This document provides an introduction to Bayesian linear mixed models. It begins by motivating the use of Bayesian methods as an alternative to frequentist approaches for analyzing psycholinguistic data. It then reviews the components of linear mixed models, including how they can account for repeated measures data. Through a simulation study, it demonstrates that the lmer package may not reliably estimate correlation parameters unless the dataset is large enough. Overall, the document argues that Bayesian linear mixed models allow for more flexible modeling of complex data structures and incorporate prior knowledge compared to frequentist methods.
This document discusses tensor decompositions for learning latent variable models. It presents five applications of latent variable modeling: topic modeling, understanding human communities, recommender systems, feature learning, and computational biology. It describes the statistical framework of discovering hidden structure through latent variable models and graphical modeling. A key challenge is efficiently learning latent variable models, as maximum likelihood is NP-hard. The document proposes using tensor decompositions to efficiently and consistently learn latent variable models from data in a guaranteed manner. It outlines experimental results on social networks that demonstrate the tensor approach outperforms variational methods in accuracy and speed.
The document discusses using machine learning methods to estimate heterogeneous causal effects. It proposes an approach of using regression trees on a transformed outcome variable to estimate individual treatment effects. However, this approach is critiqued as it can introduce noise. An improved approach is presented that uses the sample average treatment effect within each leaf as the estimator, and uses the variance of predictions for model fitting criteria and a matching estimator for out-of-sample evaluation. The approach separates the tasks of model selection and treatment effect estimation to enable valid statistical inference on estimated effects in subgroups.
Binary Classification with Models and Data Density Distribution by Xuan ChenXuan Chen
This document discusses binary classification using probabilistic labels. It introduces the problem of determining whether models can achieve more accurate predictions using probabilistic labels that indicate the probability a sample belongs to a class, rather than deterministic 0-1 labels. The author conducted experiments on several common models using different data density distributions. The results showed Gaussian Process Regression generally had the best performance and achieved error bounds of O(n-2+γ4) when using probabilistic labels under certain data distributions, outperforming existing error bounds for deterministic labels. However, under some distributions the existing bounds were not exceeded.
A COMPARATIVE STUDY ON DISTANCE MEASURING APPROACHES FOR CLUSTERINGIJORCS
Clustering plays a vital role in the various areas of research like Data Mining, Image Retrieval, Bio-computing and many a lot. Distance measure plays an important role in clustering data points. Choosing the right distance measure for a given dataset is a biggest challenge. In this paper, we study various distance measures and their effect on different clustering. This paper surveys existing distance measures for clustering and present a comparison between them based on application domain, efficiency, benefits and drawbacks. This comparison helps the researchers to take quick decision about which distance measure to use for clustering. We conclude this work by identifying trends and challenges of research and development towards clustering.
This presentation is about how global terminology can evolve without a centralized organisation. The simple idea is, that everybody has to disclose the identity of at least two identifiers for the same think. These local semantic handshakes will have the effect of global terminological alignment.
FUZZY ROUGH INFORMATION MEASURES AND THEIR APPLICATIONSijcsity
The degree of roughness characterizes the uncertainty contained in a rough set. The rough entropy was
defined to measure the roughness of a rough set. Though, it was effective and useful, but not accurate
enough. Some authors use information measure in place of entropy for better understanding which
measures the amount of uncertainty contained in fuzzy rough set .In this paper three new fuzzy rough
information measures are proposed and their validity is verified. The application of these proposed
information measures in decision making problems is studied and also compared with other existing
information measures.
Dimensionality Reduction Techniques for Document Clustering- A SurveyIJTET Journal
Abstract— Dimensionality reduction technique is applied to get rid of the inessential terms like redundant and noisy terms in documents. In this paper a systematic study is conducted for seven dimensionality reduction methods such as Latent Semantic Indexing (LSI), Random Projection (RP), Principle Component Analysis (PCA) and CUR decomposition, Latent Dirichlet Allocation(LDA), Singular value decomposition (SVD). Linear Discriminant Analysis(LDA)
Mapping Subsets of Scholarly InformationPaul Houle
The document discusses using machine learning techniques like support vector machines (SVMs) to analyze and classify academic literature from a large online corpus like arXiv. It finds that SVMs can accurately identify documents belonging to large categories with over 10,000 documents but struggles with smaller categories of under 500 documents. To improve recall for small categories, the SVM outputs are converted to probabilities using a sigmoid function rather than relying on signed distances from the hyperplane alone.
This document discusses tensor decompositions for learning latent variable models. It presents five applications of latent variable modeling: topic modeling, understanding human communities, recommender systems, feature learning, and computational biology. It describes the statistical framework of discovering hidden structure through latent variable models and graphical modeling. A key challenge is efficiently learning latent variable models, as maximum likelihood is NP-hard. The document proposes using tensor decompositions to efficiently and consistently learn latent variable models from data in a guaranteed manner. It outlines experimental results on social networks that demonstrate the tensor approach outperforms variational methods in accuracy and speed.
The document discusses using machine learning methods to estimate heterogeneous causal effects. It proposes an approach of using regression trees on a transformed outcome variable to estimate individual treatment effects. However, this approach is critiqued as it can introduce noise. An improved approach is presented that uses the sample average treatment effect within each leaf as the estimator, and uses the variance of predictions for model fitting criteria and a matching estimator for out-of-sample evaluation. The approach separates the tasks of model selection and treatment effect estimation to enable valid statistical inference on estimated effects in subgroups.
Binary Classification with Models and Data Density Distribution by Xuan ChenXuan Chen
This document discusses binary classification using probabilistic labels. It introduces the problem of determining whether models can achieve more accurate predictions using probabilistic labels that indicate the probability a sample belongs to a class, rather than deterministic 0-1 labels. The author conducted experiments on several common models using different data density distributions. The results showed Gaussian Process Regression generally had the best performance and achieved error bounds of O(n-2+γ4) when using probabilistic labels under certain data distributions, outperforming existing error bounds for deterministic labels. However, under some distributions the existing bounds were not exceeded.
A COMPARATIVE STUDY ON DISTANCE MEASURING APPROACHES FOR CLUSTERINGIJORCS
Clustering plays a vital role in the various areas of research like Data Mining, Image Retrieval, Bio-computing and many a lot. Distance measure plays an important role in clustering data points. Choosing the right distance measure for a given dataset is a biggest challenge. In this paper, we study various distance measures and their effect on different clustering. This paper surveys existing distance measures for clustering and present a comparison between them based on application domain, efficiency, benefits and drawbacks. This comparison helps the researchers to take quick decision about which distance measure to use for clustering. We conclude this work by identifying trends and challenges of research and development towards clustering.
This presentation is about how global terminology can evolve without a centralized organisation. The simple idea is, that everybody has to disclose the identity of at least two identifiers for the same think. These local semantic handshakes will have the effect of global terminological alignment.
FUZZY ROUGH INFORMATION MEASURES AND THEIR APPLICATIONSijcsity
The degree of roughness characterizes the uncertainty contained in a rough set. The rough entropy was
defined to measure the roughness of a rough set. Though, it was effective and useful, but not accurate
enough. Some authors use information measure in place of entropy for better understanding which
measures the amount of uncertainty contained in fuzzy rough set .In this paper three new fuzzy rough
information measures are proposed and their validity is verified. The application of these proposed
information measures in decision making problems is studied and also compared with other existing
information measures.
Dimensionality Reduction Techniques for Document Clustering- A SurveyIJTET Journal
Abstract— Dimensionality reduction technique is applied to get rid of the inessential terms like redundant and noisy terms in documents. In this paper a systematic study is conducted for seven dimensionality reduction methods such as Latent Semantic Indexing (LSI), Random Projection (RP), Principle Component Analysis (PCA) and CUR decomposition, Latent Dirichlet Allocation(LDA), Singular value decomposition (SVD). Linear Discriminant Analysis(LDA)
Mapping Subsets of Scholarly InformationPaul Houle
The document discusses using machine learning techniques like support vector machines (SVMs) to analyze and classify academic literature from a large online corpus like arXiv. It finds that SVMs can accurately identify documents belonging to large categories with over 10,000 documents but struggles with smaller categories of under 500 documents. To improve recall for small categories, the SVM outputs are converted to probabilities using a sigmoid function rather than relying on signed distances from the hyperplane alone.
Introduction to Some Tree based Learning MethodHonglin Yu
Random Forest, Boosted Trees and other ensemble learning methods build multiple models to improve predictive performance over single models. They combine "weak learners" like decision trees into a "strong learner". Random Forest adds randomness by selecting a random subset of features at each split. Boosting trains trees sequentially on weighted data from previous trees. Both reduce variance compared to bagging. Random Forest often outperforms Boosting while being faster to train. Neural networks can also be viewed as an ensemble method by combining simple units.
Cluster analysis is a technique used to group objects based on characteristics they possess. It involves measuring the distance or similarity between objects and grouping those that are most similar together. There are two main types: hierarchical cluster analysis, which groups objects sequentially into clusters; and nonhierarchical cluster analysis, which directly assigns objects to pre-specified clusters. The choice of method depends on factors like sample size and research objectives.
K-means clustering is an algorithm that groups data points into k clusters based on their attributes and distances from initial cluster center points. It works by first randomly selecting k data points as initial centroids, then assigning all other points to the closest centroid and recalculating the centroids. This process repeats until the centroids are stable or a maximum number of iterations is reached. K-means clustering is widely used for machine learning applications like image segmentation and speech recognition due to its efficiency, but it is sensitive to initialization and assumes spherical clusters of similar size and density.
This document discusses different approaches to multivariate data analysis and clustering, including nearest neighbor methods, hierarchical clustering, and k-means clustering. It provides examples of using Ward's method, average linkage, and k-means clustering on poverty data to identify potential clusters of countries based on variables like birth rate, death rate, and infant mortality rate. Key lessons are that different linkage methods, distance measures, and data normalizations should be tested and that higher-dimensional data may require different variable spaces or transformations to identify meaningful clusters.
The document provides information about cluster analysis and hierarchical agglomerative clustering. It discusses how hierarchical agglomerative clustering works by successively merging the most similar clusters together based on a distance measure between clusters. It begins with each object as its own cluster and merges them into larger clusters, forming a dendrogram, until all objects are in a single cluster. Different linkage criteria like single, complete, and average linkage can be used to define the distance between clusters.
The document provides an overview of various machine learning algorithms and methods. It begins with an introduction to predictive modeling and supervised vs. unsupervised learning. It then describes several supervised learning algorithms in detail including linear regression, K-nearest neighbors (KNN), decision trees, random forest, logistic regression, support vector machines (SVM), and naive Bayes. It also briefly discusses unsupervised learning techniques like clustering and dimensionality reduction methods.
This document discusses combinatorial design problems and approaches to solving them. It introduces combinatorial problems and challenges like huge search spaces. Methods covered include modeling problems as constraint satisfaction problems (CSPs) and using constraint propagation to reduce the search space. Specific problems discussed include ternary Steiner systems, Hamming distance optimization, and modeling the Data Encryption Standard (DES) cipher as a SAT problem for cryptanalysis.
The document discusses smoothing parameter selection for density estimation from length-biased data using asymmetric kernels. It reviews recent work on applying Bayesian criteria for this purpose. The key points are: 1) Asymmetric kernels are better for density estimation of non-negative data as they avoid positive mass in negative regions. 2) Length-biased data arises in situations where observations are weighted by their values. 3) Estimating the underlying density from length-biased data requires adjusting for the bias. 4) Bayesian methods provide an approach for selecting the smoothing parameter for asymmetric kernel density estimators applied to length-biased data.
Latent Semantic Word Sense Disambiguation Using Global Co-Occurrence Informationcsandit
The document describes a novel word sense disambiguation method using global co-occurrence information and non-negative matrix factorization. It proposes using global co-occurrence frequencies between words in a large corpus rather than local context windows to address data sparseness issues. An experiment compares the method to two baselines and shows it achieves the highest and most stable precision for word sense disambiguation.
Introduction to Machine Learning Aristotelis Tsirigos butest
This document provides an introduction to machine learning, covering several key concepts:
- Machine learning aims to build models from data to make predictions without being explicitly programmed.
- There are different types of learning problems including supervised, unsupervised, and reinforcement learning.
- Popular machine learning algorithms discussed include Bayesian learning, nearest neighbors, decision trees, linear classifiers, and ensembles.
- Proper evaluation of machine learning models is important using techniques like cross-validation.
This technical report explores using set-valued attributes for decision tree induction algorithms. Conventional algorithms use single-valued attributes, but the authors argue set-valued attributes can improve accuracy and speed. They describe modifying decision tree algorithms for splitting, pruning, and classification when attributes can have set values. Experiments show the proposed approach works well with only simple pre-pruning needed to limit excessive instance replication across tree branches. The set-valued approach is intended to better handle noise and variability in data values.
Machine learning basics using trees algorithm (Random forest, Gradient Boosting)Parth Khare
This document provides an overview of machine learning classification and decision trees. It discusses key concepts like supervised vs. unsupervised learning, and how decision trees work by recursively partitioning data into nodes. Random forest and gradient boosted trees are introduced as ensemble methods that combine multiple decision trees. Random forest grows trees independently in parallel while gradient boosted trees grow sequentially by minimizing error from previous trees. While both benefit from ensembling, gradient boosted trees are more prone to overfitting and random forests are better at generalizing to new data.
This document discusses vector space retrieval models. It describes how documents and queries are represented as vectors in a common vector space based on terms. Terms are weighted using metrics like term frequency (TF) and inverse document frequency (IDF) to determine importance. The cosine similarity measure is used to calculate similarity between document and query vectors and rank results by relevance. While simple and effective in practice, vector space models have limitations like missing semantic and syntactic information.
Maximizing the Representation Gap between In-domain & OOD examplesJay Nandy
Paper presented in ICML 2020 Workshop on Uncertainty & Robustness in Deep Learning
Paper link: http://www.gatsby.ucl.ac.uk/~balaji/udl2020/accepted-papers/UDL2020-paper-134.pdf
An Automatic Medical Image Segmentation using Teaching Learning Based Optimiz...idescitation
Nature inspired population based evolutionary algorithms are very popular with
their competitive solutions for a wide variety of applications. Teaching Learning based
Optimization (TLBO) is a very recent population based evolutionary algorithm evolved
on the basis of Teaching Learning process of a class room. TLBO does not require any
algorithmic specific parameters. This paper proposes an automatic grouping of pixels into
different homogeneous regions using the TLBO. The experimental results have
demonstrated the effectiveness of TLBO in image segmentation.
Learning to Rank - From pairwise approach to listwiseHasan H Topcu
The document discusses pairwise and listwise approaches to learning to rank. In the pairwise approach, training instances are document pairs and the objective is to classify pairs as correctly or incorrectly ranked. In the listwise approach, training instances are entire document lists and the objective is to minimize listwise loss functions like permutation probability and top one probability. The listwise approach directly optimizes ranking performance while the pairwise approach has drawbacks like treating all document pairs equally.
Classifiers are algorithms that map input data to categories in order to build models for predicting unknown data. There are several types of classifiers that can be used including logistic regression, decision trees, random forests, support vector machines, Naive Bayes, and neural networks. Each uses different techniques such as splitting data, averaging predictions, or maximizing margins to classify data. The best classifier depends on the problem and achieving high accuracy, sensitivity, and specificity.
This document provides an introduction to statistical model selection. It discusses various approaches to model selection including predictive risk, Bayesian methods, information theoretic measures like AIC and MDL, and adaptive methods. The key goals of model selection are to understand the bias-variance tradeoff and select models that offer the best guaranteed predictive performance on new data. Model selection aims to find the right level of complexity to explain patterns in available data while avoiding overfitting.
Discovering Novel Information with sentence Level clustering From Multi-docu...irjes
The document presents a novel fuzzy clustering algorithm called FRECCA that clusters sentences from multi-documents to discover new information. FRECCA uses fuzzy relational eigenvector centrality to calculate page rank scores for sentences within clusters, treating the scores as likelihoods. It uses expectation maximization to optimize cluster membership values and mixing coefficients without a parameterized likelihood function. An evaluation shows FRECCA achieves superior performance to other clustering algorithms on a quotations dataset, identifying overlapping clusters of semantically related sentences.
This document provides an overview of various techniques for text categorization, including decision trees, maximum entropy modeling, perceptrons, and K-nearest neighbor classification. It discusses the data representation, model class, and training procedure for each technique. Key aspects covered include feature selection, parameter estimation, convergence criteria, and the advantages/limitations of each approach.
1) The document describes a unit on repeated measures designs, including a review of standard repeated measures analyses using linear models and multi-level modeling, as well as an alternative approach.
2) Key features of repeated measures designs are discussed, such as having more than one observation per participant. Advantages and challenges like order effects are also reviewed.
3) Methods for analyzing repeated measures data using linear models by first transforming the data into wide format using differences and averages are described and compared to a multi-level modeling approach.
Introduction to Some Tree based Learning MethodHonglin Yu
Random Forest, Boosted Trees and other ensemble learning methods build multiple models to improve predictive performance over single models. They combine "weak learners" like decision trees into a "strong learner". Random Forest adds randomness by selecting a random subset of features at each split. Boosting trains trees sequentially on weighted data from previous trees. Both reduce variance compared to bagging. Random Forest often outperforms Boosting while being faster to train. Neural networks can also be viewed as an ensemble method by combining simple units.
Cluster analysis is a technique used to group objects based on characteristics they possess. It involves measuring the distance or similarity between objects and grouping those that are most similar together. There are two main types: hierarchical cluster analysis, which groups objects sequentially into clusters; and nonhierarchical cluster analysis, which directly assigns objects to pre-specified clusters. The choice of method depends on factors like sample size and research objectives.
K-means clustering is an algorithm that groups data points into k clusters based on their attributes and distances from initial cluster center points. It works by first randomly selecting k data points as initial centroids, then assigning all other points to the closest centroid and recalculating the centroids. This process repeats until the centroids are stable or a maximum number of iterations is reached. K-means clustering is widely used for machine learning applications like image segmentation and speech recognition due to its efficiency, but it is sensitive to initialization and assumes spherical clusters of similar size and density.
This document discusses different approaches to multivariate data analysis and clustering, including nearest neighbor methods, hierarchical clustering, and k-means clustering. It provides examples of using Ward's method, average linkage, and k-means clustering on poverty data to identify potential clusters of countries based on variables like birth rate, death rate, and infant mortality rate. Key lessons are that different linkage methods, distance measures, and data normalizations should be tested and that higher-dimensional data may require different variable spaces or transformations to identify meaningful clusters.
The document provides information about cluster analysis and hierarchical agglomerative clustering. It discusses how hierarchical agglomerative clustering works by successively merging the most similar clusters together based on a distance measure between clusters. It begins with each object as its own cluster and merges them into larger clusters, forming a dendrogram, until all objects are in a single cluster. Different linkage criteria like single, complete, and average linkage can be used to define the distance between clusters.
The document provides an overview of various machine learning algorithms and methods. It begins with an introduction to predictive modeling and supervised vs. unsupervised learning. It then describes several supervised learning algorithms in detail including linear regression, K-nearest neighbors (KNN), decision trees, random forest, logistic regression, support vector machines (SVM), and naive Bayes. It also briefly discusses unsupervised learning techniques like clustering and dimensionality reduction methods.
This document discusses combinatorial design problems and approaches to solving them. It introduces combinatorial problems and challenges like huge search spaces. Methods covered include modeling problems as constraint satisfaction problems (CSPs) and using constraint propagation to reduce the search space. Specific problems discussed include ternary Steiner systems, Hamming distance optimization, and modeling the Data Encryption Standard (DES) cipher as a SAT problem for cryptanalysis.
The document discusses smoothing parameter selection for density estimation from length-biased data using asymmetric kernels. It reviews recent work on applying Bayesian criteria for this purpose. The key points are: 1) Asymmetric kernels are better for density estimation of non-negative data as they avoid positive mass in negative regions. 2) Length-biased data arises in situations where observations are weighted by their values. 3) Estimating the underlying density from length-biased data requires adjusting for the bias. 4) Bayesian methods provide an approach for selecting the smoothing parameter for asymmetric kernel density estimators applied to length-biased data.
Latent Semantic Word Sense Disambiguation Using Global Co-Occurrence Informationcsandit
The document describes a novel word sense disambiguation method using global co-occurrence information and non-negative matrix factorization. It proposes using global co-occurrence frequencies between words in a large corpus rather than local context windows to address data sparseness issues. An experiment compares the method to two baselines and shows it achieves the highest and most stable precision for word sense disambiguation.
Introduction to Machine Learning Aristotelis Tsirigos butest
This document provides an introduction to machine learning, covering several key concepts:
- Machine learning aims to build models from data to make predictions without being explicitly programmed.
- There are different types of learning problems including supervised, unsupervised, and reinforcement learning.
- Popular machine learning algorithms discussed include Bayesian learning, nearest neighbors, decision trees, linear classifiers, and ensembles.
- Proper evaluation of machine learning models is important using techniques like cross-validation.
This technical report explores using set-valued attributes for decision tree induction algorithms. Conventional algorithms use single-valued attributes, but the authors argue set-valued attributes can improve accuracy and speed. They describe modifying decision tree algorithms for splitting, pruning, and classification when attributes can have set values. Experiments show the proposed approach works well with only simple pre-pruning needed to limit excessive instance replication across tree branches. The set-valued approach is intended to better handle noise and variability in data values.
Machine learning basics using trees algorithm (Random forest, Gradient Boosting)Parth Khare
This document provides an overview of machine learning classification and decision trees. It discusses key concepts like supervised vs. unsupervised learning, and how decision trees work by recursively partitioning data into nodes. Random forest and gradient boosted trees are introduced as ensemble methods that combine multiple decision trees. Random forest grows trees independently in parallel while gradient boosted trees grow sequentially by minimizing error from previous trees. While both benefit from ensembling, gradient boosted trees are more prone to overfitting and random forests are better at generalizing to new data.
This document discusses vector space retrieval models. It describes how documents and queries are represented as vectors in a common vector space based on terms. Terms are weighted using metrics like term frequency (TF) and inverse document frequency (IDF) to determine importance. The cosine similarity measure is used to calculate similarity between document and query vectors and rank results by relevance. While simple and effective in practice, vector space models have limitations like missing semantic and syntactic information.
Maximizing the Representation Gap between In-domain & OOD examplesJay Nandy
Paper presented in ICML 2020 Workshop on Uncertainty & Robustness in Deep Learning
Paper link: http://www.gatsby.ucl.ac.uk/~balaji/udl2020/accepted-papers/UDL2020-paper-134.pdf
An Automatic Medical Image Segmentation using Teaching Learning Based Optimiz...idescitation
Nature inspired population based evolutionary algorithms are very popular with
their competitive solutions for a wide variety of applications. Teaching Learning based
Optimization (TLBO) is a very recent population based evolutionary algorithm evolved
on the basis of Teaching Learning process of a class room. TLBO does not require any
algorithmic specific parameters. This paper proposes an automatic grouping of pixels into
different homogeneous regions using the TLBO. The experimental results have
demonstrated the effectiveness of TLBO in image segmentation.
Learning to Rank - From pairwise approach to listwiseHasan H Topcu
The document discusses pairwise and listwise approaches to learning to rank. In the pairwise approach, training instances are document pairs and the objective is to classify pairs as correctly or incorrectly ranked. In the listwise approach, training instances are entire document lists and the objective is to minimize listwise loss functions like permutation probability and top one probability. The listwise approach directly optimizes ranking performance while the pairwise approach has drawbacks like treating all document pairs equally.
Classifiers are algorithms that map input data to categories in order to build models for predicting unknown data. There are several types of classifiers that can be used including logistic regression, decision trees, random forests, support vector machines, Naive Bayes, and neural networks. Each uses different techniques such as splitting data, averaging predictions, or maximizing margins to classify data. The best classifier depends on the problem and achieving high accuracy, sensitivity, and specificity.
This document provides an introduction to statistical model selection. It discusses various approaches to model selection including predictive risk, Bayesian methods, information theoretic measures like AIC and MDL, and adaptive methods. The key goals of model selection are to understand the bias-variance tradeoff and select models that offer the best guaranteed predictive performance on new data. Model selection aims to find the right level of complexity to explain patterns in available data while avoiding overfitting.
Discovering Novel Information with sentence Level clustering From Multi-docu...irjes
The document presents a novel fuzzy clustering algorithm called FRECCA that clusters sentences from multi-documents to discover new information. FRECCA uses fuzzy relational eigenvector centrality to calculate page rank scores for sentences within clusters, treating the scores as likelihoods. It uses expectation maximization to optimize cluster membership values and mixing coefficients without a parameterized likelihood function. An evaluation shows FRECCA achieves superior performance to other clustering algorithms on a quotations dataset, identifying overlapping clusters of semantically related sentences.
This document provides an overview of various techniques for text categorization, including decision trees, maximum entropy modeling, perceptrons, and K-nearest neighbor classification. It discusses the data representation, model class, and training procedure for each technique. Key aspects covered include feature selection, parameter estimation, convergence criteria, and the advantages/limitations of each approach.
1) The document describes a unit on repeated measures designs, including a review of standard repeated measures analyses using linear models and multi-level modeling, as well as an alternative approach.
2) Key features of repeated measures designs are discussed, such as having more than one observation per participant. Advantages and challenges like order effects are also reviewed.
3) Methods for analyzing repeated measures data using linear models by first transforming the data into wide format using differences and averages are described and compared to a multi-level modeling approach.
Methodological Study Of Opinion Mining And Sentiment Analysis Techniques ijsc
Decision making both on individual and organizational level is always accompanied by the search of other’s opinion on the same. With tremendous establishment of opinion rich resources like, reviews, forum discussions, blogs, micro-blogs, Twitter etc provide a rich anthology of sentiments. This user generated content can serve as a benefaction to market if the semantic orientations are deliberated. Opinion mining and sentiment analysis are the formalization for studying and construing opinions and sentiments. The digital ecosystem has itself paved way for use of huge volume of opinionated data recorded. This paper is an attempt to review and evaluate the various techniques used for opinion and sentiment analysis.
Machine Learning and Artificial Neural Networks.pptAnshika865276
Machine learning and neural networks are discussed. Machine learning investigates how knowledge is acquired through experience. A machine learning model includes what is learned (the domain), who is learning (the computer program), and the information source. Techniques discussed include k-nearest neighbors algorithm, Winnow algorithm, naive Bayes classifier, decision trees, and reinforcement learning. Reinforcement learning involves an agent interacting with an environment to optimize outcomes through trial and error.
The document discusses machine learning and provides an overview of the field. It defines machine learning, explains why it is useful, and gives a brief tour of different machine learning techniques including decision trees, neural networks, Bayesian networks, and more. It also discusses some issues in machine learning like how learning problem characteristics and algorithms influence accuracy.
Lazy learning methods store training data and wait until test data is received to perform classification, taking less time to train but more time to predict. Eager learning methods construct a classification model during training. Lazy methods like k-nearest neighbors use a richer hypothesis space while eager methods commit to a single hypothesis. The k-nearest neighbor algorithm classifies new examples based on the labels of its k closest training examples. Case-based reasoning uses a symbolic case database for classification while genetic algorithms evolve rule populations through crossover and mutation to classify data.
This document discusses algorithm-independent machine learning techniques. It introduces concepts like bias and variance, which can quantify how well a learning algorithm matches a problem without depending on a specific algorithm. Methods like cross-validation, bootstrapping, and resampling can be used with different algorithms. While no algorithm is inherently superior, such techniques provide guidance on algorithm use and help integrate multiple classifiers.
The document discusses algorithm-independent machine learning and some fundamental problems in machine learning. It introduces concepts like bias and variance, the no free lunch theorem, and minimum description length principle. Key ideas are that no learning algorithm is inherently superior, algorithms can be evaluated based on how well they match the learning problem, and assumptions are needed to determine similarity between patterns or features.
This document discusses algorithm-independent machine learning techniques. It introduces concepts like bias and variance which can be used to quantify how well a learning algorithm matches a problem, regardless of the specific algorithm used. It discusses techniques like cross-validation, resampling, and combining multiple classifiers that can improve performance in a way that is independent of the learning algorithm. The document also covers principles like minimum description length and no free lunch which provide theoretical foundations for algorithm-independent machine learning.
This document provides an overview of machine learning and neural network techniques. It defines machine learning as the field that focuses on algorithms that can learn. The document discusses several key components of a machine learning model, including what is being learned (the domain) and from what information the learner is learning. It then summarizes several common machine learning algorithms like k-NN, Naive Bayes classifiers, decision trees, reinforcement learning, and the Rocchio algorithm for relevance feedback in information retrieval. For each technique, it provides a brief definition and examples of applications.
Methodological study of opinion mining and sentiment analysis techniquesijsc
Decision making both on individual and organizational level is always accompanied by the search of
other’s opinion on the same. With tremendous establishment of opinion rich resources like, reviews, forum
discussions, blogs, micro-blogs, Twitter etc provide a rich anthology of sentiments. This user generated
content can serve as a benefaction to market if the semantic orientations are deliberated. Opinion mining
and sentiment analysis are the formalization for studying and construing opinions and sentiments. The
digital ecosystem has itself paved way for use of huge volume of opinionated data recorded. This paper is
an attempt to review and evaluate the various techniques used for opinion and sentiment analysis.
Analytical study of feature extraction techniques in opinion miningcsandit
Although opinion mining is in a nascent stage of development but still the ground is set for
dense growth of researches in the field. One of the important activities of opinion mining is to
extract opinions of people based on characteristics of the object under study. Feature extraction
in opinion mining can be done by various ways like that of clustering, support vector machines
etc. This paper is an attempt to appraise the various techniques of feature extraction. The first
part discusses various techniques and second part makes a detailed appraisal of the major
techniques used for feature extraction
ANALYTICAL STUDY OF FEATURE EXTRACTION TECHNIQUES IN OPINION MININGcsandit
Although opinion mining is in a nascent stage of development but still the ground is set for dense growth of researches in the field. One of the important activities of opinion mining is to extract opinions of people based on characteristics of the object under study. Feature extraction in opinion mining can be done by various ways like that of clustering, support vector machines
etc. This paper is an attempt to appraise the various techniques of feature extraction. The first part discusses various techniques and second part makes a detailed appraisal of the major techniques used for feature extraction
Radial Basis Function Neural Network (RBFNN), Induction Motor, Vector control...cscpconf
Although opinion mining is in a nascent stage of development but still the ground is set for dense growth of researches in the field. One of the important activities of opinion mining is to
extract opinions of people based on characteristics of the object under study. Feature extraction in opinion mining can be done by various ways like that of clustering, support vector machines
etc. This paper is an attempt to appraise the various techniques of feature extraction. The first part discusses various techniques and second part makes a detailed appraisal of the major techniques used for feature extraction.
This document discusses using unsupervised support vector analysis to increase the efficiency of simulation-based functional verification. It describes applying an unsupervised machine learning technique called support vector analysis to filter redundant tests from a set of verification tests. By clustering similar tests into regions of a similarity metric space, it aims to select the most important tests to verify a design while removing redundant tests, improving verification efficiency. The approach trains an unsupervised support vector model on an initial set of simulated tests and uses it to filter future tests by comparing them to support vectors that define regions in the similarity space.
DIRECTIONS READ THE FOLLOWING STUDENT POST AND RESPOND EVALUATE I.docxlynettearnold46882
DIRECTIONS: READ THE FOLLOWING STUDENT POST AND RESPOND EVALUATE ITS CONTENT. PLEASE CITE ALL REFERENCES
Katie Kessler
Unit 2 Discussion 1
Top of Form
The word “noir” is used to remember the scaling of measurement in psychology (Embretson, 2004). In short, the letters stand for nominal, ordinal, interval and ratio (Embretson, 2004). To give a brief introduction of what each scale measures, “nominal is the simplest way to measure” because it focuses on categorizing measurements on a scale of category, according to Embretson (2004). An example of nominal is eye color. “Ordinal measures in terms of ranking, interval measures scores of tests that focus on unobservable mental functioning and ratio focuses on measuring activities in the physical world, such as someone’s running time” (Embretson, 2004). With different scales of measurement, there are two methods to compare sets of data. These include norm-referenced and criterion-referenced testing. According to Embretson (2004) norm-referenced testing “yields information on a testtaker’s standing or ranking relative to some comparison group of testtakers.” In other words, it focuses on the performance of peers. Criterion-referenced testing is a little different because it focuses on examining individual’s scores to a set standard (Embretson, 2004).
The ability for ordinal measurement scale to be utilized on a standardized test as a norm-referenced test is high since an ordinal scale is based upon ranking and norm-referenced testing gathers information on the examinees ranking compared to a group of testtakers. For example, a study conducted on decision making with the use of ordinal variables states that ordinal measurement scales has the ability to be utilized by norm-referenced testing (Barua, Kademane, Das, Gubbiyappa, Verma, & Al-Dubai, 2014).On the other hand, ordinal scaling would not be a strong measurement for criterion-referenced testing because it focuses on the ranking rather than the measurement of the scores to be close to a set standard.
Ratio scaling directs its focus on measuring objects and activities in the physical world which would be beneficial for criterion-referenced testing instead of norm-referenced testing. Imagine a marathon runner who was trying to beat the world’s fastest time running a marathon. Criterion-referenced testing allows the runner to be aware of the set standard the marathon runner needs to beat to be the best and set a new standard. Norm-referenced testing would not be as useful because the marathon runner would not have the standard measurement he or she needs to beat. However, the marathon runner would be aware of the relative time he or she needs to beat to be the best. That is not as helpful as the criterion-referenced testing because runners need an exact number instead of a relative number in comparison to other runners.
Norm-referenced data would be collected by “the standards relative to a group, such as means and standard deviation.
The document discusses the vector space model used in information retrieval. It explains that documents and queries are represented as weighted vectors in a multidimensional space. Similar vectors are close to each other. The weights used are usually tf-idf, which considers both the frequency of a term within a document and its rarity across documents. Documents are ranked based on the similarity between their vector representation and the query vector.
The document discusses the vector space model used in information retrieval. It explains that documents and queries are represented as weighted vectors in a high dimensional vector space. Similarities between queries and documents are calculated to rank documents by relevance. Weights are often calculated using TF-IDF, which considers the frequency of terms within documents and across collections. Documents with vector representations closer to the query vector are considered more relevant.
This dissertation consists of three chapters that study identification and inference in econometric models.
Chapter 1 considers identification robust inference when the moment variance matrix is singular. It develops a novel asymptotic approach based on higher order expansions of the eigensystem to show that the Generalized Anderson-Rubin statistic possesses a chi-squared limit under additional regularity conditions. When these conditions are violated, the statistic is shown to be Op(n) and exhibit "moment-singularity bias".
Chapter 2 provides a method called "Normalized Principal Components" to minimize many weak instrument bias in linear IV settings. It derives an asymptotically valid ranking of instruments in terms of correlation and selects instruments to minimize MSE approximations.
Chapter
Enhanced Enterprise Intelligence with your personal AI Data Copilot.pdfGetInData
Recently we have observed the rise of open-source Large Language Models (LLMs) that are community-driven or developed by the AI market leaders, such as Meta (Llama3), Databricks (DBRX) and Snowflake (Arctic). On the other hand, there is a growth in interest in specialized, carefully fine-tuned yet relatively small models that can efficiently assist programmers in day-to-day tasks. Finally, Retrieval-Augmented Generation (RAG) architectures have gained a lot of traction as the preferred approach for LLMs context and prompt augmentation for building conversational SQL data copilots, code copilots and chatbots.
In this presentation, we will show how we built upon these three concepts a robust Data Copilot that can help to democratize access to company data assets and boost performance of everyone working with data platforms.
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How can we build one?
Architecture and evaluation
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1. Lecture 1: Review of Linear Mixed Models
Lecture 1: Review of Linear Mixed Models
Shravan Vasishth
Department of Linguistics
University of Potsdam, Germany
October 7, 2014
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2. Lecture 1: Review of Linear Mixed Models
Introduction
Motivating this course
In psycholinguistics, we usually use frequentist methods to
analyze our data.
The most common tool I use is lme4.
In recent years, very powerful programming languages have
become available that make Bayesian modeling relatively easy.
Bayesian tools have several important advantages over
frequentist methods, but they require some very specific
background knowledge.
My goal in this course is to try to provide that background
knowledge.
Note: I will teach a more detailed (2-week) course in ESSLLI in
Barcelona, Aug 3-14, 2015.
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3. Lecture 1: Review of Linear Mixed Models
Introduction
Motivating this course
In this introductory lecture, my goals are to
1 motivate you look at Bayesian Linear Mixed Models as an
alternative to using frequentist methods.
2 make sure we are all on the same page regarding the moving
parts of a linear mixed model.
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4. Lecture 1: Review of Linear Mixed Models
Preliminaries
Prerequisites
1 Familiarity with fitting standard LMMs such as:
lmer(rt~cond+(1+cond|subj)+(1+cond|item),dat)
2 Basic knowledge of R.
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5. Lecture 1: Review of Linear Mixed Models
Preliminaries
A bit about my background
1999: Discovered the word“ANOVA”in a chance conversation
with a psycholinguist.
1999: Did my first self-paced reading experiment. Fit a
repeated measures ANOVA.
2000: Went to statisticans at Ohio State’ statistical
consulting unit, and they said: “why are you fitting ANOVA?
You need linear mixed models.”
2000-2010: Kept fitting and publishing linear mixed models.
2010: Realized I didn’t really know what I was doing, and
started a part-time MSc in Statistics at Sheffield’s School of
Math and Statistics (2011-2015).
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6. Lecture 1: Review of Linear Mixed Models
Repeated measures data
Linear mixed models
Example: Gibson and Wu data, Language and Cognitive Processes, 2012
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7. Lecture 1: Review of Linear Mixed Models
Repeated measures data
Linear mixed models
Example: Gibson and Wu data, Language and Cognitive Processes, 2012
Subject vs object relative clauses in Chinese, self-paced
reading.
The critical region is the head noun.
The goal is to find out whether SRs are harder to process than
ORs at the head noun.
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8. Lecture 1: Review of Linear Mixed Models
Repeated measures data
Linear mixed models
Example: Gibson and Wu 2012 data
> head(data[,c(1,2,3,4,7,10,11)])
subj item type pos rt rrt x
7 1 13 obj-ext 6 1140 -0.8771930 0.5
20 1 6 subj-ext 6 1197 -0.8354219 -0.5
32 1 5 obj-ext 6 756 -1.3227513 0.5
44 1 9 obj-ext 6 643 -1.5552100 0.5
60 1 14 subj-ext 6 860 -1.1627907 -0.5
73 1 4 subj-ext 6 868 -1.1520737 -0.5
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9. Lecture 1: Review of Linear Mixed Models
Repeated measures data
lme4 model of Gibson and Wu data
Crossed varying intercepts and slopes model, with correlation
This is the type of“maximal”model that most people fit nowadays
(citing Barr et al 2012):
> m1 <- lmer(rrt~x+(1+x|subj)+(1+x|item),
+ subset(data,region=="headnoun"))
I will now show two major (related) problems that occur with the
small datasets we usually have in psycholinguistics:
The correlation estimates either lead to degenerate variance
covariance matrices, and/or
The correlation estimates are wild estimates that have no
bearing with reality.
[This is not a failing of lmer, but rather that the user is
demanding too much of lmer.]
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10. Lecture 1: Review of Linear Mixed Models
Repeated measures data
lme4 model of Gibson and Wu data
Typical data analysis: Crossed varying intercepts and slopes model, with correlation
> summary(m1)
Linear mixed model fit by REML [lmerMod]
Formula: rrt ~ x + (1 + x | subj) + (1 + x | item)
Data: subset(data, region == "headnoun")
REML criterion at convergence: 1595.6
Scaled residuals:
Min 1Q Median 3Q Max
-2.5441 -0.6430 -0.1237 0.5996 3.2501
Random effects:
Groups Name Variance Std.Dev. Corr
subj (Intercept) 0.371228 0.60928
x 0.053241 0.23074 -0.51
item (Intercept) 0.110034 0.33171
x 0.009218 0.09601 1.00
Residual 0.891577 0.94423
Number of obs: 547, groups: subj, 37; item, 15
Fixed effects:
Estimate Std. Error t value
(Intercept) -2.67151 0.13793 -19.369
x -0.07758 0.09289 -0.835
Correlation of Fixed Effects:
(Intr)
x 0.012
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11. Lecture 1: Review of Linear Mixed Models
Repeated measures data
The“best”model
The way to decide on the“best”model is to find the simplest
model using the Generalized Likelihood Ratio Test (Pinheiro and
Bates 2000). Here, this is the varying intercepts model, not the
maximal model.
> m1<- lmer(rrt~x+(1+x|subj)+(1+x|item),
+ headnoun)
> m1a<- lmer(rrt~x+(1|subj)+(1|item),
+ headnoun)
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12. Lecture 1: Review of Linear Mixed Models
Repeated measures data
The“best”model
> anova(m1,m1a)
Data: headnoun
Models:
m1a: rrt ~ x + (1 | subj) + (1 | item)
m1: rrt ~ x + (1 + x | subj) + (1 + x | item)
Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
m1a 5 1603.5 1625.0 -796.76 1593.5
m1 9 1608.5 1647.3 -795.27 1590.5 2.9742 4 0.5622
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13. Lecture 1: Review of Linear Mixed Models
Repeated measures data
How meaningful were the lmer estimates of correlations in
the maximal model m1?
Simulated data
Here, we simulate data with the same structure, sample size, and
parameter values as the Gibson and Wu data, except that we
assume that the correlations are 0.6. Then we analyze the data
using lmer (maximal model). Can lmer recover the
correlations?
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14. Lecture 1: Review of Linear Mixed Models
Repeated measures data
How meaningful were the lmer estimates of correlations in
the maximal model m1?
Simulated data
We define a function called new.df that generates data similar to
the Gibson and Wu data-set. For code, see accompanying .R file.
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15. Lecture 1: Review of Linear Mixed Models
Repeated measures data
How meaningful were the lmer estimates of correlations in
the maximal model m1?
Simulated data
Next, we write a function that generates data for us repeatedly
with the following specifications: sample size for subjects and
items, and some correlation between subject intercept and slope,
and item intercept and slope.
> gendata<-function(subjects=37,items=15){
+ dat<-new.df(nsubj=subjects,nitems=items,
+ rho.u=0.6,rho.w=0.6)
+ dat <- dat[[1]]
+ dat<-dat[,c(1,2,3,9)]
+ dat$x<-ifelse(dat$cond==1,-0.5,0.5)
+
+ return(dat)
+ }
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16. Lecture 1: Review of Linear Mixed Models
Repeated measures data
How meaningful were the lmer estimates of correlations in
the maximal model m1?
Simulated data
Set number of simulations:
> nsim<-100
Next, we generate simulated data 100 times, and then store the
estimated subject and item level correlations in the random effects,
and plot their distributions.
We do this for two settings: Gibson and Wu sample sizes (37
subjects, 15 items), and 50 subjects and 30 items.
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17. Lecture 1: Review of Linear Mixed Models
Repeated measures data
How meaningful were the lmer estimates of correlations in
the maximal model m1?
Simulated data
37 subjects and 15 items
> library(lme4)
> subjcorr<-rep(NA,nsim)
> itemcorr<-rep(NA,nsim)
> for(i in 1:nsim){
+ dat<-gendata()
+ m3<-lmer(rt~x+(1+x|subj)+(1+x|item),dat)
+ subjcorr[i]<-attr(VarCorr(m3)$subj,"correlation")[1,2]
+ itemcorr[i]<-attr(VarCorr(m3)$item,"correlation")[1,2]
+ }
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18. Lecture 1: Review of Linear Mixed Models
Repeated measures data
How meaningful were the lmer estimates of correlations in
the maximal model m1?
Simulated data
Distribution of subj. corr.
ρ^
u
Density
−0.2 0.2 0.6 1.0
0.00.51.01.5
Distribution of item corr.
ρ^
w
Density
−1.0 0.0 0.5 1.0
0.00.51.01.5
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19. Lecture 1: Review of Linear Mixed Models
Repeated measures data
How meaningful were the lmer estimates of correlations in
the maximal model m1?
Simulated data
50 subjects and 30 items
> subjcorr<-rep(NA,nsim)
> itemcorr<-rep(NA,nsim)
> for(i in 1:nsim){
+ #print(i)
+ dat<-gendata(subjects=50,items=30)
+ m3<-lmer(rt~x+(1+x|subj)+(1+x|item),dat)
+ subjcorr[i]<-attr(VarCorr(m3)$subj,"correlation")[1,2]
+ itemcorr[i]<-attr(VarCorr(m3)$item,"correlation")[1,2]
+ }
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20. Lecture 1: Review of Linear Mixed Models
Repeated measures data
How meaningful were the lmer estimates of correlations in
the maximal model m1?
Simulated data
Distribution of subj. corr.
ρ^
u
Density
0.2 0.6 1.0
0.01.02.0
Distribution of item corr.
ρ^
w
Density
0.0 0.4 0.8
0.01.02.0
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21. Lecture 1: Review of Linear Mixed Models
Repeated measures data
How meaningful were the lmer estimates of correlations in
the maximal model m1?
Simulated data
Conclusion:
1 It seems that lmer can estimate the correlation parameters just
in case sample size for items and subjects is“large enough”
(this can be established using simulation, as done above).
2 Barr et al’s recommendation to fit a maximal model makes
sense as a general rule only if it’s already clear that we have
enough data to estimate all the variance components and
parameters.
3 In my experience, that is rarely the case at least in
psycholinguistics, especially when we go to more complex
designs than a simple two-condition study.
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22. Lecture 1: Review of Linear Mixed Models
Repeated measures data
Keep it maximal?
Gelman and Hill (2007, p. 549) make a more tempered
recommendation than Barr et al:
Don’t get hung up on whether a coefficient“should”vary
by group. Just allow it to vary in the model, and then, if
the estimated scale of variation is small . . . , maybe you
can ignore it if that would be more convenient. Practical
concerns sometimes limit the feasible complexity of a
model–for example, we might fit a varying-intercept
model first, then allow slopes to vary, then add
group-level predictors, and so forth. Generally, however,
it is only the difficulties of fitting and, especially,
understanding the models that keeps us from adding even
more complexity, more varying coefficients, and more
interactions.
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23. Lecture 1: Review of Linear Mixed Models
Why fit Bayesian LMMs?
Advantages of fitting a Bayesian LMM
1 For such data, there can be situations where you really need
to or want to fit full variance-covariance matrices for random
effects. Bayesian LMMs will let you fit them even in cases
where lmer would fail to converge or return nonsensical
estimates (due to too little data).
The way we will set them up, Bayesian LMMs will typically
underestimate correlation, unless there is enough data.
2 A direct answer to the research question can be obtained by
examining the posterior distribution given data.
3 We can avoid the traditional hard binary decision associated
with frequentist methods: p < 0.05 implies reject null, and
p > 0.05 implies“accept”null. We are more interested in
quantifying our uncertainty about the scientific claim.
4 Prior knowledge can be included in the model.
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24. Lecture 1: Review of Linear Mixed Models
Why fit Bayesian LMMs?
Disadvantages of doing a Bayesian analysis
You have to invest effort into specifying a model; unlike lmer,
which involves a single line of code, JAGS and Stan model
specifications can extend to 20-30 lines.
A lot of decisions have to be made.
There is a steep learning curve; you have to know a bit about
probability distributions, MCMC methods, and of course
Bayes’ Theorem.
It takes much more time to fit a complicated model in a
Bayesian setting than with lmer.
But I will try to demonstrate to you in this course that it’s worth
the effort, especially when you don’t have a lot of data (usually the
case in psycholinguistics).
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25. Lecture 1: Review of Linear Mixed Models
Brief review of linear (mixed) models
Linear models
yi
↑
response
=
parameter
↓
β0 +
parameter
↓
β1xi
↑
predictor
+ εi
↑
error
(1)
where
εi is the residual error, assumed to be normally distributed:
εi ∼ N(0,σ2).
Each response yi (i ranging from 1 to I) is independently and
identically distributed as yi ∼ N(β0 +β1xi ,σ2).
Point values for parameters: β0 and β1 are the parameters
to be estimated. In the frequentist setting, these are point
values, they have no distribution.
Hypothesis test: Usually, β1 is the parameter of interest; in
the frequentist setting, we test the null hypothesis that β1 = 0.
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26. Lecture 1: Review of Linear Mixed Models
Brief review of linear (mixed) models
Linear models and repeated measures data
Repeated measures data
Linear mixed models are useful for correlated data (e.g.,
repeated measures) where the responses y are not
independently distributed.
A key difference from linear models is that the intercept
and/or slope vary by subject j = 1,...,J (and possibly also by
item k = 1,...,K):
yi
↑
response
=
varying intercepts
↓
[β0 +u0j +w0k] +
varying slopes
↓
[β1 +u1j +w1k]xi
↑
predictor
+ εi
↑
error
(2)
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27. Lecture 1: Review of Linear Mixed Models
Brief review of linear (mixed) models
Linear models and repeated measures data
Unpacking the lme4 model
Crossed varying intercepts and slopes model, with correlation
yi
↑
response
=
varying intercepts
↓
[β0 +u0j +w0k] +
varying slopes
↓
[β1 +u1j +w1k]xi
↑
predictor
+ εi
↑
error
(3)
This is the“maximal”model we saw earlier:
> m1 <- lmer(rrt~x+(1+x|subj)+(1+x|item),
+ headnoun)
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28. Lecture 1: Review of Linear Mixed Models
Brief review of linear (mixed) models
Linear models and repeated measures data
Unpacking the lme4 model
Crossed varying intercepts and slopes model, with correlation
> summary(m1)
Linear mixed model fit by REML [lmerMod]
Formula: rrt ~ x + (1 + x | subj) + (1 + x | item)
Data: headnoun
REML criterion at convergence: 1595.6
Scaled residuals:
Min 1Q Median 3Q Max
-2.5441 -0.6430 -0.1237 0.5996 3.2501
Random effects:
Groups Name Variance Std.Dev. Corr
subj (Intercept) 0.371228 0.60928
x 0.053241 0.23074 -0.51
item (Intercept) 0.110034 0.33171
x 0.009218 0.09601 1.00
Residual 0.891577 0.94423
Number of obs: 547, groups: subj, 37; item, 15
Fixed effects:
Estimate Std. Error t value
(Intercept) -2.67151 0.13793 -19.369
x -0.07758 0.09289 -0.835
Correlation of Fixed Effects:
(Intr)
x 0.012
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29. Lecture 1: Review of Linear Mixed Models
Brief review of linear (mixed) models
Linear models and repeated measures data
Unpacking the lme4 model
Crossed varying intercepts and slopes model, with correlation
rrti = (β0 +u0j +w0k)+(β1 +u1j +w1k)xi +εi (4)
1 i = 1,...,547 data points; j = 1,...,37 items; k = 1,...,15 data
points
2 xi is coded −0.5 (SR) and 0.5 (OR).
3 εi ∼ N(0,σ2).
4 u0j ∼ N(0,σ0j ) and u1j ∼ N(0,σ1j ).
5 w0k ∼ N(0,σ0k) and w1k ∼ N(0,σ1k).
with a multivariate normal distribution for the varying slopes and
intercepts
u0j
u1j
∼ N
0
0
,Σu
w0k
w1k
∼ N
0
0
,Σw (5)
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30. Lecture 1: Review of Linear Mixed Models
Brief review of linear (mixed) models
Linear models and repeated measures data
The variance components associated with subjects
Random effects:
Groups Name Variance Std.Dev. Corr
subj (Intercept) 0.37 0.61
so 0.05 0.23 -0.51
Σu =
σ2
u0 ρu σu0σu1
ρu σu0σu1 σ2
u1
=
0.612 −.51×0.61×0.23
−.51×0.61×0.23 0.232
(6)
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31. Lecture 1: Review of Linear Mixed Models
Brief review of linear (mixed) models
Linear models and repeated measures data
The variance components associated with items
Random effects:
Groups Name Variance Std.Dev. Corr
item (Intercept) 0.11 0.33
so 0.01 0.10 1.00
Note the by items intercept-slope correlation of +1.00.
Σw =
σ2
w0 ρw σw0σw1
ρw σw0σw1 σ2
w1
=
0.332 1×0.33×0.10
1×0.33×0.10 0.102
(7)
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32. Lecture 1: Review of Linear Mixed Models
Brief review of linear (mixed) models
Linear models and repeated measures data
The variance components of the“maximal”linear mixed
model
RTi = β0 +u0j +w0k +(β1 +u1j +w1k)xi +εi (8)
u0j
u1j
∼ N
0
0
,Σu
w0k
w1k
∼ N
0
0
,Σw (9)
εi ∼ N(0,σ2
) (10)
The parameters are β0,β1,Σu,Σw ,σ. Each of the matrices Σ has
three parameters. So we have 9 parameters.
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33. Lecture 1: Review of Linear Mixed Models
Brief review of linear (mixed) models
Linear models and repeated measures data
Summary so far
Linear mixed models allow us to take all relevant variance
components into account; LMMs allow us to describe how the
data were generated.
However, maximal models should not be fit blindly, especially
when there is not enough data to estimate parameters.
For small datasets we often see degenerate variance
covariance estimates (with correlation ±1). Many
psycholinguists ignore this degeneracy.
If one cares about the correlation, one should not ignore the
degeneracy.
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34. Lecture 1: Review of Linear Mixed Models
Frequentist vs Bayesian methods
The frequentist approach
1 In the frequentist setting, we start with a dependent measure
y, for which we assume a probability model.
2 In the above example, we have reading time data, rt, which
we assume is generated from a normal distribution with some
mean µ and variance σ2; we write this rt ∼ N(µ,σ2).
3 Given a particular set of parameter values µ and σ2, we could
state the probability distribution of rt given the parameters.
We can write this as p(rt | µ,σ2).
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35. Lecture 1: Review of Linear Mixed Models
Frequentist vs Bayesian methods
The frequentist approach
1 In reality, we know neither µ nor σ2. The goal of fitting a
model to such data is to estimate the two parameters, and
then to draw inferences about what the true value of µ is.
2 The frequentist method relies on the fact that, under repeated
sampling and with a large enough sample size, the sampling
distribution the sample mean ¯X is distributed as N(µ,σ2/n).
3 The standard method is to use the sample mean ¯x as an
estimate of µ and given a large enough sample size n, we can
compute an approximate 95% confidence interval
¯x ±2×(ˆσ2/n).
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36. Lecture 1: Review of Linear Mixed Models
Frequentist vs Bayesian methods
The frequentist approach
The 95% confidence interval has a slightly complicated
interpretation:
If we were to repeatedly carry out the experiment and compute a
confidence interval each time using the above procedure, 95% of
those confidence intervals would contain the true parameter value
µ (assuming, of course, that all our model assumptions are
satisfied).
The particular confidence interval we calculated for our particular
sample does not give us a range such that we are 95% certain that
the true µ lies within it, although this is how most users of
statistics seem to (mis)interpret the confidence interval.
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37. Lecture 1: Review of Linear Mixed Models
Frequentist vs Bayesian methods
The 95% CI
0 20 40 60 80 100
5658606264
95% CIs in 100 repeated samples
Scores
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38. Lecture 1: Review of Linear Mixed Models
Frequentist vs Bayesian methods
The Bayesian approach
1 The Bayesian approach starts with a probability model that
defines our prior belief about the possible values that the
parameter µ and σ2 might have.
2 This probability model expresses what we know so far about
these two parameters (we may not know much, but in
practical situations, it is not the case that we don’t know
anything about their possible values).
3 Given this prior distribution, the probability model p(y | µ,σ2)
and the data y allow us to compute the probability
distribution of the parameters given the data, p(µ,σ2 | y).
4 This probability distribution, called the posterior distribution,
is what we use for inference.
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39. Lecture 1: Review of Linear Mixed Models
Frequentist vs Bayesian methods
The Bayesian approach
1 Unlike the 95% confidence interval, we can define a 95%
credible interval that represents the range within which we are
95% certain that the true value of the parameter lies, given
the data at hand.
2 Note that in the frequentist setting, the parameters are point
values: µ is assumed to have a particular value in nature.
3 In the Bayesian setting, µ is a random variable with a
probability distribution; it has a mean, but there is also some
uncertainty associated with its true value.
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40. Lecture 1: Review of Linear Mixed Models
Frequentist vs Bayesian methods
The Bayesian approach
Bayes’ theorem makes it possible to derive the posterior distribution
given the prior and the data. The conditional probability rule in
probability theory (see Kerns) is that the joint distribution of two
random variables p(θ,y) is equal to p(θ | y)p(y). It follows that:
p(θ,y) =p(θ | y)p(y)
=p(y,θ) (because p(θ,y) = p(y,θ))
=p(y | θ)p(θ).
(11)
The first and third lines in the equalities above imply that
p(θ | y)p(y) = p(y | θ)p(θ). (12)
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41. Lecture 1: Review of Linear Mixed Models
Frequentist vs Bayesian methods
The Bayesian approach
Dividing both sides by p(y) we get:
p(θ | y) =
p(y | θ)p(θ)
p(y)
(13)
The term p(y | θ) is the probability of the data given θ. If we treat
this as a function of θ, we have the likelihood function.
Since p(θ | y) is the posterior distribution of θ given y, and
p(y | θ) the likelihood, and p(θ) the prior, the following
relationship is established:
Posterior ∝ Likelihood×Prior (14)
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42. Lecture 1: Review of Linear Mixed Models
Frequentist vs Bayesian methods
The Bayesian approach
Posterior ∝ Likelihood×Prior (15)
We ignore the denominator p(y) here because it only serves as a
normalizing constant that renders the left-hand side (the posterior)
a probability distribution.
The above is Bayes’ theorem, and is the basis for determining the
posterior distribution given a prior and the likelihood.
The rest of this course simply unpacks this idea.
Next week, we will look at some simple examples of the application
of Bayes’ Theorem.
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