Detection theory involves making decisions based on observations. There are two main hypotheses: the null hypothesis (H0, e.g. target is absent) and alternative hypothesis (H1, e.g. target is present). A likelihood ratio test compares the likelihood ratio to a threshold to determine which hypothesis is chosen. There are two types of errors: Type I errors (false alarms) occur when H0 is rejected when it is true, and Type II errors (misses) occur when H1 is rejected when it is true. Bayesian decision theory aims to minimize the average risk by choosing the hypothesis with minimum cost based on prior probabilities and costs of different errors.
This document presents research on using the Homotopy Analysis Method (HAM) to solve a time-fractional diffusion equation with a moving boundary condition. HAM is a semi-analytic technique used to solve nonlinear differential equations by generating a convergent series solution. The author applies HAM to obtain approximate analytic solutions for the concentration of a drug in a matrix and the diffusion front over time. Comparisons with exact solutions show good agreement for different parameter values. The author concludes that HAM can accurately predict drug distribution and the diffusion front in this problem.
This chapter discusses hypothesis testing based on a single sample. It covers the basics of hypotheses, test statistics, types of errors, and significance levels. Specific tests are presented for the population mean (both normal and large sample cases) and population proportion (large sample tests and small sample tests). The chapter also discusses p-values and selecting an appropriate test procedure based on the situation. Key considerations include statistical versus practical significance and the likelihood ratio principle for comparing tests.
This document provides an introduction to random variables. It defines random variables as functions that assign real numbers to outcomes of an experiment. Random variables can be either discrete or continuous depending on whether their possible values are countable or uncountable. The document also defines probability mass functions (pmf) which describe the probabilities of discrete random variables taking on particular values. Expectation is introduced as a way to summarize random variables using a single number by taking a weighted average of all possible outcomes.
The document provides an introduction to probability theory, including definitions of key concepts like sample space, events, probabilities of events, conditional probabilities, independent events, and Bayes' formula. It gives examples of sample spaces for experiments like coin flips and dice rolls. It explains that the probability of an event is a number between 0 and 1 that satisfies three conditions. It also describes how to calculate probabilities of unions, intersections, and complements of events.
- The document proposes a new class of coherent risk measures called spectral measures of risk. These measures are generated by expanding on expected shortfall measures using a risk aversion function φ.
- The risk aversion function φ assigns weights to different confidence levels of portfolio losses, allowing for a more flexible representation of subjective risk preferences than expected shortfall alone.
- φ must be positive, decreasing, and normalized to define a coherent spectral risk measure. This provides an intuitive interpretation of coherence as assigning larger weights to worse outcomes. φ allows investors to express their individual risk aversion profile.
This document discusses quantile estimation techniques, including parametric, semiparametric, and nonparametric approaches. Parametric estimation assumes a distribution like Gaussian and estimates quantiles based on parameters of that distribution. Semiparametric estimation uses extreme value theory to model upper tails with a generalized Pareto distribution. Nonparametric estimation estimates quantiles directly from the data without assuming a particular distribution. The document presents several techniques for quantile estimation and compares their performance.
The document provides information about probability and statistics concepts including:
1) Mathematical, statistical, and axiomatic definitions of probability are given along with examples of mutually exclusive, equally likely, and independent events.
2) Laws of probability such as addition law, multiplication law, and total probability theorem are defined and formulas are provided.
3) Concepts of random variables, discrete and continuous random variables, probability mass functions, probability density functions, and expected value are introduced.
This document presents research on using the Homotopy Analysis Method (HAM) to solve a time-fractional diffusion equation with a moving boundary condition. HAM is a semi-analytic technique used to solve nonlinear differential equations by generating a convergent series solution. The author applies HAM to obtain approximate analytic solutions for the concentration of a drug in a matrix and the diffusion front over time. Comparisons with exact solutions show good agreement for different parameter values. The author concludes that HAM can accurately predict drug distribution and the diffusion front in this problem.
This chapter discusses hypothesis testing based on a single sample. It covers the basics of hypotheses, test statistics, types of errors, and significance levels. Specific tests are presented for the population mean (both normal and large sample cases) and population proportion (large sample tests and small sample tests). The chapter also discusses p-values and selecting an appropriate test procedure based on the situation. Key considerations include statistical versus practical significance and the likelihood ratio principle for comparing tests.
This document provides an introduction to random variables. It defines random variables as functions that assign real numbers to outcomes of an experiment. Random variables can be either discrete or continuous depending on whether their possible values are countable or uncountable. The document also defines probability mass functions (pmf) which describe the probabilities of discrete random variables taking on particular values. Expectation is introduced as a way to summarize random variables using a single number by taking a weighted average of all possible outcomes.
The document provides an introduction to probability theory, including definitions of key concepts like sample space, events, probabilities of events, conditional probabilities, independent events, and Bayes' formula. It gives examples of sample spaces for experiments like coin flips and dice rolls. It explains that the probability of an event is a number between 0 and 1 that satisfies three conditions. It also describes how to calculate probabilities of unions, intersections, and complements of events.
- The document proposes a new class of coherent risk measures called spectral measures of risk. These measures are generated by expanding on expected shortfall measures using a risk aversion function φ.
- The risk aversion function φ assigns weights to different confidence levels of portfolio losses, allowing for a more flexible representation of subjective risk preferences than expected shortfall alone.
- φ must be positive, decreasing, and normalized to define a coherent spectral risk measure. This provides an intuitive interpretation of coherence as assigning larger weights to worse outcomes. φ allows investors to express their individual risk aversion profile.
This document discusses quantile estimation techniques, including parametric, semiparametric, and nonparametric approaches. Parametric estimation assumes a distribution like Gaussian and estimates quantiles based on parameters of that distribution. Semiparametric estimation uses extreme value theory to model upper tails with a generalized Pareto distribution. Nonparametric estimation estimates quantiles directly from the data without assuming a particular distribution. The document presents several techniques for quantile estimation and compares their performance.
The document provides information about probability and statistics concepts including:
1) Mathematical, statistical, and axiomatic definitions of probability are given along with examples of mutually exclusive, equally likely, and independent events.
2) Laws of probability such as addition law, multiplication law, and total probability theorem are defined and formulas are provided.
3) Concepts of random variables, discrete and continuous random variables, probability mass functions, probability density functions, and expected value are introduced.
The document discusses cumulative distribution functions (CDFs) and probability density functions (PDFs) for continuous random variables. It provides definitions and properties of CDFs and PDFs. For CDFs, it describes how they give the probability that a random variable is less than or equal to a value. For PDFs, it explains how they provide the probability of a random variable taking on a particular value. The document also gives examples of CDFs and PDFs for exponential and uniform random variables.
This document discusses several numerical methods for finding the roots or zeros of nonlinear equations, including bracketing methods like bisection that repeatedly decrease an interval containing the solution, open methods like Newton-Raphson that require a good initial guess, and fixed-point iteration that rewrites the equation as x=g(x) and iteratively applies the function. Examples are provided to illustrate applying bisection, false position, Newton's method, secant method, and fixed-point iteration to solve specific equations numerically.
This document provides a concise probability cheatsheet compiled by William Chen and others. It covers key probability concepts like counting rules, sampling tables, definitions of probability, independence, unions and intersections, joint/marginal/conditional probabilities, Bayes' rule, random variables and their distributions, expected value, variance, indicators, moment generating functions, and independence of random variables. The cheatsheet is licensed under CC BY-NC-SA 4.0 and the last updated date is March 20, 2015.
This document outlines an introduction to Bayesian estimation. It discusses key concepts like the likelihood principle, sufficiency, and Bayesian inference. The likelihood principle states that all experimental information about an unknown parameter is contained within the likelihood function. An example is provided testing the fairness of a coin using different data collection scenarios to illustrate how the likelihood function remains the same. The document also discusses the history of the likelihood principle and provides an outline of topics to be covered.
Computational language have been used in physics research
for many years and there is a plethora of programs and packages on the Web which can be used to solve dierent problems. In this report I trying to use as many of these available solutions as possible and not reinvent the wheel. Some of these packages have been written in C program. As I stated above, physics relies heavily on graphical representations. Usually,the scientist would save the results
from some calculations into a file, which then can be read and used for display by a graphics package like Gnuplot.
This document summarizes a presentation on analyzing the tail behavior of Archimedean copulas. It discusses how extreme value theory can be used to approximate risk measures like Value-at-Risk and Tail Value-at-Risk by modeling the distribution of losses exceeding a high threshold using generalized Pareto distributions. The presentation also examines how to extend this univariate extreme value analysis to higher dimensions to model the dependence structure between losses in the tails.
1) The document provides four probability problems involving combinations and permutations to further understanding of combinatorics. It gives the solutions and explanations for each problem involving probabilities of card hands.
2) It then introduces conditional probability and uses examples like the Monty Hall problem to illustrate how conditioning on additional information can change a probability. It provides the definition of conditional probability and proves Bayes' theorem.
3) The document discusses independence of events and uses examples like coin flips and natural disasters to demonstrate independence. It also introduces the continuity of probability and uses examples like the Cantor set to illustrate how it allows calculating probabilities of infinite sets.
This document provides a probability cheatsheet compiled by William Chen and Joe Blitzstein with contributions from others. It is licensed under CC BY-NC-SA 4.0 and contains information on topics like counting rules, probability definitions, random variables, expectations, independence, and more. The cheatsheet is designed to summarize essential concepts in probability.
"Upper: woolen yarn+3 mm padded upper with polyester
Lining: Polar fleece
Innersole: Polar fleece+3 mm padded upper with polyester+7mmEPE filler
Out sole: TPR
Size:36-41
Package: Paper shaping,silica fee,hook, poly bag,master carton(1 kg)(42 cm*28.5 cm*42 cm,24 pairs)
Small MOQ, waiting for your inquiry, please contact at yolanda@wftimeline.com"
Este documento proporciona guías para el diseño de medianas en carreteras. Explica que las medianas mejoran la seguridad y el flujo de tránsito al controlar los giros a la izquierda. Recomienda el uso de medianas no traspasables para carreteras de alto tránsito y carriles de giro a la izquierda o medianas direccionales para tránsito menor. También cubre temas como el espaciamiento adecuado de aberturas en la mediana y el ancho necesario para permitir giros en U
This document describes a student's submission of a project on simulating and validating a cascade refrigeration system. The student introduces the project, which involves theoretically simulating and experimentally validating a cascade refrigeration system that can operate at low temperatures around -30°C using CO2 as the refrigerant. The working principle and benefits of cascade systems using two conventional vapor compression cycles are explained. The simulation approach, which models the low and high temperature systems individually, is outlined. Finally, the experimental analysis that was conducted using a high temperature system with varying compressor capacities and condensers is summarized before the document concludes.
We propose to formulate an absolutely new paradigm for the solid-state power engineering in the field of the electricity conversion into cold based on the “Freezing Chip” technology. It is the creation of the novel thin-film materials (electro freezing/heating foil) based on the nanostructured dielectrics and semiconductors with the preset optimal parameters and changing a cooling process into a heating process simply by switching the polarity of the supply voltage. We have developed nanostructured semiconductors with a non-conventional mechanism of electron conductivity 10 – 10^3 Ohm-1*cm-1. In this case the material heat conduction is of a purely lattice type and is 0,1- 1,0 W/(m*K). Actually, we have developed some kind of an electron heat pipe, but in a nanostructured solid. It allows the principal constraint imposed by the Wiedemann-Franz’s law on the efficiency in Peltier effect-based thermo electrical refrigerators to be eliminated. The heat conduction of nanostructured semiconductors having the same electrical conductance as that in the best Bi:Te:Se:-based electrical refrigerators is more than 10-100 times less that offers promise to achieve the energy efficiencies 50%-70 % at 100-300 K and 15-50 % at 10-100 K temperature ranges per one refrigerator stage.
Composites on the Move: The Need for Dynamic Testing Instron
- Dynamic testing of composites is critical for design but still lags behind metals testing. While fatigue testing of composites has begun, it has mostly involved simple tension-tension testing without temperature control. More realistic testing is needed that involves compression, reversed loading, and fixed strain rates.
- Strain rate testing is also important as materials behave differently at higher strain rates. Recent work has developed high strain rate compression testing up to 100/s but challenges remain in reducing noise and interpreting results.
- New analysis tools like digital image correlation and thermoelastic stress analysis allow more data-rich testing of failure modes like crack propagation but composites testing still has headroom for development.
Wrigley has dominated the global gum market for over 100 years but faces new challenges adapting to changing consumer preferences and increasing competition. To strengthen its position, Wrigley is exploring social media marketing, focusing on health benefits to appeal to developing markets like China, and diversifying its product portfolio through the Mars acquisition. Wrigley's strengths include strong brands and R&D but it relies heavily on the US market and faces threats from low-cost competitors. Adapting to marketing trends and leveraging partnerships will be important to sustain growth in China's expanding confectionery industry.
La fuente de poder convierte la corriente alterna de la red eléctrica en corrientes continuas de bajo voltaje que alimentan los componentes internos de la computadora. Realiza un proceso de transformación, rectificación, filtrado y estabilización de la corriente utilizando componentes como transformadores, diodos y capacitores. Suministra voltajes estandar de 3.3V, 5V y 12V a la tarjeta madre y otros dispositivos para su correcto funcionamiento.
A shock absorber uses fluid and valves to damp vibrations from vehicle suspension springs. It consists of an inner cylinder inside an outer reservoir, with a piston connecting the cylinder to the vehicle frame. Fluid passes through the piston and valves as it moves up and down, with resistance from small holes and viscosity of the fluid absorbing energy from vibrations. Telescopic shock absorbers have this basic design but also include a piston rod connecting the inner cylinder to the axle, and valves in the piston and cylinder foot that control fluid flow to provide damping in both compression and rebound strokes.
SIMULATION OF THERMODYNAMIC ANALYSIS OF CASCADE REFRIGERATION SYSTEM WITH ALT...IAEME Publication
This document summarizes a study that analyzes the thermodynamic performance of a cascade refrigeration system using various alternative refrigerant pairs. The study evaluates 15 different refrigerant pairs for the higher and lower temperature circuits. It assumes 5°C subcooling and 10°C superheating, varies the higher circuit condenser temperature from 30-50°C and lower circuit evaporator from -70 to -50°C. The analysis finds that COP increases with higher evaporator temperature but decreases with higher condenser temperature for all pairs. It also finds that mass flow and compressor work increase with both higher evaporator and condenser temperatures. The best performing pair is R134a-R170 with the highest COP and lowest mass flow,
Компания «Дмитрий Чуприна & Партнеры» приглашает на трансформационный бизнес-курс «Лето со смыслом. Перезагрузка» на Иссык-Куле
Этот курс – правильное решение для Вас:
Если Вы владелец компании, собственник, руководитель, человек, на чьих плечах судьба компании и порой Вам кажется, что Вы как Герой-одиночка, тянете всё на себе, а сотрудники не оправдывают Ваших ожиданий.
Иногда Вам не хватает сил и энергии осуществить все планы и реализовать свои мечты. А иногда, такое ощущение, что Вы упёрлись в стену и будущее туманно и нет прежней концентрации и чёткости восприятия.
Возможно Вы энергичны и амбициозны и планомерно движетесь к своим целям, но хотите узнать как сделать “квантовый скачок” и вывести свою компанию на новый уровень, обретя такое качество внутренней энергетики, которым владеют лишь единицы проснувшихся от “футляра обусловленности” людей.
Если хоть одно из этих описаний Вам близко наш проект “Лето со смыслом.Перезагрузка” для Вас!
Регистрация обязательна. Все вопросы к модератору – Ксения Ермолова, менеджер проекта “Лето со смыслом.Перезагрузка”.
«Дмитрий Чуприна & Партнеры»
e-mail: info.chuprina@gmail.com
+7 727 357 20 39, +7 727 329 98 30 (Казахстан)
+996 559 21 83 21, +996 312 87 00 88 (Кыргызстан)
+792 846 59769 (Россия)
The document discusses Bayes' rule and entropy in data mining. It provides step-by-step derivations of Bayes' rule from definitions of conditional probability and the chain rule. It then gives examples of calculating entropy for variables with different probability distributions, noting that maximum entropy occurs with a uniform distribution where all outcomes are equally likely, while minimum entropy occurs when the probability of one outcome is 1.
The document discusses cumulative distribution functions (CDFs) and probability density functions (PDFs) for continuous random variables. It provides definitions and properties of CDFs and PDFs. For CDFs, it describes how they give the probability that a random variable is less than or equal to a value. For PDFs, it explains how they provide the probability of a random variable taking on a particular value. The document also gives examples of CDFs and PDFs for exponential and uniform random variables.
This document discusses several numerical methods for finding the roots or zeros of nonlinear equations, including bracketing methods like bisection that repeatedly decrease an interval containing the solution, open methods like Newton-Raphson that require a good initial guess, and fixed-point iteration that rewrites the equation as x=g(x) and iteratively applies the function. Examples are provided to illustrate applying bisection, false position, Newton's method, secant method, and fixed-point iteration to solve specific equations numerically.
This document provides a concise probability cheatsheet compiled by William Chen and others. It covers key probability concepts like counting rules, sampling tables, definitions of probability, independence, unions and intersections, joint/marginal/conditional probabilities, Bayes' rule, random variables and their distributions, expected value, variance, indicators, moment generating functions, and independence of random variables. The cheatsheet is licensed under CC BY-NC-SA 4.0 and the last updated date is March 20, 2015.
This document outlines an introduction to Bayesian estimation. It discusses key concepts like the likelihood principle, sufficiency, and Bayesian inference. The likelihood principle states that all experimental information about an unknown parameter is contained within the likelihood function. An example is provided testing the fairness of a coin using different data collection scenarios to illustrate how the likelihood function remains the same. The document also discusses the history of the likelihood principle and provides an outline of topics to be covered.
Computational language have been used in physics research
for many years and there is a plethora of programs and packages on the Web which can be used to solve dierent problems. In this report I trying to use as many of these available solutions as possible and not reinvent the wheel. Some of these packages have been written in C program. As I stated above, physics relies heavily on graphical representations. Usually,the scientist would save the results
from some calculations into a file, which then can be read and used for display by a graphics package like Gnuplot.
This document summarizes a presentation on analyzing the tail behavior of Archimedean copulas. It discusses how extreme value theory can be used to approximate risk measures like Value-at-Risk and Tail Value-at-Risk by modeling the distribution of losses exceeding a high threshold using generalized Pareto distributions. The presentation also examines how to extend this univariate extreme value analysis to higher dimensions to model the dependence structure between losses in the tails.
1) The document provides four probability problems involving combinations and permutations to further understanding of combinatorics. It gives the solutions and explanations for each problem involving probabilities of card hands.
2) It then introduces conditional probability and uses examples like the Monty Hall problem to illustrate how conditioning on additional information can change a probability. It provides the definition of conditional probability and proves Bayes' theorem.
3) The document discusses independence of events and uses examples like coin flips and natural disasters to demonstrate independence. It also introduces the continuity of probability and uses examples like the Cantor set to illustrate how it allows calculating probabilities of infinite sets.
This document provides a probability cheatsheet compiled by William Chen and Joe Blitzstein with contributions from others. It is licensed under CC BY-NC-SA 4.0 and contains information on topics like counting rules, probability definitions, random variables, expectations, independence, and more. The cheatsheet is designed to summarize essential concepts in probability.
"Upper: woolen yarn+3 mm padded upper with polyester
Lining: Polar fleece
Innersole: Polar fleece+3 mm padded upper with polyester+7mmEPE filler
Out sole: TPR
Size:36-41
Package: Paper shaping,silica fee,hook, poly bag,master carton(1 kg)(42 cm*28.5 cm*42 cm,24 pairs)
Small MOQ, waiting for your inquiry, please contact at yolanda@wftimeline.com"
Este documento proporciona guías para el diseño de medianas en carreteras. Explica que las medianas mejoran la seguridad y el flujo de tránsito al controlar los giros a la izquierda. Recomienda el uso de medianas no traspasables para carreteras de alto tránsito y carriles de giro a la izquierda o medianas direccionales para tránsito menor. También cubre temas como el espaciamiento adecuado de aberturas en la mediana y el ancho necesario para permitir giros en U
This document describes a student's submission of a project on simulating and validating a cascade refrigeration system. The student introduces the project, which involves theoretically simulating and experimentally validating a cascade refrigeration system that can operate at low temperatures around -30°C using CO2 as the refrigerant. The working principle and benefits of cascade systems using two conventional vapor compression cycles are explained. The simulation approach, which models the low and high temperature systems individually, is outlined. Finally, the experimental analysis that was conducted using a high temperature system with varying compressor capacities and condensers is summarized before the document concludes.
We propose to formulate an absolutely new paradigm for the solid-state power engineering in the field of the electricity conversion into cold based on the “Freezing Chip” technology. It is the creation of the novel thin-film materials (electro freezing/heating foil) based on the nanostructured dielectrics and semiconductors with the preset optimal parameters and changing a cooling process into a heating process simply by switching the polarity of the supply voltage. We have developed nanostructured semiconductors with a non-conventional mechanism of electron conductivity 10 – 10^3 Ohm-1*cm-1. In this case the material heat conduction is of a purely lattice type and is 0,1- 1,0 W/(m*K). Actually, we have developed some kind of an electron heat pipe, but in a nanostructured solid. It allows the principal constraint imposed by the Wiedemann-Franz’s law on the efficiency in Peltier effect-based thermo electrical refrigerators to be eliminated. The heat conduction of nanostructured semiconductors having the same electrical conductance as that in the best Bi:Te:Se:-based electrical refrigerators is more than 10-100 times less that offers promise to achieve the energy efficiencies 50%-70 % at 100-300 K and 15-50 % at 10-100 K temperature ranges per one refrigerator stage.
Composites on the Move: The Need for Dynamic Testing Instron
- Dynamic testing of composites is critical for design but still lags behind metals testing. While fatigue testing of composites has begun, it has mostly involved simple tension-tension testing without temperature control. More realistic testing is needed that involves compression, reversed loading, and fixed strain rates.
- Strain rate testing is also important as materials behave differently at higher strain rates. Recent work has developed high strain rate compression testing up to 100/s but challenges remain in reducing noise and interpreting results.
- New analysis tools like digital image correlation and thermoelastic stress analysis allow more data-rich testing of failure modes like crack propagation but composites testing still has headroom for development.
Wrigley has dominated the global gum market for over 100 years but faces new challenges adapting to changing consumer preferences and increasing competition. To strengthen its position, Wrigley is exploring social media marketing, focusing on health benefits to appeal to developing markets like China, and diversifying its product portfolio through the Mars acquisition. Wrigley's strengths include strong brands and R&D but it relies heavily on the US market and faces threats from low-cost competitors. Adapting to marketing trends and leveraging partnerships will be important to sustain growth in China's expanding confectionery industry.
La fuente de poder convierte la corriente alterna de la red eléctrica en corrientes continuas de bajo voltaje que alimentan los componentes internos de la computadora. Realiza un proceso de transformación, rectificación, filtrado y estabilización de la corriente utilizando componentes como transformadores, diodos y capacitores. Suministra voltajes estandar de 3.3V, 5V y 12V a la tarjeta madre y otros dispositivos para su correcto funcionamiento.
A shock absorber uses fluid and valves to damp vibrations from vehicle suspension springs. It consists of an inner cylinder inside an outer reservoir, with a piston connecting the cylinder to the vehicle frame. Fluid passes through the piston and valves as it moves up and down, with resistance from small holes and viscosity of the fluid absorbing energy from vibrations. Telescopic shock absorbers have this basic design but also include a piston rod connecting the inner cylinder to the axle, and valves in the piston and cylinder foot that control fluid flow to provide damping in both compression and rebound strokes.
SIMULATION OF THERMODYNAMIC ANALYSIS OF CASCADE REFRIGERATION SYSTEM WITH ALT...IAEME Publication
This document summarizes a study that analyzes the thermodynamic performance of a cascade refrigeration system using various alternative refrigerant pairs. The study evaluates 15 different refrigerant pairs for the higher and lower temperature circuits. It assumes 5°C subcooling and 10°C superheating, varies the higher circuit condenser temperature from 30-50°C and lower circuit evaporator from -70 to -50°C. The analysis finds that COP increases with higher evaporator temperature but decreases with higher condenser temperature for all pairs. It also finds that mass flow and compressor work increase with both higher evaporator and condenser temperatures. The best performing pair is R134a-R170 with the highest COP and lowest mass flow,
Компания «Дмитрий Чуприна & Партнеры» приглашает на трансформационный бизнес-курс «Лето со смыслом. Перезагрузка» на Иссык-Куле
Этот курс – правильное решение для Вас:
Если Вы владелец компании, собственник, руководитель, человек, на чьих плечах судьба компании и порой Вам кажется, что Вы как Герой-одиночка, тянете всё на себе, а сотрудники не оправдывают Ваших ожиданий.
Иногда Вам не хватает сил и энергии осуществить все планы и реализовать свои мечты. А иногда, такое ощущение, что Вы упёрлись в стену и будущее туманно и нет прежней концентрации и чёткости восприятия.
Возможно Вы энергичны и амбициозны и планомерно движетесь к своим целям, но хотите узнать как сделать “квантовый скачок” и вывести свою компанию на новый уровень, обретя такое качество внутренней энергетики, которым владеют лишь единицы проснувшихся от “футляра обусловленности” людей.
Если хоть одно из этих описаний Вам близко наш проект “Лето со смыслом.Перезагрузка” для Вас!
Регистрация обязательна. Все вопросы к модератору – Ксения Ермолова, менеджер проекта “Лето со смыслом.Перезагрузка”.
«Дмитрий Чуприна & Партнеры»
e-mail: info.chuprina@gmail.com
+7 727 357 20 39, +7 727 329 98 30 (Казахстан)
+996 559 21 83 21, +996 312 87 00 88 (Кыргызстан)
+792 846 59769 (Россия)
The document discusses Bayes' rule and entropy in data mining. It provides step-by-step derivations of Bayes' rule from definitions of conditional probability and the chain rule. It then gives examples of calculating entropy for variables with different probability distributions, noting that maximum entropy occurs with a uniform distribution where all outcomes are equally likely, while minimum entropy occurs when the probability of one outcome is 1.
The document simulates Bernoulli random variables and signal detection theory. For Bernoulli random variables, it generates matrices of 0s and 1s using different probabilities and plots the distributions of results over many simulations to show they match the theoretical Bernoulli distribution. For signal detection theory, it defines functions to calculate the probability of a hit, miss, false alarm, and correct rejection given the d-prime and criterion value, by generating the cumulative distribution functions for the signal and noise and calculating the relevant areas. It discusses how the results change for different d-prime and criterion values.
Double Robustness: Theory and Applications with Missing DataLu Mao
When data are missing at random (MAR), complete-case analysis with the full-data estimating equation is in general not valid. To correct the bias, we can employ the inverse probability weighting (IPW) technique on the complete cases. This requires modeling the missing pattern on the observed data (call it the $\pi$ model). The resulting IPW estimator, however, ignores information contained in cases with missing components, and is thus statistically inefficient. Efficiency can be improved by modifying the estimating equation along the lines of the semiparametric efficiency theory of Bickel et al. (1993). This modification usually requires modeling the distribution of the missing component on the observed ones (call it the $\mu$ model). Hence, when both the $\pi$ and the $\mu$ models are correct, the modified estimator is valid and is more efficient than the IPW one. In addition, the modified estimator is "doubly robust" in the sense that it is valid when either the $\pi$ model or the $\mu$ model is correct.
Essential materials of the slides are extracted from the book "Semiparametric Theory and Missing Data" (Tsiatis, 2006). The slides were originally presented in the class BIOS 773 Statistical Analysis with Missing Data in Spring 2013 at UNC Chapel Hill as a final project.
The document summarizes key concepts in hypothesis testing including:
- The null and alternative hypotheses are formulated, with the null hypothesis stating the parameter equals a specific value and the alternative allowing other values.
- There are two types of errors - type I rejects the null when true, type II accepts when false. Tests aim to minimize both.
- The power of a test is the probability it correctly rejects the null when an alternative is true.
- One-tailed tests have critical regions in one tail, two-tailed in both. P-values are used to determine if results are significant.
- Steps of hypothesis testing are outlined along with examples of tests for single and two means/proportions.
Algorithms For Solving The Variational Inequality Problem Over The Triple Hie...Sarah Marie
This summary provides the key details about the document in 3 sentences:
The document discusses algorithms for solving variational inequality problems over the solution set of other variational inequality problems and fixed point sets of mappings. It proposes a new iterative algorithm to solve the "triple hierarchical problem" and proves that the algorithm generates a sequence that converges strongly to the solution. The algorithm involves projections, compositions of mappings, and satisfies certain conditions on the parameters.
This document provides a probability cheatsheet compiled by William Chen and Joe Blitzstein with contributions from others. It is licensed under CC BY-NC-SA 4.0 and contains information on topics like counting rules, probability definitions, random variables, moments, and more. The cheatsheet is regularly updated with comments and suggestions submitted through a GitHub repository.
This document provides an introduction to machine learning and Bayesian decision theory. It discusses key concepts like empirical risk minimization, overfitting, generalization gap, Bayes theorem, risk, Bayesian inference, and classification and regression using Bayesian decision theory. Specific loss functions and decision rules are defined for tasks like binary classification, cost-sensitive classification, and regression. Bayesian approaches to model fitting and probabilistic prediction problems are also covered.
The document discusses different types of random variables including discrete, continuous, Bernoulli, binomial, geometric, Poisson, and uniform random variables. It provides the definitions and probability mass/density functions for each type. Examples are also given to illustrate concepts such as calculating probabilities for different random variables.
This document discusses predicates and quantifiers in predicate logic. Predicate logic can express statements about objects and their properties, while propositional logic cannot. Predicates assign properties to variables, and quantifiers specify whether a predicate applies to all or some variables in a domain. There are two types of quantifiers: universal quantification with ∀ and existential quantification with ∃. Quantified statements involve predicates, variables ranging over a domain, and quantifiers to specify the scope of the predicate.
Accounting for uncertainty is a crucial component in decision making (e.g., classification) because of ambiguity in our measurements.
Probability theory is the proper mechanism for accounting for uncertainty.
Noise is unwanted sound considered unpleasant, loud, or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference arises when the brain receives and perceives a sound.
This document presents a new model of decision making under risk and uncertainty called the Harmonic Probability Weighting Function (HPWF) model. The HPWF model incorporates mental states using a weak harmonic transitivity axiom and an abstract harmonic representation of noise. It explains phenomena like the conjunction fallacy and preference reversal. The HPWF uses a harmonic component controlled by a phase function to characterize how a decision maker's mental states influence probability weighting. Maximum entropy methods can be used to derive a coherent harmonic probability weighting function from the HPWF model.
Categorical data analysis full lecture note PPT.pptxMinilikDerseh1
This document provides an overview of categorical data analysis techniques. It discusses categorical and quantitative variables, different types of categorical variables, and common distributions for categorical data like binomial and multinomial. Methods for categorical data like chi-square tests, logistic regression, and Poisson regression are presented. Examples are provided to illustrate hypothesis testing, confidence intervals, and likelihood ratio tests for categorical proportions.
Multiple regression analysis allows modeling of a dependent variable (y) as a function of multiple independent variables (x1, x2,...xk). The model takes the form y = β0 + β1x1 + β2x2 +...+ βkxk + u. For classical hypothesis testing of the coefficients (β1, β2,...βk), the model assumes u is independent of the x's and normally distributed. The t-test can be used to test hypotheses about individual coefficients, such as H0: βj = 0, while the F-test allows jointly testing hypotheses about multiple coefficients, such as exclusion restrictions that several coefficients equal zero. P-values indicate the probability of observing test statistics at least
This document discusses modeling and estimating extreme risks and quantiles in non-life insurance. It introduces the generalized extreme value distribution and Pickands–Balkema–de Haan theorem, which state that maximums of iid random variables converge to one of three extreme value distributions. It also discusses estimators for the shape parameter of these distributions, such as the Hill estimator, and using the generalized Pareto distribution above a threshold to estimate value-at-risk quantiles. Examples are given applying these methods to Danish fire insurance loss data.
Probability formula sheet
Set theory, sample space, events, concepts of randomness and uncertainty, basic principles of probability, axioms and properties of probability, conditional probability, independent events, Baye’s formula, Bernoulli trails, sequential experiments, discrete and continuous random variable, distribution and density functions, one and two dimensional random variables, marginal and joint distributions and density functions. Expectations, probability distribution families (binomial, poisson, hyper geometric, geometric distribution, normal, uniform and exponential), mean, variance, standard deviations, moments and moment generating functions, law of large numbers, limits theorems
for more visit http://tricntip.blogspot.com/
This document discusses modeling and estimating extreme risks and quantiles in non-life insurance. It introduces the generalized extreme value distribution and three limiting distributions used to model extreme values. It also discusses estimators like the Hill estimator that are used to estimate the shape parameter of distributions modeling extreme risks. Methods for estimating value-at-risk and tail-value-at-risk based on the generalized Pareto distribution above a threshold are also presented.
The document discusses random processes and probability theory concepts relevant to communication systems. It defines key terms like random variables, sample space, events, probability, and distributions. It describes different types of random variables like discrete and continuous, and their probability mass functions and density functions. It also discusses statistical measures like mean, variance, and covariance that are used to characterize random signals and compare signals. Specific random variables discussed include binomial and uniform. The document provides foundations for analyzing random signals in communication systems.
The Use of Fuzzy Optimization Methods for Radiation.ppteslameslam18
This document discusses different approaches to solving linear programming problems under uncertainty, including deterministic, stochastic, and fuzzy optimization methods. It begins by defining types of uncertainty like deterministic, probabilistic, possibilistic, and fuzzy errors. It then outlines fuzzy optimization approaches proposed by Bellman and Zadeh using flexible constraints and goals, and by Tanaka using fuzzy coefficients. The document uses a farmer's crop planting problem as an example deterministic problem, and formulates it as a stochastic problem with uncertain yields and as a fuzzy linear program with flexible constraints. The goal is to compare solutions from these different uncertainty modeling techniques.
Talk at CIRM on Poisson equation and debiasing techniquesPierre Jacob
- The document discusses debiasing techniques for Markov chain Monte Carlo (MCMC) algorithms.
- It introduces the concept of "fishy functions" which are solutions to Poisson's equation and can be used for control variates to reduce bias and variance in MCMC estimators.
- The document outlines different sections including revisiting unbiased estimation through Poisson's equation, asymptotic variance estimation using a novel "fishy function" estimator, and experiments on different examples.
Tree data structures can be represented as nodes connected in a parent-child relationship. Binary trees restrict each node to having at most two children. Binary search trees organize nodes so that all left descendants of a node are less than the node and all right descendants are greater. They allow efficient lookup, insertion, and deletion operations that take O(log n) time on balanced trees. Other tree types include parse trees for representing code structure and XML trees for hierarchical data storage.
The document discusses sorting and searching algorithms. It introduces sorting as a way to make searching faster. Linear search and binary search are described as ways to search through unsorted and sorted lists, respectively. Common sorting algorithms like selection sort, insertion sort, and Shellsort are explained. Selection sort and insertion sort have quadratic runtimes while Shellsort is subquadratic. Maintaining stability is also discussed as a property of some sorting algorithms.
Heapsort is an O(n log n) sorting algorithm that uses a heap data structure. It works by first turning the input array into a max heap, where the largest element is stored at the root. It then repeatedly removes the root element and replaces it with the last element of the heap, and sifts the new root element down to maintain the heap property. This produces a sorted array from largest to smallest in O(n log n) time.
This document discusses hashing techniques for data storage and retrieval. Static hashing stores data in buckets accessed via a hash function, with solutions for bucket overflow. Dynamic hashing uses extendable hashing to adjust the hash table size as the database grows or shrinks. Queries and updates in extendable hashing follow the hash value to a bucket. The structure allows splitting and merging buckets efficiently. Compared to ordered indexing, hashing is more efficient for lookups by specific values rather than ranges.
This document contains a quiz for the course EE693 Data Structures and Algorithms. It has 16 questions testing knowledge of data structures and algorithms concepts like matrix operations, polynomial evaluation, quicksort analysis, hashing, heaps, binary trees, and balanced search trees. The questions are worth 2 marks each, with an optional bonus question worth an additional 2 marks. Students are instructed to show their work, include their name and student ID, and are notified that electronic devices other than calculators are prohibited during the exam.
This document contains a 10 question quiz on data structures and algorithms. It provides instructions that the quiz is for the EE693 course, is 1 hour, and contains multiple choice questions worth 2 marks each with a -1 mark penalty for incorrect answers. Students are instructed to write their name and roll number and choose the single correct answer for each question. The questions cover topics like binary trees, heaps, stacks, queues, and binary search trees.
This document contains a 20 question quiz on data structures and algorithms for the course EE693 at IIT Guwahati. The quiz covers topics like algorithm analysis, sorting algorithms, stacks, queues, and linked lists. Students are instructed to attempt all questions, which range from multiple choice to short answer. Correct answers will be awarded partial marks. Use of electronic devices other than calculators is prohibited. The document provides the date, time limit, and other instructions for the quiz.
This document contains a 30 question mid-semester exam for a data structures and algorithms course. The exam covers topics like asymptotic analysis, sorting algorithms, hashing, binary search trees, and recursion. It provides multiple choice questions to test understanding of algorithm time complexities, worst-case inputs, and recursive functions. Students are instructed to attempt all questions in the 2 hour time limit and notify the proctor if any electronic devices other than calculators are used.
This document contains 38 multiple choice questions related to data structures and algorithms. The questions cover topics such as trees, graphs, hashing, sorting, algorithm analysis, and complexity classes. They assess knowledge of concepts like tree traversals, shortest paths, minimum spanning trees, asymptotic analysis, and resolving collisions in hash tables.
This document discusses the concept of dynamic programming. It provides examples of dynamic programming problems including assembly line scheduling and matrix chain multiplication. The key steps of a dynamic programming problem are: (1) characterize the optimal structure of a solution, (2) define the problem recursively, (3) compute the optimal solution in a bottom-up manner by solving subproblems only once and storing results, and (4) construct an optimal solution from the computed information.
The document discusses bucket sort and radix sort algorithms. Bucket sort works by distributing elements into buckets based on their values, then outputting the buckets in order. Radix sort improves upon bucket sort for integers by sorting based on individual digits from least to most significant. Both algorithms run in O(n) time when values are restricted, outperforming general comparison-based sorting which requires O(n log n) time.
The document discusses and compares several sorting algorithms. It describes selection sort, bubble sort, insertion sort, merge sort, quicksort, and radix sort. For each algorithm, it provides pseudocode examples and analyzes their time complexity, which ranges from O(n^2) for simpler algorithms like selection and bubble sort, to O(n log n) for more advanced algorithms like merge sort and quicksort, to O(n) for radix sort. The document concludes by comparing the approximate growth rates of running time for various sorting algorithms as input size increases.
The document contains 16 multiple choice questions about algorithms, data structures, and graph theory. Each question has 4 possible answers and the correct answer is provided. The maximum number of comparisons needed to merge sorted sequences is 358, and depth first search on a graph represented with an adjacency matrix has a worst case time complexity of O(n^2).
The document discusses sorting algorithms. It begins by defining the sorting problem as taking an unsorted sequence of numbers and outputting a permutation of the numbers in ascending order. It then discusses different types of sorts like internal versus external sorts and stable versus unstable sorts. Specific algorithms covered include insertion sort, bubble sort, and selection sort. Analysis is provided on the best, average, and worst case time complexity of insertion sort.
This document provides the solutions to selected problems from the textbook "Introduction to Parallel Computing". The solutions are supplemented with figures where needed. Figure and equation numbers are represented in roman numerals to differentiate them from the textbook. The document contains solutions to problems from 13 chapters of the textbook covering topics in parallel computing models, algorithms, and applications.
This document provides an introduction to POSIX threads (Pthreads) programming. It discusses what threads are, how they differ from processes, and how Pthreads provide a standardized threading interface for UNIX systems. The key benefits of Pthreads for parallel programming are improved performance from overlapping CPU and I/O work and priority-based scheduling. Pthreads are well-suited for applications that can break work into independent tasks or respond to asynchronous events. The document outlines common threading models and emphasizes that programmers are responsible for synchronizing access to shared memory in multithreaded programs.
This document discusses shared-memory parallel programming using OpenMP. It begins with an overview of OpenMP and the shared-memory programming model. It then covers key OpenMP constructs for parallelizing loops, including the parallel for pragma and clauses for declaring private variables. It also discusses managing shared data with critical sections and reductions. The document provides several techniques for improving performance, such as loop inversions, if clauses, and dynamic scheduling.
The document provides an overview of computational complexity and discusses the Clay Mathematics Institute's Millennium Prize Problems. It notes that the Clay Mathematics Institute selected seven important unsolved problems in mathematics and is offering a $1 million prize for solving each one. The problems were announced at a 2000 meeting in Paris to celebrate their importance. The rules for awarding the prizes were developed by the Institute's Scientific Advisory Board and approved by its Directors. Inquiries about the Millennium Prize Problems can be sent to the listed email address.
This document provides an introduction to the CUDA parallel computing platform from NVIDIA. It discusses the CUDA hardware capabilities including GPUDirect, Dynamic Parallelism, and HyperQ. It then outlines three main programming approaches for CUDA: using libraries, OpenACC directives, and programming languages. It provides examples of libraries like cuBLAS and cuRAND. For OpenACC, it shows how to add directives to existing Fortran/C code to parallelize loops. And for languages, it lists supports like CUDA C/C++, CUDA Fortran, Python with PyCUDA etc. The document aims to provide developers with maximum flexibility in choosing the best approach to accelerate their applications using CUDA and GPUs.
Vector space interpretation_of_random_variablesGopi Saiteja
This document discusses vector space interpretation of random variables. It begins by introducing vector spaces and their properties such as closure under addition and scalar multiplication. Random variables can be interpreted as elements of a vector space. Inner products, norms, orthogonality and projections are discussed in the context of both vector spaces and random variables. Interpreting expectations as inner products allows treating random variables as vectors in an inner product space.
4th Modern Marketing Reckoner by MMA Global India & Group M: 60+ experts on W...Social Samosa
The Modern Marketing Reckoner (MMR) is a comprehensive resource packed with POVs from 60+ industry leaders on how AI is transforming the 4 key pillars of marketing – product, place, price and promotions.
06-04-2024 - NYC Tech Week - Discussion on Vector Databases, Unstructured Data and AI
Discussion on Vector Databases, Unstructured Data and AI
https://www.meetup.com/unstructured-data-meetup-new-york/
This meetup is for people working in unstructured data. Speakers will come present about related topics such as vector databases, LLMs, and managing data at scale. The intended audience of this group includes roles like machine learning engineers, data scientists, data engineers, software engineers, and PMs.This meetup was formerly Milvus Meetup, and is sponsored by Zilliz maintainers of Milvus.
Beyond the Basics of A/B Tests: Highly Innovative Experimentation Tactics You...Aggregage
This webinar will explore cutting-edge, less familiar but powerful experimentation methodologies which address well-known limitations of standard A/B Testing. Designed for data and product leaders, this session aims to inspire the embrace of innovative approaches and provide insights into the frontiers of experimentation!
Predictably Improve Your B2B Tech Company's Performance by Leveraging DataKiwi Creative
Harness the power of AI-backed reports, benchmarking and data analysis to predict trends and detect anomalies in your marketing efforts.
Peter Caputa, CEO at Databox, reveals how you can discover the strategies and tools to increase your growth rate (and margins!).
From metrics to track to data habits to pick up, enhance your reporting for powerful insights to improve your B2B tech company's marketing.
- - -
This is the webinar recording from the June 2024 HubSpot User Group (HUG) for B2B Technology USA.
Watch the video recording at https://youtu.be/5vjwGfPN9lw
Sign up for future HUG events at https://events.hubspot.com/b2b-technology-usa/
End-to-end pipeline agility - Berlin Buzzwords 2024Lars Albertsson
We describe how we achieve high change agility in data engineering by eliminating the fear of breaking downstream data pipelines through end-to-end pipeline testing, and by using schema metaprogramming to safely eliminate boilerplate involved in changes that affect whole pipelines.
A quick poll on agility in changing pipelines from end to end indicated a huge span in capabilities. For the question "How long time does it take for all downstream pipelines to be adapted to an upstream change," the median response was 6 months, but some respondents could do it in less than a day. When quantitative data engineering differences between the best and worst are measured, the span is often 100x-1000x, sometimes even more.
A long time ago, we suffered at Spotify from fear of changing pipelines due to not knowing what the impact might be downstream. We made plans for a technical solution to test pipelines end-to-end to mitigate that fear, but the effort failed for cultural reasons. We eventually solved this challenge, but in a different context. In this presentation we will describe how we test full pipelines effectively by manipulating workflow orchestration, which enables us to make changes in pipelines without fear of breaking downstream.
Making schema changes that affect many jobs also involves a lot of toil and boilerplate. Using schema-on-read mitigates some of it, but has drawbacks since it makes it more difficult to detect errors early. We will describe how we have rejected this tradeoff by applying schema metaprogramming, eliminating boilerplate but keeping the protection of static typing, thereby further improving agility to quickly modify data pipelines without fear.
Global Situational Awareness of A.I. and where its headedvikram sood
You can see the future first in San Francisco.
Over the past year, the talk of the town has shifted from $10 billion compute clusters to $100 billion clusters to trillion-dollar clusters. Every six months another zero is added to the boardroom plans. Behind the scenes, there’s a fierce scramble to secure every power contract still available for the rest of the decade, every voltage transformer that can possibly be procured. American big business is gearing up to pour trillions of dollars into a long-unseen mobilization of American industrial might. By the end of the decade, American electricity production will have grown tens of percent; from the shale fields of Pennsylvania to the solar farms of Nevada, hundreds of millions of GPUs will hum.
The AGI race has begun. We are building machines that can think and reason. By 2025/26, these machines will outpace college graduates. By the end of the decade, they will be smarter than you or I; we will have superintelligence, in the true sense of the word. Along the way, national security forces not seen in half a century will be un-leashed, and before long, The Project will be on. If we’re lucky, we’ll be in an all-out race with the CCP; if we’re unlucky, an all-out war.
Everyone is now talking about AI, but few have the faintest glimmer of what is about to hit them. Nvidia analysts still think 2024 might be close to the peak. Mainstream pundits are stuck on the wilful blindness of “it’s just predicting the next word”. They see only hype and business-as-usual; at most they entertain another internet-scale technological change.
Before long, the world will wake up. But right now, there are perhaps a few hundred people, most of them in San Francisco and the AI labs, that have situational awareness. Through whatever peculiar forces of fate, I have found myself amongst them. A few years ago, these people were derided as crazy—but they trusted the trendlines, which allowed them to correctly predict the AI advances of the past few years. Whether these people are also right about the next few years remains to be seen. But these are very smart people—the smartest people I have ever met—and they are the ones building this technology. Perhaps they will be an odd footnote in history, or perhaps they will go down in history like Szilard and Oppenheimer and Teller. If they are seeing the future even close to correctly, we are in for a wild ride.
Let me tell you what we see.
Natural Language Processing (NLP), RAG and its applications .pptxfkyes25
1. In the realm of Natural Language Processing (NLP), knowledge-intensive tasks such as question answering, fact verification, and open-domain dialogue generation require the integration of vast and up-to-date information. Traditional neural models, though powerful, struggle with encoding all necessary knowledge within their parameters, leading to limitations in generalization and scalability. The paper "Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks" introduces RAG (Retrieval-Augmented Generation), a novel framework that synergizes retrieval mechanisms with generative models, enhancing performance by dynamically incorporating external knowledge during inference.
06-04-2024 - NYC Tech Week - Discussion on Vector Databases, Unstructured Data and AI
Round table discussion of vector databases, unstructured data, ai, big data, real-time, robots and Milvus.
A lively discussion with NJ Gen AI Meetup Lead, Prasad and Procure.FYI's Co-Found
1. Detection theory
Detection theory involves making a decision based on set of measurements. Given a set of
observations, a decision has to be made regarding the source of observations.
Hypothesis –A statement about the possible source of the observations.
Simplest –binary hypothesis testing chooses one of two hypotheses, namely
0H Null hypothesis which is the usually true statement.
1H Alternative hypothesis.
In radar application, these two hypotheses denote
0H Target is absent
1H Target is present
M-ary hypothesis testing chooses one of M alternatives:
0 1 1, ........, MH H H .
A set of observations denoted by
1 2[ ..., ]'nx x xx
We can associate an a prior probability to each of hypothesis as
0 1, ,..., MP H P H P H
Given hypothesis iH ,the observations are determined by the conditional PDF / ( )iHfX x
The hypothesis may be about some parameter that determines / ( )iHfX x . is chosen
from a parametric space .
Simple and composite hypotheses:
For a simple hypothesis, the parameter is a distinct point in while for a composite hypothesis
the parameter is specified in a region. For example, 0 : 0H is a simple hypothesis while
1 : 0H is a composite one.
Bayesian Decision theory for simple binary hypothesis testing
The decision process D X partitions the observation space in to the region 0 1
n
Z Z
such that 0D X H if 0x Z and 1H otherwise
A cost ,ij j iC C H D H x is assigned to each ,j iH D Hx pair. Thus
00 0 0, ( )C C H D H x , and so on 1,0 0 1,C C H D H x .The objective is to minimize
the average risk
2.
,
,
,
Bayesian decison minimize over D , 0,1
Equivalently minimize , over , 0,1
X X x
X x X
j
X x j
C R D E C H D X
E E C H D X
E C H d X f x dx
R D X H j
E C H D X d X H j
We can assign 00 11 10 010 and C 1C C C but cost function need not be symmetric
Likelyhood ratio Test
Suppose 0D X H
1
0
1
0
00 0 01 1
1
1 10 0 11 1
0 11
00 0 01 1 10 0 11
, /
similarly if
, , ( )
The decision rule will be
, , ( ) , , ( )
H
H
H
H
X x X xE C H D X X x C P H C P H
D X H
E C H D X X x D X H C P H X x C P H X x
E C H D X X x D X H E C H D X X x D X H
C P H X x C P H X x C P H X x C P
1
0
1
| 01
0
| 1
| 0
0
1
00 01 11 10
01 11 1 10 00 0
|
1
|
0
01 11 1 10 00 0 |
since and ,we can simplify
Note that
, 0,1.
we can write
H
H
j
i
H
X H
H
H
X H
X H
H
X x X x
j X H
j X x
i X H
i
X H
H X x
C C C C
C C P H C C P H
P H f x
P H j
P H f x
C C P H f x C C P H f x
f x
L X
f x
1
10 00 0
01 11 1
C C P H
C C P H
This decision rule is known as likelihood ratio test LRT .
Errors in decision
3. The decision space partitions the observation space into two regions: In 0Z , 0H is decided
to be true and in 1Z , 1H is decided to be true. If the decisions are wrong, two errors are
committed:
Type I error=probability false alarm given by
0
1
1 0
/
( ( ) | )
( )
FA
X H
Z
P P D X H H
f x dx
Type II error given by
1
0
0 1
/
( ( ) | )
( )
M
X H
Z
P P D X H H
f x dx
4. Error Analysis:
Two types of errors Type
0
1
1
0
1 0
11 0 1
0 1
/
/
/
/
/
False alarm probability
Type II error probability:
/
miss detction probability
probability of error
=P P
We observe that
P
( )
I
FA
M
e
F M
F X H
Z
X H
X H
X
P P D x H H
P
P P D x H H
P
P
H P H P
f x
f x
f L x dL x L x
f
0
( )H x
Similarly,
1
0
1
/
/
0
M X H
Z
X H
P f x dx
f L x dL x
00 0 0 0
10 0 1 0
01 1 0 1
11 1 1 1
00 0 10 0
01 1 11 1
0 1
00
Now Bayesian riskis given by
,
/
/
/
/
1
1
if we substitute 1 , can be wriiten as
F F
M M
R D C EC H D x
C P H P D x H H
C P H P D x H H
C P H P D x H H
C P H P D x H H
C P H P C P H P
C P H P C P H P
P H P H R D
R D C
10 1 11 00 01 11 10 00
1
1
=function of and the threshold
F F M FP C P P H C C C C P C C P
P H
I
5.
00 11 10 01
0 1
0 01 00 0
1 10 11 1
0, 1
The threshold is given by
=
F M
e
C C C C
thenR d isgivenby
P H P P H P
P
LR
P C C P
P C C P
Minimum probability of error criterion :
MinMax Decision Rule
Recall the Baysesian risk function is given by
00 10 1 11 00 01 11 10 001 F F M FR D C P C P P H C C C C P C C P which is function
of 1P H . For a given 1P H we can determine the other parameters in the above expression
using the minimum Baye’s risk criterion.
R D
0
1P H
Suppose the parameters are designed using the Baye’s minimum risk at 1P H p .If 1P H is
now varied,the modified risk curve will be a straight line tangential to the Baye’s risk curve at
, ( )p R D p .The decision will no longer optimal. To overcome this difficulty, Baye’s minimax
criterion is used. According to this criterion, decide by
6.
1
1
Min Max ,
j
R D P H
H P H
Under mild conditions, we can write
1 1
Min Max ( ) Max Min ( )
j jH P H P H H
R D R D
Assuming differentabilty, we get
00 10 1 11 00 01 11 10 00
1 1
11 00 01 11 10 00
1 0
( ) ( )
0
F F M F
M F
d d
R D C P C P P H C C C C P C C P
dP H dP H
C C C C P C C P
The above equation is known as the minimax equation and can be solved to find the threshold.
Example Suppose
0
1
00 11
01 10
: ~ exp(1)
: ~ exp(2)
0
2, 1
H X
H X
C C
C C
Then,
0
1
0
2
1
11 00 01 11 10 00
2
0
2
2
We have to solve the min max
0
Now
1
2
2
1
H
H
x
X H
x
X H
M FA
M F
x
FA
x
M
f e u x
f e u x
L x
equation
C C C C P C C P
P P
P e dx
e
P e dx
e
Substituting FAP and MP in the minimax equation, we can find
7. Receiver Operating Characteristics
The performance of a test is analyzed in terms of graph showing vsD FAP P . Note that
( )1
1 ( )
D X H
Z
P f x dx
η
∞
= ∫ and ( )0
1 ( )
FA X H
Z
P f x dx
η
∞
= ∫ where 1
( )Z η is a point corresponding to the
likelihood ratio threshold η and 1
Z represents the region corresponding to the decision of
1H .
=D FAP P
In general, we will like to select a FAP that results in a DP near the knee of the ROC. If we
increase FAP beyond that value, there is a very small increase in DP . For a continuous DP ,the
ROC has the following properties.
1.ROC is a non-decreasing function of FAP . This is because to increase FAP , we have to
expand 1Z and hence DP will increase.
2.ROC is on or above the line =D FAP P
3.For the likely- hood ratio test, the slope of the roc gives the threshold
Recall that
1
DP
1FAP
8. ( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( )
1 1
1
0 0
1
0
1
0 0
1
1( )
1
1( )
( )
1 1 1
1
1
1
( )
( )
( )
( )
( ) ( ) ( )
( )
( )
( )
D
D X H X H
Z
FA
FA X H X H
Z
D X H
Z
D
X H X H
D
D
FAFA
dP
P f x dx f Z
dZ
dP
P f x dx f Z
dZ
P L x f x dx
dP
L Z f Z f Z
dZ
dP
dP dZ
dPdP
dZ
η
η
η
η
η
η
η
η η η η
η
η
η
η
∞
∞
∞
= ⇒ = −
= ⇒ = −
=
∴ = − = −
∴ = =
∫
∫
∫
Neyman-Pearson (NP) Hypothesis Testing
The Bayesean approach requires the knowledge of the a priori probabilities 0( )P H and .
Finding these is a problem in many cases. In such cases, the NP hypothesis testing can be
applied.
The NP method maximizes the detection probability while keeping the probability of false
alarm with a limit. The problem can be mathematically written as
( )
maximize
, 0,1
subject to
D
j
FA
P
D x H j
P α
= =
≤
I statistical parlance, α is called the size of the test.
We observe that ROC curve is non-decreasing one. Decreasing FAP will decrease DP also.
Therefore for optimal performance FAP should be kept fixed at α .Hence the modified
optimized problem is
( )
maximize
, 0,1
subject to
D
j
FA
P
D x H j
P α
= =
=
We can solve the problem by the Lagrange multiplier method.
9. ( )
( )
( ) ( )
( ) ( )( )
1 0
1 1
1 0
1
, 0,1
Now
j
D FA
D x H j
X H X H
Z Z
X H X H
Z
Maximize J P P
J f x dx f x dx
f x f x dx
λ α
λ
αλ
= =
= − −
= −
= − +
∫ ∫
∫
To maximize we should select 1Z such that
( ) ( )
( )
( )
( )
1 0
1
1
0
0
0
1
0
will give thethreshold in terms of .
can be found out from
H
H
X H X H
X H
X H
X H
Z
f x f x
f x
f x
f x dx
λ
λ
λ
λ α
>
<
− >
=∫
Example
( )
( )
( )
( )
( )
1
0
1
0
0
1
2 1
2
2 1
2
; 0,1
; 1,1
0.25
H
H
FA
x
X H
X H
x
H X N
H X N
P
f x
L x e
f x
e λ
−
−
>
<
∼
∼
=
= =
Taking logarithm,
( )
( )( )
( )( )
1
0
1
0
2
2
1
1 2ln
2
2 1 2ln
1
1 2ln
2
1
0.25
2
0.675
H
H
H
H
x
x
x
e dx
λ
λ
λ η
π
η
>
<
>
<
∞ −
+
−
⇒ + =
=
⇒ =
∫
10. Composite Hypothesis Testing
Uncertainty of the parameter under the hypothesis .Suppose
0
1
0 / 0 0
1 / 1 1 1
: ~ ,
: ~ ,
X H
X H
H X f x
H X f x
0
θθ
If 0
θ
1or
θ
contains single element it is called a simple hypothesis; otherwise it is called a
composite Hypothesis.
Example:
Suppose
0
1
: ~ N 0,1
: ~ N ,1 , 0
H X
H X
These two hypotheses may represent the absence and presence of dc signal in presence of a
0mean Gaussian noise of known variance 1.
Clearly 0H is a simple and 1H is a composite algorithm. We will consider how to deal
with the decision problems.
Uniformly Most Powerful (UMP) Test
Consider the example
2
2
0
1
1
2
0
1
2
0
21
0
: , 0,1,..., 1 , ~ 0,1
: , 0,1,..., 1 , ~ ,1 , 0
Likelyhood ratio
1
2ln ln
1
2
2
i
i
i i
i i
xN
i
xN
i
N
i
i
H x i N X iid N
H x i N X iid N
e
L x
e
n
x
If 0 ,then we can have
1
00
H
H
N
i
i
T x x
Now we have the modified hypothysis follows:
11.
0
1
:T ~ N 0,N
: t ~ N , N
H X
H
DP
2
1
2
1
2
x
N
FA
v
P e dx
v
Q
N
Example
FAP
1
2
1
2
1
2
2
0
2
1
2 2
2
2
2
: ~ N 0,
: ~ N 1, , 0
1
ln
2
1
2
1
2
if we take 0,then
1
2
H
H
H
H
H
H
H X
H X
x x
L x
x
x
x
With this threshold we can determine the probability of detection DP and the possibility of
false alarm FAP .However, the detection is not optimal any sense. The UMP may not generally
exist. The following result is particularly useful for the UMP test.
Karlin-Rubin Theorem
Suppose 0 0 1 1 0: , :H H
Let T T x be test statistic . If the likelihood ratio
1
0
1
0
L
t H
t H
f t
t
f t
is a non-decreasing function of t , then the test
1
0
H
H
T
maximizes the detection probability DP for a given FAP Thus the test is UMP for a
fixed FAP .
12. Example:
1 0
0 0
1 1 0
1
0
: ~
: ~
Then
x
H X Poi
H X Poi
L x e
is a non-decreasing function of x.
Therefore the threshold test for a given PFA is UMP.
13. Generalized likelihood ratio test(GLRT):
In this approach to composite hypothesis testing, the non –random parameters are replaced by
their MLE in the decision rule.
Suppose
0
1
0
1
: ~
: ~
X
X
H X f x
H X f x
Then according to GLRT, the models under the two hypotheses are compared. Thus the
decision rule is
11
1
0
0
0
max
max
H
H
X H
X
f x
f x
This approach also provides the value of the unknown parameter. It is not optimal but works
well in practical situations.
Example:
2
1
2
0
0
1
2
1
2
1
2
1
1
2
1
: 0
: 0
~ ( , )
.
.
1
2
1
2
i
i
i
n
xn
X
i
n x
X
i
H
H
X iid N
x
x
=
x
f x e
f x e
X
The MLE of 0 is given by
14.
2
2
1
2
2
2
2
1 1
2
2
1
2
2
1
1
1 1
2
1
1
2
1
1 1
2
1
2
1
2
2
1
1
ˆ
Under GLRT, the likelihood ratio becomes
1
2L x =
1
2
1
ln L
2
n
i i
i
i
n n
i i
i i
n
i
i
n
i
i
n
i
i
x xn
n
i
n x
i
x x
n
x
x
n
n
i
i
x
n
e
e
e
e
e
x x
n
2
2
2
1
2
1
2
1
0
2 2
2 2
1 1
2 2 2
1
Using the GLRT
2 ln
~ ,
~ chi-squre distributed
Particularly, under
1
n
i
i
n
i
i
n
i
i
n n
i i
i i
x n
X N n
x
H
x n x
n
n X
where 2
1X is a chi-square random variable with degree 1.
we can find the FAP
Multiple Hypothesis testing:
Decide of 0 1 2 1, , ,...... MH H H H on the basis of observed that
,i 0,1,2,....M 1iP H are assumed to be known, Associate cost ijC associated with the
decision iH and defined by
( , ( ) )ij j iC C H D X H .
The average cost then given by
1 1
0 0
/
M M
ij i j j
i j
C C P D H H P H
15. Z
The decision process will partition the observation space n
Z (or a subset of it) into M
subsets 0 1 1, ,..., MZ Z Z
1
0
1
0
1
1
j j
M
i
i
i j
j
i j X H X H
Z
Z
M
X H
j
j i
P D H H f x dx f x dx
f x dx
We can write
1 1 1
0 0 1
j
i
M M M
ii i j ij jj X H
i i jZ
j i
C C P H P H C C f x
Minimization is achieved by placing x in the region such that the above integral is
minimum. Choose the region of x corresponding to the minimum value of
1
1
j
M
i j ij jj X H
j
j i
C x P H C C f x
Thus decision rule based on minimizing over iH
0 0
1
0
1
0
1
0
, 0,1,2,..., 1
1, , 0
then
j
j
M
X Hi
i j ij jj
jX H X H
j i
M
i ij jj j
j
ij jj
M
i j X H j
j
j i
f xC x
J x P H C C
f x f x
P H C C L x i M
C i j C
J x P H f x H
The above minimization corresponds to the minimization of the probability of error.
Z0 Z1
.
.
:
ZM-1
16. Rewriting iJ x we get,
1
0
1
M
i j X
j
j i
i X
J x P H x f x
P H x f x
and arrive at MAP criterion
If the hypotheses are equally likely i.e 0 1 1...... MP H P H P H P then we can write
1
0
j i
M
i X H X X H
j
j i
J x Pf x f x Pf x
Therefore the minimization is equivalent to the minimization of the likelihood iX Hf x .
Example
1
2
2
0
1 1
2 2
2 2
0 0
0
1
2
1
2
1 1
2
2 2
, 0,1,...niid Gaussian
: 1
:
:
1
2
1
.
2
n
i
i
i
n n
i i
i i
i
x
X H n
x x
n
X i
H
H
H
f x e
e e
We have to decide on the basis
1 1
2
0 0
2
2
0
1
2
0
1
2
2
: 2 1
: 4 4
: 2 1
3
:0
2
3
:
2
: 0
n n
i
i i
x
T x x
H T x x
H T x x
H T x x
H x
H x
H x
17. Sequential Detection and Wald’s test
In many applications of decision making observations are sequential in nature and the
decision can be made sequentially on the basis of available data.
We discuss the simple case of sequential binary hypothesis testing by modification of the NP
test.This test is called the sequential probability ratio test(SPRT) or the Wald’s test.
In NP test, we had only one threshold for the likelihood ratio L x given by
1
0
H
H
L x
The threshold is determined from the given level of significance
In SPRT, L x is computed recursively and two thresholds 0 1and are used. The simple
decision rule is:
If 1 1, decideL Hx
If 0 0, decideL Hx
If 0 1L x wait for the next sample to decide.
The algorithm stops when we get 1 0orL x L x .
Consider iX to be iid.
1
0
1
0
0
1 2
1 2
0 1 2
1
1
1
1
1
Then for .....
, ,....,
, ,....,
i i
i
i
i
i
n
n
X H n
X H n
n
X H i
i
n
X H i
i
n
X H i
X H ni
n
X H n
X H i
i
n n
X X X
L L
f x x x
f x x x
f x
f x
f x
f x
f x
f x
L L x
n
n
X
X
In terms of logarithm
1ln ln lnn n nL L L x
18. Suppose the test requirement is
1 andM D FAP P P
We have to fix 0 1and on the basis of and .
Relation between 0 1, and ,
We have
1
1
0
1
0
1
1
0
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
Similarly
1
1
1
D X H
Z
X H
Z
X H
Z
M X H
Z
X H
Z
X H
Z
P f x dx
L x f x dx
f x dx
P f x dx
L x f x dx
f x dx
Average stopping timefor the SPRT is optimal in the sense that for a given error levels no
test can perform better than the SPRT with the average number of samples less than that
required for the test. We may take the conservative values of 0 1and as
1
1
and
0
1
19. Example Suppose
2
0
2
1
~ ~ 0,
~ ~ ,
H X N
H X N
and and
2
2
1
2
1
1
2
1
i
i
xn
i
xn
i
e
L x
e
We can compute 1
1
and
0
1
and design the SPRT.