This document discusses correlation and regression analysis. It covers calculating Pearson's correlation coefficient (r) to determine the strength and direction of the linear relationship between two variables. r ranges from -1 to 1, with values closer to these extremes indicating a stronger linear relationship. The document also discusses using linear regression to model the relationship between variables and predict one from the other by fitting a best-fit line that minimizes the residual sum of squares. Spearman's rank correlation coefficient is also introduced as a nonparametric alternative to Pearson's r.
This document defines correlation and correlation analysis. It provides examples of how to construct scatter plots to explore relationships between two variables. Positive correlation is shown by points sloping upwards to the right on a scatter plot, while negative correlation is shown by points sloping downwards to the right. The Pearson correlation coefficient measures the strength and direction of linear relationships between variables and ranges from -1 to 1. A value close to 0 indicates a weak relationship, while values close to 1 or -1 indicate a strong positive or negative relationship, respectively. Hypothesis tests can determine if observed correlation coefficients are statistically significant. Nonparametric methods like the Spearman rank correlation can be used if the data is not interval scaled.
Correlation by Neeraj Bhandari ( Surkhet.Nepal )Neeraj Bhandari
The regression coefficients are 0.8 and 0.2.
The coefficient of correlation r is the geometric mean of the regression coefficients, which is:
√(0.8 × 0.2) = 0.4
Therefore, the value of the coefficient of correlation is 0.4.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses linear regression and can analyze effects across multiple dependent variables.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r2, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both linear regression and multiple regression.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses linear regression and can analyze effects across multiple dependent variables.
Correlation & Regression for Statistics Social Sciencessuser71ac73
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both simple and multiple regression.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both simple and multiple regression.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both simple and multiple regression.
This document defines correlation and correlation analysis. It provides examples of how to construct scatter plots to explore relationships between two variables. Positive correlation is shown by points sloping upwards to the right on a scatter plot, while negative correlation is shown by points sloping downwards to the right. The Pearson correlation coefficient measures the strength and direction of linear relationships between variables and ranges from -1 to 1. A value close to 0 indicates a weak relationship, while values close to 1 or -1 indicate a strong positive or negative relationship, respectively. Hypothesis tests can determine if observed correlation coefficients are statistically significant. Nonparametric methods like the Spearman rank correlation can be used if the data is not interval scaled.
Correlation by Neeraj Bhandari ( Surkhet.Nepal )Neeraj Bhandari
The regression coefficients are 0.8 and 0.2.
The coefficient of correlation r is the geometric mean of the regression coefficients, which is:
√(0.8 × 0.2) = 0.4
Therefore, the value of the coefficient of correlation is 0.4.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses linear regression and can analyze effects across multiple dependent variables.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r2, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both linear regression and multiple regression.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses linear regression and can analyze effects across multiple dependent variables.
Correlation & Regression for Statistics Social Sciencessuser71ac73
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both simple and multiple regression.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both simple and multiple regression.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r-squared, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both simple and multiple regression.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r2, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both linear regression and multiple regression.
This document discusses correlation and regression analysis. It defines correlation as assessing the relationship between two variables, while regression determines how well one variable can predict another. Correlation does not imply causation. Pearson's r standardizes the covariance between variables and ranges from -1 to 1, indicating the strength and direction of their linear relationship. Regression finds the best-fitting linear relationship through the least squares method to minimize residuals and predict one variable from another. It provides the slope and intercept of the regression line. The coefficient of determination, r-squared, indicates how well the regression model fits the data.
The document provides additional information on correlation analysis. It discusses various examples of correlation between variables like sugar consumption and activity level. It explains the characteristics of a relationship such as the direction, form, and degree of correlation. Correlations can be used for prediction, validity, and reliability. The document also discusses the difference between correlation and causation. It then provides examples to test the reader's understanding of correlation through multiple choice questions. Finally, it covers topics like probable error, coefficient of correlation, coefficient of determination, Spearman's rank correlation method, and concurrent deviation method for calculating correlation.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 10: Correlation and Regression
10.1: Correlation
Galton invented the concept of correlation to measure how variables vary together. Pearson later formalized this method. Correlation measures the strength and direction of association between two variables on a scale of -1 to 1. A value of 0 indicates no association, 1 indicates perfect positive association, and -1 indicates perfect negative association. Correlation is widely used to study relationships between variables in fields such as business, psychology, and engineering.
This document presents a nonparametric approach to multiple regression that uses ranks instead of raw values for both the dependent and independent variables. The key points are:
1. It develops a nonparametric multiple regression model using the ranks of observations on the dependent variable and ranks of observations on the independent variables.
2. The method of least squares is applied to the rank-based model to obtain estimates of the regression coefficients.
3. Prediction equations are presented that allow predicting dependent variable ranks based on independent variable ranks.
This document discusses correlation and regression analysis. It defines correlation as a mutual relationship between two or more variables, and identifies positive, negative, simple, partial and multiple correlation. Regression is defined as determining the statistical relationship between a dependent variable and one or more independent variables. Methods for calculating correlation coefficients like Pearson's r and Spearman's rank correlation coefficient are presented. Steps for determining the regression equation and calculating the slope and intercept are also outlined.
The document provides information on correlation and linear regression. It defines correlation as the association between two variables and discusses how the correlation coefficient r measures the strength of this linear association. It then discusses:
- Computing r from sample data
- Testing the hypothesis that r = 0 using a t-test
- Computing the linear regression equation and coefficient of determination
- Using the regression equation to make predictions when there is a significant linear correlation
Two examples are then provided to demonstrate computing r from data, testing for a significant correlation, finding the regression equation, and making a prediction.
This document provides an introduction to regression and correlation analysis. It discusses simple and multiple linear regression models, how to interpret regression coefficients, and how to check the assumptions and adequacy of regression models. Key aspects covered include computing the regression line using the least squares method, interpreting the slope and intercept, checking the normality of residuals, and examining residual plots to validate the model. The goal of regression analysis is to model the relationship between a dependent variable and one or more independent variables.
This document provides information about determinants of square matrices:
- It defines the determinant of a matrix as a scalar value associated with the matrix. Determinants are computed using minors and cofactors.
- Properties of determinants are described, such as how determinants change with row/column operations or identical rows/columns.
- Examples are provided to demonstrate computing determinants by expanding along rows or columns and using cofactors and minors.
- Applications of determinants include finding the area of triangles and solving systems of linear equations.
The document discusses correlation and regression, explaining that correlation describes the strength of a linear relationship between two variables, while regression tells us how to draw the straight line described by the correlation. It provides examples of using correlation coefficients to determine the strength and direction of relationships between independent and dependent variables, and discusses calculating correlation coefficients and using regression analysis to predict variable relationships and outcomes.
This document provides an overview of regression analysis. It defines regression as a statistical technique for finding the best-fitting straight line for a set of data. Regression allows predictions to be made based on correlations between two variables. The relationship between correlation and regression is examined, noting that correlation determines the relationship between variables while regression is used to make predictions. Various aspects of the linear regression equation are described, including computing predictions, graphing lines, and determining how well data fits the regression line.
Correlation _ Regression Analysis statistics.pptxkrunal soni
This document discusses correlation and related statistical concepts. Correlation measures the strength and direction of association between two quantitative variables. A correlation of 0 means no association, 1 means perfect positive association, and -1 means perfect negative association. Correlation is independent of measurement units and scaling of variables. Hypothesis testing is used to make inferences about the population correlation based on a sample correlation. The null hypothesis is that the population correlation is 0, and alternative hypotheses specify a non-zero correlation. The test statistic used is Student's t distribution. The null is rejected if the calculated t exceeds the critical value or if the p-value is less than the significance level.
This document discusses correlation analysis and different methods of studying correlation. It begins by defining correlation as the association between two or more variables. There are different types of correlation such as positive, negative, linear, and curvilinear correlation. The degree of correlation can be determined using the correlation coefficient, with values ranging from -1 to 1. Common methods for studying correlation discussed include scatter diagrams, Karl Pearson's coefficient, Spearman's rank correlation, and concurrent deviation method. The properties and interpretation of the correlation coefficient are also outlined.
This document provides an overview of regression models and analysis techniques. It introduces simple and multiple linear regression, as well as logistic regression. It discusses assessing regression models, cross-validation, model selection, and using regression models for prediction. Additionally, it covers the similarities and differences between linear and logistic regression, and assessing correlation without inferring causation. Scatter plots, correlation coefficients, and computing regression equations are also summarized.
Correlation analysis establishes relationships between two variables. The correlation coefficient, r, measures the linear relationship between variables on a scale from -1 to 1, where -1 is total negative correlation, 1 is total positive correlation, and 0 is no correlation. A scatterplot diagrams the relationship between two variables based on their x and y values, showing the strength, shape, direction, and outliers of the relationship to help interpret the correlation coefficient.
This document provides an overview of various statistical methods for summarizing and analyzing biological data, including:
- Calculating the mean, median, and mode to summarize sample data
- Using distribution curves like histograms to visualize patterns in data and identify if the distribution is normal or skewed
- Calculating standard deviation to quantify the variation of data from the mean
- Using t-tests to compare two normally distributed samples and determine if differences are statistically significant
- Using non-parametric tests like the Mann-Whitney U test for small or skewed sample comparisons
- Applying the chi-squared test to analyze relationships between categorical variables
- Using the Spearman rank correlation coefficient to identify monotonic relationships between two variable sets
This document discusses correlation and regression analysis. It begins by outlining the chapter's objectives and providing an introduction to investigating relationships between variables using statistical analysis. The document then presents examples of collecting data to study potential relationships between variables like stone dimensions, human heights and weights, and sprint and long jump performances. It introduces various statistical measures for quantifying relationships in data, including covariance, Pearson's product moment correlation coefficient, and Spearman's rank correlation coefficient. Examples are provided to demonstrate calculating and interpreting these statistics. Limitations of correlation analysis are also noted.
This document discusses correlation and regression. Correlation describes the strength and direction of a linear relationship between two variables, while regression allows predicting a dependent variable from an independent variable. It provides examples of calculating the correlation coefficient r to determine the strength and direction of relationships between variables like education and self-esteem or family income and number of children. The regression equation describes the linear regression line and can be used to predict values of the dependent variable from known values of the independent variable.
This document discusses correlation, regression, and the general linear model. It defines correlation as assessing the relationship between two variables, while regression describes how well one variable can predict another. Pearson's r standardizes the covariance between variables. Linear regression finds the best-fitting line that minimizes the residuals through the least squares method. The coefficient of determination, r2, indicates how much variance in the dependent variable is explained by the independent variable. Multiple regression extends this to include multiple independent variables. The general linear model encompasses both linear regression and multiple regression.
This document discusses correlation and regression analysis. It defines correlation as assessing the relationship between two variables, while regression determines how well one variable can predict another. Correlation does not imply causation. Pearson's r standardizes the covariance between variables and ranges from -1 to 1, indicating the strength and direction of their linear relationship. Regression finds the best-fitting linear relationship through the least squares method to minimize residuals and predict one variable from another. It provides the slope and intercept of the regression line. The coefficient of determination, r-squared, indicates how well the regression model fits the data.
The document provides additional information on correlation analysis. It discusses various examples of correlation between variables like sugar consumption and activity level. It explains the characteristics of a relationship such as the direction, form, and degree of correlation. Correlations can be used for prediction, validity, and reliability. The document also discusses the difference between correlation and causation. It then provides examples to test the reader's understanding of correlation through multiple choice questions. Finally, it covers topics like probable error, coefficient of correlation, coefficient of determination, Spearman's rank correlation method, and concurrent deviation method for calculating correlation.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 10: Correlation and Regression
10.1: Correlation
Galton invented the concept of correlation to measure how variables vary together. Pearson later formalized this method. Correlation measures the strength and direction of association between two variables on a scale of -1 to 1. A value of 0 indicates no association, 1 indicates perfect positive association, and -1 indicates perfect negative association. Correlation is widely used to study relationships between variables in fields such as business, psychology, and engineering.
This document presents a nonparametric approach to multiple regression that uses ranks instead of raw values for both the dependent and independent variables. The key points are:
1. It develops a nonparametric multiple regression model using the ranks of observations on the dependent variable and ranks of observations on the independent variables.
2. The method of least squares is applied to the rank-based model to obtain estimates of the regression coefficients.
3. Prediction equations are presented that allow predicting dependent variable ranks based on independent variable ranks.
This document discusses correlation and regression analysis. It defines correlation as a mutual relationship between two or more variables, and identifies positive, negative, simple, partial and multiple correlation. Regression is defined as determining the statistical relationship between a dependent variable and one or more independent variables. Methods for calculating correlation coefficients like Pearson's r and Spearman's rank correlation coefficient are presented. Steps for determining the regression equation and calculating the slope and intercept are also outlined.
The document provides information on correlation and linear regression. It defines correlation as the association between two variables and discusses how the correlation coefficient r measures the strength of this linear association. It then discusses:
- Computing r from sample data
- Testing the hypothesis that r = 0 using a t-test
- Computing the linear regression equation and coefficient of determination
- Using the regression equation to make predictions when there is a significant linear correlation
Two examples are then provided to demonstrate computing r from data, testing for a significant correlation, finding the regression equation, and making a prediction.
This document provides an introduction to regression and correlation analysis. It discusses simple and multiple linear regression models, how to interpret regression coefficients, and how to check the assumptions and adequacy of regression models. Key aspects covered include computing the regression line using the least squares method, interpreting the slope and intercept, checking the normality of residuals, and examining residual plots to validate the model. The goal of regression analysis is to model the relationship between a dependent variable and one or more independent variables.
This document provides information about determinants of square matrices:
- It defines the determinant of a matrix as a scalar value associated with the matrix. Determinants are computed using minors and cofactors.
- Properties of determinants are described, such as how determinants change with row/column operations or identical rows/columns.
- Examples are provided to demonstrate computing determinants by expanding along rows or columns and using cofactors and minors.
- Applications of determinants include finding the area of triangles and solving systems of linear equations.
The document discusses correlation and regression, explaining that correlation describes the strength of a linear relationship between two variables, while regression tells us how to draw the straight line described by the correlation. It provides examples of using correlation coefficients to determine the strength and direction of relationships between independent and dependent variables, and discusses calculating correlation coefficients and using regression analysis to predict variable relationships and outcomes.
This document provides an overview of regression analysis. It defines regression as a statistical technique for finding the best-fitting straight line for a set of data. Regression allows predictions to be made based on correlations between two variables. The relationship between correlation and regression is examined, noting that correlation determines the relationship between variables while regression is used to make predictions. Various aspects of the linear regression equation are described, including computing predictions, graphing lines, and determining how well data fits the regression line.
Correlation _ Regression Analysis statistics.pptxkrunal soni
This document discusses correlation and related statistical concepts. Correlation measures the strength and direction of association between two quantitative variables. A correlation of 0 means no association, 1 means perfect positive association, and -1 means perfect negative association. Correlation is independent of measurement units and scaling of variables. Hypothesis testing is used to make inferences about the population correlation based on a sample correlation. The null hypothesis is that the population correlation is 0, and alternative hypotheses specify a non-zero correlation. The test statistic used is Student's t distribution. The null is rejected if the calculated t exceeds the critical value or if the p-value is less than the significance level.
This document discusses correlation analysis and different methods of studying correlation. It begins by defining correlation as the association between two or more variables. There are different types of correlation such as positive, negative, linear, and curvilinear correlation. The degree of correlation can be determined using the correlation coefficient, with values ranging from -1 to 1. Common methods for studying correlation discussed include scatter diagrams, Karl Pearson's coefficient, Spearman's rank correlation, and concurrent deviation method. The properties and interpretation of the correlation coefficient are also outlined.
This document provides an overview of regression models and analysis techniques. It introduces simple and multiple linear regression, as well as logistic regression. It discusses assessing regression models, cross-validation, model selection, and using regression models for prediction. Additionally, it covers the similarities and differences between linear and logistic regression, and assessing correlation without inferring causation. Scatter plots, correlation coefficients, and computing regression equations are also summarized.
Correlation analysis establishes relationships between two variables. The correlation coefficient, r, measures the linear relationship between variables on a scale from -1 to 1, where -1 is total negative correlation, 1 is total positive correlation, and 0 is no correlation. A scatterplot diagrams the relationship between two variables based on their x and y values, showing the strength, shape, direction, and outliers of the relationship to help interpret the correlation coefficient.
This document provides an overview of various statistical methods for summarizing and analyzing biological data, including:
- Calculating the mean, median, and mode to summarize sample data
- Using distribution curves like histograms to visualize patterns in data and identify if the distribution is normal or skewed
- Calculating standard deviation to quantify the variation of data from the mean
- Using t-tests to compare two normally distributed samples and determine if differences are statistically significant
- Using non-parametric tests like the Mann-Whitney U test for small or skewed sample comparisons
- Applying the chi-squared test to analyze relationships between categorical variables
- Using the Spearman rank correlation coefficient to identify monotonic relationships between two variable sets
This document discusses correlation and regression analysis. It begins by outlining the chapter's objectives and providing an introduction to investigating relationships between variables using statistical analysis. The document then presents examples of collecting data to study potential relationships between variables like stone dimensions, human heights and weights, and sprint and long jump performances. It introduces various statistical measures for quantifying relationships in data, including covariance, Pearson's product moment correlation coefficient, and Spearman's rank correlation coefficient. Examples are provided to demonstrate calculating and interpreting these statistics. Limitations of correlation analysis are also noted.
This document discusses correlation and regression. Correlation describes the strength and direction of a linear relationship between two variables, while regression allows predicting a dependent variable from an independent variable. It provides examples of calculating the correlation coefficient r to determine the strength and direction of relationships between variables like education and self-esteem or family income and number of children. The regression equation describes the linear regression line and can be used to predict values of the dependent variable from known values of the independent variable.
Similar to Unit 4_3 Correlation Regression.pptx (20)
Codeless Generative AI Pipelines
(GenAI with Milvus)
https://ml.dssconf.pl/user.html#!/lecture/DSSML24-041a/rate
Discover the potential of real-time streaming in the context of GenAI as we delve into the intricacies of Apache NiFi and its capabilities. Learn how this tool can significantly simplify the data engineering workflow for GenAI applications, allowing you to focus on the creative aspects rather than the technical complexities. I will guide you through practical examples and use cases, showing the impact of automation on prompt building. From data ingestion to transformation and delivery, witness how Apache NiFi streamlines the entire pipeline, ensuring a smooth and hassle-free experience.
Timothy Spann
https://www.youtube.com/@FLaNK-Stack
https://medium.com/@tspann
https://www.datainmotion.dev/
milvus, unstructured data, vector database, zilliz, cloud, vectors, python, deep learning, generative ai, genai, nifi, kafka, flink, streaming, iot, edge
The Building Blocks of QuestDB, a Time Series Databasejavier ramirez
Talk Delivered at Valencia Codes Meetup 2024-06.
Traditionally, databases have treated timestamps just as another data type. However, when performing real-time analytics, timestamps should be first class citizens and we need rich time semantics to get the most out of our data. We also need to deal with ever growing datasets while keeping performant, which is as fun as it sounds.
It is no wonder time-series databases are now more popular than ever before. Join me in this session to learn about the internal architecture and building blocks of QuestDB, an open source time-series database designed for speed. We will also review a history of some of the changes we have gone over the past two years to deal with late and unordered data, non-blocking writes, read-replicas, or faster batch ingestion.
State of Artificial intelligence Report 2023kuntobimo2016
Artificial intelligence (AI) is a multidisciplinary field of science and engineering whose goal is to create intelligent machines.
We believe that AI will be a force multiplier on technological progress in our increasingly digital, data-driven world. This is because everything around us today, ranging from culture to consumer products, is a product of intelligence.
The State of AI Report is now in its sixth year. Consider this report as a compilation of the most interesting things we’ve seen with a goal of triggering an informed conversation about the state of AI and its implication for the future.
We consider the following key dimensions in our report:
Research: Technology breakthroughs and their capabilities.
Industry: Areas of commercial application for AI and its business impact.
Politics: Regulation of AI, its economic implications and the evolving geopolitics of AI.
Safety: Identifying and mitigating catastrophic risks that highly-capable future AI systems could pose to us.
Predictions: What we believe will happen in the next 12 months and a 2022 performance review to keep us honest.
06-04-2024 - NYC Tech Week - Discussion on Vector Databases, Unstructured Data and AI
Discussion on Vector Databases, Unstructured Data and AI
https://www.meetup.com/unstructured-data-meetup-new-york/
This meetup is for people working in unstructured data. Speakers will come present about related topics such as vector databases, LLMs, and managing data at scale. The intended audience of this group includes roles like machine learning engineers, data scientists, data engineers, software engineers, and PMs.This meetup was formerly Milvus Meetup, and is sponsored by Zilliz maintainers of Milvus.
Global Situational Awareness of A.I. and where its headedvikram sood
You can see the future first in San Francisco.
Over the past year, the talk of the town has shifted from $10 billion compute clusters to $100 billion clusters to trillion-dollar clusters. Every six months another zero is added to the boardroom plans. Behind the scenes, there’s a fierce scramble to secure every power contract still available for the rest of the decade, every voltage transformer that can possibly be procured. American big business is gearing up to pour trillions of dollars into a long-unseen mobilization of American industrial might. By the end of the decade, American electricity production will have grown tens of percent; from the shale fields of Pennsylvania to the solar farms of Nevada, hundreds of millions of GPUs will hum.
The AGI race has begun. We are building machines that can think and reason. By 2025/26, these machines will outpace college graduates. By the end of the decade, they will be smarter than you or I; we will have superintelligence, in the true sense of the word. Along the way, national security forces not seen in half a century will be un-leashed, and before long, The Project will be on. If we’re lucky, we’ll be in an all-out race with the CCP; if we’re unlucky, an all-out war.
Everyone is now talking about AI, but few have the faintest glimmer of what is about to hit them. Nvidia analysts still think 2024 might be close to the peak. Mainstream pundits are stuck on the wilful blindness of “it’s just predicting the next word”. They see only hype and business-as-usual; at most they entertain another internet-scale technological change.
Before long, the world will wake up. But right now, there are perhaps a few hundred people, most of them in San Francisco and the AI labs, that have situational awareness. Through whatever peculiar forces of fate, I have found myself amongst them. A few years ago, these people were derided as crazy—but they trusted the trendlines, which allowed them to correctly predict the AI advances of the past few years. Whether these people are also right about the next few years remains to be seen. But these are very smart people—the smartest people I have ever met—and they are the ones building this technology. Perhaps they will be an odd footnote in history, or perhaps they will go down in history like Szilard and Oppenheimer and Teller. If they are seeing the future even close to correctly, we are in for a wild ride.
Let me tell you what we see.
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Dive into the world of data analysis with our comprehensive guide on mastering SQL! This presentation offers a practical approach to learning SQL, focusing on real-world applications and hands-on practice. Whether you're a beginner or looking to sharpen your skills, this guide provides the tools you need to extract, analyze, and interpret data effectively.
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Beyond the Basics of A/B Tests: Highly Innovative Experimentation Tactics You...Aggregage
This webinar will explore cutting-edge, less familiar but powerful experimentation methodologies which address well-known limitations of standard A/B Testing. Designed for data and product leaders, this session aims to inspire the embrace of innovative approaches and provide insights into the frontiers of experimentation!
2. Topics Covered:
• Is there a relationship between x and y?
• What is the strength of this relationship
– Pearson’s correlation coefficient r.
– Rank correlation coefficient
• Spearman's rank
• Kendall tau rank
• Goodman and Kruskal's gamma
• Can we describe this relationship and use this to predict y from x?
– Regression
• Is the relationship we have described statistically significant?
– t test
11/14/2022 2
NITT / CA
3. The relationship between x and y
• Correlation: Is there a relationship between 2
variables?
• Regression: How well a certain independent
variable predict dependent variable?
• CORRELATION CAUSATION
– In order to infer causality: manipulate independent
variable and observe effect on dependent variable
11/14/2022 3
NITT / CA
6. Variance vs Covariance
• Note on your sample:
• If you’re wishing to assume that your sample is
representative of the general population (RANDOM
EFFECTS MODEL), use the degrees of freedom (n – 1)
in your calculations of variance or covariance.
• But if you’re simply wanting to assess your current
sample (FIXED EFFECTS MODEL), substitute n for
the degrees of freedom.
11/14/2022 6
NITT / CA
7. Variance vs Covariance
• Do two variables change together?
1
)
)(
(
)
,
cov( 1
n
y
y
x
x
y
x
i
n
i
i
Covariance:
• Gives information on the degree to which
two variables vary together.
• Note how similar the covariance is to
variance: the equation simply multiplies x’s
error scores by y’s error scores as opposed
to squaring x’s error scores.
1
)
( 2
1
2
n
x
x
S
n
i
i
x
Variance:
• Gives information on variability of a
single variable.
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8. Covariance
When X and Y : cov (x,y) = pos.
When X and Y : cov (x,y) = neg.
When no constant relationship: cov (x,y) = 0
1
)
)(
(
)
,
cov( 1
n
y
y
x
x
y
x
i
n
i
i
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9. Example Covariance
x y x
xi
y
yi
( x
i
x )( y
i
y )
0 3 -3 0 0
2 2 -1 -1 1
3 4 0 1 0
4 0 1 -3 -3
6 6 3 3 9
3
x 3
y 7
75
.
1
4
7
1
))
)(
(
)
,
cov( 1
n
y
y
x
x
y
x
i
n
i
i What does this
number tell us?
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7
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10. Problem with Covariance:
• The value obtained by covariance is dependent on the size of
the data’s standard deviations: if large, the value will be
greater than if small… even if the relationship between x and y
is exactly the same in the large versus small standard
deviation datasets.
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11. Example of how covariance value
relies on variance
High variance data Low variance data
Subject x y x error * y
error
x y X error * y
error
1 101 100 2500 54 53 9
2 81 80 900 53 52 4
3 61 60 100 52 51 1
4 51 50 0 51 50 0
5 41 40 100 50 49 1
6 21 20 900 49 48 4
7 1 0 2500 48 47 9
Mean 51 50 51 50
Sum of x error * y error : 7000 Sum of x error * y error : 28
Covariance: 1166.67 Covariance: 4.67
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12. Solution: Pearson’s r
Covariance does not really tell us anything
Solution: standardise this measure
Pearson’s R: standardises the covariance value.
Divides the covariance by the multiplied standard deviations of X
and Y:
y
x
xy
s
s
y
x
r
)
,
cov(
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13. Pearson’s R continued
1
)
)(
(
)
,
cov( 1
n
y
y
x
x
y
x
i
n
i
i
y
x
i
n
i
i
xy
s
s
n
y
y
x
x
r
)
1
(
)
)(
(
1
1
*
1
n
Z
Z
r
n
i
y
x
xy
i
i
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14. Limitations of r
• When r = 1 or r = -1:
– We can predict y from x with certainty
– All data points are on a straight line: y = ax + b
• r is actually
– r = true r of whole population
– = estimate of r based on data
• r is very sensitive to extreme values:
0
1
2
3
4
5
0 1 2 3 4 5 6
r̂
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15. The Pearson's correlation coefficient:
• Varies between -1 and +1
• r = 1 means the data is perfectly linear with a
positive slope (i.e., both variables tend to change in the same direction)
• r = -1 means the data is perfectly linear with a
negative slope ( i.e., both variables tend to change in different directions)
• r = 0 means there is no linear association
• 0 < r < 0.4 means there is a weak association
• 0.4 < r < 0.8 means there is a moderate association
• r > 0.8 means there is a strong association
18. Pearson Correlation Coefficient Formula r:
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r = Pearson Coefficient.
x = First variable
y = Second variable
n= number of the pairs of the data values
∑xy = sum of products of the paired data values
∑x = sum of the x scores
∑y= sum of the y scores
∑x2 = sum of the squared x scores
∑y2 = sum of the squared y scores
19. Example:
• Calculate of the value of the Pearson’s correlation
coefficient r with the help of the following details for
the six people who have different ages and weights.
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S. No Age (x) Weight (y)
1 40 78
2 21 70
3 25 60
4 31 55
5 38 80
6 47 66
22. Example:
• There are 2 stocks
– A and B. Their
share prices on
particular days are
as follows:
Stock A (x) Stock B (y)
45 9
50 8
53 8
58 7
60 5
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23. Calculate the following values (xy, x2,and y2)
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r = (5*1935-266*37)/((5*14298-(266)^2)*(5*283-(37)^2))^0.5
r = -0.9088
26. Advantages
• It helps to know, the stengthness of the
relationship between the two variables.
• It also determines the exact extent to which
those variables are correlated.
• Using this method, one can ascertain the
direction of correlation,
– negative or positive.
• It is independent of the unit of measurement
of the variables (Eg: cm and inch).
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27. Disadvantages
• The Pearson correlation coefficient r is insufficient to
tell the difference between the dependent and
independent variables (r is symmetric for 2 variables).
• We cannot get information about the slope of the line
as it only states whether any relationship between the
two variables exists or not.
• The Pearson correlation coefficient may likely be
misinterpreted, especially in the case of
homogeneous data.
• Compared with the other calculation methods, this
method takes more time to arrive at the results.
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28. Spearman correlation coefficient: ρ(rho)
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Here,
n= number of data points of the two variables
di= difference in ranks of the “ith” element
The Spearman Coefficient, ⍴, can take a value between +1 to -1 where,
A ⍴ value of +1 means a perfect association of rank
A ⍴ value of 0 means no association of ranks
A ⍴ value of -1 means a perfect negative association between ranks.
29. Example:
Maths Science
35 30
23 33
47 45
17 23
10 8
43 49
9 12
6 4
28 31
Calculate the Spearman correlation coefficient for the following data
34. Example: Calculate Spearman's rank
coefficient
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Suppose we have ranks of 8 students in Statistics and Mathematics. On the basis of rank we
would like to know that to what extent the knowledge of the student in Statistics and
Mathematics is related
Rank in Stat 1 2 3 4 5 6 7 8
Rank in Math 2 4 1 5 3 8 7 6
Rank in Stat
Rank in
Math
Difference of
Ranks = d
d2
1 2 -1 1
2 4 -2 4
3 1 2 4
4 5 -1 1
5 3 2 4
6 8 -2 4
7 7 0 0
8 6 2 4
22
Here,
n = number of paired observations = 8
36. Regression
• Correlation tells you if there is an association
between x and y but it doesn’t allow you to
predict one variable from the other.
• To do this we need REGRESSION!
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37. Best-fit Line
= ŷ, predicted value
• Aim of linear regression is to fit a straight line, ŷ = ax + b, to data that
gives best prediction of y for any value of x
• This will be the line that
minimises distance between
data and fitted line, i.e.
the residuals
intercept
ε
ŷ = ax + b
ε = residual error
= y i , true value
slope
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38. Least Squares Regression
To find the best line we must minimise the sum of
the squares of the residuals (the vertical distances
from the data points to our line)
Residual (ε) = y - ŷ
Sum of squares of residuals = Σ (y – ŷ)2
Model line: ŷ = ax + b
We must find values of a and b that minimise
Σ (y – ŷ)2
a = slope, b = intercept
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39. Finding b
• First we find the value of b that gives the min
sum of squares
ε ε
b
b
b
Trying different values of b is equivalent to
shifting the line up and down the scatter plot
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40. Finding a
• Now we find the value of a that gives the min
sum of squares
b b b
Trying out different values of a is equivalent to
changing the slope of the line, while b stays
constant
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41. Minimising sums of squares
Need to minimise Σ(y–ŷ)2
But ŷ = ax + b
So Wee need to minimise:
Σ(y - ax - b)2
If we plot the sums of squares
for all different values of a and b
we get a parabola, because it is a
squared term
So the min sum of squares is at
the bottom of the curve, where
the gradient is zero.
Values of a and b
sums
of
squares
(S)
Gradient = 0
min S
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42. 11/14/2022 NITT / CA 42
Recall that the expression for the correlation coefficient is
2
2
2
2
2
2
)
(
)
(
)
(
)
(
)
)(
(
y
y
n
x
x
n
y
x
xy
n
r
y
y
x
x
y
y
x
x
r
43. Regression Line
• A simple linear relationship between two
variables.
• An estimated regression line based on sample
data as
• Least squares method give us the “best”
estimated line. It chooses the best values for b0,
and b1 to minimize the sum of squared errors
x
y 1
0
x
b
b
y 1
0
ˆ
2
1
1
0
1
2
)
ˆ
(
n
i
n
i
i
i x
b
b
y
y
y
SSE
n
i
n
i
i
i
n
i
n
i
n
i
i
i
i
i
n
i
i
n
i
i
i
x
x
n
y
x
y
x
n
x
x
y
y
x
x
b
1 1
2
2
1 1 1
1
2
1
1
)
(
)
(
)
)(
(
or
x
y
S
S
r
b
1
x
b
y
b 1
0
44. Example
• The weekly advertising expenditure
(x) and weekly sales (y) are presented
in the following table
• Use fitted regression line to estimate
the mean value of y for a given value
of x=50.
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S. no. x y xy x2
1 41 1250 51250 1681
2 54 1380 74520 2916
3 63 1425 89775 3969
4 54 1425 76950 2916
5 48 1450 69600 2304
6 46 1300 59800 2116
7 62 1400 86800 3844
8 61 1510 92110 3721
9 64 1575 100800 4096
10 71 1650 117150 5041
Total 564 14365 818755 32604
45. • From previous table we have:
• The least squares estimates of the regression coefficients are:
• The estimated regression function is:
• If the advertising expenditure is Rs 50, then the estimated Sales is:
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818755
14365
32604
564
10 2
xy
y
x
x
n
8
.
10
)
564
(
)
32604
(
10
)
14365
)(
564
(
)
818755
(
10
)
( 2
2
2
1
x
x
n
y
x
xy
n
b
828
)
4
.
56
(
8
.
10
5
.
1436
0
b
e
Expenditur
8
.
10
828
Sales
10.8x
828
ŷ
1368
)
50
(
8
.
10
828
Sales
46. Residual
• The difference between the observed value yi and
the corresponding fitted value .,
• Residuals are highly useful for studying whether a
given regression model is appropriate for the data
at hand.
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i
ŷ i
i
i y
y
e ˆ
y x y-hat Residual (e)
1250 41 1270.8 -20.8
1380 54 1411.2 -31.2
1425 63 1508.4 -83.4
1425 54 1411.2 13.8
1450 48 1346.4 103.6
1300 46 1324.8 -24.8
1400 62 1497.6 -97.6
1510 61 1486.8 23.2
1575 64 1519.2 55.8
1650 71 1594.8 55.2
47. Regression Standard Error
• For simple linear regression the estimate of 2
is the average squared residual.
• To estimate , use
• This estimates the standard deviation of the
error term in the statistical model for simple
linear regression.
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2
2
2
. )
ˆ
(
2
1
2
1
i
i
i
x
y y
y
n
e
n
s
2
.
. x
y
x
y s
s
49. The maths bit
• The min sum of squares is at the bottom of the curve
where the gradient = 0
• So we can find a and b that give min sum of squares
by taking partial derivatives of Σ(y - ax - b)2 with
respect to a and b separately
• Then we solve these for 0 to give us the values of a
and b that give the min sum of squares
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50. The solution
• Doing this gives the following equations for a and b:
a =
r sy
sx
r = correlation coefficient of x and y
sy = standard deviation of y
sx = standard deviation of x
From you can see that:
A low correlation coefficient gives a flatter slope (small value of a)
Large spread of y, i.e. high standard deviation, results in a steeper
slope (high value of a)
Large spread of x, i.e. high standard deviation, results in a flatter
slope (high value of a)
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51. The solution cont.
• Our model equation is ŷ = ax + b
• This line must pass through the mean so:
y = ax + b b = y – ax
We can put our equation for a into this giving:
b = y – ax
b = y -
r sy
sx
r = correlation coefficient of x and y
sy = standard deviation of y
sx = standard deviation of x
x
The smaller the correlation, the closer the
intercept is to the mean of y
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52. Back to the model
If the correlation is zero, we will simply predict the mean of y for every
value of x, and our regression line is just a flat straight line crossing the
x-axis at y
But this isn’t very useful.
We can calculate the regression line for any data, but the important
question is how well does this line fit the data, or how good is it at
predicting y from x
ŷ = ax + b =
r sy
sx
r sy
sx
x + y - x
r sy
sx
ŷ = (x – x) + y
Rearranges to:
a b
a a
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53. How good is our model?
• Total variance of y: sy
2 =
∑(y – y)2
n - 1
SSy
dfy
=
Variance of predicted y values (ŷ):
Error variance:
sŷ
2 =
∑(ŷ – y)2
n - 1
SSpred
dfŷ
=
This is the variance
explained by our
regression model
serror
2 =
∑(y – ŷ)2
n - 2
SSer
dfer
=
This is the variance of the error
between our predicted y values and
the actual y values, and thus is the
variance in y that is NOT explained
by the regression model
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54. • Total variance = predicted variance + error variance
sy
2 = sŷ
2 + ser
2
• Conveniently, via some complicated rearranging
sŷ
2 = r2 sy
2
r2 = sŷ
2 / sy
2
• so r2 is the proportion of the variance in y that is explained by
our regression model
How good is our model cont.
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55. How good is our model cont.
• Insert r2 sy
2 into sy
2 = sŷ
2 + ser
2 and rearrange to get:
ser
2 = sy
2 – r2sy
2
= sy
2 (1 – r2)
• From this we can see that the greater the correlation
the smaller the error variance, so the better our
prediction
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56. Is the model significant?
• i.e. do we get a significantly better prediction of y
from our regression equation than by just predicting
the mean?
• F-statistic:
F(dfŷ,dfer) =
sŷ
2
ser
2
=......=
r2 (n - 2)2
1 – r2
complicated
rearranging
And it follows that:
t(n-2) =
r (n - 2)
√1 – r2
(because F = t2)
So all we need to
know are r and n
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57. General Linear Model
• Linear regression is actually a form of the
General Linear Model where the parameters
are a, the slope of the line, and b, the intercept.
y = ax + b +ε
• A General Linear Model is just any model that
describes the data in terms of a straight line
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58. Multiple regression
• Multiple regression is used to determine the effect of a number
of independent variables, x1, x2, x3 etc, on a single dependent
variable, y
• The different x variables are combined in a linear way and
each has its own regression coefficient:
y = a1x1+ a2x2 +…..+ anxn + b + ε
• The a parameters reflect the independent contribution of each
independent variable, x, to the value of the dependent variable,
y.
• i.e. the amount of variance in y that is accounted for by each x
variable after all the other x variables have been accounted for
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