on interpolation, but one must use extreme caution in estimating output values based on extrapolation because the regression model is an explanatory model, not a predictive one.

We hear the term "machine learning" a lot these days, usually in the context of predictive analysis and artificial intelligence. Machine learning is, more or less, a way for computers to learn things without being specifically programmed. But how does that actually happen? The answer is, in one word, algorithms. Algorithms are sets of rules that a computer is able to follow.

I consider unsupervised extensions of the fast stepwise linear regression algorithm \cite{efroymson1960multiple}. These extensions allow one to efficiently identify highly-representative feature variable subsets within a given set of jointly distributed variables. This in turn allows for the efficient dimensional reduction of large data sets via the removal of redundant features. Fast search is effected here through the avoidance of repeat computations across trial fits, allowing for a full representative-importance ranking of a set of feature variables to be carried out in $O(n^2 m)$ time, where $n$ is the number of variables and $m$ is the number of data samples available. This runtime complexity matches that needed to carry out a single regression and is $O(n^2)$ faster than that of naive implementations. I present pseudocode suitable for efficient forward, reverse, and forward-reverse unsupervised feature selection. To illustrate the algorithm's application, I apply it to the problem of identifying representative stocks within a given financial market index -- a challenge relevant to the design of Exchange Traded Funds (ETFs). I also characterize the growth of numerical error with iteration step in these algorithms, and finally demonstrate and rationalize the observation that the forward and reverse algorithms return exactly inverted feature orderings in the weakly-correlated feature set regime.

Softmax Regression (synonyms: Multinomial Logistic, Maximum Entropy Classifier, or just Multi-class Logistic Regression) is a generalization of logistic regression that we can use for multi-class classification (under the assumption that the classes are mutually exclusive). In contrast, we use the (standard) Logistic Regression model in binary classification tasks. Now, let me briefly explain how that works and how softmax regression differs from logistic regression. Now, this softmax function computes the probability that this training sample x(i) belongs to class j given the weight and net input z(i). So, we compute the probability p(y j x(i); wj) for each class label in j 1, ..., k.

Get the code: To follow along, all the code is also available as an iPython notebook on Github. In this post we will implement a simple 3-layer neural network from scratch. We won't derive all the math that's required, but I will try to give an intuitive explanation of what we are doing. I will also point to resources for you read up on the details. Here I'm assuming that you are familiar with basic Calculus and Machine Learning concepts, e.g.

This article on going deeper into regression analysis with assumptions, plots & solutions, was posted by Manish Saraswat. Manish who works in marketing and Data Science at Analytics Vidhya believes that education can change this world. R, Data Science and Machine Learning keep him busy. Regression analysis marks the first step in predictive modeling. No doubt, it's fairly easy to implement.

Analyzing time series data is important to predict future events and changes in finance, manufacturing, and administrative decisions. Gaussian processes (GPs) solve regression and classification problems by choosing appropriate kernels capturing covariance structure of data. In time series analysis, GP based regression methods recently demonstrate competitive performance by decomposing temporal covariance structure. Such covariance structure decomposition allows exploiting shared parameters over a set of multiple but selected time series. In this paper, we handle multiple time series by placing an Indian Buffet Process (IBP) prior on the presence of shared kernels. We investigate the validity of model when infinite latent components are introduced. We also propose an improved search algorithm to find interpretable kernels among multiple time series along with comparison reports. Experiments are conducted on both synthetic data sets and real world data sets, showing promising results in term of structure discoveries and predictive performances.