In time series analysis, when fitting an autoregressive model, one must solve a Toeplitz ordinary least squares problem numerous times to find an appropriate model, which can severely affect computational times with large data sets. Two recent algorithms (LSAR and Repeated Halving) have applied randomized numerical linear algebra (RandNLA) techniques to fitting an autoregressive model to big time-series data. We investigate and compare the quality of these two approximation algorithms on large-scale synthetic and real-world data. While both algorithms display comparable results for synthetic datasets, the LSAR algorithm appears to be more robust when applied to real-world time series data. We conclude that RandNLA is effective in the context of big-data time series.

AI Researcher, Cognitive Technologist Inventor - AI Thinking, Think Chain Innovator - AIOT, XAI, Autonomous Cars, IIOT Founder Fisheyebox Spatial Computing Savant, Transformative Leader, Industry X.0 Practitioner Why Real #BigData is impossible without Data Ontology? Mathematics is key to DO. It is dealing with ontological entities but as mathematical objects, as quantities, changes, and relationships (numbers, magnitudes, multitudes, spaces, manifolds) and their functional relationships, as listed below: Number theory: numbers, operations Combinatorics: permutations, derangements, combinations Set theory: sets, set partitions; functions, and relations Geometry: points, lines, line segments, polygons, circles, ellipses, parabolas, hyperbolas, polyhedra, spheres, ellipsoids, paraboloids, hyperboloids, cylinders, cones Graph theory: graphs, trees, nodes, edges Topology: topological spaces and manifolds Linear algebra: scalars, vectors, matrices, tensors Abstract algebra: groups, rings, modules, fields, vector spaces, group-theoretic lattices, and order-theoretic lattices Category Theory (a general theory of functions): categories, objects, edges Datalogy, as #DataScience and technology summarized as follows: Easy Architectural Changes: Applying structural changes to relational databases is a cumbersome process. Something as simple as changing a property from being single-valued to multi-valued could mean having to add a new table and foreign key reference to the original table, possibly compromising existing queries to it. With an ontology, you could simply modify the semantic concept underpinning the property.

When 3D animated balls on a computer screen defy certain laws of physics, dogs act in a way that suggests they feel like their eyes are deceiving them. Pet dogs stare for longer and their pupils widen if virtual balls start rolling on their own rather than being set in motion by a collision with another ball. This suggests that the animals are surprised that the balls didn't move the way they had expected them to, says Christoph Völter at the University of Veterinary Medicine, Vienna. "This is the starting point for learning," says Völter. "You have expectations about the environment – regularities in your environment that are connected to physics – and then something happens that doesn't fit. And now you pay attention. And now you try to see what's going on."

We study a multi-leader single-follower congestion game where multiple users (leaders) choose one resource out of a set of resources and, after observing the realized loads, an adversary (single-follower) attacks the resources with maximum loads, causing additional costs for the leaders. For the resulting strategic game among the leaders, we show that pure Nash equilibria may fail to exist and therefore, we consider approximate equilibria instead. As our first main result, we show that the existence of a $K$-approximate equilibrium can always be guaranteed, where $K \approx 1.1974$ is the unique solution of a cubic polynomial equation. To this end, we give a polynomial time combinatorial algorithm which computes a $K$-approximate equilibrium. The factor $K$ is tight, meaning that there is an instance that does not admit an $\alpha$-approximate equilibrium for any $\alpha

We present a numerical method to model dynamical systems from data. We use the recently introduced method Scalable Probabilistic Approximation (SPA) to project points from a Euclidean space to convex polytopes and represent these projected states of a system in new, lower-dimensional coordinates denoting their position in the polytope. We then introduce a specific nonlinear transformation to construct a model of the dynamics in the polytope and to transform back into the original state space. To overcome the potential loss of information from the projection to a lower-dimensional polytope, we use memory in the sense of the delay-embedding theorem of Takens. By construction, our method produces stable models. We illustrate the capacity of the method to reproduce even chaotic dynamics and attractors with multiple connected components on various examples.

You've found the right Statistics and Probability with Excel course! This course will teach you the skill to apply statistics and data analysis tools to various business applications. How this course will help you? A Verifiable Certificate of Completion is presented to all students who undertake this course on Probability and Statistics in Excel. If you are a business manager, or business analyst or an executive, or a student who wants to learn Probability and Statistics concepts and apply these techniques to real-world problems of the business function, this course will give you a solid base for Probability and Statistics by teaching you the most important concepts of Probability and Statistics and how to implement them in MS Excel.

VIP Cheat Sheets - Deep Learning by Stanford's CS 229 Students Download whole PDF of Supervised Learning Cheatsheet: From Here VIP Cheat Sheets - Machine Learning Tips by Stanford's CS 229 Students Download whole PDF of Supervised Learning Cheatsheet: From Here VIP Refresher: Probabilities and Statistics Cheatsheet Download whole PDF of Probability and Statistics Cheatsheets: From Here VIP Refresher: Linear Algebra and Calculus Cheat Sheets Download whole PDF of Linear Algebra and Calculus Cheatsheets: From Here You may like this: 100 Free Machine Learning Books Super VIP Cheat Sheet: Machine Learning Download whole PDF of Super VIP Machine Learning Cheat Sheet: From Here

If you've started looking into behind the scenes of popular machine learning algorithms, you might have come across the term "linear algebra". The term seems scary, but it isn't really so. Many of the machine learning algorithms rely on linear algebra because it provides the ability to "vectorize" them, making them computationally fast and efficient. Linear algebra is a vast branch of Mathematics, and not all of its knowledge is required in understanding and building machine learning algorithms, so our focus will be on the basic topics related to machine learning. NumPy implementations for each of the operations are also included at the end of each topic.

Wasserstein gradient flows on probability measures have found a host of applications in various optimization problems. They typically arise as the continuum limit of exchangeable particle systems evolving by some mean-field interaction involving a gradient-type potential. However, in many problems, such as in multi-layer neural networks, the so-called particles are edge weights on large graphs whose nodes are exchangeable. Such large graphs are known to converge to continuum limits called graphons as their size grow to infinity. We show that the Euclidean gradient flow of a suitable function of the edge-weights converges to a novel continuum limit given by a curve on the space of graphons that can be appropriately described as a gradient flow or, more technically, a curve of maximal slope. Several natural functions on graphons, such as homomorphism functions and the scalar entropy, are covered by our set-up, and the examples have been worked out in detail.

We solve university level probability and statistics questions by program synthesis using OpenAI's Codex, a Transformer trained on text and fine-tuned on code. We transform course problems from MIT's 18.05 Introduction to Probability and Statistics and Harvard's STAT110 Probability into programming tasks. We then execute the generated code to get a solution. Since these course questions are grounded in probability, we often aim to have Codex generate probabilistic programs that simulate a large number of probabilistic dependencies to compute its solution. Our approach requires prompt engineering to transform the question from its original form to an explicit, tractable form that results in a correct program and solution. To estimate the amount of work needed to translate an original question into its tractable form, we measure the similarity between original and transformed questions. Our work is the first to introduce a new dataset of university-level probability and statistics problems and solve these problems in a scalable fashion using the program synthesis capabilities of large language models.