Discriminant analysis is a technique that is used by the researcher to analyze the research data when the criterion or the dependent variable is categorical and the predictor or the independent variable is the interval in nature. The term categorical variable means that the predictor variable is divided into a number of categories.
DA is typically used when the groups are already defined prior to the study.
The end result of DA is a model that can be used for the prediction of group memberships. This model allows us to understand the relationship between the set of selected variables and the observations. Furthermore, this model will enable one to assess the contributions of different variables.
1. Discriminant analysis is a statistical technique used to discriminate between two or more groups based on a set of predictor variables when the dependent variable is categorical.
2. It creates a discriminant function that provides weights for the predictor variables to maximize differences between groups based on centroids.
3. Key outputs include canonical correlations, classification matrices, discriminant function coefficients, and Wilk's lambda, which is used to assess how well the functions separate cases into groups.
An Overview and Application of Discriminant Analysis in Data AnalysisIOSR Journals
This document provides an overview of discriminant analysis, including its history, key assumptions, and different types (e.g. linear, quadratic). It discusses advantages of discriminant analysis compared to logistic regression, such as its ability to handle small sample sizes. The document also describes steps to develop a discriminant model, including variable selection, assumptions checking, and evaluation. It then presents an application of discriminant analysis to classify failed vs successful companies in Nigeria based on financial ratios. The model was able to predict company failure up to 3 years in advance.
- Discriminant analysis is a statistical technique used to discriminate between two or more groups based on multiple predictor variables.
- A study analyzed data on effective and ineffective extension agents to identify variables that best discriminate between the two groups. Variables like years of experience, communication skills, and positive attitude to work significantly differed between the groups.
- Discriminant analysis generated a function to maximize differences between the groups based on predictor variables. The function was statistically significant based on a small Wilks' lambda value, indicating most variability was explained.
April Heyward Research Methods Class Session - 8-5-2021April Heyward
This document provides an overview of key concepts in research methods for public administration, including:
1. Levels of measurement for variables, including nominal, ordinal, interval, and ratio levels. Examples are provided for each level.
2. Common research designs such as experimental, quasi-experimental, cross-sectional, and longitudinal designs.
3. Quantitative data analysis techniques including descriptive statistics, inferential statistics like ANOVA and regression, and correlation analysis. Frequency distributions, measures of central tendency and variability are covered.
4. Confidence intervals and how they are used to estimate population parameters more accurately than point estimates, by providing a probability assessment through setting a confidence level. Common confidence levels like 90%, 95%,
This summary analyzes a document describing the use of discriminant analysis in SPSS to predict whether graduate students will successfully complete their PhD programs based on information collected before acceptance. It identifies key outputs of the discriminant analysis including tests showing one predictor variable significantly differs between groups, coefficients of the discriminant function, group centroids, and a classification results table showing around 80% of students were correctly classified.
- Multinomial logistic regression predicts categorical membership in a dependent variable based on multiple independent variables. It is an extension of binary logistic regression that allows for more than two categories.
- Careful data analysis including checking for outliers and multicollinearity is important. A minimum sample size of 10 cases per independent variable is recommended.
- Multinomial logistic regression does not assume normality, linearity or homoscedasticity like discriminant function analysis does, making it more flexible and commonly used. It does assume independence between dependent variable categories.
This document provides an overview of quantitative data analysis techniques for hypothesis testing, including types of errors, statistical power, and tests for single and multiple sample means. It also discusses regression analysis, issues of multicollinearity, and other multivariate tests such as discriminant analysis, logistic regression, and canonical correlation.
The document discusses multiple linear regression analysis. It defines multiple regression as exploring the relationship between one continuous dependent variable and multiple independent variables. It provides examples of multiple regression models with one and two predictors. It also discusses assumptions of multiple regression like sample size, multicollinearity, outliers, and normality of residuals. Key steps in multiple regression like estimating parameters, assessing model fit and diagnosing assumptions are outlined.
1. Discriminant analysis is a statistical technique used to discriminate between two or more groups based on a set of predictor variables when the dependent variable is categorical.
2. It creates a discriminant function that provides weights for the predictor variables to maximize differences between groups based on centroids.
3. Key outputs include canonical correlations, classification matrices, discriminant function coefficients, and Wilk's lambda, which is used to assess how well the functions separate cases into groups.
An Overview and Application of Discriminant Analysis in Data AnalysisIOSR Journals
This document provides an overview of discriminant analysis, including its history, key assumptions, and different types (e.g. linear, quadratic). It discusses advantages of discriminant analysis compared to logistic regression, such as its ability to handle small sample sizes. The document also describes steps to develop a discriminant model, including variable selection, assumptions checking, and evaluation. It then presents an application of discriminant analysis to classify failed vs successful companies in Nigeria based on financial ratios. The model was able to predict company failure up to 3 years in advance.
- Discriminant analysis is a statistical technique used to discriminate between two or more groups based on multiple predictor variables.
- A study analyzed data on effective and ineffective extension agents to identify variables that best discriminate between the two groups. Variables like years of experience, communication skills, and positive attitude to work significantly differed between the groups.
- Discriminant analysis generated a function to maximize differences between the groups based on predictor variables. The function was statistically significant based on a small Wilks' lambda value, indicating most variability was explained.
April Heyward Research Methods Class Session - 8-5-2021April Heyward
This document provides an overview of key concepts in research methods for public administration, including:
1. Levels of measurement for variables, including nominal, ordinal, interval, and ratio levels. Examples are provided for each level.
2. Common research designs such as experimental, quasi-experimental, cross-sectional, and longitudinal designs.
3. Quantitative data analysis techniques including descriptive statistics, inferential statistics like ANOVA and regression, and correlation analysis. Frequency distributions, measures of central tendency and variability are covered.
4. Confidence intervals and how they are used to estimate population parameters more accurately than point estimates, by providing a probability assessment through setting a confidence level. Common confidence levels like 90%, 95%,
This summary analyzes a document describing the use of discriminant analysis in SPSS to predict whether graduate students will successfully complete their PhD programs based on information collected before acceptance. It identifies key outputs of the discriminant analysis including tests showing one predictor variable significantly differs between groups, coefficients of the discriminant function, group centroids, and a classification results table showing around 80% of students were correctly classified.
- Multinomial logistic regression predicts categorical membership in a dependent variable based on multiple independent variables. It is an extension of binary logistic regression that allows for more than two categories.
- Careful data analysis including checking for outliers and multicollinearity is important. A minimum sample size of 10 cases per independent variable is recommended.
- Multinomial logistic regression does not assume normality, linearity or homoscedasticity like discriminant function analysis does, making it more flexible and commonly used. It does assume independence between dependent variable categories.
This document provides an overview of quantitative data analysis techniques for hypothesis testing, including types of errors, statistical power, and tests for single and multiple sample means. It also discusses regression analysis, issues of multicollinearity, and other multivariate tests such as discriminant analysis, logistic regression, and canonical correlation.
The document discusses multiple linear regression analysis. It defines multiple regression as exploring the relationship between one continuous dependent variable and multiple independent variables. It provides examples of multiple regression models with one and two predictors. It also discusses assumptions of multiple regression like sample size, multicollinearity, outliers, and normality of residuals. Key steps in multiple regression like estimating parameters, assessing model fit and diagnosing assumptions are outlined.
Multiple discriminant analysis (MDA) is used to classify cases into groups when there are more than two categories. MDA derives multiple discriminant functions to discriminate between groups, with the first function accounting for the most variation between groups. The number of functions derived is usually equal to the number of groups minus one or the number of predictor variables, whichever is smaller. MDA outputs include standardized discriminant function coefficients, structure correlations, group centroids, and a classification matrix assessing prediction accuracy.
Discriminant analysis is a statistical technique used to classify individuals or cases into groups based on a set of predictor variables. It can be used to determine which variables discriminate between two or more naturally occurring groups and to classify new observations into one of the existing groups. The key steps involve developing a discriminant function using a linear combination of predictors, evaluating the accuracy of classification, and determining the relative importance of predictors in discriminating between groups. Discriminant analysis requires certain assumptions about the data such as normality and equality of group variances to be valid.
Discriminant function analysis (DFA) is a statistical technique used to determine which variables discriminate between two or more naturally occurring groups. It creates linear combinations of predictor variables that maximize differences between groups. The document outlines the purpose of DFA, the steps involved including developing discriminant functions and testing significance, assumptions of the analysis, types of DFA including linear and multiple DFA, and applications in fields like agriculture, marketing, and face recognition.
The document discusses various methods for analyzing data, including descriptive, statistical, and multivariate analyses. Statistical analysis makes raw data meaningful by testing hypotheses, obtaining significant results, and drawing inferences. The appropriate analysis depends on the type of measurement, number of variables, and type of statistical inference required. Correlation analysis studies relationships between variables while causal analysis examines how independent variables affect dependents. Multivariate techniques include multiple regression, discriminant analysis, ANOVA, and canonical analysis.
Discriminant function analysis (DFA) is a statistical technique used to determine which variables discriminate between two or more naturally occurring groups. It creates linear combinations of predictor variables that maximize differences between groups. The presentation outlines the purpose of DFA, the basic model involving discriminant functions, assumptions of the analysis, and types including linear and multiple DFA. Applications of DFA discussed include prediction, description, agriculture/crop studies, marketing, and more.
Discriminant analysis is a statistical technique used to classify individuals or cases into groups based on a set of predictor variables. It aims to determine which variables discriminate between two or more naturally occurring groups and build a model to predict group membership. The key steps involve developing discriminant functions using linear combinations of predictors, examining differences between groups on predictors, and evaluating the accuracy of classification. Discriminant analysis is commonly used to classify individuals into categories based on characteristics like athletic ability, performance level, or other attributes.
This document provides an introduction to regression analysis and statistical methods. It discusses that regression analysis estimates the linear relationship between dependent and independent variables. Multiple linear regression allows studying the relationship between one dependent variable and two or more independent variables. The accuracy of regression models can be evaluated using measures like R-squared and testing overall model significance. Diagnostic tests of assumptions like independence of errors, normality, homoscedasticity and absence of multicollinearity/influential outliers are important.
This document provides an overview of discriminant analysis, including its definition, objectives, assumptions, and steps. Discriminant analysis is a statistical technique used to classify observations into predefined groups based on independent variables. It can be used to understand group differences and predict the likelihood an entity belongs to a particular class. Key assumptions include normal distributions, homogeneity of variances, absence of outliers and multicollinearity. The steps involve selecting discriminating variables, developing a discriminant function model, and classifying observations into groups. Outputs include group statistics, Box's M test of equality of covariance matrices, canonical discriminant functions, eigenvalues, Wilks' lambda, and classification results.
Canonical correlation analysis was used to detect potential bias in faculty promotion scoring at American University of Nigeria (AUN). Three committees independently scored candidates based on teaching, research, and service. CCA discriminated between promotable and non-promotable candidates at the 90% confidence level, rejecting the hypothesis that it could not do so. CCA also found no significant differences in scoring between committees or evidence that individual assessors' scores overbearingly influenced outcomes, rejecting the hypotheses that it could not detect bias. The results suggest CCA is an effective tool for AUN to analyze scoring and ensure fairness in its promotion process.
Canonical correlation analysis was used to detect potential bias in faculty promotion scoring at the American University of Nigeria (AUN). The analysis compared scores from three promotion committees and tested whether any committee showed bias that influenced candidates' promotability. The analysis found:
1) It could discriminate between candidates deemed promotable versus non-promotable, rejecting the hypothesis that it couldn't do so.
2) There were no significant differences in scoring between committees, rejecting the hypothesis that it couldn't detect bias.
3) Only the president's committee showed significant score weight influence on promotability, rejecting the hypothesis that it couldn't detect overbearing influences.
The study demonstrated canonical correlation analysis can be an effective tool for unbiased faculty
The document discusses different types of variables in experimental research:
- Independent variable: Factor manipulated by researcher to determine its effect
- Dependent variable: Factor observed and measured to determine effect of independent variable
- Moderator variable: Factor that modifies relationship between independent and dependent variables
- Control variable: Factors controlled by researcher to neutralize their effects
- Intervening variable: Factor that theoretically affects phenomena but cannot be directly observed
It also discusses data types, central tendency measures, data variability measures, and statistical techniques like correlation analysis, t-tests, ANOVA that are used for quantitative analysis.
This presentation discusses the application of discriminant analysis in sports research. One can understand the steps involved in the analysis and testing its assumptions.
Selection of appropriate data analysis techniqueRajaKrishnan M
- The document discusses choosing the right statistical method for data analysis, which depends on factors like the number and measurement level of variables, the distribution of variables, the dependence/independence structure, the nature of the hypotheses, and sample size.
- It presents flowcharts for choosing a statistical method based on whether the hypothesis involves one variable (univariate), two variables (bivariate), or more than two variables (multivariate).
- For univariate data, descriptive statistics or a one-sample t-test can be used depending on whether description or inference is the goal; for bivariate data, the choice depends on the nature of the hypothesis (difference or association) and the level of measurement (parametric or nonparame
Mba2216 week 11 data analysis part 03 appendixStephen Ong
Multivariate analysis involves simultaneously analyzing multiple variables to understand relationships. This document discusses key concepts in multivariate analysis including:
1. Defining multivariate analysis and when it is appropriate to use.
2. Describing specific techniques like multiple regression, discriminant analysis, logistic regression, MANOVA, canonical correlation analysis, conjoint analysis, factor analysis, cluster analysis, multidimensional scaling, and correspondence analysis.
3. Providing guidelines for selecting the appropriate technique based on the measurement scales and relationship between variables.
It also covers important considerations like measurement error, statistical power, and a structured approach to multivariate model building.
Discriminant function analysis (DFA) is a statistical technique used to determine which variables discriminate between two or more naturally occurring groups. It creates linear combinations of predictor variables that discriminate between the groups of a categorical dependent variable. DFA is useful for predicting group membership and understanding the relationship between predictors and groups. It works by developing discriminant functions, which are linear combinations of predictors that maximize differences between groups. Common applications of DFA include classification, prediction, and understanding differences between groups.
This document provides an overview of parametric and nonparametric statistical methods. It defines key concepts like standard error, degrees of freedom, critical values, and one-tailed versus two-tailed hypotheses. Common parametric tests discussed include t-tests, ANOVA, ANCOVA, and MANOVA. Nonparametric tests covered are chi-square, Mann-Whitney U, Kruskal-Wallis, and Friedman. The document explains when to use parametric versus nonparametric methods and how measures like effect size can quantify the strength of relationships found.
1) ANOVA is used to compare the means of more than two populations and determine if observed differences are due to chance or actual differences in the population means.
2) The document provides an example of using a one-way single factor ANOVA to analyze the effects of different teaching formats on student exam scores.
3) The ANOVA compares the between-treatment variability to the within-treatment variability using an F-test. If the between-treatment variability is significantly larger, it suggests the population means differ. In this example, the F-test showed no significant difference between the teaching formats.
Clustering is the process of grouping objects into clusters based on similarities. There are different types of clustering including hierarchical, k-means, and two stage clustering. Factor analysis reduces a large number of variables into fewer factors that capture maximum common variance. Classification sorts data into predefined categories or classes while clustering does not predefine categories, allowing structure in the data to determine the grouping. Clustering and classification are both used for data analysis but differ in how groups are determined.
Chapter 11 KNN Naive Bayes and LDA.pptxkiitlabsbsc
This document discusses three supervised machine learning algorithms: K-nearest neighbors (KNN), Naive Bayes, and Linear Discriminant Analysis (LDA). KNN performs classification or regression based on distance between data points. Naive Bayes is a classification technique based on Bayes' theorem that assumes independence between features. LDA estimates relationships between dependent and independent variables to classify objects into groups based on continuous variables. The document outlines the concepts, formulas, applications, and steps to implement each algorithm using R and Python.
Generative Classifiers: Classifying with Bayesian decision theory, Bayes’ rule, Naïve Bayes classifier.
Discriminative Classifiers: Logistic Regression, Decision Trees: Training and Visualizing a Decision Tree, Making Predictions, Estimating Class Probabilities, The CART Training Algorithm, Attribute selection measures- Gini impurity; Entropy, Regularization Hyperparameters, Regression Trees, Linear Support vector machines.
Build applications with generative AI on Google CloudMárton Kodok
We will explore Vertex AI - Model Garden powered experiences, we are going to learn more about the integration of these generative AI APIs. We are going to see in action what the Gemini family of generative models are for developers to build and deploy AI-driven applications. Vertex AI includes a suite of foundation models, these are referred to as the PaLM and Gemini family of generative ai models, and they come in different versions. We are going to cover how to use via API to: - execute prompts in text and chat - cover multimodal use cases with image prompts. - finetune and distill to improve knowledge domains - run function calls with foundation models to optimize them for specific tasks. At the end of the session, developers will understand how to innovate with generative AI and develop apps using the generative ai industry trends.
Multiple discriminant analysis (MDA) is used to classify cases into groups when there are more than two categories. MDA derives multiple discriminant functions to discriminate between groups, with the first function accounting for the most variation between groups. The number of functions derived is usually equal to the number of groups minus one or the number of predictor variables, whichever is smaller. MDA outputs include standardized discriminant function coefficients, structure correlations, group centroids, and a classification matrix assessing prediction accuracy.
Discriminant analysis is a statistical technique used to classify individuals or cases into groups based on a set of predictor variables. It can be used to determine which variables discriminate between two or more naturally occurring groups and to classify new observations into one of the existing groups. The key steps involve developing a discriminant function using a linear combination of predictors, evaluating the accuracy of classification, and determining the relative importance of predictors in discriminating between groups. Discriminant analysis requires certain assumptions about the data such as normality and equality of group variances to be valid.
Discriminant function analysis (DFA) is a statistical technique used to determine which variables discriminate between two or more naturally occurring groups. It creates linear combinations of predictor variables that maximize differences between groups. The document outlines the purpose of DFA, the steps involved including developing discriminant functions and testing significance, assumptions of the analysis, types of DFA including linear and multiple DFA, and applications in fields like agriculture, marketing, and face recognition.
The document discusses various methods for analyzing data, including descriptive, statistical, and multivariate analyses. Statistical analysis makes raw data meaningful by testing hypotheses, obtaining significant results, and drawing inferences. The appropriate analysis depends on the type of measurement, number of variables, and type of statistical inference required. Correlation analysis studies relationships between variables while causal analysis examines how independent variables affect dependents. Multivariate techniques include multiple regression, discriminant analysis, ANOVA, and canonical analysis.
Discriminant function analysis (DFA) is a statistical technique used to determine which variables discriminate between two or more naturally occurring groups. It creates linear combinations of predictor variables that maximize differences between groups. The presentation outlines the purpose of DFA, the basic model involving discriminant functions, assumptions of the analysis, and types including linear and multiple DFA. Applications of DFA discussed include prediction, description, agriculture/crop studies, marketing, and more.
Discriminant analysis is a statistical technique used to classify individuals or cases into groups based on a set of predictor variables. It aims to determine which variables discriminate between two or more naturally occurring groups and build a model to predict group membership. The key steps involve developing discriminant functions using linear combinations of predictors, examining differences between groups on predictors, and evaluating the accuracy of classification. Discriminant analysis is commonly used to classify individuals into categories based on characteristics like athletic ability, performance level, or other attributes.
This document provides an introduction to regression analysis and statistical methods. It discusses that regression analysis estimates the linear relationship between dependent and independent variables. Multiple linear regression allows studying the relationship between one dependent variable and two or more independent variables. The accuracy of regression models can be evaluated using measures like R-squared and testing overall model significance. Diagnostic tests of assumptions like independence of errors, normality, homoscedasticity and absence of multicollinearity/influential outliers are important.
This document provides an overview of discriminant analysis, including its definition, objectives, assumptions, and steps. Discriminant analysis is a statistical technique used to classify observations into predefined groups based on independent variables. It can be used to understand group differences and predict the likelihood an entity belongs to a particular class. Key assumptions include normal distributions, homogeneity of variances, absence of outliers and multicollinearity. The steps involve selecting discriminating variables, developing a discriminant function model, and classifying observations into groups. Outputs include group statistics, Box's M test of equality of covariance matrices, canonical discriminant functions, eigenvalues, Wilks' lambda, and classification results.
Canonical correlation analysis was used to detect potential bias in faculty promotion scoring at American University of Nigeria (AUN). Three committees independently scored candidates based on teaching, research, and service. CCA discriminated between promotable and non-promotable candidates at the 90% confidence level, rejecting the hypothesis that it could not do so. CCA also found no significant differences in scoring between committees or evidence that individual assessors' scores overbearingly influenced outcomes, rejecting the hypotheses that it could not detect bias. The results suggest CCA is an effective tool for AUN to analyze scoring and ensure fairness in its promotion process.
Canonical correlation analysis was used to detect potential bias in faculty promotion scoring at the American University of Nigeria (AUN). The analysis compared scores from three promotion committees and tested whether any committee showed bias that influenced candidates' promotability. The analysis found:
1) It could discriminate between candidates deemed promotable versus non-promotable, rejecting the hypothesis that it couldn't do so.
2) There were no significant differences in scoring between committees, rejecting the hypothesis that it couldn't detect bias.
3) Only the president's committee showed significant score weight influence on promotability, rejecting the hypothesis that it couldn't detect overbearing influences.
The study demonstrated canonical correlation analysis can be an effective tool for unbiased faculty
The document discusses different types of variables in experimental research:
- Independent variable: Factor manipulated by researcher to determine its effect
- Dependent variable: Factor observed and measured to determine effect of independent variable
- Moderator variable: Factor that modifies relationship between independent and dependent variables
- Control variable: Factors controlled by researcher to neutralize their effects
- Intervening variable: Factor that theoretically affects phenomena but cannot be directly observed
It also discusses data types, central tendency measures, data variability measures, and statistical techniques like correlation analysis, t-tests, ANOVA that are used for quantitative analysis.
This presentation discusses the application of discriminant analysis in sports research. One can understand the steps involved in the analysis and testing its assumptions.
Selection of appropriate data analysis techniqueRajaKrishnan M
- The document discusses choosing the right statistical method for data analysis, which depends on factors like the number and measurement level of variables, the distribution of variables, the dependence/independence structure, the nature of the hypotheses, and sample size.
- It presents flowcharts for choosing a statistical method based on whether the hypothesis involves one variable (univariate), two variables (bivariate), or more than two variables (multivariate).
- For univariate data, descriptive statistics or a one-sample t-test can be used depending on whether description or inference is the goal; for bivariate data, the choice depends on the nature of the hypothesis (difference or association) and the level of measurement (parametric or nonparame
Mba2216 week 11 data analysis part 03 appendixStephen Ong
Multivariate analysis involves simultaneously analyzing multiple variables to understand relationships. This document discusses key concepts in multivariate analysis including:
1. Defining multivariate analysis and when it is appropriate to use.
2. Describing specific techniques like multiple regression, discriminant analysis, logistic regression, MANOVA, canonical correlation analysis, conjoint analysis, factor analysis, cluster analysis, multidimensional scaling, and correspondence analysis.
3. Providing guidelines for selecting the appropriate technique based on the measurement scales and relationship between variables.
It also covers important considerations like measurement error, statistical power, and a structured approach to multivariate model building.
Discriminant function analysis (DFA) is a statistical technique used to determine which variables discriminate between two or more naturally occurring groups. It creates linear combinations of predictor variables that discriminate between the groups of a categorical dependent variable. DFA is useful for predicting group membership and understanding the relationship between predictors and groups. It works by developing discriminant functions, which are linear combinations of predictors that maximize differences between groups. Common applications of DFA include classification, prediction, and understanding differences between groups.
This document provides an overview of parametric and nonparametric statistical methods. It defines key concepts like standard error, degrees of freedom, critical values, and one-tailed versus two-tailed hypotheses. Common parametric tests discussed include t-tests, ANOVA, ANCOVA, and MANOVA. Nonparametric tests covered are chi-square, Mann-Whitney U, Kruskal-Wallis, and Friedman. The document explains when to use parametric versus nonparametric methods and how measures like effect size can quantify the strength of relationships found.
1) ANOVA is used to compare the means of more than two populations and determine if observed differences are due to chance or actual differences in the population means.
2) The document provides an example of using a one-way single factor ANOVA to analyze the effects of different teaching formats on student exam scores.
3) The ANOVA compares the between-treatment variability to the within-treatment variability using an F-test. If the between-treatment variability is significantly larger, it suggests the population means differ. In this example, the F-test showed no significant difference between the teaching formats.
Clustering is the process of grouping objects into clusters based on similarities. There are different types of clustering including hierarchical, k-means, and two stage clustering. Factor analysis reduces a large number of variables into fewer factors that capture maximum common variance. Classification sorts data into predefined categories or classes while clustering does not predefine categories, allowing structure in the data to determine the grouping. Clustering and classification are both used for data analysis but differ in how groups are determined.
Chapter 11 KNN Naive Bayes and LDA.pptxkiitlabsbsc
This document discusses three supervised machine learning algorithms: K-nearest neighbors (KNN), Naive Bayes, and Linear Discriminant Analysis (LDA). KNN performs classification or regression based on distance between data points. Naive Bayes is a classification technique based on Bayes' theorem that assumes independence between features. LDA estimates relationships between dependent and independent variables to classify objects into groups based on continuous variables. The document outlines the concepts, formulas, applications, and steps to implement each algorithm using R and Python.
Generative Classifiers: Classifying with Bayesian decision theory, Bayes’ rule, Naïve Bayes classifier.
Discriminative Classifiers: Logistic Regression, Decision Trees: Training and Visualizing a Decision Tree, Making Predictions, Estimating Class Probabilities, The CART Training Algorithm, Attribute selection measures- Gini impurity; Entropy, Regularization Hyperparameters, Regression Trees, Linear Support vector machines.
Build applications with generative AI on Google CloudMárton Kodok
We will explore Vertex AI - Model Garden powered experiences, we are going to learn more about the integration of these generative AI APIs. We are going to see in action what the Gemini family of generative models are for developers to build and deploy AI-driven applications. Vertex AI includes a suite of foundation models, these are referred to as the PaLM and Gemini family of generative ai models, and they come in different versions. We are going to cover how to use via API to: - execute prompts in text and chat - cover multimodal use cases with image prompts. - finetune and distill to improve knowledge domains - run function calls with foundation models to optimize them for specific tasks. At the end of the session, developers will understand how to innovate with generative AI and develop apps using the generative ai industry trends.
Enhanced data collection methods can help uncover the true extent of child abuse and neglect. This includes Integrated Data Systems from various sources (e.g., schools, healthcare providers, social services) to identify patterns and potential cases of abuse and neglect.
Discovering Digital Process Twins for What-if Analysis: a Process Mining Appr...Marlon Dumas
This webinar discusses the limitations of traditional approaches for business process simulation based on had-crafted model with restrictive assumptions. It shows how process mining techniques can be assembled together to discover high-fidelity digital twins of end-to-end processes from event data.
We are pleased to share with you the latest VCOSA statistical report on the cotton and yarn industry for the month of March 2024.
Starting from January 2024, the full weekly and monthly reports will only be available for free to VCOSA members. To access the complete weekly report with figures, charts, and detailed analysis of the cotton fiber market in the past week, interested parties are kindly requested to contact VCOSA to subscribe to the newsletter.
Did you know that drowning is a leading cause of unintentional death among young children? According to recent data, children aged 1-4 years are at the highest risk. Let's raise awareness and take steps to prevent these tragic incidents. Supervision, barriers around pools, and learning CPR can make a difference. Stay safe this summer!
2. INTRODUCTION
Discriminant analysis is a technique that is used by the researcher to
analyze the research data when the criterion or the dependent variable is
categorical and the predictor or the independent variable is the interval
in nature. The term categorical variable means that the predictor variable
is divided into a number of categories.
DA is typically used when the groups are already defined prior to the
prior to the study.
The end result of DA is a model that can be used for the prediction of
group memberships. This model allows us to understand the relationship
between the set of selected variables and the observations.
Furthermore, this model will enable one to assess the contributions of
different variables.
3. DISCRIMINANT ANALYSIS AND
BINARY LOGISTIC REGRESSION
Discriminant Analysis and Binary Logistic Regression
although they do the same thing same job but
discriminant is more powerful in comparison to logistic
because logistics is generally done with 0 and 1 case
yes or no case but discriminant analysis can take up
more than 2,3,4 also categories but larger too large
number of categories also is not very advisable.
5. HOMOGENEOUS WITHIN-GROUP VARIANCES.
Variances among group variables are the same across levels of predictors. It has been suggested,
however, that linear discriminant analysis be used when covariances are equal, and that quadratic
discriminant analysis may be used when covariances are not equal.
DA is very sensitive to the heterogeneity of variance-covariance matrices. Before accepting the final
conclusions for an important study, review the within-group variances and correlation matrices.
Homoscedasticity is evaluated through scatterplots and corrected by the transformation of variables.
The heterogeneity may arise due to the non-normality of data. Another one is due to large sample
since the significance probability becomes smaller even for almost homogeneous covariance matrices
if the sample size is large.
NO MULTI-COLLINEARITY.
Predictive power can decrease with an increased correlation between predictor variables.
6. BOX’s M-Test
𝑯𝑶: 𝜮𝟏 = 𝜮𝟐=….= 𝜮𝑳
𝑯𝟏: 𝜮𝒍 ≠ 𝜮𝒎 for at least one pair of (l,m) is statistically different [ 𝒍 ≠ m]
Statistic D=(1-u)M
M= -2ln[ 𝒊=𝑙
𝑳
(
|𝑺𝒍|
|𝑺𝒑𝒐𝒐𝒍𝒆𝒅|
)
(𝒏𝒍−𝟏)
𝟐 ] (Log we used here for our convenience.)
u=[ 𝑙
1
(𝑛𝑙−1)
−
1
𝑙(𝑛𝑙−1)
] [
2𝑝2+3𝑝−1
6(𝑝+1)(𝐿−1)
]
Reject 𝑯𝒐 when D > 𝝌𝟐
𝜶,𝒗
D.F.= v
v =
1
2
p(p + 1)(L − 1)
7. PRIOR PROBABILITIES.
The prior probability is the probability of an observation coming from a particular group in a
simple random sample with replacement.
If the prior probabilities are the same for all three of the groups (also known as equal priors), then
the function is only based on the squared MAHALANOBIS distance.
MULTIVARIATE NORMALITY WITHIN GROUPS.
The independent variables should be multivariate normal; in other words, when all other
independent variables are held constant, the independent variable under examination should
have a normal distribution.
Mahalanobis procedure: a stepwise procedure used in discriminant analysis to maximize a
generalized measure of the distance between the two closest groups.
8. OBJECTIVES
• To find the linear combinations of variables that discriminate
between categories of dependent variables in the best possible
manner.
• To find out which independent variables are relatively better
in discriminating between groups.
• To determine the statistical significance of the discriminant
function and whether any statistical difference exists among
groups in terms of the predictor variable.
• To evaluate the accuracy of classification, i.e., the percentage
of customers that is able to classify correctly.
9. DISCRIMINANT ANALYSIS & MANOVA
• Discriminant analysis is a lot like MANOVA.
• In MANOVA the criterion is metric and the predictor is categorical. However, in discriminant analysis the
criterion is categorical and the predictor is metric.
In MANOVA, D1,D2 = Continuous Variables ; IV1,IV2= Categorical Variables
In DA, D1,D2 = Categorical Variables; IV1,IV2= Continuous Variables
• The multiple index values for the multiple linear discriminant function has been discussed by Hyberty
(1994). The approach is to conduct 𝒑 MANOVAs, each involving (𝒑 − 𝟏) variables. That is, delete each
variable, in turn, and conduct a MANOVA using the remaining 𝒑 − 𝟏 variables.
• The important variable is the one for which the MANOVA on the remaining variables provides the largest
Wilks lambda. The second most important variable is the one for which the Wilks lambda value is the
second largest one. Thus the variables can be ranked according to their importance depending on the ranks
of 𝚲 values.
11. DISCRIMINANT ANALYSIS
The linear combination can be represented by D=b’X, where D is the discriminant
score of order (1 x n), b is a (p x 1)vector of discriminant weight and X is the
(n x p) data matrix.
In two groups of discriminant problems, the sample objects are classified with the help of a binary or
indicator variable with values zero and one. Now, corresponding to this binary variable the discriminant score D=b’X is
calculated using the data matrix X . This calculated discriminant score looks like the fitted multiple regression line when the
binary variable is considered as dependent one. In such situations, Y=b’X is a linear probability model where Y is the binary
variable and X is the matrix of the explanatory variables.
However, multiple regression analysis is not similar to discriminant analysis. The predictor variable in
multiple linear regression analysis is assumed to be normally distributed, whereas the binary variable in the discriminant
analysis does not follow any statistical distribution.
The explanatory variables in regression analysis do not follow any statistical distribution but in discriminant
analysis follow a multivariate normal distribution.
The objective of regression analysis is to predict response variables on the basis of predictors, whereas the
objective of discriminant analysis is to classify the sample objects with minimum classification error.
12. DISCRIMINANT ANALYSIS MODEL
• Discriminant analysis model is defined as the statistical model on which discriminant analysis
is based.
• The discriminant analysis model involves linear combinations of the following form:
D= b0 +b1X1+b₂X₂+b3X3+………………..+bkXk
Where,
D=discriminant score
b’s=discriminant coefficient or weight
X’s=predictor or independent variable
• Coefficient or weights (b) are estimated so that the group differ as much as possible on the
values of the discriminant function.
• This occurs when the ratio of the between-group sum of squares to the within-group sum of
squares for discriminant scores is at a maximum.
• There are as many linear combinations as there are groups and the prediction rule enables us
to determine the group with which an object is identified.
13. Canonical correlation: It measures the extent of association
between the discriminant score and the group.
Centroid: It is the mean value for the discriminant scores for a
particular group.
Classification matrix: It contains the number of correctly classified
and misclassified cases.
Hit Ratio: In the classification matrix, the sum of diagonal elements
divided by the total number of cases represents the hit ratio. It is the
percentage of cases correctly classified by discriminant analysis.
Discriminant function coefficients:
1)Discriminant function coefficients (unstandardized) are the
multipliers of variables. When the variables are in the original units
of measurement.
2)They are the discriminant function coefficients that are used as the
multipliers when the variables have been standardized to a mean 0
and variance 1.
14. Discriminant scores: The unstandardized coefficients are multiplied by the
values of the variables. These products are summed and added to the
constant term to obtain the discriminant scores.
Eigenvalue: For each discriminant function, the eigenvalue is the ratio of
between-group to within-group sums of squares.
• Wilks’ Lambda is the ratio of within-group sums of squares to the total sums
of squares. This is the proportion of the total variance in the discriminant
scores not explained by differences among groups.
• Wilks’ lambda takes a value between 0 and 1 and the lower the value of Wilks’
lambda, the higher the significance of the discriminant function as the
decrease in error of classification leads to a decrease in the amount of
variance.
15. Let 𝑿𝒍 𝒏𝒍 × 𝒑 be the 𝑙-th data matrix [𝒍 = 𝟏, 𝟐, … , 𝒌] from 𝑵𝒑 𝝁𝒍, 𝜮𝒍 .
Assume that 𝜮𝟏 = 𝜮𝟐 = ⋯ = 𝜮𝒌. If 𝑿𝒍 = 𝑿𝟏𝒍𝑿𝟐𝒍 ⋯ 𝑿𝒑𝒍
′
be the data vector
and 𝒇𝒍 𝑿𝒍 be the density function of 𝑿𝒍, then the objective of the
discriminant analysis is to identify the 𝒇𝒍 𝑿𝒍 of an object on the basis of the
values of 𝒑 variables of 𝑿. The identification is done in such a way that the
error of identification is minimum.
Let us explain the technique by an example. Consider that a doctor needs to
examine many patients to diagnose their diseases. Different patients are
suffering from different diseases and the symptoms of the diseases are also
different. The symptoms help the doctor to diagnose the disease correctly
which in turn makes the patient cure. The treatment of the patient
becomes easier if the diagnosis of the disease is made correctly.
Justification of Discriminant Analysis and Selection of
Variables
16. 𝑫 = 𝜷𝟎 + 𝜷𝟏𝒙𝟏 + 𝜷𝟐𝒙𝟐 + ⋯ + 𝜷𝒑𝒙𝒑
Let us consider that the total sample objects of size 𝒏 are to be divided into two
groups of sizes 𝒏𝟏 and 𝒏𝟐 such that 𝒏 = 𝒏𝟏 + 𝒏𝟐. Let us assume that 𝒍-th [𝒍 = 𝟏, 𝟐]
group of sample observations have the p.d.f. 𝒇𝒍(𝒙), where 𝒍-th population has mean
vector 𝝁𝒍. Now, if it is observed that the null hypothesis 𝑯𝟎: 𝝁𝟏 = 𝝁𝟐 is rejected, the
discriminant analysis can be performed.
The rejection of 𝑯𝟎: 𝝁𝟏 = 𝝁𝟐 = ⋯ = 𝝁𝒌 does not mean that the means of 𝒋-th variable [𝒋 =
𝟏, 𝟐, … , 𝒑] for all 𝒌 samples are heterogeneous. If some of the means, assume that the means of
𝒑𝟏 < 𝒑 variables, are homogeneous, the above hypothesis may be rejected and decision will be
made in favor of discriminant analysis. However, the homogeneity in the variables in 𝒌 groups
has nothing to contribute to discriminate among groups.
Thus, even if the hypothesis of equality of group means is rejected, it needs a decision regarding
the inclusion of variables for discriminant analysis. Let 𝝁𝒍𝒋(𝒍 = 𝟏, 𝟐, … , 𝒌; 𝒋 = 𝟏, 𝟐, … , 𝒑) be the
mean of 𝑗-th variable of 𝑙-th sample. The 𝑗-th variable should be included in the analysis if the
null hypothesis
𝑯𝟎: 𝝁𝟏𝒋 = 𝝁𝟐𝒋 = ⋯ = 𝝁𝒌𝒋
17. is rejected, otherwise 𝒋-th variable is deleted from the analysis. This hypothesis is tested by univariate
analysis of variance 𝑭-test and it can be judged for all of the 𝒑 variables.
The decision regarding the deletion of some variables from discriminant analysis can be made using
the McCabe (1975) FORTRAN program. The program searches all possible subsets of a given set of
variables. A subset is selected if it provides lowest Wilks Lambda value, where Wilks Lambda is the
test statistic in testing
𝑯𝟎: 𝝁𝟏 = 𝝁𝟐 = ⋯ = 𝝁𝒌,
with a subset of variables. A subset is selected from the plot of Wilks Lambda value versus the subset
size. The plot takes the shape as shown in Fig. It is seen from the graph that increasing the size of
subset of variables there is no sharp decrease in the value of Wilks lambda at a certain stage. This can
be decided if the points representing Wilks lambda values for some subset size touch a straight line as
is shown in the graph. The cut point of subset size is that one which does not touch the straight line
but produces minimum Wilks lambda value.
18.
19. The correlation coefficient of 𝑫 values and 𝒙𝒋(𝒋 = 𝟏, 𝟐, … , 𝒑) values. This correlation is used to
measure the contribution of 𝒋-th variable in discriminating the groups. The most contributing
variable is one for which the above-mentioned correlation coefficient is maximum.
If any pair of variables are highly correlated, which one has more discriminating power when both of
these are highly correlated with 𝑫. The amount of correlation of 𝑫 and 𝒙𝒋 and the sign of correlation
coefficient will be affected if 𝒙𝒋 and 𝒙𝒋′ 𝒋 ≠ 𝒋′
are highly correlated. thus if 𝒙𝒋 and 𝒙𝒋′ are
correlated, their correlation with 𝑫 will not provide any fruitful information about discriminating
power of the variable.
To avoid this, pooled within-groups correlation of all variables for all sample points are studied. If any
pair of variables is highly correlated, these are linearly related. The linear relationship may exist
among different variables. Let 𝒙𝒋 is linearly related with other 𝒙𝒋′ 's 𝒋′
≠ 𝒋 = 𝟏, 𝟐, … , 𝒑 and the
multiple correlation coefficient of 𝒋-th variable with other variables be 𝑹𝒋. Then 𝟏 − 𝑹𝒋
𝟐
is known
as tolerance. Now, if tolerance of any of the 𝒋-th variable is small, the inclusion of that variable in
discriminant analysis will not be fruitful.