Regression analysis is a predictive modeling technique used to investigate relationships between variables. It allows one to estimate the effects of independent variables on a dependent variable. Regression analysis can be used for forecasting, time series modeling, and determining causal relationships. There are different types of regression depending on the number of variables and the shape of the regression line. Linear regression models the linear relationship between two variables using an equation with parameters estimated to minimize error. Correlation and covariance measures the strength and direction of association between variables. Analysis of variance (ANOVA) compares the means of groups within data. Heteroskedasticity refers to unequal variability of a dependent variable across the range of independent variable values.
The document discusses goodness-of-fit tests which are used to determine if observed frequency counts fit a claimed distribution. It provides examples of how to calculate expected frequencies under different assumptions and how to perform a chi-square goodness-of-fit test. The test statistic is calculated as the sum of the squared differences between observed and expected frequencies divided by the expected frequencies. The test statistic is then compared to critical values from the chi-square distribution to determine if there is a significant lack of fit. Examples demonstrate how to state hypotheses, calculate test statistics, compare to critical values, and make conclusions about goodness of fit.
The document discusses comparing measurements from two physicians who measured tumor volumes. A paired t-test was used to test if the measurements significantly differed. The t-test showed no significant difference between physicians. Additionally, two sample t-tests were used to compare tumor volumes between different cancer types, finding significant differences between brain and breast cancers and brain and liver cancers.
Normal Distribution
Properties of Normal Distribution
Empirical rule of normal distribution
Normality limits
Standard normal distribution(z-score/ SND)
Properties of SND
Use of z/normal table
Solved examples
This document discusses key concepts in statistics for engineers and scientists such as point estimates, properties of good estimators, confidence intervals, and the t-distribution. A point estimate is a single numerical value used to estimate a population parameter from a sample. A good estimator must be unbiased, consistent, and relatively efficient. A confidence interval provides a range of values that is likely to contain the true population parameter based on the sample data and confidence level. The t-distribution is similar to the normal distribution but has greater variance and depends on degrees of freedom. Examples are provided to demonstrate how to calculate confidence intervals for means using the normal and t-distributions.
There are three main probability distributions: binomial, Poisson, and normal. The binomial distribution calculates the probability of a certain number of successes in a fixed number of trials when the probability of success is constant. The Poisson distribution calculates the probability of a number of random events occurring in an interval. It applies when occurrences are independent and the average number of occurrences in an interval is known. The normal distribution is the most important continuous probability distribution and describes variables that can take any value within a range.
This document discusses the concept of probability. It defines probability as a measure of how likely an event is to occur. Probabilities can be described using terms like certain, likely, unlikely, and impossible. Mathematically, probabilities are often expressed as fractions, with the numerator representing the number of possible outcomes for an event and the denominator representing the total number of possible outcomes. The document provides examples to illustrate concepts like independent and conditional probabilities, as well as complementary events and the gambler's fallacy.
Random Variable
Discrete Probability Distribution
continuous Probability Distribution
Probability Mass Function
Probability Density Function
Expected value
variance
Binomial Distribution
poisson distribution
normal distribution
The document provides information about binomial probability distributions including:
- Binomial experiments have a fixed number (n) of independent trials with two possible outcomes and a constant probability (p) of success.
- The binomial probability distribution gives the probability of getting exactly x successes in n trials. It is calculated using the binomial coefficient and p and q=1-p.
- The mean, variance and standard deviation of a binomial distribution are np, npq, and √npq respectively.
- Examples demonstrate calculating probabilities of outcomes for binomial experiments and determining if results are significantly low or high using the range rule of μ ± 2σ.
The document discusses goodness-of-fit tests which are used to determine if observed frequency counts fit a claimed distribution. It provides examples of how to calculate expected frequencies under different assumptions and how to perform a chi-square goodness-of-fit test. The test statistic is calculated as the sum of the squared differences between observed and expected frequencies divided by the expected frequencies. The test statistic is then compared to critical values from the chi-square distribution to determine if there is a significant lack of fit. Examples demonstrate how to state hypotheses, calculate test statistics, compare to critical values, and make conclusions about goodness of fit.
The document discusses comparing measurements from two physicians who measured tumor volumes. A paired t-test was used to test if the measurements significantly differed. The t-test showed no significant difference between physicians. Additionally, two sample t-tests were used to compare tumor volumes between different cancer types, finding significant differences between brain and breast cancers and brain and liver cancers.
Normal Distribution
Properties of Normal Distribution
Empirical rule of normal distribution
Normality limits
Standard normal distribution(z-score/ SND)
Properties of SND
Use of z/normal table
Solved examples
This document discusses key concepts in statistics for engineers and scientists such as point estimates, properties of good estimators, confidence intervals, and the t-distribution. A point estimate is a single numerical value used to estimate a population parameter from a sample. A good estimator must be unbiased, consistent, and relatively efficient. A confidence interval provides a range of values that is likely to contain the true population parameter based on the sample data and confidence level. The t-distribution is similar to the normal distribution but has greater variance and depends on degrees of freedom. Examples are provided to demonstrate how to calculate confidence intervals for means using the normal and t-distributions.
There are three main probability distributions: binomial, Poisson, and normal. The binomial distribution calculates the probability of a certain number of successes in a fixed number of trials when the probability of success is constant. The Poisson distribution calculates the probability of a number of random events occurring in an interval. It applies when occurrences are independent and the average number of occurrences in an interval is known. The normal distribution is the most important continuous probability distribution and describes variables that can take any value within a range.
This document discusses the concept of probability. It defines probability as a measure of how likely an event is to occur. Probabilities can be described using terms like certain, likely, unlikely, and impossible. Mathematically, probabilities are often expressed as fractions, with the numerator representing the number of possible outcomes for an event and the denominator representing the total number of possible outcomes. The document provides examples to illustrate concepts like independent and conditional probabilities, as well as complementary events and the gambler's fallacy.
Random Variable
Discrete Probability Distribution
continuous Probability Distribution
Probability Mass Function
Probability Density Function
Expected value
variance
Binomial Distribution
poisson distribution
normal distribution
The document provides information about binomial probability distributions including:
- Binomial experiments have a fixed number (n) of independent trials with two possible outcomes and a constant probability (p) of success.
- The binomial probability distribution gives the probability of getting exactly x successes in n trials. It is calculated using the binomial coefficient and p and q=1-p.
- The mean, variance and standard deviation of a binomial distribution are np, npq, and √npq respectively.
- Examples demonstrate calculating probabilities of outcomes for binomial experiments and determining if results are significantly low or high using the range rule of μ ± 2σ.
The document discusses maximum likelihood estimation. It begins by explaining that maximum likelihood chooses parameter values that make the observed data most probable given a statistical model. This provides a justification for estimation techniques like least squares regression. The document provides an example of estimating a population proportion from a sample. It then generalizes maximum likelihood to cover a wide range of models and estimation problems. It discusses properties like consistency, efficiency, and how to conduct hypothesis tests based on maximum likelihood. Numerical optimization techniques are often required to find maximum likelihood estimates for complex models.
A gentle introduction to survival analysisAngelo Tinazzi
This document provides an introduction to survival analysis techniques for statistical programmers. It discusses key concepts in survival analysis including censoring, the Kaplan-Meier method for estimating survival probabilities, and assumptions of survival models. Programming aspects like creating time-to-event datasets and using SAS procedures for survival analysis are also covered.
The ppt cover General Introduction to the topic,
Description of CHI-SQUARE TEST, Contingency table, Degree of Freedom, Determination of Chi – square test, Assumption for validity of chi - square test, Characteristics , Applications, Limitations
Monte Carlo simulations involve running models multiple times with random inputs to determine probabilities of various outcomes. For each run, random values are selected from ranges for uncertain factors, the model is calculated, and the result recorded. Thousands of runs are typically done to build a pool of results describing the likelihood of different outcomes. The method assumes variables are not influenced by each other. It is useful when probabilities are known but results are hard to determine directly.
The document discusses the standard normal distribution. It defines the standard normal distribution as having a mean of 0, a standard deviation of 1, and a bell-shaped curve. It provides examples of how to find probabilities and z-scores using the standard normal distribution table or calculator. For example, it shows how to find the probability of an event being below or above a given z-score, or between two z-scores. It also shows how to find the z-score corresponding to a given cumulative probability.
This document discusses point and interval estimation. It defines an estimator as a function used to infer an unknown population parameter based on sample data. Point estimation provides a single value, while interval estimation provides a range of values with a certain confidence level, such as 95%. Common point estimators include the sample mean and proportion. Interval estimators account for variability in samples and provide more information than point estimators. The document provides examples of how to construct confidence intervals using point estimates, confidence levels, and standard errors or deviations.
The central limit theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, even if the population is not normally distributed. It provides the mean and standard deviation of the sampling distribution of the sample mean. The document gives the definition of the central limit theorem and provides an example of how to use it to calculate probabilities related to the sample mean of a large normally distributed population.
The document discusses the geometric distribution, a discrete probability distribution that models the number of Bernoulli trials needed to get one success. It defines the geometric distribution and gives its probability mass function. Some key properties and applications are discussed, including: the mean is 1/p, the variance is q/p^2, where q is 1-p. It is used in situations like modeling the probability of events occurring after repeated independent trials with a constant probability of success each trial. Examples given include analyzing success rates in sports and deciding when to stop research trials.
The exponential probability distribution is useful for describing the time it takes to complete random tasks. It can model the time between events like vehicle arrivals at a toll booth, time to complete a survey, or distance between defects on a highway. The distribution is defined by a probability density function that uses the mean time or rate of the process. It can calculate the probability that an event will occur within a certain time threshold, like the chance a car will arrive at a gas pump within 2 minutes. The mean and standard deviation of the exponential distribution are equal, and it is an extremely skewed distribution without a defined mode.
1) The document discusses number systems and properties of numbers including natural numbers, whole numbers, integers, even/odd numbers, and prime/composite numbers.
2) Formulas for expanding expressions with numbers are provided along with tests for divisibility of numbers by various factors.
3) Types of numbers such as integers, even, odd, prime and composite numbers are defined along with their key properties.
The t distribution is used when sample sizes are small to determine the probability of obtaining a given sample mean. It is similar to the normal distribution but has fatter tails. Properties include having a mean of 0 and a variance that decreases and approaches 1 as the degrees of freedom increase. The t distribution approaches the normal distribution as the sample size increases to infinity or the degrees of freedom become very large. Examples show how to find t-scores, critical values, and confidence intervals using a t-table based on the sample size and desired confidence level.
This document provides information on chi-square tests and other statistical tests for qualitative data analysis. It discusses the chi-square test for goodness of fit and independence. It also covers Fisher's exact test and McNemar's test. Examples are provided to illustrate chi-square calculations and how to determine statistical significance based on degrees of freedom and critical values. Assumptions and criteria for applying different tests are outlined.
1) The t-test is a statistical test used to determine if there are any statistically significant differences between the means of two groups, and was developed by William Gosset under the pseudonym "Student".
2) The t-distribution is used for calculating t-tests when sample sizes are small and/or variances are unknown. It has a mean of zero and variance greater than one.
3) Paired t-tests are used to compare the means of two related groups when samples are paired, while unpaired t-tests are used to compare unrelated groups or independent samples.
This document discusses probability and Bayes' theorem. It provides examples of basic probability concepts like the probability of a coin toss. It then defines conditional probability as the probability of an event given another event. Bayes' theorem is introduced as a way to revise a probability based on new information. An example problem demonstrates how to calculate the probability of rain given a weather forecast using Bayes' theorem.
This document discusses various measures of central tendency including arithmetic mean, median, mode, and quartiles. It provides definitions and formulas for calculating each measure, and describes how to calculate the mean and median for different types of data distributions including raw data, continuous series, and less than/more than/inclusive series. It also covers weighted mean, combined mean, and properties and limitations of the arithmetic mean.
This document provides an overview of statistical tests and hypothesis testing. It discusses the four steps of hypothesis testing, including stating hypotheses, setting decision criteria, computing test statistics, and making a decision. It also describes different types of statistical analyses, common descriptive statistics, and forms of statistical relationships. Finally, it provides examples of various parametric and nonparametric statistical tests, including t-tests, ANOVA, chi-square tests, correlation, regression, and decision trees.
This document describes how to conduct a chi-square goodness of fit test. The test involves:
1) Stating the null and alternative hypotheses. The null hypothesis specifies the expected probabilities, while the alternative is that at least one expected probability is incorrect.
2) Developing an analysis plan specifying the significance level and test to be used.
3) Analyzing sample data to calculate degrees of freedom, expected frequencies, the test statistic, and p-value.
4) Interpreting the results by comparing the p-value to the significance level and rejecting or failing to reject the null hypothesis. An example problem demonstrates applying the test to determine if observed outcomes match a casino's claimed probabilities.
The document discusses key concepts related to the normal distribution, including:
- The normal distribution is characterized by two parameters: the mean (μ) and standard deviation (σ).
- Many real-world variables, like heights and test scores, are approximately normally distributed.
- Z-scores allow comparison of observations across different normal distributions by expressing them in units of standard deviations from the mean.
Hypothesis testing involves making an assumption about an unknown population parameter, called the null hypothesis (H0). A hypothesis is tested by collecting a sample from the population and comparing sample statistics to the hypothesized parameter value. If the sample value differs significantly from the hypothesized value based on a predetermined significance level, then the null hypothesis is rejected. There are two types of errors that can occur - type 1 errors occur when a true null hypothesis is rejected, and type 2 errors occur when a false null hypothesis is not rejected. Hypothesis tests can be one-tailed, testing if the sample value is greater than or less than the hypothesized value, or two-tailed, testing if the sample value is significantly different from the hypothesized value.
The document defines a sampling distribution of sample means as a distribution of means from random samples of a population. The mean of sample means equals the population mean, and the standard deviation of sample means is smaller than the population standard deviation, equaling it divided by the square root of the sample size. As sample size increases, the distribution of sample means approaches a normal distribution according to the Central Limit Theorem.
Correation, Linear Regression and Multilinear Regression using R softwareshrikrishna kesharwani
The document describes performing correlation, linear regression, and multilinear regression analysis on transportation-related data using R software. It provides theory on correlation, linear regression, and multilinear regression. The procedures section outlines the steps to perform correlation analysis, simple linear regression, and multiple linear regression. The results and analysis section shows the output of applying these techniques to variables in a transportation data set and interpreting the correlation coefficients, p-values, and regression results.
Regression analysis models the relationship between a dependent (target) variable and one or more independent (predictor) variables. Linear regression predicts continuous variables using a linear equation. Simple linear regression uses one independent variable, while multiple linear regression uses more than one. The goal is to find the "best fit" line that minimizes error between predicted and actual values. Feature selection identifies important predictors by removing irrelevant or redundant features. Techniques include wrapper, filter, and embedded methods. Overfitting and underfitting occur when models are too complex or simple, respectively. Dimensionality reduction through techniques like principal component analysis (PCA) transform correlated variables into linearly uncorrelated components.
The document discusses maximum likelihood estimation. It begins by explaining that maximum likelihood chooses parameter values that make the observed data most probable given a statistical model. This provides a justification for estimation techniques like least squares regression. The document provides an example of estimating a population proportion from a sample. It then generalizes maximum likelihood to cover a wide range of models and estimation problems. It discusses properties like consistency, efficiency, and how to conduct hypothesis tests based on maximum likelihood. Numerical optimization techniques are often required to find maximum likelihood estimates for complex models.
A gentle introduction to survival analysisAngelo Tinazzi
This document provides an introduction to survival analysis techniques for statistical programmers. It discusses key concepts in survival analysis including censoring, the Kaplan-Meier method for estimating survival probabilities, and assumptions of survival models. Programming aspects like creating time-to-event datasets and using SAS procedures for survival analysis are also covered.
The ppt cover General Introduction to the topic,
Description of CHI-SQUARE TEST, Contingency table, Degree of Freedom, Determination of Chi – square test, Assumption for validity of chi - square test, Characteristics , Applications, Limitations
Monte Carlo simulations involve running models multiple times with random inputs to determine probabilities of various outcomes. For each run, random values are selected from ranges for uncertain factors, the model is calculated, and the result recorded. Thousands of runs are typically done to build a pool of results describing the likelihood of different outcomes. The method assumes variables are not influenced by each other. It is useful when probabilities are known but results are hard to determine directly.
The document discusses the standard normal distribution. It defines the standard normal distribution as having a mean of 0, a standard deviation of 1, and a bell-shaped curve. It provides examples of how to find probabilities and z-scores using the standard normal distribution table or calculator. For example, it shows how to find the probability of an event being below or above a given z-score, or between two z-scores. It also shows how to find the z-score corresponding to a given cumulative probability.
This document discusses point and interval estimation. It defines an estimator as a function used to infer an unknown population parameter based on sample data. Point estimation provides a single value, while interval estimation provides a range of values with a certain confidence level, such as 95%. Common point estimators include the sample mean and proportion. Interval estimators account for variability in samples and provide more information than point estimators. The document provides examples of how to construct confidence intervals using point estimates, confidence levels, and standard errors or deviations.
The central limit theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, even if the population is not normally distributed. It provides the mean and standard deviation of the sampling distribution of the sample mean. The document gives the definition of the central limit theorem and provides an example of how to use it to calculate probabilities related to the sample mean of a large normally distributed population.
The document discusses the geometric distribution, a discrete probability distribution that models the number of Bernoulli trials needed to get one success. It defines the geometric distribution and gives its probability mass function. Some key properties and applications are discussed, including: the mean is 1/p, the variance is q/p^2, where q is 1-p. It is used in situations like modeling the probability of events occurring after repeated independent trials with a constant probability of success each trial. Examples given include analyzing success rates in sports and deciding when to stop research trials.
The exponential probability distribution is useful for describing the time it takes to complete random tasks. It can model the time between events like vehicle arrivals at a toll booth, time to complete a survey, or distance between defects on a highway. The distribution is defined by a probability density function that uses the mean time or rate of the process. It can calculate the probability that an event will occur within a certain time threshold, like the chance a car will arrive at a gas pump within 2 minutes. The mean and standard deviation of the exponential distribution are equal, and it is an extremely skewed distribution without a defined mode.
1) The document discusses number systems and properties of numbers including natural numbers, whole numbers, integers, even/odd numbers, and prime/composite numbers.
2) Formulas for expanding expressions with numbers are provided along with tests for divisibility of numbers by various factors.
3) Types of numbers such as integers, even, odd, prime and composite numbers are defined along with their key properties.
The t distribution is used when sample sizes are small to determine the probability of obtaining a given sample mean. It is similar to the normal distribution but has fatter tails. Properties include having a mean of 0 and a variance that decreases and approaches 1 as the degrees of freedom increase. The t distribution approaches the normal distribution as the sample size increases to infinity or the degrees of freedom become very large. Examples show how to find t-scores, critical values, and confidence intervals using a t-table based on the sample size and desired confidence level.
This document provides information on chi-square tests and other statistical tests for qualitative data analysis. It discusses the chi-square test for goodness of fit and independence. It also covers Fisher's exact test and McNemar's test. Examples are provided to illustrate chi-square calculations and how to determine statistical significance based on degrees of freedom and critical values. Assumptions and criteria for applying different tests are outlined.
1) The t-test is a statistical test used to determine if there are any statistically significant differences between the means of two groups, and was developed by William Gosset under the pseudonym "Student".
2) The t-distribution is used for calculating t-tests when sample sizes are small and/or variances are unknown. It has a mean of zero and variance greater than one.
3) Paired t-tests are used to compare the means of two related groups when samples are paired, while unpaired t-tests are used to compare unrelated groups or independent samples.
This document discusses probability and Bayes' theorem. It provides examples of basic probability concepts like the probability of a coin toss. It then defines conditional probability as the probability of an event given another event. Bayes' theorem is introduced as a way to revise a probability based on new information. An example problem demonstrates how to calculate the probability of rain given a weather forecast using Bayes' theorem.
This document discusses various measures of central tendency including arithmetic mean, median, mode, and quartiles. It provides definitions and formulas for calculating each measure, and describes how to calculate the mean and median for different types of data distributions including raw data, continuous series, and less than/more than/inclusive series. It also covers weighted mean, combined mean, and properties and limitations of the arithmetic mean.
This document provides an overview of statistical tests and hypothesis testing. It discusses the four steps of hypothesis testing, including stating hypotheses, setting decision criteria, computing test statistics, and making a decision. It also describes different types of statistical analyses, common descriptive statistics, and forms of statistical relationships. Finally, it provides examples of various parametric and nonparametric statistical tests, including t-tests, ANOVA, chi-square tests, correlation, regression, and decision trees.
This document describes how to conduct a chi-square goodness of fit test. The test involves:
1) Stating the null and alternative hypotheses. The null hypothesis specifies the expected probabilities, while the alternative is that at least one expected probability is incorrect.
2) Developing an analysis plan specifying the significance level and test to be used.
3) Analyzing sample data to calculate degrees of freedom, expected frequencies, the test statistic, and p-value.
4) Interpreting the results by comparing the p-value to the significance level and rejecting or failing to reject the null hypothesis. An example problem demonstrates applying the test to determine if observed outcomes match a casino's claimed probabilities.
The document discusses key concepts related to the normal distribution, including:
- The normal distribution is characterized by two parameters: the mean (μ) and standard deviation (σ).
- Many real-world variables, like heights and test scores, are approximately normally distributed.
- Z-scores allow comparison of observations across different normal distributions by expressing them in units of standard deviations from the mean.
Hypothesis testing involves making an assumption about an unknown population parameter, called the null hypothesis (H0). A hypothesis is tested by collecting a sample from the population and comparing sample statistics to the hypothesized parameter value. If the sample value differs significantly from the hypothesized value based on a predetermined significance level, then the null hypothesis is rejected. There are two types of errors that can occur - type 1 errors occur when a true null hypothesis is rejected, and type 2 errors occur when a false null hypothesis is not rejected. Hypothesis tests can be one-tailed, testing if the sample value is greater than or less than the hypothesized value, or two-tailed, testing if the sample value is significantly different from the hypothesized value.
The document defines a sampling distribution of sample means as a distribution of means from random samples of a population. The mean of sample means equals the population mean, and the standard deviation of sample means is smaller than the population standard deviation, equaling it divided by the square root of the sample size. As sample size increases, the distribution of sample means approaches a normal distribution according to the Central Limit Theorem.
Correation, Linear Regression and Multilinear Regression using R softwareshrikrishna kesharwani
The document describes performing correlation, linear regression, and multilinear regression analysis on transportation-related data using R software. It provides theory on correlation, linear regression, and multilinear regression. The procedures section outlines the steps to perform correlation analysis, simple linear regression, and multiple linear regression. The results and analysis section shows the output of applying these techniques to variables in a transportation data set and interpreting the correlation coefficients, p-values, and regression results.
Regression analysis models the relationship between a dependent (target) variable and one or more independent (predictor) variables. Linear regression predicts continuous variables using a linear equation. Simple linear regression uses one independent variable, while multiple linear regression uses more than one. The goal is to find the "best fit" line that minimizes error between predicted and actual values. Feature selection identifies important predictors by removing irrelevant or redundant features. Techniques include wrapper, filter, and embedded methods. Overfitting and underfitting occur when models are too complex or simple, respectively. Dimensionality reduction through techniques like principal component analysis (PCA) transform correlated variables into linearly uncorrelated components.
linear regression is a linear approach for modelling a predictive relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables), which are measured without error. The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable. If the explanatory variables are measured with error then errors-in-variables models are required, also known as measurement error models.
In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters are estimated from the data. Such models are called linear models. Most commonly, the conditional mean of the response given the values of the explanatory variables (or predictors) is assumed to be an affine function of those values; less commonly, the conditional median or some other quantile is used. Like all forms of regression analysis, linear regression focuses on the conditional probability distribution of the response given the values of the predictors, rather than on the joint probability distribution of all of these variables, which is the domain of multivariate analysis.
Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications.[4] This is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and because the statistical properties of the resulting estimators are easier to determine.
Linear regression has many practical uses. Most applications fall into one of the following two broad categories:
If the goal is error reduction in prediction or forecasting, linear regression can be used to fit a predictive model to an observed data set of values of the response and explanatory variables. After developing such a model, if additional values of the explanatory variables are collected without an accompanying response value, the fitted model can be used to make a prediction of the response.
If the goal is to explain variation in the response variable that can be attributed to variation in the explanatory variables, linear regression analysis can be applied to quantify the strength of the relationship between the response and the explanatory variables, and in particular to determine whether some explanatory variables may have no linear relationship with the response at all, or to identify which subsets of explanatory variables may contain redundant information about the response.
This document provides an overview of forecasting using Eviews 2.0 software. It distinguishes between ex post and ex ante forecasting. Ex post forecasts use known data to evaluate a forecasting model, while ex ante forecasts predict values using uncertain explanatory variables. The document then discusses univariate forecasting methods in Eviews, including trend extrapolation, modeling trend behavior, and analyzing residuals to check assumptions. It provides examples of estimating a trend model, viewing residuals, and making forecasts in Eviews.
Linear regression is a supervised machine learning technique used to model the relationship between a continuous dependent variable and one or more independent variables. It is commonly used for prediction and forecasting. The regression line represents the best fit line for the data using the least squares method to minimize the distance between the observed data points and the regression line. R-squared measures how well the regression line represents the data, on a scale of 0-100%. Linear regression performs well when data is linearly separable but has limitations such as assuming linear relationships and being sensitive to outliers and multicollinearity.
Linear regression and logistic regression are two machine learning algorithms that can be implemented in Python. Linear regression is used for predictive analysis to find relationships between variables, while logistic regression is used for classification with binary dependent variables. Support vector machines (SVMs) are another algorithm that finds the optimal hyperplane to separate data points and maximize the margin between the classes. Key terms discussed include cost functions, gradient descent, confusion matrices, and ROC curves. Code examples are provided to demonstrate implementing linear regression, logistic regression, and SVM in Python using scikit-learn.
Linear regression is a supervised machine learning technique used to model the relationship between a continuous dependent variable and one or more independent variables. It finds the line of best fit that minimizes the distance between the observed data points and the regression line. The slope of the regression line is determined using the least squares method. R-squared measures how well the regression line represents the data, with values closer to 1 indicating a stronger relationship. The standard error of the estimate quantifies the accuracy of predictions made by the linear regression model. Linear regression performs well when data is linearly separable, but has limitations such as an assumption of linear relationships and sensitivity to outliers and multicollinearity.
This document provides an introduction to regression analysis. It discusses that regression analysis investigates the relationship between dependent and independent variables to model and analyze data. The document outlines different types of regressions including linear, polynomial, stepwise, ridge, lasso, and elastic net regressions. It explains that regression analysis is used for predictive modeling, forecasting, and determining the impact of variables. The benefits of regression analysis are that it indicates significant relationships and the strength of impact between variables.
CHPTER 3: Multiple Linear Regression
Introduction
In simple regression we study the relationship between a dependent variable and a single explanatory (independent variable); assume that a dependent variable is influenced by only one explanatory variable.
The document discusses regression analysis and different types of regression models. It defines regression analysis as a statistical method to model the relationship between a dependent variable and one or more independent variables. It explains linear regression, multiple linear regression, and polynomial regression. Linear regression finds the linear relationship between two variables, multiple linear regression handles multiple independent variables, and polynomial regression models nonlinear relationships using polynomial functions. Examples and code snippets in Python are provided to illustrate simple and multiple linear regression analysis.
Machine learning is a type of artificial intelligence that allows systems to learn from data and improve automatically without being explicitly programmed. There are several types of machine learning algorithms, including supervised learning which uses labeled training data to predict outcomes, unsupervised learning which finds patterns in unlabeled data, and reinforcement learning which interacts with its environment to discover rewards or errors. Linear regression is an example machine learning model that fits a linear equation to describe the relationship between a dependent variable and one or more independent variables. It works by minimizing the residual sum of squares to find the coefficients that produce the best fitting line.
This document discusses different types of regression analysis techniques including linear regression, polynomial regression, support vector regression, decision tree regression, ridge regression, lasso regression, and logistic regression. Linear regression finds the relationship between a continuous dependent variable and one or more independent variables. Polynomial regression handles nonlinear relationships through higher-order terms. Support vector regression and decision tree regression can handle both linear and nonlinear data. Ridge and lasso regression are regularization techniques used to prevent overfitting. Logistic regression is for classification rather than regression problems.
Linear regression is a statistical method used to model the relationship between variables. It finds the line of best fit for the data and uses this to predict the value of the dependent variable based on the independent variable. Simple linear regression involves one independent variable, while multiple linear regression can have multiple independent variables. In Python, the Scikit-learn library can be used to perform linear regression on data and evaluate the model performance using metrics like R-squared and root mean square error.
The document provides information about the syllabus for the Data Analytics (KIT-601) course. It includes 5 units that will be covered: Introduction to Data Analytics, Data Analysis techniques including regression modeling and multivariate analysis, Mining Data Streams, Frequent Itemsets and Clustering, and Frameworks and Visualization. It lists the course outcomes and Bloom's taxonomy levels. It also provides details on the topics to be covered in each unit, including proposed lecture hours, textbooks, and an evaluation scheme. The syllabus aims to discuss concepts of data analytics and apply techniques such as classification, regression, clustering, and frequent pattern mining on data.
Regression analysis is a statistical method used to model relationships between variables. It involves plotting data points and finding the line of best fit that minimizes the distance between the points and the line. This line can then be used to predict future outcomes. Simple linear regression uses one independent variable to predict a continuous dependent variable. Multiple linear regression extends this to use multiple independent variables to better capture complex relationships between factors.
The document discusses regression analysis and random forest machine learning algorithms. It explains that regression analysis is used to predict continuous variables like sales based on related predictor variables like advertisement spending. Regression finds the correlation between variables to enable predictions. Random forest is an ensemble technique that creates multiple decision trees on subsets of data and takes a majority vote of the trees' predictions to improve accuracy. It provides a two-phase working process for random forest involving creating trees on random data samples and then making predictions based on the trees' votes.
This document discusses supervised learning. Supervised learning uses labeled training data to train models to predict outputs for new data. Examples given include weather prediction apps, spam filters, and Netflix recommendations. Supervised learning algorithms are selected based on whether the target variable is categorical or continuous. Classification algorithms are used when the target is categorical while regression is used for continuous targets. Common regression algorithms discussed include linear regression, logistic regression, ridge regression, lasso regression, and elastic net. Metrics for evaluating supervised learning models include accuracy, R-squared, adjusted R-squared, mean squared error, and coefficients/p-values. The document also covers challenges like overfitting and regularization techniques to address it.
The document discusses nonlinear least squares regression in R. It explains that real-world data is rarely linear and instead follows curves and higher-order mathematical functions. Nonlinear regression aims to find the curve that best fits the data by adjusting the model's parameters. In R, the nls() function is used to estimate the parameters and their confidence intervals for a defined nonlinear model, with the basic syntax being nls(formula, data, start). An example is provided using a quadratic model and nls() to output the sum of squared residuals and confidence intervals for the coefficients.
The document provides an overview of regression analysis. It defines regression analysis as a technique used to estimate the relationship between a dependent variable and one or more independent variables. The key purposes of regression are to estimate relationships between variables, determine the effect of each independent variable on the dependent variable, and predict the dependent variable given values of the independent variables. The document also outlines the assumptions of the linear regression model, introduces simple and multiple regression, and describes methods for model building including variable selection procedures.
Natural Language Processing (NLP), RAG and its applications .pptxfkyes25
1. In the realm of Natural Language Processing (NLP), knowledge-intensive tasks such as question answering, fact verification, and open-domain dialogue generation require the integration of vast and up-to-date information. Traditional neural models, though powerful, struggle with encoding all necessary knowledge within their parameters, leading to limitations in generalization and scalability. The paper "Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks" introduces RAG (Retrieval-Augmented Generation), a novel framework that synergizes retrieval mechanisms with generative models, enhancing performance by dynamically incorporating external knowledge during inference.
Analysis insight about a Flyball dog competition team's performanceroli9797
Insight of my analysis about a Flyball dog competition team's last year performance. Find more: https://github.com/rolandnagy-ds/flyball_race_analysis/tree/main
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.
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.
4th Modern Marketing Reckoner by MMA Global India & Group M: 60+ experts on W...Social Samosa
The Modern Marketing Reckoner (MMR) is a comprehensive resource packed with POVs from 60+ industry leaders on how AI is transforming the 4 key pillars of marketing – product, place, price and promotions.
4th Modern Marketing Reckoner by MMA Global India & Group M: 60+ experts on W...
Correlation and regression in r
1. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
UNIT IV
Correlation and Regression Analysis
(NOS 9001)
Regression Analysis and Modeling –
Introduction:
Regression analysis is a form of predictive modeling technique
which investigates the relationship between a dependent (target)
and independent variable(s) (predictor).
This technique is used for forecasting, time series modeling and
finding the causal effect relationship between the variables. For
example, relationship between rash driving and number of road
accidents by a driver is best studied through regression.
Regression analysis is an important tool for modeling and analyzing
data. Here, we fit a curve / line to the data points, in such a
manner that the differences between the distances of data points
from the curve or line is minimized
Regression analysis estimates the relationship between two or more
variables.
example:
Let’s say, if we want to estimate growth in sales of a company based
on current economic conditions. we have the recent company data
which indicates that the growth in sales is around two and a half
times the growth in the economy. Using this insight, we can predict
future sales of the company based on current & past information.
There are multiple benefits of using regression analysis. They are as
follows:
* It indicates the significant relationships between dependent
variable and independent variable.
2. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
* It indicates the strength of impact of multiple independent
variables on a dependent variable.
Regression analysis also allows us to compare the effects of
variables measured on different scales, such as the effect of price
changes and the number of promotional activities. These benefits
help market researchers / data analysts / data scientists to
eliminate and evaluate the best set of variables to be used for
building predictive models.
There are various kinds of regression techniques available to make
predictions.
These techniques are mostly driven by three metrics.
1. Number of independent variables,
2. Type of dependent variables and
3. Shape of regression line
Linear Regression:
A simple linear regression model describes the relationship between
two variables x and y can be expressed by the following equation.
The numbers α and β are called parameters, and ϵ is the error term.
If we choose the parameters α and β in the simple linear regression
model so as to minimize the sum of squares of the error term ϵ, we
will have the so called estimated simple regression equation. It
allows us to compute fitted values of y based on values of x.
In R we use lm () function to do simple regression modeling.
Apply the simple linear regression model for the data set cars. The
cars dataset as two variables (attributes) speed and dist and has 50
values.
> head(cars)
speed dist
3. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
1 4 2
2 4 10
3 7 4
4 7 22
5 8 16
6 9 10
> attach(cars)
By using the attach( ) function the database is attached to the R
search path. This means that the database is searched by R when
evaluating a variable, so objects in the database can be accessed by
simply giving their names.
> speed
[1] 4 4 7 7 8 9 10 10 10 11 11 12 12 12 12 13 13 13 13 14 14 14 14
15 15 15 16
[28] 16 17 17 17 18 18 18 18 19 19 19 20 20 20 20 20 22 23 24 24
24 24 25
> plot(cars)
> plot(dist,speed)
The plot() function gives a scatterplot whenever we give two numeric
variables.
The first variable listed will be plotted on the horizontal axis.
4. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
Now apply the regression analysis on the dataset using lm( )
function.
> speed.lm=lm(speed ~ dist, data = cars)
lm function that describes the variable speed by the variable dist,
and save the linear regression model in a new variable speed.lm. In
the above function y variables or dependent variable is speed and x
variable or independent variable is dist.
We get the intercept “C” and the slope “m” of the equation –
Y=mX+C
> speed.lm
Call:
lm(formula = speed ~ dist, data = cars)
5. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
Coefficients:
(Intercept) dist
8.2839 0.1656
> abline(speed.lm)
This function adds one or more straight lines through the current
plot.
> plot(speed.lm)
The plot function displays four charts: Residuals vs. Fitted, Normal
QQ, ScaleLocation, and Residuals vs. Leverage.
8. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
What is a quantile in statistics?
In statistics and the theory of probability, quantiles are cutpoints
dividing the range of a probability distribution into contiguous
intervals with equal probabilities, or dividing the observations in a
sample in the same way. There is one less quantilethan the
number of groups created.
9. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
The residual data of the simple linear regression model is the
difference between the observed data of the dependent
variable y and the fitted values ŷ.
> eruption.lm = lm(eruptions ~ waiting, data=faithful)
> eruption.res = resid(eruption.lm)
We now plot the residual against the observed values of the
variable waiting.
> plot(faithful$waiting, eruption.res,
+ ylab="Residuals", xlab="Waiting Time",
+ main="Old Faithful Eruptions")
> abline(0, 0) # the horizon
10. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
Residual: The difference between the predicted value (based on the
regression equation) and the actual, observed value.
Outlier: In linear regression, an outlier is an observation with large
residual. In other words, it is an observation whose
dependent¬variable value is unusual given its value on the
predictor variables. An outlier may indicate a sample peculiarity or
may indicate a data entry error or other problem.
Leverage: An observation with an extreme value on a predictor
variable is a point with high leverage. Leverage is a measure of how
far an independent variable deviates from its mean. High leverage
points can have a great amount of effect on the estimate of
regression coefficients.
Influence: An observation is said to be influential if removing the
observation substantially changes the estimate of the regression
coefficients. Influence can be thought of as the product of leverage
and outlierness.
Cook's distance (or Cook's D): A measure that combines the
information of leverage and residual of the observation.
Estimated simple regression equation:
Apply we will use the above simple linear regression model, and
estimate the next speed if the distance covered is 80.
Extract the parameters of the estimated regression equation with
the coefficients function.
> coeffs = coefficients(speed.lm)
> coeffs
11. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
(Intercept) dist
8.2839056 0.1655676
Forecasting/Prediction:
We now fit the speed using the estimated regression equation.
> newdist = 80
> distance = coeffs[1] + coeffs[2]*newdist
> distance
(Intercept)
21.52931
To create a summary of the fitted model:
> summary (speed.lm)
Call:
lm(formula = speed ~ dist, data = cars)
Residuals:
Min 1Q Median 3Q Max
-7.5293 -2.1550 0.3615 2.4377 6.4179
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 8.28391 0.87438 9.474 1.44e-12 ***
dist 0.16557 0.01749 9.464 1.49e-12 ***
---
12. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 3.156 on 48 degrees of freedom Multiple
R-squared: 0.6511, Adjusted R-squared: 0.6438
F-statistic: 89.57 on 1 and 48 DF, p-value: 1.49e-12
OLS Regression:
ordinary least squares (OLS) or linear least squares is a method for
estimating the unknown parameters in a linear regression model,
with the goal of minimizing the differences between the observed
responses in a dataset and the responses predicted by the linear
approximation of the data.
This is applied in both simple linear and multiple regression where
the common assumptions are
(1) The model is linear in the coefficients of the predictor with an
additive random
error term
(2) The random error terms are
* normally distributed with 0 mean and
* a variance that doesn't change as the values of the predictor
covariates change.
Correlation:
Correlation is a statistical measure that indicates the extent to
which two or more variables fluctuate together. That can show
whether and how strongly pairs of variables are related. It Measure
the association between variables. Positive and
negative correlation, ranging between +1 and -1.
13. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
For example, height and weight are related; taller people tend to be
heavier than shorter people. A positive correlation indicates the
extent to which those variables increase or decrease in parallel; a
negative correlation indicates the extent to which one variable
increases as the other decreases.
When the fluctuation of one variable reliably predicts a similar
fluctuation in another variable, there’s often a tendency to think
that means that the change in one causes the change in the other.
However, correlation does not imply causation.
There may be, for example, an unknown factor that influences both
variables similarly.
An intelligent correlation analysis can lead to a greater
understanding of your data.
Correlation in R:
We use the cor( ) function to produce correlations.
A simplified format of cor(x, use=, method= ) where
Option Description
x Matrix or data frame
use Specifies the handling of missing data. Options are
all.obs (assumes no missing data - missing data will
produce an error), complete.obs (listwise deletion), and
pairwise.complete.obs (pairwise deletion)
method Specifies the type of correlation. Options are pearson,
spearman or kendall.
> cor(cars)
speed dist
speed 1.0000000 0.8068949
14. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
dist 0.8068949 1.0000000
> cor(cars, use="complete.obs", method="kendall")
speed dist
speed 1.0000000 0.6689901
dist 0.6689901 1.0000000
> cor(cars, use="complete.obs", method="pearson")
speed dist
speed 1.0000000 0.8068949
dist 0.8068949 1.0000000
Correlation Coefficient:
The correlation coefficient of two variables in a data sample is their
covariance divided by the product of their individual standard
deviations. It is a normalized measurement of how the two are
linearly related.
Formally, the sample correlation coefficient is defined by the
following formula, where s x and sy are the sample standard
deviations, and sxy is the sample covariance.
Similarly, the population correlation coefficient is defined as follows,
where σ x and σy are the population standard deviations, and σxy is
the population covariance.
15. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
If the correlation coefficient is close to 1, it would indicates that the
variables are positively linearly related and the scatter plot falls
almost along a straight line with positive slope. For -1, it indicates
that the variables are negatively linearly related and the scatter plot
almost falls along a straight line with negative slope. And for zero, it
would indicates a weak linear relationship between the variables.
* r : correlation coefficient
* +1 : Perfectly positive
* -1 : Perfectly negative
* 0 – 0.2 : No or very weak association
* 0.2 – 0.4 : Weak association
* 0.4 – 0.6 : Moderate association
* 0.6 – 0.8 : Strong association
* 0.8 – 1 : Very strong to perfect association
Covariance:
Covariance provides a measure of the strength of the correlation
between two or more sets of random variates. Correlation is defined
in terms of the variance of x, the variance of y, and the covariance
of x and y (the way the two vary together; the way they co-vary) on
the assumption that both variables are normally distributed.
Covariance in R:
We apply the cov function to compute the covariance of eruptions
and waiting in faithful dataset
16. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
> duration = faithful$eruptions # the eruption durations
> waiting = faithful$waiting # the waiting period
> cov(duration, waiting) # apply the cov function
[1] 13.978
ANOVA:
Analysis of Variance (ANOVA) is a commonly used statistical
technique for investigating data by comparing the means of subsets
of the data. The base case is the one-way ANOVA which is an
extension of two-sample t test for independent groups covering
situations where there are more than two groups being compared.
In one-way ANOVA the data is sub-divided into groups based on a
single classification factor and the standard terminology used to
describe the set of factor levels is treatment even though this might
not always have meaning for the particular application. There is
variation in the measurements taken on the individual components
of the data set and ANOVA investigates whether this variation can
be explained by the grouping introduced by the classification factor.
To investigate these differences we fit the one-way ANOVA model
using the lm function and look at the parameter estimates and
standard errors for the treatment effects.
> anova(speed.lm)
Analysis of Variance Table
Response: speed
Df Sum Sq Mean Sq F value Pr(>F)
dist 1 891.98 891.98 89.567 1.49e-12 ***
Residuals 48 478.02 9.96
17. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
This table confirms that there are differences between the groups
which were highlighted in the model summary. The function confint
is used to calculate confidence intervals on the treatment
parameters, by default 95% confidence intervals
> confint(speed.lm)
2.5 % 97.5 %
(Intercept) 6.5258378 10.0419735
dist 0.1303926 0.2007426
Heteroscedasticity:
Heteroscedasticity (also spelled heteroskedasticity) refers to the
circumstance in which the variability of a variable is unequal across
the range of values of a second variable that predicts it.
A scatterplot of these variables will often create a cone-like shape,
as the scatter (or variability) of the dependent variable (DV) widens
or narrows as the value of the independent variable (IV) increases.
The inverse of heteroscedasticity is homoscedasticity, which
indicates that a DV's variability is equal across values of an IV.
Hetero (different or unequal) is the opposite of Homo (same or
equal).
Skedastic means spread or scatter.
Homoskedasticity = equal spread.
Heteroskedasticity = unequal spread.
20. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
Detecting Heteroskedasticity
There are two ways in general.
The first is the informal way which is done through graphs and
therefore we call it the graphical method.
The second is through formal tests for heteroskedasticity, like the
following ones:
1. The Breusch-Pagan LM Test
2. The Glesjer LM Test
21. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
3. The Harvey-Godfrey LM Test
4. The Park LM Test
5. The Goldfeld-Quandt Tets
6. White’s Test
Heteroscedasticity test in R:
bptest(p) does the Breuch Pagan test to formally check presence of
heteroscedasticity. To use bptest, you will have to call lmtest
library.
> install.packages("lmtest")
> library(lmtest)
> bptest(speed.lm)
studentized Breusch-Pagan test
data: speed.lm
BP = 0.71522, df = 1, p-value = 0.3977
If the test is positive (low p value), you should see if any
transformation of the dependent variable helps you eliminate
heteroscedasticity.
Autocorrelation:
Autocorrelation, also known as serial correlation or
cross-autocorrelation, is the cross-correlation of a signal with itself
at different points in time. Informally, it is the similarity between
observations as a function of the time lag between them.
It is a mathematical tool for finding repeating patterns, such as the
presence of a periodic signal obscured by noise, or identifying the
missing fundamental frequency in a signal implied by its harmonic
22. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
frequencies. It is often used in signal processing for analyzing
functions or series of values, such as time domain signals.
Autocorrelation is a mathematical representation of the degree of
similarity between a given time series and a lagged version of itself
over successive time intervals.
In statistics, the autocorrelation of a random process is
the correlation between values of the process at different times, as a
function of the two times or of the time lag. Let X be a stochastic
process, and t be any point in time. (t may be an integer for
a discrete-time process or a real number for a continuous-
time process.) Then Xt is the value (or realization) produced by a
given run of the process at time t. Suppose that the process
has mean μt and variance σt
2 at time t, for each t. Then the
definition of the autocorrelation between times s and t is
where "E" is the expected value operator. Note that this expression
is not well-defined for all-time series or processes, because the
mean may not exist, or the variance may be zero (for a constant
process) or infinite (for processes with distribution lacking well-
behaved moments, such as certain types of power law). If the
function R is well-defined, its value must lie in the range [−1, 1],
with 1 indicating perfect correlation and −1 indicating perfect anti-
correlation.
23. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
Above: A plot of a series of 100 random numbers concealing
a sine function. Below: The sine function revealed in
a correlogram produced by autocorrelation.
24. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
Visual comparison of convolution, cross-correlation and
autocorrelation.
The function acf ( ) in R computes estimates of the autocovariance
or autocorrelation function.
Test: -
The traditional test for the presence of first-order autocorrelation is
the Durbin– Watson statistic or, if the explanatory variables include
a lagged dependent variable, Durbin's h statistic. The
Durbin-Watson can be linearly mapped however to the Pearson
correlation between values and their lags.
A more flexible test, covering autocorrelation of higher orders and
applicable whether or not the regressors include lags of the
dependent variable, is the Breusch–Godfrey test. This involves an
auxiliary regression, wherein the residuals obtained from estimating
the model of interest are regressed on (a) the original regressors and
25. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
(b) k lags of the residuals, where k is the order of the test. The
simplest version of the test statistic from this auxiliary regression is
TR2, where T is the sample size and R2 is the coefficient of
determination. Under the null hypothesis of no autocorrelation, this
statistic is asymptotically distributed as x2 with k degrees of
freedom.
Introduction to Multiple Regression:
Multiple regression is a flexible method of data analysis that may be
appropriate whenever a quantitative variable (the dependent
variable) is to be examined in relationship to any other factors
(expressed as independent or predictor variables).
Relationships may be nonlinear, independent variables may be
quantitative or qualitative, and one can examine the effects of a
single variable or multiple variables with or without the effects of
other variables taken into account.
Many practical questions involve the relationship between a
dependent variable of interest (call it Y) and a set of k independent
variables or potential predictor
variables (call them X1, X2, X3,..., Xk), where the scores on all
variables are measured for N cases. For example, you might be
interested in predicting performance on a job (Y) using information
on years of experience (X1), performance in a training program (X2),
and performance on an aptitude test (X3).
A multiple regression equation for predicting Y can be expressed a
follows:
To apply the equation, each Xj score for an individual case is
multiplied by the corresponding Bj value, the products are added
together, and the constant A is added to the sum. The result is Y',
the predicted Y value for the case. Multiple Regression in R:
YEAR ROLL UNEM HGRAD INC
11 5501 8.1 9552 1923
27. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
23 23 15107 10.1 17813 2983
24 24 14831 7.5 17304 3069
25 25 15081 8.8 16756 3151
26 26 15127 9.1 16749 3127
27 27 15856 8.8 16925 3179
28 28 15938 7.8 17231 3207
29 29 16081 7.0 16816 3345
> #read data into variable
> datavar <- read.csv("dataset_enrollmentForecast.csv")
> #attach data variable
> attach(datavar)
> #two predictor model
> #create a linear model using lm(FORMULA, DATAVAR)
> #predict the fall enrollment (ROLL) using the unemployment rate
(UNEM) and number of spring high school graduates (HGRAD)
> twoPredictorModel <- lm(ROLL ~ UNEM + HGRAD, datavar)
> #display model
> twoPredictorModel
Call:
lm(formula = ROLL ~ UNEM + HGRAD, data = datavar)
Coefficients:
28. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
(Intercept) UNEM HGRAD
-8255.7511 698.2681 0.9423
> #what is the expected fall enrollment (ROLL) given this year's
unemployment rate (UNEM) of 9% and spring high school
graduating class (HGRAD) of 100,000
> -8255.8 + 698.2 * 9 + 0.9 * 100000
[1] 88028
> #the predicted fall enrollment, given a 9% unemployment rate and
100,000 student spring high school graduating class, is 88,028
students.
> #three predictor model
> #create a linear model using lm(FORMULA, DATAVAR)
> #predict the fall enrollment (ROLL) using the unemployment rate
(UNEM), number of spring high school graduates (HGRAD), and per
capita income (INC)
> threePredictorModel <- lm(ROLL ~ UNEM + HGRAD + INC,
datavar)
> #display model
> threePredictorModel
Call:
lm(formula = ROLL ~ UNEM + HGRAD + INC, data = datavar)
Coefficients:
(Intercept) UNEM HGRAD INC
-9153.2545 450.1245 0.4065 4.2749
29. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
Multicollinearity
In statistics, multicollinearity (also collinearity) is a phenomenon in
which two or more predictor variables in a multiple regression
model are highly correlated, meaning that one can be linearly
predicted from the others with a substantial degree of accuracy. In
this situation the coefficient estimates of the multiple regressions
may change erratically in response to small changes in the model or
the data. Multicollinearity does not reduce the predictive power or
reliability of the model as a whole, at least within the sample data
set; it only affects calculations regarding individual predictors. That
is, a multiple regression model with correlated predictors can
indicate how well the entire bundle of predictors predicts the
outcome variable, but it may not give valid results about any
individual predictor, or about which predictors are redundant with
respect to others.
Key Assumptions of OLS:
Introduction
Linear regression models find several uses in real-life problems. For example, a
multi-national corporation wanting to identify factors that can affect the sales
of its product can run a linear regression to find out which factors are
important. In econometrics, Ordinary Least Squares (OLS) method is widely
used to estimate the parameter of a linear regression model. OLS estimators
minimize the sum of the squared errors (a difference between observed values
and predicted values). While OLS is computationally feasible and can be easily
used while doing any econometrics test, it is important to know the underlying
assumptions of OLS regression. This is because a lack of knowledge of OLS
assumptions would result in its misuse and give incorrect results for the
30. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
econometrics test completed. The importance of OLS assumptions cannot be
overemphasized. The next section describes the assumptions of OLS
regression.
Assumptions of OLS Regression
The necessary OLS assumptions, which are used to derive the OLS estimators
in linear regression models, are discussed below.
OLS Assumption 1: The linear regression model is “linear in parameters.”
When the dependent variable (Y)(Y) is a linear function of independent
variables (X's)(X′s) and the error term, the regression is linear in parameters
and not necessarily linear in X'sX′s. For example, consider the following:
A1. The linear regression model is “linear in parameters.”
A2. There is a random sampling of observations.
A3. The conditional mean should be zero.
A4. There is no multi-collinearity (or perfect collinearity).
A5. Spherical errors: There is homoscedasticity and no autocorrelation
A6: Optional Assumption: Error terms should be normally distributed.
a)Y=β0+β1X1+β2X2+ε
b)Y=β0+β1X12+β2X2+ε
c)Y=β0+β12X1+β2X2+ε
31. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
In the above three examples, for a) and b) OLS assumption 1 is satisfied. For c)
OLS assumption 1 is not satisfied because it is not linear in parameter { beta
}_{ 1 }β1.
OLS Assumption 2: There is a random sampling of observations
This assumption of OLS regression says that:
• The sample taken for the linear regression model must be drawn randomly
from the population. For example, if you have to run a regression model to
study the factors that impact the scores of students in the final exam, then you
must select students randomly from the university during your data collection
process, rather than adopting a convenient sampling procedure.
• The number of observations taken in the sample for making the linear
regression model should be greater than the number of parameters to be
estimated. This makes sense mathematically too. If a number of parameters to
be estimated (unknowns) are more than the number of observations, then
estimation is not possible. If a number of parameters to be estimated
(unknowns) equal the number of observations, then OLS is not required. You
can simply use algebra.
• The X'sX′s should be fixed (e. independent variables should impact
dependent variables). It should not be the case that dependent variables impact
independent variables. This is because, in regression models, the causal
relationship is studied and there is not a correlation between the two variables.
For example, if you run the regression with inflation as your dependent
variable and unemployment as the independent variable, the OLS
estimators are likely to be incorrect because with inflation and unemployment,
we expect correlation rather than a causal relationship.
• The error terms are random. This makes the dependent variable random.
OLS Assumption 3: The conditional mean should be zero.
32. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
The expected value of the mean of the error terms of OLS regression should be
zero given the values of independent variables.
Mathematically, E(ε∣X)=0 This is sometimes just written as E(ε)=0
In other words, the distribution of error terms has zero mean and doesn’t
depend on the independent variables X'sX′s. Thus, there must be no
relationship between the X'sX′s and the error term.
OLS Assumption 4: There is no multi-collinearity (or perfect collinearity).
In a simple linear regression model, there is only one independent variable and
hence, by default, this assumption will hold true. However, in the case of
multiple linear regression models, there are more than one independent
variable. The OLS assumption of no multi-collinearity says that there should be
no linear relationship between the independent variables. For example,
suppose you spend your 24 hours in a day on three things – sleeping, studying,
or playing. Now, if you run a regression with dependent variable as exam
score/performance and independent variables as time spent sleeping, time
spent studying, and time spent playing, then this assumption will not hold.
This is because there is perfect collinearity between the three independent
variables.
Time spent sleeping = 24 – Time spent studying – Time spent playing.
In such a situation, it is better to drop one of the three independent variables
from the linear regression model. If the relationship (correlation) between
independent variables is strong (but not exactly perfect), it still causes
problems in OLS estimators. Hence, this OLS assumption says that you should
select independent variables that are not correlated with each other.
33. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
An important implication of this assumption of OLS regression is that there
should be sufficient variation in the X's. More the variability in X's, better are
the OLS estimates in determining the impact of X's on Y.
OLS Assumption 5: Spherical errors: There is homoscedasticity and no
autocorrelation.
According to this OLS assumption, the error terms in the regression should all
have the same variance.
Mathematically, Var(ε∣X)=σ2
If this variance is not constant (i.e. dependent on X’s), then the linear
regression model has heteroscedastic errors and likely to give incorrect
estimates.
This OLS assumption of no autocorrelation says that the error terms of
different observations should not be correlated with each other.
Mathematically, Cov(εiεj∣X)=0fori≠j
For example, when we have time series data (e.g. yearly data of
unemployment), then the regression is likely to suffer from autocorrelation
because unemployment next year will certainly be dependent on
unemployment this year. Hence, error terms in different observations will
surely be correlated with each other.
In simple terms, this OLS assumption means that the error terms should be
IID (Independent and Identically Distributed).
34. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
Image Source: Laerd Statistics
The above diagram shows the difference between Homoscedasticity and
Heteroscedasticity. The variance of errors is constant in case of homoscedasticity
while it’s not the case if errors are heteroscedastic.
OLS Assumption 6: Error terms should be normally distributed.
This assumption states that the errors are normally distributed, conditional
upon the independent variables. This OLS assumption is not required for the
validity of OLS method; however, it becomes important when one needs to
define some additional finite-sample properties. Note that only the error terms
need to be normally distributed. The dependent variable Y need not be
normally distributed.
The Use of OLS Assumptions
OLS assumptions are extremely important. If the OLS assumptions 1 to 5 hold,
then according to Gauss-Markov Theorem, OLS estimator is Best Linear
Unbiased Estimator (BLUE). These are desirable properties of OLS estimators
and require separate discussion in detail. However, below the focus is on the
importance of OLS assumptions by discussing what happens when they fail
and how can you look out for potential errors when assumptions are not
outlined.
35. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
1. The Assumption of Linearity (OLS Assumption 1) – If you fit a linear model to
a data that is non-linearly related, the model will be incorrect and hence
unreliable. When you use the model for extrapolation, you are likely to get
erroneous results. Hence, you should always plot a graph of observed predicted
values. If this graph is symmetrically distributed along the 45-degree line, then
you can be sure that the linearity assumption holds. If linearity assumptions
don’t hold, then you need to change the functional form of the regression,
which can be done by taking non-linear transformations of independent
variables (i.e. you can take log X instead of X as your independent variable)
and then check for linearity.
2. The Assumption of Homoscedasticity (OLS Assumption 5) – If errors are
heteroscedastic (i.e. OLS assumption is violated), then it will be difficult to trust
the standard errors of the OLS estimates. Hence, the confidence intervals will
be either too narrow or too wide. Also, violation of this assumption has a
tendency to give too much weight on some portion (subsection) of the data.
Hence, it is important to fix this if error variances are not constant. You can
easily check if error variances are constant or not. Examine the plot
of residuals predicted values or residuals vs. time (for time series models).
Typically, if the data set is large, then errors are more or less homoscedastic. If
your data set is small, check for this assumption.
3. The Assumption of Independence/No Autocorrelation (OLS Assumption 5) –
As discussed previously, this assumption is most likely to be violated in time
series regression models and, hence, intuition says that there is no need to
investigate it. However, you can still check for autocorrelation by viewing
the residual time series plot. If autocorrelation is present in the model, you can
try taking lags of independent variables to correct for the trend component. If
36. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
you do not correct for autocorrelation, then OLS estimates won’t be BLUE, and
they won’t be reliable enough.
4. The Assumption of Normality of Errors (OLS Assumption 6) – If error terms
are not normal, then the standard errors of OLS estimates won’t be reliable,
which means the confidence intervals would be too wide or narrow. Also, OLS
estimators won’t have the desirable BLUE property. A normal probability plot or
a normal quantile plot can be used to check if the error terms are normally
distributed or not. A bow-shaped deviated pattern in these plots reveals that
the errors are not normally distributed. Sometimes errors are not normal
because the linearity assumption is not holding. So, it is worthwhile to check
for linearity assumption again if this assumption fails.
5. Assumption of No Multicollinearity (OLS assumption 4) – You can check for
multicollinearity by making a correlation matrix (though there are other
complex ways of checking them like Variance Inflation Factor, etc.). Almost a
sure indication of the presence of multi-collinearity is when you get opposite
(unexpected) signs for your regression coefficients (e. if you expect that the
independent variable positively impacts your dependent variable but you get a
negative sign of the coefficient from the regression model). It is highly likely
that the regression suffers from multi-collinearity. If the variable is not that
important intuitively, then dropping that variable or any of the correlated
variables can fix the problem.
6. OLS assumptions 1, 2, and 4 are necessary for the setup of the OLS problem
and its derivation. Random sampling, observations being greater than the
number of parameters, and regression being linear in parameters are all part of
the setup of OLS regression. The assumption of no perfect collinearity allows
one to solve for first order conditions in the derivation of OLS estimates.
37. Dr.K.Sreenivasa Rao B.Tech, M.Tech, Ph.D VBIT, Hyderabad
Conclusion
Linear regression models are extremely useful and have a wide range of
applications. When you use them, be careful that all the assumptions of OLS
regression are satisfied while doing an econometrics test so that your efforts
don’t go wasted. These assumptions are extremely important, and one cannot
just neglect them. Having said that, many times these OLS assumptions will be
violated. However, that should not stop you from conducting your econometric
test. Rather, when the assumption is violated, applying the correct fixes and
then running the linear regression model should be the way out for a reliable
econometric test.