Introduction to statistics ii
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This document discusses approaches for analyzing microarray gene expression data with few observations. It suggests assuming a canonical distribution based on prior knowledge, like the log-normal distribution often seen in this type of data. Parameters of this distribution can then be estimated from the available observations using techniques like estimating the mean and variance. A t-statistic can be used to determine if two gene conditions have the same distribution parameters and thus identify differentially expressed genes. Quality control plots of the data are recommended to validate distribution assumptions.
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Multiplying fractions
Multiplying fractions involves multiplying the numerators and denominators, then simplifying the resulting fraction. When multiplying a fraction by a whole number, change the whole number to an equivalent fraction. It is often easier to simplify fractions before multiplying them. To multiply mixed numbers, first convert them to improper fractions, then multiply the numerators and denominators and simplify. Practice problems are provided for the learner to try.
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2) Multiply the denominators together to get the new denominator
3) Simplify the resulting fraction if possible by cancelling common factors in the numerator and denominator.
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This document provides a step-by-step explanation of the long division process using a family metaphor. It describes the five steps of long division as represented by different family members: Dad (divide), Mom (multiply), Sister (subtract), Brother (bring down), and Rover (repeat or remainder). It then works through an example problem of 4947 divided by 2 to demonstrate each step. In the end, the student is congratulated on completing the long division problem correctly with a remainder of 1.
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Multiplying fractions
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The document discusses multiplying decimals. It provides steps for multiplying decimals which are: 1) write the problem vertically, 2) ignore the decimal points and multiply, 3) determine where the decimal point goes in the product by counting how many places the decimal point has moved. Examples are given showing the application of these steps. The key points are that decimals are multiplied like whole numbers and the decimal point is placed in the product by counting how many places it moves to the left.
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Introduction to statistics
The document discusses modeling unknown distributions with few observations by assuming they come from a canonical distribution family with unknown parameters. It explains that canonical distributions like the normal, binomial, and Poisson distributions can describe data using just a few parameters. The document then discusses how to estimate these unknown parameters, like the mean and variance, from the limited observations. Specifically, it addresses estimating the mean using the sample mean as an unbiased estimate, and estimating the variance, noting the need for an unbiased estimate that becomes more precise as the number of observations increases.
T Test For Two Independent Samples
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A cognição neural é o estudo fascinante da maneira como o cérebro processa informações e gera pensamentos, emoções e comportamentos. Nos últimos anos, avanços significativos em tecnologias de neuroimagem e neurociência computacional têm nos permitido investigar os intricados processos neurais subjacentes à cognição com uma precisão sem precedentes.
Os neurônios, as células fundamentais do sistema nervoso, formam redes complexas que se comunicam por meio de impulsos elétricos e neurotransmissores. Essas redes neurais são altamente adaptáveis e capazes de reorganização dinâmica, o que permite ao cérebro aprender, lembrar e tomar decisões.
A cognição neural examina como essas redes neurais se organizam e interagem para processar informações sensoriais, resolver problemas, planejar ações e criar memórias. Ao estudar padrões de atividade cerebral e conectividade funcional, os cientistas podem desvendar os mecanismos subjacentes a funções cognitivas complexas, como atenção, linguagem, memória e tomada de decisão.
Além disso, a cognição neural também investiga como fatores genéticos, experiências de vida e transtornos neurológicos influenciam o funcionamento do cérebro e a cognição. Com uma compreensão mais profunda desses processos, esperamos abrir novas oportunidades para o desenvolvimento de tratamentos inovadores para distúrbios cerebrais e para aprimorar a cognição e o bem-estar humano.
Microarray Analysis
Microarrays allow researchers to analyze gene expression across thousands of genes simultaneously. DNA probes are arrayed on a small glass or nylon slide, and labeled mRNA from samples is hybridized to the probes. Fluorescent scanning detects which genes are expressed. Data analysis includes normalization, distance metrics, clustering, and visualization to group genes with similar expression profiles and identify patterns of co-regulated genes. Microarrays enable functional genomics studies of development, disease, response to drugs or environmental factors, and more.
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DESeq2 is used to analyze differential expression from RNA-seq count data using a generalized linear model. It models counts using a gamma-Poisson distribution and estimates dispersion using empirical Bayes shrinkage. Key steps include normalizing counts, estimating dispersion, fitting the linear model, and using Wald and likelihood ratio tests to identify differentially expressed genes while controlling the false discovery rate. Results can be explored using plots of p-values, mean-variance trends, ordination plots, and heatmaps to visualize sample relationships and differentially expressed genes.
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This document discusses suffix trees, including their definition, important contributors, construction, implementation, and applications. Key points:
- A suffix tree is a compressed trie representing all suffixes of a string. It allows efficient pattern matching in linear time.
- Important contributors include Weiner (1973), McCreight (1976), Ukkonen (1995), and Farach (1997) who developed faster construction algorithms.
- There are multiple ways to implement a suffix tree, including using sibling lists, hash maps, balanced search trees, or sorted arrays. External memory may be needed for very large trees.
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Packet forwarding in wan.46
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Trie tree
A trie, or prefix tree, is an ordered tree data structure used to store associative arrays where keys are usually strings. It shows a trie with words like "tree", "trie", "algo", and "assoc". Functions of a trie include counting prefixes and matching words exactly. It is used to check consistency of phone numbers by ensuring no number is a prefix of another in the data set.
Suffix Tree and Suffix Array
Suffix trees and suffix arrays are data structures used to solve problems related to string matching and text indexing in an efficient manner. Suffix trees allow finding patterns in text in O(m) time where m is the pattern length, by traversing the tree. Suffix arrays store suffixes in sorted order and allow pattern searching in O(m+logn) time where n is text length. Both structures take O(n) time and space to construct where n is text length. They find applications in bioinformatics, data compression, and other string algorithms.
Data structure tries
Tries are a data structure for storing strings that allow for fast pattern matching. A trie is a tree where each edge represents a character and each path from the root node to a leaf spells out a key. Standard tries insert strings by adding nodes for each character. Compressed tries reduce redundant nodes by compressing chains. Suffix tries store all suffixes of a text in a compressed trie to enable quick string queries. Tries support faster insertion and lookup compared to hash tables, with no collisions between keys.
Lec18
The document discusses different types of tries data structures and their applications. It describes standard tries, compressed tries, and suffix tries. Standard tries support operations like finding, inserting, and removing strings in time proportional to the string length and alphabet size. Compressed tries reduce space by compressing chains of redundant nodes. Suffix tries store all suffixes of a text in linear space and support fast pattern matching queries in time proportional to the pattern length plus the number of matches. The document provides examples of using tries for text processing, web search indexing, internet routing, and other applications.
Fundamentals
The document discusses the fundamentals of computer systems, including definitions, components, and how they work together. It defines a computer as an electronic device that accepts input, processes it, and provides output. The key components are the input and output units, memory unit, CPU (consisting of the ALU and control unit), and secondary storage. The input and output units send and receive data, the memory unit temporarily stores programs and data, the CPU performs arithmetic/logical operations and coordinates tasks, and secondary storage provides long-term storage. Together these components work to accept user input, process the data, and provide the results.
Tries - Tree Based Structures for Strings
Trie (aka radix tree or prefix tree), is an ordered tree data structure where the keys are usually strings. Tries have tremendous applications from all sorts of things like dictionary to
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Application of tries
Tries are tree data structures that are useful for storing and retrieving data with associative keys like strings, especially variable-length strings. Tries have applications in auto-complete suggestions, longest prefix matching for IP routing, spell checking, phone book contact searching, and predictive text entry systems like T9. The keys are stored based on prefix matching in a trie, allowing fast lookup and retrieval of all keys that start with a given prefix in near-constant time.
Digital Search Tree
This presentation is about digital search tree. Here we discuss about implementation, application and algorithm of digital search tree.
Trie Data Structure
Trie is an efficient data structure for storing and retrieving strings. It stores strings in a tree structure, with each node representing a character. Common operations on a trie like insertion, deletion and searching of strings can be performed in O(M) time where M is the length of the string. The document then provides details on the node structure used to implement a trie, along with pseudocode for inserting strings like "Apple" and "Army" into an empty trie.
Multi ways trees
1. A multi-way search tree allows nodes to have up to m children, where keys in each node are ordered and divide the search space.
2. B-trees are a generalization of binary search trees where all leaves are at the same depth and internal nodes have at least m/2 children.
3. Searching and inserting keys in a B-tree starts at the root and proceeds by comparing keys to guide traversal to the appropriate child node. Insertion may require splitting full nodes to balance the tree.
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T Test For Two Independent Samples
An independent t-test is used to compare the means of two independent groups on a continuous dependent variable. It tests if there is a statistically significant difference between the population means of the two groups. The test assumes the groups are independent, the dependent variable is normally distributed for each group, and the groups have equal variances. To perform the test, the researcher states the hypotheses, sets an alpha level, calculates the t-statistic and degrees of freedom, and determines whether to reject or fail to reject the null hypothesis by comparing the t-statistic to the critical value.
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DNA microarrays, also known as DNA chips, allow simultaneous measurement of gene expression levels for every gene in a genome. Microarrays detect messenger RNA (mRNA) or complementary DNA (cDNA). They are manufactured by amplifying individual genes using PCR and spotting them on a medium like a glass slide. When fluorescently labeled cDNA from two samples are hybridized to the array, the level of fluorescence indicates gene expression differences between the samples. Microarray data is analyzed to identify genes that are up-regulated or down-regulated under different experimental conditions.
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A cognição neural examina como essas redes neurais se organizam e interagem para processar informações sensoriais, resolver problemas, planejar ações e criar memórias. Ao estudar padrões de atividade cerebral e conectividade funcional, os cientistas podem desvendar os mecanismos subjacentes a funções cognitivas complexas, como atenção, linguagem, memória e tomada de decisão.
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みなさんこんにちはこれ何文字まで入るの？40文字以下不可とか本当に意味わからないけどこれ限界文字数書いてないからマジでやばい文字数いけるんじゃないの？えこ...
ここ3000字までしか入らないけどタイトルの方がたくさん文字入ると思います。
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みなさんこんにちはこれ何文字まで入るの？40文字以下不可とか本当に意味わからないけどこれ限界文字数書いてないからマジでやばい文字数いけるんじゃないの？えこ...
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Introduction to statistics ii
- 1. Statistics for Microarray Data
- 2. Background μ, σ2 • Few observations made by a black box • What is the distribution behind the black box? • E.g., with what probability will it output a number bigger than 5?
- 3. Approach • Easy to determine with many observations • With few observations.. • Assume a canonical distribution based on prior knowledge • Determine parameters of this distribution using the observations, e.g., mean, variance
- 5. Estimating the variance σ2 Chi-Square if the original distribution was Normal
- 6. Microarray Data • Many genes, 25000 • 2 conditions (or more), many replicates within each condition • Which genes are differentially expressed between the two conditions?
- 7. More Specifically • For a particular gene – Each condition is a black box – Say 3 observations from each black box • Do both black boxes have the same distribution? – Assume same canonical distribution – Do both have the same parameters?
- 8. Which Canonical Distribution • Use data with many replicates • 418.0294, 295.8019, 272.1220, 315.2978, 294.2242, 379.8320, 392.1817, 450.4758, 335.8242, 265.2478, 196.6982, 289.6532, 274.4035, 246.6807, 254.8710, 165.9416, 281.9463, 246.6434, 259.0019, 242.1968 • Distribution??
- 9. What is a QQ Plot
- 10. Distribution of log raw intensities across genes on a single array
- 11. The QQ plot of log scale intensities (i.e., actual vs simulated from normal)
- 12. QQ Plot against a Normal Distribution • 10 + 10 replicates in two groups • Single group QQ plot • Combined 2 groups QQ plot • Combined log-scale QQ plot Shapiro- Wilk Test
- 13. Which Canonical Distribution • Assume log normal distribution
- 14. Benford’s Law • Frequency distribution of first significant digit Pr(d<=x<d+1 )= log10(1+d)-log10(d), log10(x) is uniformly distributed in [0,1]
- 15. Differential Expression μ1,σ12 μ2,σ22 Group 1 Group 2 Is μ1= μ2? σ1 = σ2 ? Is variance a function of mean?
- 16. SD increases linearly with Mean SD vs Mean across 3 replicates plotted for all genes
- 17. SD is flat now, except for very low values Another reason to work on the log scale SD vs Mean across 3 replicates computed for all genes after log-transformation
- 18. Differential Expression μ1,σ12 μ2,σ22 Group 1 Group 2 Is μ1= μ2? σ1 = σ2 ? Sort-of YES
- 19. The T-Statistic
- 20. The T-Statistic
- 21. The T-Statistic
- 22. The T-Statistic Flattened Normal or T- Distribution
- 23. A Problem
- 24. The curve fit here may be a better estimate Lots of false positives can Not much be avoided difference here here SD vs Mean across 3 replicates computed for all genes after log-transformattion
- 25. Thank You