Correlation dimension

Chaos theory | Fractals | Dimension theory | Dynamical systems

In chaos theory, the correlation dimension (denoted by ν) is a measure of the dimensionality of the space occupied by a set of random points, often referred to as a type of fractal dimension. For example, if we have a set of random points on the real number line between 0 and 1, the correlation dimension will be ν = 1, while if they are distributed on say, a triangle embedded in three-dimensional space (or m-dimensional space), the correlation dimension will be ν = 2. This is what we would intuitively expect from a measure of dimension. The real utility of the correlation dimension is in determining the (possibly fractional) dimensions of fractal objects. There are other methods of measuring dimension (e.g. the Hausdorff dimension, the box-counting dimension, and theinformation dimension) but the correlation dimension has the advantage of being straightforwardly and quickly calculated, of being less noisy when only a small number of points is available, and is often in agreement with other calculations of dimension. For any set of N points in an m-dimensional space then the correlation integral C(ε) is calculated by: where g is the total number of pairs of points which have a distance between them that is less than distance ε (a graphical representation of such close pairs is the recurrence plot). As the number of points tends to infinity, and the distance between them tends to zero, the correlation integral, for small values of ε, will take the form: If the number of points is sufficiently large, and evenly distributed, a log-log graph of the correlation integral versus ε will yield an estimate of ν. This idea can be qualitatively understood by realizing that for higher-dimensional objects, there will be more ways for points to be close to each other, and so the number of pairs close to each other will rise more rapidly for higher dimensions. Grassberger and Procaccia introduced the technique in 1983; the article gives the results of such estimates for a number of fractal objects, as well as comparing the values to other measures of fractal dimension. The technique can be used to distinguish between (deterministic) chaotic and truly random behavior, although it may not be good at detecting deterministic behavior if the deterministic generating mechanism is very complex. As an example, in the "Sun in Time" article, the method was used to show that the number of sunspots on the sun, after accounting for the known cycles such as the daily and 11-year cycles, is very likely not random noise, but rather chaotic noise, with a low-dimensional fractal attractor. (Wikipedia).

RELATIONSHIPS Between Variables: Standardized Covariance (7-1)

Correlation is a way of measuring the extent to which two variables are related. The term correlation is synonymous with “relationship.” Variables are related when changes in one variable are consistently associated with changes in another variable. Dr. Daniel reviews Variance, Covariance,

From playlist Correlation And Regression in Statistics (WK 07 - QBA 237)

EFFECT Size for Correlation: Coefficient of Determination (7-3)

The Correlation Coefficient is also an Effect Size. An r value can be squared to calculate an effect size. The r-squared is the Coefficient of Determination, expressing the proportion of variance in the dependent variable (Y) explained by variance in the independent variable (X). The rever

From playlist Correlation And Regression in Statistics (WK 07 - QBA 237)

Limits of correlation (applied)

Correlation is a standardized covariance (i.e., translated into unit-less form with volatilities). It cannot be used alone: (i) it can be "distorted" by low volatilities, and (ii) it does not give information revealed by the scatter (in this example, both hedge fund series are similarly co

From playlist Statistics: Introduction

Covariance Definition and Example

What is covariance? How do I find it? Step by step example of a solved covariance problem for a sample, along with an explanation of what the results mean and how it compares to correlation. 00:00 Overview 03:01 Positive, Negative, Zero Correlation 03:19 Covariance for a Sample Example

From playlist Correlation

Covariance (8 of 17) What is the Correlation Coefficient?

Visit http://ilectureonline.com for more math and science lectures! To donate:a http://www.ilectureonline.com/donate https://www.patreon.com/user?u=3236071 We will learn what is and how to find the correlation coefficient of 2 data sets and see how it corresponds to the graph of the data

From playlist COVARIANCE AND VARIANCE

Covariance (1 of 17) What is Covariance? in Relation to Variance and Correlation

Visit http://ilectureonline.com for more math and science lectures! To donate:a http://www.ilectureonline.com/donate https://www.patreon.com/user?u=3236071 We will learn the difference between the variance and the covariance. A variance (s^2) is a measure of how spread out the numbers of

From playlist COVARIANCE AND VARIANCE

Intro to the Correlation Coefficient

Brief intro to the correlation coefficient. What it means to have negative correlation, positive correlation or zero correlation. Pearson's, sample and population formulas.

From playlist Correlation

Covariance (12 of 17) Covariance Matrix wth 3 Data Sets and Correlation Coefficients

Visit http://ilectureonline.com for more math and science lectures! To donate:a http://www.ilectureonline.com/donate https://www.patreon.com/user?u=3236071 We will find the correlation coefficients of the 3 data sets form the previous 2 videos. Next video in this series can be seen at:

From playlist COVARIANCE AND VARIANCE

Pearson's Correlation Coefficient (1 of 3: Unpacking the formula)

More resources available at www.misterwootube.com

From playlist Descriptive Statistics & Bivariate Data Analysis

Ian McCulloch: "Finite-entanglement scaling functions at quantum critical points"

Tensor Methods and Emerging Applications to the Physical and Data Sciences 2021 Workshop II: Tensor Network States and Applications "Finite-entanglement scaling functions at quantum critical points" Ian McCulloch - University of Queensland Abstract: For translationally invariant infinite

From playlist Tensor Methods and Emerging Applications to the Physical and Data Sciences 2021

Vector Space of CFTs by Abhijit Gadde

ORGANIZERS : Pallab Basu, Avinash Dhar, Rajesh Gopakumar, R. Loganayagam, Gautam Mandal, Shiraz Minwalla, Suvrat Raju, Sandip Trivedi and Spenta Wadia DATE : 21 May 2018 to 02 June 2018 VENUE : Ramanujan Lecture Hall, ICTS Bangalore In the past twenty years, the discovery of the AdS/C

From playlist AdS/CFT at 20 and Beyond

Local Drive (a pump or a battery) in Interacting Diffusive Systems by David Mukamel

DISCUSSION MEETING : STATISTICAL PHYSICS OF COMPLEX SYSTEMS ORGANIZERS : Sumedha (NISER, India), Abhishek Dhar (ICTS-TIFR, India), Satya Majumdar (University of Paris-Saclay, France), R Rajesh (IMSc, India), Sanjib Sabhapandit (RRI, India) and Tridib Sadhu (TIFR, India) DATE : 19 December

From playlist Statistical Physics of Complex Systems - 2022

Statistical Rethinking Fall 2017 - week09 lecture16

Week 09, lecture 16 for Statistical Rethinking: A Bayesian Course with Examples in R and Stan, taught at MPI-EVA in Fall 2017. This lecture covers Chapter 13. Slides are available here: https://speakerdeck.com/rmcelreath/statistical-rethinking-fall-2017-lecture-16 Additional informatio

From playlist Statistical Rethinking Fall 2017

Neuroscience source separation 2a: Spatial separation

This is part two of a three-part lecture series I taught in a masters-level neuroscience course in fall of 2020 at the Donders Institute (the Netherlands). The lectures were all online in order to minimize the spread of the coronavirus. That's good for you, because now you can watch the en

From playlist Neuroscience source separation (3-part lecture series)

R - Register Variation with Exploratory Factor Analysis

Lecturer: Dr. Erin M. Buchanan Summer 2019 https://www.patreon.com/statisticsofdoom This video is part of my human language modeling class. On our last video, we will cover exploratory factor analysis to capture the dimensional data that words can present. Note: these videos are part of

From playlist Human Language (ANLY 540)

Critical dynamics (Lecture - 05) by Uwe C Täuber

Bangalore School on Statistical Physics - VIII DATE: 28 June 2017 to 14 July 2017 VENUE: Ramanujan Lecture Hall, ICTS, Bengaluru This advanced level school is the eighth in the series. This is a pedagogical school, aimed at bridging the gap between masters-level courses and topics in s

From playlist Bangalore School on Statistical Physics - VIII

The Long-Ranged Influence of Disorder on Active Systems by Yariv Kafri

From playlist Statistical Physics of Complex Systems - 2022

Zohar KOMARGODSKI - 2/4 Extremal correlators

From playlist Summer School 2018: Supersymmetric Localization and Exact Results

Hugo Duminil-Copin: Lecture #1

First lecture on "Marginal triviality of the scaling limits of critical 4D Ising and ϕ^4_4 models" by Professor Hugo Duminil-Copin. Unfortunately due to the technical difficulties (read organisers did not start the recording on time) first 10 minutes of the lecture are missing. These 10 m

From playlist Summer School on PDE & Randomness

Correlation Coefficient

This video explains how to find the correlation coefficient which describes the strength of the linear relationship between two variables x and y. My Website: https://www.video-tutor.net Patreon: https://www.patreon.com/MathScienceTutor Amazon Store: https://www.amazon.com/shop/theorga

From playlist Statistics

Correlation dimension

Related pages