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1. 4 0. Berikut ini akan disinggungsedikit tentang penaksiran parameter ini. An alternative decision rule, the maximum average normalised likelihood (MANL), is proposed in this paper. 38. Postal study course - https://gatea Choose <None> for the non-parametric decision rule. -Other decision problems represent one-time decisions where historical data for estimating probabilities don't exist. Jan 28, 2005 · Bayesian phylogenetic inference holds promise as an alternative to maximum likelihood, particularly for large molecular-sequence data sets. ML (maximum likelihood) Classifier. Lectures 5 & 6: Classifiers Hilary Term 2007 A. g. time of year, fishing area, …) Simple decision Rule: Make a decision without seeing the fish. edu/rtd/9039 Non-probabilistic approaches to decision making have been proposed for situations in which an individual does not have enough information to assess probabilities over an uncertainty. 7 and 8 show the results of either procedure on diﬀerent databases. The loss function tells us how bad we feel about our decision once we nd out the true value of the parameter chosen by nature. The Rule Classifier automatically finds the corresponding rule image Chi Squared value. 1) as 1-1M Ap(ý). shop GATE ACADEMY launches its products for GATE/ESE/UGC-NET aspirants. For the classification threshold, enter the probability threshold used in the maximum likelihood classification as a percentage (for example, 95%). its likelihood. While data mining techniques such as decision trees, neural networks, to using the classical maximum likelihood allocation rule (MLAR). (But it may not be displayed – you might have to open it in another Viewer. • Formal decision rules. 3 The Maximum Likelihood Model of Multisensory Enhancement 3. Note that if T= 1 , then we maximize the number of true positives at the cost of Such a decision rule is called a maximum likelihood (ML) decision rule, the resulting receiver is a maximum likelihood receiver. The approach used here converts the decoding problem into a search problem through a graph that is a trellis for an equivalent code of the transmitted code. Since each observation is meant to be independent of each other one, the probability of observed data is the probability of the observed class (for a binary class: 0's and 1's) May 16, 2018 · This is referred to as the Maximum A Posteriori decision rule. One non-probabilistic method is to use intervals in which an uncertainty has a minimum and maximum but nothing is assumed about the relative likelihood of any value within the interval. 3 Amemiya [1] gives a rigorous development of maximum likelihood estimation for one class of limited dependent variable regressions. The model solution is a ‘decision rule’ Ω=ttt++11FX(, ,),ε ξ where X t is a vector of state variables (predetermined endogenous variables and exogenous variables), i. ” We shall utilize the maximum likelihood (ML) principle. The HMM filtering algorithm . X tt =ΛΩ, where Λ is a matrix that picks the state variables among the elements Mar 11, 2013 · Maximum likelihood adalah teknik yang sangat luas dipakai dalampenaksiran suatu parameter distribusi data dan tetap dominan dipakai dalampengembangan uji -uji yang baru (Lehmann, 1986). >>”The standard Bayesian argument against the use of p-values in this scenario is that we do not know how the 500 trials were conducted and that the researcher may have capitalized on chance by stopping whenever the result was significant. 1) is that the MAP rule depends only on the conditional prob ability p U|V and thus is completely determined by the joint distribution of U and V. More formally: IF size>100 AND garden=1 THEN value=high. The maximum likelihood principle builds on the intuition that the probability for a variable X to take place around a given observation x is proportional to the pdf evaluated at that point Maximum likelihood and two-step estimation of an ordered-probit selection model Richard Chiburis Princeton University Princeton, NJ <EMAIL>. 2. OK if deciding for one fish The prior probabilities are incorporated by modifying the maximum likelihood decision rule employed in a Bayesian-type classifier to calculate a posteriori probabilities of class membership which are based not only on the resemblance of a pixel to the class signature, but also on the weight of the class which is estimated for the final output Moreover, due to the fact that Soft Decision Trees redirect instances by probabilities, Maximum Likelihood Estimation (MLE) could provide a way of extracting sparse multivariate rules from Soft acreage estimation, feature selection, and signature extension dependent upon the maximum likelihood decision rule by j. The maximum likelihood estimator is a function from proﬁles to alternatives (more accurately, sub-sets of alternatives, since there may be ties), and as such is a voting rule (more accurately, a correspondence). Maximum Likelihood (ML) updating indicates one example of such a refinement, whereas Full Bayesian (FB) updating does not allow for any refinement. Abstract. Maximum Likelihood Rule Ensembles. 3 As such, the ability of clinicians to assess severity of acute asthma exacerbations is variable and limited in Dec 08, 2016 · “CHAPTER 3” Maximum Likelihood and Bayesian Estimation Oleh : <NAME> (156150100111022) 1 2. List the payoff or profit or reward 4. 6447. Maximum Likelihood Estimation (MLE) and Maximum A Posteriori (MAP), are both a method for estimating some variable in the setting of probability distributions or graphical models. the classification. An alternative way of estimating parameters: Maximum likelihood estimation ( MLE) Remember the rules of exponents, in particular eaeb = ea+b. Below we consider the problem of choosing the best alternative from the finite set of alternatives X = {x 1, x 2,…, x n}, n ≥ 2. = yxg yxg. The likelihood of a simple hypothesis ˆis de ned as L(ˆ) = Pr(Djˆ). 4. ! The likelihood of θ is L(θ;D) = Prob(D|θ). It's natural to think about the job of the likelihood function in this direction: given a fixed value of model parameters, what i maximum likelihood decision rule is by far the most common supervised classification method used for analyzing satellite im age data (Richards, 1986). 26 Mar 2020 According to the Bayes rule, the posterior can be decomposed into the product of the likelihood and prior. We propose a new rule induction algorithm for solving classification problems via probability estimation. MLEπ(P) = argmaxV ∗∈L(X) Q V ∈P π(V|V ∗) In this paper, we require that all such conditional probabilities to be positive for technical reasons. Choose maximum likelihood for the parametric rule. Maximum likelihood is an intuitive criterion to build estimators or to learn the structure of data. Fisher [2] for the use of mathematical statistics, and more particularly for its part concerned with point estimation, i. What can we say about this decision rule? Mar 15, 2016 · Hence, for sufficiently large r min, the Bias from any adaptive sample increase will not have a substantial negative effect on the precision of the overall maximum likelihood estimate. The estimates for the two shape parameters c and k of the Burr Type XII distribution are 3. Abstract: Maximum-likelihood soft-decision decoding of linear block codes is addressed. ⎧. -For decision problems that occur more than once, we can often estimate these probabilities from historical data. = ≠. MLRules [11] are derived from the 4. As this is the value of h associated with the maximum value of the likelihood, it is known as the maximum likelihood estimate (MLE) of the phrase that was spoken. Solution For details please refer to this awesome article: MLE vs MAP: the connection between Maximum Likelihood and Maximum A Posteriori Estimation. 6: L(P) = logPr(X 1 = x 1)+ X i,j n ij logP ij (41) This is the equation for the log-likelihood of an exponential family, in which the canonical sufﬁcient statistics are the n ij and x 1, and the natural parameters are the logP ij and the log probabilities of the initial states The operations manager for a community bus company wants to decide whether he should purchase a small, medium, or large new bus for his company. , 2000) and MART (Fried-man, 2001), however, here we show a modiﬁed proce-dure, adapted to the case when the decision rule is a base classiﬁer in the ensemble. ]),([. X tt =ΛΩ, where Λ is a matrix that picks the state variables among the elements The model solution is a ‘decision rule’ Ω=ttt++11FX(, ,),ε ξ where X t is a vector of state variables (predetermined endogenous variables and exogenous variables), i. Maximum likelihood estimation is a method that determines values for the parameters of a model. Of all the probabilities that a given model might have produced, those obtained by MLE yield maximum in-sample betting return to a log utility investor. Last, Pr{y|1}≥Pr{y|0}⟺ln(Pr{y|1})≥ln(Pr{y|0}). Han is with the School of Computer and Information Science at Syracuse University, Syracuse, NY 13244-4100 (e-mail: <EMAIL>. De nition 2 (Admissibility). S. poznan. Look at the maximum possible gain from each choice and select the option that offers the biggest gain. 6 describes decision making. This rule requires the receiver to determine the probability of transmission of a message  Maximum-Likelihood (ML) Decision Rule. iastate. 1, here we discuss the special case of estimation applied to a time series of invariants. 10 for i=1…a and then select the action a i for which R(a i |x) is minimum. 7898 and 3. A decision rule (X) is called \inadmissible" if there exists some decision rule 0(X) such that 1. Canonical maximum likelihood I For a p q-dimensional space !, we de ne the likelihood ratio = max 2! L( ) max 2 L( ) I As before, large (close to 1) values of suggest that H 0 is true while small ones are the evidence in favor of H 1 I For the signi cance level the decision rule is to reject H 0 if c where c is the solution of = max 2! P [ c] Levine STAT 517:Maximum The AIC c, BIC, and DT are normally applied in a maximum likelihood framework, but we have also used them in a Bayesian framework under the assumption that the HML i s under alternative partitioning strategies exhibit a similar relationship to one another as would maximum likelihood estimates (see Castoe et al. e. Understanding MLE with an example While studying stats and probability, you must have come across problems like – What is the probability of x > 100, given that x follows a normal distribution with mean 50 and standard deviation (sd) 10. Finally, Secs. To convert between the rule image’s data space and probability, use the Rule Classifier. In Bayesian statistics, a maximum a posteriori probability ( MAP) estimate is an estimate of an unknown quantity, that equals the mode of the posterior distribution. 1. We Bayesians like posterior distributions. )(,1. Y. Under the assumption that the samples are i. 9039. Deﬁnition 1 ([4]) A voting rule (correspondence) r is a maximum likelihood estimator for winners under i. garden=1 is the second condition in the IF-part. ! Normally a hypothesis will be defined by a set of parameters θ. They provide a practical way to begin and carry out an analysis or experiment. The conditional PDF P(r|sm) or any monotonic function of it is usually called the likelihood function. When a suﬃcient and consistent In estimation theory and decision theory, a Bayes estimator or a Bayes action is an estimator or decision rule that minimizes the posterior expected value of a loss function (i. Bayes’ Decision Rule • How is the decision affected by the probability values? • What is the minimum probability of oil such that we choose to drill the land under Bayesʼ decision rule? State of Nature Expected Action Oil Dry Payoff Drill for oil 700 -100 700p - 100(1-p) = 800p - 100 Sell the land 90 90 90p - 90(1-p) = 90 Maximum Likelihood Estimation I The maximum likelihood estimate (MLE) of is, by deﬁnition, the value ^that maximizes L( jD) and can be computed as ^= argmax L( jD): I It is often easier to work with the logarithm of the likelihood function (log-likelihood function) that gives ^= argmax logL( jD) = argmax Xn i=1 logp(x ij ): CS 551, Fall 2019 Steps in Decision Theory 1. 1 Maximum Likelihood Estimation 4 15. 19. Lease (Bayes' Decision Rule is calculating the expected payoff EX: . Introduction to graphical models and inference . In practice, values for T Thanks for contributing an answer to Signal Processing Stack Exchange! Please be sure to answer the question. Two centuries later, Young (1988) showed that a correct application of the maximum likelihood principle leads to the selection of rankings called Kemeny orders, which have the minimal total number of disagreements with those of the voters. 2 Hu Guangdao2, <NAME> 3, 1. The term Λ is called the likelihood ratio, and the decision rule is known as the likelihood Bayes criterion, and seeks to minimize the maximum Bayes Risk. It was originally formulated by <NAME>. Now classify your image using a different rule (e. INTRODUCTION. Dec 18, 2017 · Pre-book Pen Drive and G Drive at www. Question: A Binary Communication System Employs The Maximum Likelihood Decision Rule To Distinguish Between The Two Equally Likely Hypotheses Of X = 0 And X = 1. A computer listing is provided in [12]. 1) is called the maximum a posteriori probability (MAP) rule. Your solutions for this assignment need to be in a pdf format and should be submitted to the blackboard and a webpage (to be speci ed later) for peer-reviewing. Assessment Of Land Use Cover Change Detection Using Supervised Maximum Likelihood Decision Rule And Its Post-Classification Technique In Puer-Simao Counties, China Diallo Yacouba 1, Xu Yuanjin 2, BAH Amadou Apho1, Bokhari Abdulah. 1 Maximum likelihood. Should we use maximum likelihood, method of moments, or some other procedure to estimate g( )? Decision theory allows us to rule out certain inadmissible procedures. <NAME>. This rule selects an alternative that maximizes the probability of realizing individual orderings, conditional on the alternative being the top according to a true social ordering. The goal of maximum likelihood estimation is to make inferences about the population that is most likely to have generated the sample, specifically the joint probability distribution of the random variables {,, …}, not necessarily independent and identically distributed. For the “0-1” loss the optimal decision rule is the maximum a-posteriori probability rule. 4: Maximum Likelihood (ML) product of the probabilities associated with the. An important consequence of (8. gateacademy. 14 May 2001 Formally, the maximum likelihood estimator, Now, by the chain rule A statistical test is a decision rule based on the observed data to either  9 Feb 2017 ML: maximum likelihood X ∼ N(1, 1). ” Thus in 1925 the theory said that if there is an eﬃcient statistic, then the maximum likelihood es-timate is eﬃcient. Discover bayes opimization, naive  4 Jan 2017 Θ with the help of the Bayes' Rule prob(Θ|X). Though LS, especially in small samples, is sus-ceptible to the effects of outlying data points, the procedure does not have to discern between the probabilities that different lines best ﬁt the data. )(,0. c) Now suppose that P(H0) = 2/3, P(H1) = 1/3. trichel earth observations division After reviewing the performance of a single-element antenna (scalar) maximum likelihood (ML) detection for CDMA signals, we analyze two ML detection receiver structures, employing interference suppression and desired signal enhancement, and compare the performance of the receivers with that of a single-antenna ML detection receiver. List the possible alternatives (actions/decisions) 2. 1 Introduction. If we only used P(B|A), the likelihood, we would be using a Maximum Likelihood decision rule. cis. https://lib. "training" sa~ples representing the feature types to be mapped (Llllesand and Kiefer, 1987). ⎨. Maximum A Posteriori Probability ( MAP) Decision Rule. We have investigated the performance of Bayesian inference with empirical and simulated protein-sequence data under conditions of relative branch-length differences and model violation. Hartmann. _1$as the decision instead of the The Likelihood Function ! Assume we have a set of hypotheses to choose from. Now, like I said earlier, all phylogenetic trees will rely on some level of assumptions. , 2005; Castoe and Parkinson, 2006 Maximum Likelihood So, using maximum parsimony we have grown a phylogenetic tree. I The maximum-likelihood decision rule disregards priors and decides for the hypothesis with higher likelihood. Idea Check: What is a reasonable Decision Rule if the only available information is the prior, and the cost of any incorrect classi cation is equal? Decide ! 1 if P(! 1) >P(! 2); otherwise decide ! 2. Since both Viterbi algorithm and MLSDA use the ML decoding rule, the only factor remaining for the choice of decoders is their computational complexity. If we let the decision makers be the rows and the set Warren Buffy decides that Bayes decision rule is his most reliable decision criterion. 1, 2 A challenging clinical feature of this complex environmental and genetic disease is the heterogeneity of clinical expression. A*, The decision boundary is found by solving for the roots of this quadratic, x 1 = 15 3 p 15 = 3:381 and x 2 = 15 + 3 p 15 = 26:62 d. Han. Decision Rule Using Conditional Probabilities • Using Bayes’ rule, the posterior probability of category ω j given measurement x is given by: where (i. Notice that, decision based on the MLE can be over-ruled if we had access to prior probabilities. 11 Nov 2017 Essentially, the decision rule is a function that takes a value of of the random variable X, and outputs a Maximum Likelihood Decision Rule. syr. Question: How to design the decision rule ˆH(x)?. Two classes were defined for the mangrove map: mangrove forest and non- mangrove forest (water, fishpond, built-up area). there exists A maximum likelihood decision rule method was used to extract a mangrove map. When we want to distinguish between different decision rules, we denote the MAP decision rule in (3. a. b) Find the probability of missed detection (P MD) for the maximum-likelihood decision rule, and enter the answer, up to two decimal places. d, the prior probabilities can be obtained via the maximum-likelihood estimate (i. Rearranging the preceding expression n The term Λ(x) is called the likelihood ratio, and the decision rule is known as the likelihood ratio test 2 1 2 1 else choose if P( | x) >P( | x Designing Efficient Maximum-Likelihood Soft-Decision Decoding Algorithms for Linear Block Codes Using Algorithm A* . Press OK. The classified one on the right. Haberman [5] provides a general treatment of maximum likelihood methods for contingency table analysis. The derivation of maximum-likelihood (ML) estimates for the Naive Bayes model chain rule, the following identity is exact (any joint distribution over Y,X1 Xd. Estimate Parameters of a Noncentral Chi-Square Distribution In this example, we would choose h=“fork handles”. quirein lacie verification department lockheed electronics company aerospace systems division 16811 el camino real houston, "texas 17058 m. i. The ensemble is built by 1) A module on Maximum Likelihood Estimation - Examples by Ewa Paszek 2) Lecture on Maximum Likelihood Estimation by Dr. org Abstract. It states that the decision maker should ignore all possible events except the one most likely to occur, and should select the course of action that produces the best possible result (maximum gain or minimum loss) in the given Apr 15, 2015 · Hi Dr. Funo, Eiichiro, "Proving admissibility using the stepwise Bayes technique: with applications to maximum likelihood estimation "(1989). The main advantage of decision rules is their simplicity and good interpretability. with determining the numerical value of parameter, for example, a mean. , signals that require immediate reaction, and conditional distribution is harder than learning decision rule • Generative Model –Find P(X,Y), then derive h(x) via Bayes rule –Pro: not yet committed to loss, input, or output during training; often computationally easy –Con: Needs to model dependencies in X Bayes Decision Rule •Assumption: –learning task P(X,Y)=P(Y|X) P(X) is known Feb 02, 2014 · Maximum a Posteriori (MAP) and Maximum Learn more about bayesian, pattern-recognition, ml, map, maximum likelihood, maximum a posteriori Oct 04, 2014 · Maximum-Likelihood Estimates. While the early approaches to rule induction were based on sequential covering, we follow an approach in which a single decision rule is treated as a base Jul 05, 2008 · Maximum likelihood rule ensembles Maximum likelihood rule ensembles Dembczyński, Krzysztof; <NAME>; <NAME> 2008-07-05 00:00:00 Maximum Likelihood Rule Ensembles <NAME> ski <EMAIL>. Decide w 1 if P( w 1 ) > P( w 2 ); w 2 otherwise. Retrospective Theses and Dissertations. This decision rule is known as maximum a posteriori probability rule. In terms of Bayes’ rule, the image data are equally consistent with a hill and a crater, where each interpretation corresponds to a different maximum likelihood value. . Example: P (x) = p1 2ˇ e (x ) 2 This rule is also known as the maximum likelihood decision rule and is usually presented as the maximum log-likelihood decision rule assign X to si if ∑ ∑ = = λ> λ T t t k T t log f( t | i ) log f( |) 1 1 x x for all k ≠ i (3. 3 15. We introduce the maximum likelihood principle in Section 38. Let us break down the decision rule: size>100 is the first condition in the IF-part. 31 Jul 2017 If you hang out around statisticians long enough, sooner or later someone is going to mumble "maximum likelihood" and everyone will 8 Apr 2013 Three examples of applying the maximum likelihood criterion to find an estimator: 1) Mean and variance of an iid Gaussian, 2) Linear signal 3 Jan 2018 A beginners introduction to the maximum likelihood method for parameter ( Making this sort of decision on the fly with only 10 data points is (e. pl Institute of Maximum likelihood detection of PPM signals governed by arbitrary point-process plus additive Gaussian noise Abstract: The maximum likelihood decision statistic for pulse-position modulated (PPM) signals governed by an arbitrary discrete point process in the presence of additive Gaussian noise is derived. It can be shown that this is achieved by always choosing the most likely alternative given the observations, which amounts to basing the decision on the likelihood ratio (Green & Swets, 1966 Jan 03, 2018 · Intuitive explanation of maximum likelihood estimation. 5 The maximum likelihood estimator @(x) of a parameter 6 max-. (. Maximum Likelihood estimation (MLE) Choose value that maximizes the probability of observed data Maximum a posteriori (MAP) estimation Choose value that is most probable given observed data and prior belief 34 Wald test. 1 . , scale factor – sum of probs = 1) Decide ω 1 if P(ω 1 /x) > P(ω 2 /x); otherwise decide ω 2 or Decide ω 1 if p(x/ω 1)P(ω 1)>p(x/ω 2)P(ω Jul 16, 2018 · This is where Maximum Likelihood Estimation (MLE) has such a major advantage. A shape constrained maximum likelihood variant of the kernel based empirical Bayes rule proposed by Brown and Greenshtein (2009) for the classical Gaussian compound decision problem is described and some simulation comparisons are presented. One decision rule learned by this model could be: If a house is bigger than 100 square meters and has a garden, then its value is high. The decision rule in (8. –The term Λ𝑥 is called the likelihood ratio, and the decision rule is known as the likelihood ratio test * L(𝑥) can be disregarded in the decision rule since it is constant regardless of class 𝜔 . Yunghsiang S. So the likelihood function of the parameter vector theta, which dictates the transition model given a set of observations, alpha of the states between time 0 and time t decomposes using the Markov assumption. 5) Using a mixture of multivariate Gaussian pdfs and the "winner-take-all" assumption [11] we can relate mixture MLE vs MAP: the connection between Maximum Likelihood and Maximum A Posteriori Estimation. 5 Fig. R, Thank you for reading, I really appreciate your kind compliments 🙂 Here is my long-winded reply. (1) If maximum likelihood isn't decision theory decision rule based on 0 (p). That's maximax thinking – choosing the maximum of the maximums. Decision Rule: (X). Apply the model and make your decision 4. ! We do not know θ, but we make some observations and get data D. Maximum likelihood assumes no prior distribution, We Bayesians need our priors, it could be informative or uninformative, but it need to exists Moreover, in this context, there are important respects in which the likelihood ratio formulation of the decision rule is not a true likelihood ratio test: Although a higher likelihood ratio, all else equal, still favors the result associated with H, the test is not a proper likelihood ratio test because the critical value of the likelihood (3. Parameter Estimation 1. It is closely related to the method of maximum likelihood (ML) estimation Jul 31, 2017 · Published on Jul 31, 2017. 8 Optimum Decision Rule: Likelihood Ratio Test. INTRODUCTION Max Likelihood Bayesian • Permasalahan estimasi parameter adalah salah satu hal klasik dalam statistik, yang dapat dilakukan melalui pendekatan dalam berbagai cara. 1 is just about right as the prior probability of an improving economy, but is quite uncertain about how to split the remaining probabilities between a stable economy and a worsening economy. 1) is thus called the maximum a posteriori probability (MAP) rule. The algorithm used by the Maximum Likelihood Classification tool is based on two principles: The cells in each class sample in the multidimensional space being normally distributed; Bayes' theorem of decision making CHATZIDIAMANTIS et al. Let’s look again at the equation for the log-likelihood, Eq. Lecture 9 Bayesian Decision Rules Lecture 10 Evaluating Performance Lecture 11 Parameter Estimation Lecture 12 Bayesian Prior Lecture 13 Connecting Bayesian and Linear Regression Today’s Lecture Basic Principles Likelihood Function Maximum Likelihood Estimate 1D Illustration Gaussian Distributions Examples Non-Gaussian Distributions A decision rule was developed using three predictors: a cricomental space of 1. 3/33 Odds ratio, Bayes’ Theorem, maximum likelihood We start with an “odds ratio” version of Bayes’ Theorem: take the ratio of The maximum likelihood decision rule is still one of the most widely used from GEOG 457 at Arizona State University Maximum Likelihood Rule Ensembles tree being a base classiﬁer has already been used in LogitBoost (Friedman et al. Maximum likelihood optimal alternatives. It states that the decision maker should select the course of action whose worst (maximum) loss is better than the least (minimum) loss of all other courses of action possible in given circumstances. ! When new information is added, it is assimilated in Bayesian Decision-theoretic Detection Theory yielding the following Bayes’ decision rule, called the maximum likelihood ratio H1 The introduced approach is compared with other decision rule induction algorithms such as SLIPPER, LRI and RuleFit. [36]. The Hurwicz decision rule The maximum likelihood approach addresses this issue, refer to Example 38. How hard is it to learn the Decision rule: Maximum Likelihood Estimate (MLE): choose θ that The mle function computes maximum likelihood estimates (MLEs) for a distribution specified by its name and for a custom distribution specified by its probability X0 p(x|θ)dx for θ in Θ1. For instance, let be a series of coin flips where denotes heads'' and denotes tails''. 2 Baye's Decision Rule. He believes that 0. Everything else in the probability space is irrelevant to making a MAP Nov 20, 2019 · Having covered the techniques of hard and soft decision decoding, its time to illustrate the most important concept of Maximum Likelihood Decoding. From a statistical standpoint, a given set of observations are a random sample from an unknown population. 5. time of year, fishing area, …) ▫ Simple decision Rule: ➢Make a decision without seeing the fish. 1 Likelihood Function n In this case the decision rule becomes g Or, in a more compact form n Applying Bayes Rule n P(x) does not affect the decision rule so it can be eliminated*. , the probability that a pixel with feature vector ω belongs to class i, is given by: a process based in likelihood theory, but again, relies on the simple min-imization decision rule. Human decision making behavior is believed to be controlled by multiple rule is the same between MB and MF approaches, but the first state This rule was applied to regres- sion problems with uncertain dependent variable in. Bayes rule and maximum expected utility. Classification = using Bayes rule to calculate P(Y | Xnew). <NAME> lecture notes for Purdue ECE 662 - Pattern Recognition and Decision Making Processes Decision Rule From Only Priors A decision rule prescribes what action to take based on observed input. The strategy is maximum likelihood estimation. In the general case, we assume that there exists an algorithm, or decision rule, which includes some parameter γ which takes on maximin criterion: In decision theory, the pessimistic (conservative) decision making rule under conditions of uncertainty. 75(-100)=100) Payoff table: What is the maximum that the decision maker should be willing to spend to get more information in this situation? Pattern Recognition & Classification Maximum-Likelihood decision rule Given Baye's Theorem, it is easy to define a rule by which pixels can be sorted among various classes: a pixel will be assigned to the class with the highest posterior probability given that it has the characteristics of the measurement vector x. The decision rule can be defined as. Previous work Maximum Likelihood (ML) classiﬁers for face detection and recognition have been or likelihood: (How likely is this prediction to be true?) They gives the probability of a predicted outcome (the chance of something happening) If you give me several pictures of cats and dogs – and then you ask me to classify a new cat photo – I should return a prediction with rather high confidence. We can. In contrary to RuleFit Thus, the Bayes decision rule states that to minimize the overall risk, compute the conditional risk given in Eq. Therefore, in the absence of any prior assumptions on your part, you should see the image as depicting either a hill or a crater with equal probability. <NAME>, Assistant Professor, Univeristy of Utah 3) Partially based on Dr. A canonical way to de ne model M’s likelihood is via the general method of maximum likelihood (ML), by maxi-mizing L(ˆ) over ˆ2M. Although the that for any clasification rule, we can find another based solely on sufficient statistics that has The maxigi(x) decision is a separate O(c) operation. 2 indicates two decision zones in a two-dimensional signal space. Also called maximin maximum likelihood criterion: In decision theory, one of the decision making rules under conditions of uncertainty. In Restricted maximum likelihood factor analysis, pro-grammed by Joreskog and Gruvaeus as RMLFA, is an extension of maximum likelihood factor analysis which permits the experimenter to test a factor struc-ture hypothesis. pl Institute of Computing Science, Pozna University of Technology, Pozna , 60-965, <NAME> <EMAIL>. The standard implementation of su pervised maximum likelihood classification requires the selection of. The Bayesian Decision Rule states the we have to pick the class i which minimizes the Risk associated by picking the class. , the posterior expected loss). 25(700)+. The resulting minimum overall risk is called the Bayes risk , denoted R, and is the best performance that can be achieved. A generalized 10-701 Machine Learning: Assignment 1 Due on Februrary 20, 2014 at 12 noon <NAME>, <NAME> Instructions: Failure to follow these directions may result in loss of points. g So let's look at the likelihood function in this context before we consider what maximum likelihood estimation would look like. Maximum likelihood is the most widely used statistical estimation technique. The hypothesis must specify certain elements of A, (, and P. The mapping from the observations to the response space is determined by a decision rule. May 11, 2014 · As you've probably already figured out, the data and the model parameters are both inputs to the likelihood function. edu <NAME> The World Bank Washington, DC <EMAIL>. Loss Function: L( ; ). The tool considers both the variances and covariances of the class signatures when assigning each cell to one of the classes Third key decision: structure of the error term OLS in a Maximum Likelihood Framework individual decision rule some other underlying structural process. The MAP estimator begins with this . A binary multiple-check generalization of the Wagner rule is presented, and two methods for its implementation, one of which resembles the suboptimal Forney-Chase algorithms, are described. Jan 02, 2010 · The criterion used by the decision maker will be dependent upon his risk attitude: • Risk seeker management will use maximax rule • Risk averter management will use maximin rule • Risk neutral management will use minimax regret rule ***** (1) Maximax Rule It is a strategy which maximizes the maximum gain. 6 Apr 2018 Fitting; Maximum Likelihood; Reinforcement Learning Modelling. ) ( ). The maximum likelihood decision statistic for pulse-position modulated (PPM) signals governed by an arbitrary discrete point process in the presence of additive Gaussian noise is derived. Asthma is the most prevalent chronic disease of childhood and the most frequent reason for childhood hospitalization in the United States. The Multivariate Normal Distribution: Topics 1. Bayesian Parameter Estimation. Carbone - UPMC 22 Maximum likelihood for tree identiﬁcation : the complex case When ambiguous beliefs are represented by multiple priors, a decision maker (DM)'s updating rule may also include a step of refining the initial belief. ) The original image is on the left. CS 194-10, F’11 Lect. Want to read all 11 pages? Maximum Likelihood Estimation MLE Principle: Choose parameters that maximize the likelihood function This is one of the most commonly used estimators in statistics Intuitively appealing 6 Example: MLE in Binomial Data It can be shown that the MLE for the probability of heads is given by (which coincides with what one would expect) 0 0. While the early approaches to rule induction were based on sequential covering, we follow an approach in which a single decision rule is treated as a base classifier in an ensemble. Zisserman • Bayesian Decision Theory • Bayes decision rule • Loss functions • Likelihood ratio test • Classifiers and Decision Surfaces • Discriminant function • Normal distributions • Linear Classifiers • The Perceptron • Logistic Regression Decision Theory Jun 20, 2017 · Maximum-likelihood updating (MLU) is a well-known approach for extending static ambiguity sensitive preferences to dynamic set-ups. Identify the possible outcomes 3. Maximum Likelihood Estimation MLE Ken <NAME> <NAME> ECE 175A Winter 2012 UCSD Statistical Learning Goal Given a relationship betwee… In this problem, we introduce a new maximum likelihood rule and analyse its performance. Bayesian Parameter Estimation Downloadable! Condorcet (1785) initiated the statistical approach to vote aggregation. Let be the estimate of a parameter , obtained by maximizing the log-likelihood over the whole parameter space : The Wald test is based on the following test statistic: where is the sample size and is a consistent estimate of the asymptotic covariance matrix of (see the lecture entitled Maximum likelihood - Covariance matrix estimation). After the classification, the raster classification result was then converted into vector format. 6 SVM Recap Logistic Regression Basic idea Logistic model Maximum-likelihood Solving Convexity Algorithms Logistic model We model the probability of a label Y to be equal y 2f 1;1g, given a After each lecture, you can download the video or watch it in youtube, where it is listed as undergraduate machine learning . d. votes (MLEWIV) if The Maximum Likelihood Estimator The decision rule, for input x, requires making the decision ^ (x) which minimizes the ex- Bayes decision theory is the ideal Jun 18, 2019 · When you use the maximax decision-making approach, the first step is to list your potential choices and the possible outcomes of each. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Receiver with Perfect CSI Given the received signal 𝑟[𝑘] and the channel’s instant fading state 𝐼, the decision rule of the receiver with perfect CSI will be Pr{𝑟[𝑘]∣𝑠[𝑘]=1,𝐼} maximum-likelihood soft-decision decoding algorithm for linear block codes. Select one of the decision theory models 5. 5722, respectively. Sufficient conditions are given for determining when the optimum PPM symbol detection strategy is to choose the PPM symbol corresponding to the maximum slot optimal decision rule is the maximum a posteriori. Section 9 gives a summary and outlook to future work. Bayesian Decision Theory • Bayesian decision theory is a fundamental statistical approach to the problem of pattern classification. This is because, referring back to our formulation of Bayes rule, we only use the P(B|A) and P(A) terms, which are the likelihood and prior terms, respectively. In fact, ML and FB are the two extremes of refining beliefs with likelihood in updating. 1 The decision rule The maximum likelihood model (ML model, for short) incorporates the basic deci sion problem an organism is faced with in a typical environment: to discriminate between relevant stimuli (targets), i. The maximum likelihood estimates for the scale parameter α is 34. This example was completely computable because : - JC is the simplest model of sequence evolution - the tree has a unique topology A. Formally, a decision rule is a function1 δ(x) from X into A, specifying how actions/decisions are chosen, given observation(s) x. In patients with all three predictors (17%), the decision rule had a positive predictive value of 95% (95% confidence interval [CI], 75–100%) and a negative predictive value of 49% (95% The maximum likelihood estimate is deﬁned as follows. Models, however, are com-posite hypotheses, comprising many possible values of ˆ. Towards this, the signal space is divides in M decision regions, Zi, i = 1, 2, …, M such that, vector r lies inside 'Z 'if,i ln is maximum for k = iPrmrk ⎧ ⎪ ⎨ ⎡⎤ ⎪⎩ ⎣⎦ 4. dures, maximum likelihood estimation and Bayesian estimation. Bayes' theorem of decision making. 1 When working with the quadratic loss function, a decision rule d is unbiased if and 5. 5 cm or less, a pharyngeal grade of more than II, and the presence of overbite. We discuss the estimation of a regression model with an ordered-probit selection rule. An alternative way of formulating an estimator within Bayesian statistics is maximum a posteriori estimation. This is the Baye's optimum modi es a Gaussian likelihood function whose log-likelihood ratio is used as the decision rule: If log P(xjy=1) p(xjy= 1) >T;then y= 1, otherwise, y= 1. For small . 1 General theory .$\begingroup$<NAME> has used the result that the maximum-likelihood decision rule minimizes the sum of the false-alarm and missed-detection probabilities. • optimal decision rule is the maximum a-posteriori probability Thus, the Bayes Decision Rule is stated as 5 Mar 2018 Hints: E[|y|2]=1⟹E[|yA|2]=A2. Since the MAP rule maximizes the probability of correct decision for each sample value i, it also maximizes the probability of correct decision averaged over all j. ! Decision making when all the probabilistic information is known. A ‘Maximum Likelihood Detector’ realizes the above decision rule. maximum likelihood will always provide a statistic which, if normally distributed in large samples with variance falling oﬀ inversely to the sample number, will be an eﬃcient statistic. : GENERALIZED MAXIMUM-LIKELIHOOD SEQUENCE DETECTION FOR PHOTON-COUNTING FREE SPACE OPTICAL SYSTEMS 3383 A. You've reached the end of your free preview. 21 Dec 2008 Abstract: A novel decision-aided maximum likelihood (DA ML) As for M-ary PSK (M>4), the decision rule cannot be easily simplified into This is referred to as the likelihood function. The decision rule need not be deterministic. If you hang out around statisticians long enough, sooner or later someone is going to mumble "maximum likelihood" and everyone will knowingly nod. in decision analysis, states of face detection and matching steps, while Sec. Prior probabilities reflect our prior knowledge (e. Bernoulli distributions and expectations . Interpretation of the Bayes decision rule: If the likelihood ratio of class 1 and class 2 exceeds a threshold value (independent of the input pattern x), the optimal action is: decide 1 Maximum likelihood decision rule is a special case of minimum risk decision rule: • Threshold value = 1 • 0-1 loss function • Maximum likelihood was introduced by <NAME> back in the 1920s. Find the value of the threshold in terms of the system parameters. The parameter values are found such that they maximise the likelihood that the process described by the model produced the data that were actually observed. For the decision regions in part (c), what is the numerical value of the Bayes risk? Solution: For x <15 3 p 15 the decision rule will choose ! 1 For 15 3 p 15 <x <15 + 3 p 15 the decision rule will choose ! 2 For to choose the maximum likelihood estimate of the correct decision. yxgL. 6 Maximum likelihood Estimation (MLE) •Given a parameterized pdf how should one estimate the parameters which define the pdf? •There are many techniques of “parameter estimation. The decision criterion based on the maximum of P(r|sm) over the M signals is called maximum-likelihood (ML) criterion. Maximum Likelihood Estimators 2. Maximum likelihood estimation is one of the procedures used to estimate the parameters From: Data Science for Business and Decision Making, 2019 When estimating GARCH models, a fairly good general rule of thumb is to use at least 14 Dec 2012 Maximum likelihood estimation basically chooses a value of i estimation! Going back to Bayes rule, again, we seek the value of i. In practice, the maximization Maximum Likelihood (ML) is a supervised classification method derived from the Bayes theorem, which states that the a posteriori distribution P(i|ω), i. •This has three steps: – 1) We choose a parametric model for all probabilities. And one more difference is that maximum likelihood is overfitting-prone, but if you adopt the Bayesian approach the over-fitting problem can be avoided. = by <NAME>. Explain the function of correlator receiver. Maximum Likelihood Detection of Signal Vectors in Gaussian Noise. 2 . A. 2 0. After this video, so Bayesian decision theory • recall that we have – Y – state of the world – X – observations – g(x) – decision function – L[g(x),y] – loss of predicting y with g(x) • Bayes decision rule is the rule that minimizes the risk Risk =EX Y [L(X,Y)] • for the “0-1” loss, ⎧1 ( )≠ • optimal decision rule is the maximum a Show that the Maximum Likelihood (ML) decision rule based on y is equivalent to comparing$|y|^2$with a threshold. He estimates that the annual profits (in$000) will vary depending upon whether. This paper develops an example in which MLU induces an ambiguity averse maxmin expected utility (MEU) decision-maker to (1) prefer a bet on an ambiguous over a risky urn and (2) be more willing to bet on the ambiguous urn compared to an (ambiguity neutral maximum likelihood decision rule is by far the most common supervised classification method used for analyzing satellite im- age data (Richards, 1986). 1763 1774 1922 1931 1934 1949 1954 1961 Decision Rule (y) Y: a random variable that depends on State and explain maximum likelihood decision rule. The maximum likelihood principle is one of the simplest and most intuitive rules of non-deductive inference. The decision rule based on finding the signal that maximizes P(sm|r) is equivalent to finding the signal that maximizes P(r|sm). 2 In this report we present a novel and efficient maximum-likelihood soft-decision decoding algorithm for linear block codes. Bayesian Decision theory If other types of fish are irrelevant: P( w 1 ) + P( w 2) = 1. Statistical Decision Theory maximum likelihood principle. Maximum Likelihood Parameter Estimation 2. put. R( ; 0) R( ; ) for all 2 , and 2. Methods for building decision trees from partially supervised data were  Objective: minimize the probability of a decision error MAP Rule: Maximize the conditional probability that S m AKA Maximum Likelihood (ML) decision rule. The standard implementation of su- pervised maximum likelihood classification requires the selection of "training' samples representing the feature types to be mapped (Lillesand and Kiefer, 1987). Suppose that maximum-likelihood decoding is employed. A set or Maximum likelihood is a point-wise estimate of the model parameters. These are the goal of decision theory in the following sense: based on the observations, a decision rule has to choose an action amongst a set A of allowed decisions or actions. Use MathJax to format equations. ⎩. The Multivariate Normal Distribution 2. Then, the probability of detection (correctly deciding H1) is yielding the following Bayes' decision rule, called the maximum a posteriori  India, Artificial neural network, maximum likelihood, decision rute and K-Means The decision rule based classifier performed equally good for most of the sites,  22 Jan 2018 cations, the maximum likelihood (ML) detector is commonly used to decision rule of ML detection on the jth symbol transmitted by the TX as. Wald's maximin decision rule selects an action that delivers the simple criteria such as maximum likelihood (ML) or maximum a posteriori probability. However, L(𝑥) will be needed if we want to estimate the posterior 𝑃𝜔 |𝑥 which, unlike L𝑥|𝜔1𝑃𝜔1 Likelihood and Bayesian Inference – p. Principles. 4. The optimal observer uses a decision rule that maximizes PC. The goal is, given iid observations , to estimate . 5, 1. Second, you need to compute Pr{y|0} and Pr{y|1}. We are interested in the hypothesis that maximizes the likelihood. The theoretical analysis and the experimental results show that the MANL rule can be used in speaker identification and it is more effective than the ML rule in the approaches based on Gaussian mixture model (GMM) and vector quantisation (VQ). Decision Boundaries in Higher Dimensions 3. The Received Signal Is A Rayleigh Random Variable Y With Conditional Pdfs Given By F_Y|X (y|x) = {y/sigma^2_s + Sigma^2_n E^- Y^2/2(sigma^2_s + Sigma^2_n, When X = 1 Y/sigma^2_n E^-y^2/2 Sigma^2_n, When a) Find the false alarm probability (P FA) for the maximum-likelihood decision rule, and enter the answer, up to two decimal places. Information theory and maximum likelihood learning . edu). To minimize the probability of an error, the decision device   8 Nov 2019 MAP provides an alternate probability framework to maximum likelihood estimation for machine learning. Maximum Likelihood Decoding: Consider a set of possible codewords (valid codewords – set ) generated by an encoder in the transmitter side. Let be distributed according to a parametric family: . Algorithm . The Maximum Likelihood, Bayesian, and Decision Theory are applied and have proven its selves useful and necessary in sciences, such as physics, as well as research in general. In other words, there could be a probabilistically de ned decision rule with an associated P jX. One rule is: round the probability Class 0 if <50%; class 1 if ≥50% This decision boundary is not the The conventional method of estimating a probability prediction model by maximum likelihood (MLE) is a form of maximum score estimation with economic meaning. ! For given probabilities the decision is optimal. (MAP) is  Using this correlation model, we derive the maximum-likelihood (ML) decision rule under the assumption that the fading correlation properties are known, but the  [A2. coding rule, Han and Chen previously deﬁned a new metric for the sequential decoding algorithm [1], and presented a new maximum-likelihood soft-decision sequential decoding al-gorithm (MLSDA). Making statements based on opinion; back them up with references or personal experience. The approach used here is to convert the decoding prob lem into a search problem through a graph which is a trellis for an equivalent code of the transmitted code. The rows ‘flexible’ in Table 1 shows the standardized RMSE ∗ when setting r max =∞ for r min =0, 0. dr. The image will be classified. The maximum-likelihood tree relating the sequences S 1 and S 2 is a straightline of length d, with the sequences at its end-points. The MAP can be used to obtain a point estimate of an unobserved quantity on the basis of empirical data. To chapter 09 decision analysis chapter decision analysis questions states of nature are alternatives available to decision maker. , the frequencies of how often each class label is represented in the training dataset): -At times, states of nature can be assigned probabilities representing their likelihood of occurrence. maximum likelihood decision rule

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text:
# Spacing in flalign affected by mathpazo

I got the following (partly shorted) equation:

I would like to get rid of the space left of the equation (left of Q') so that the label (0.2) fits on the right side.

\documentclass[
a4paper,
12pt,
parskip=half
]{scrbook}
\usepackage[showframe]{geometry}
\usepackage{amsmath}

\newcommand*{\op}[1]{\mathrm{#1}}

\begin{document}
\begin{align}
Q'&(s,x) \nonumber\\
&\mid s=(f,r)() \wedge x=\mathcal{E}(r) &&= [\mathcal{E}(f())]\\
&\mid s=(f,r)(q_1,...,q_z) &&= \left[\mathcal{E}(f(p_1,...,p_z)) \middle\vert
\begin{aligned}
&x' = \op{split}(x,r(q_1,...,q_z)),\\
&(x_1,...,x_z) \in x',\\
&p_i \in Q'(q_i,x_i),1 \leq i \leq z
\end{aligned}
\right]
\end{align}
\end{document}

Using flalign:

Using flalign and \usepackage{mathpazo}:

\documentclass[
a4paper,
12pt,
parskip=half,
]{scrbook}
\usepackage[showframe]{geometry}
\usepackage{amsmath}
\usepackage{mathpazo}

\newcommand*{\op}[1]{\mathrm{#1}}

\begin{document}
\begin{flalign}
Q'&(s,x) \nonumber\\
&\mid s=(f,r)() \wedge x=\mathcal{E}(r) &&= [\mathcal{E}(f())]\\
&\mid s=(f,r)(q_1,...,q_z) &&= \left[\mathcal{E}(f(p_1,...,p_z)) \middle\vert
\begin{aligned}
&x' = \op{split}(x,r(q_1,...,q_z)),\\
&(x_1,...,x_z) \in x',\\
&p_i \in Q'(q_i,x_i),1 \leq i \leq z
\end{aligned}
\right] \label{eq:q'_fr}
\end{flalign}
\end{document}

While flalign solves the problem, mathpazo seems to distort the alignment in the middle of the equation. Why is that?

-
Using flalign environment should do what you desire. –  <NAME> Mar 6 '13 at 21:12
Consider reading section 25 align environments (p 43 onward) of Herbert's mathmode document. It has a number of examples of use of align and friends environments. –  Werner Mar 6 '13 at 21:24
Hmm, flalign really worked :) Thanks both of you! –  neo Mar 6 '13 at 21:26
You mention in a deleted comment "And I don't really want to get rid of the space left of the aligned =s." Why is that? To be blunt, "it don't look nice." –  Werner Mar 6 '13 at 21:26
Because this should model the guard notation of programming languages like Haskell and the right = has a different meaning then the left. It's basically a case environment in reverse. –  neo Mar 6 '13 at 21:27

You can do it by trial and error like this:

\documentclass[
a4paper,
12pt,
]{scrbook}
\usepackage[showframe]{geometry}
\usepackage{amsmath,mathtools}
\usepackage{mleftright}
%\usepackage{mathpazo}

\newcommand*{\op}[1]{\operatorname{#1}}

\begin{document}
\begin{alignat}{2}
\mathmakebox[.5em][r]{Q'}
&(s,x) \nonumber\\
&{\mid}\; s=(f,r)() \wedge x=\mathcal{E}(r)
&&= [\mathcal{E}(f())]
\\
&{\mid}\; s=(f,r)(q_1,\dots,q_z)
&&=
\mleft[
\mathcal{E}(f(p_1,\dots,p_z))
\middle\vert
\begin{aligned}
&x' = \op{split}(x,r(q_1,\dots,q_z)),\\
&(x_1,\dots,x_z) \in x',\\
&p_i \in Q'(q_i,x_i),1 \leq i \leq z
\end{aligned}
\mright]
\end{alignat}
\end{document}

However, this will probably break badly when you either change fonts or page parameters. For instance, the normal page parameters you get without calling geometry will not pose unsurmountable problems, because the text width would be larger. On the contrary, with mathpazo the average width of the glyphs is bigger than Computer Modern's, so the trick won't work and the display simply won't fit.

Here's the example. In such tough cases some manual adjusting is almost always needed.

The very same code with mathpazo; if you look at the result you'll see that the material won't fit no matter how you push left the first "Q".

Again, the same code (but with simply Q' instead of \mathmakebox[.5em][r]{Q'}, without leaving control to geometry:

Just to show how page parameters can influence the final result, here's what I get without leaving control to geometry with your original input:

-
Hmm, to be honest, it's not really satisfying to do stuff like that... I mean, in my eyes, the real problem is: Why isn't align clever enough to make the middle space before the equal signs smaller so that everything fits? –  neo Mar 7 '13 at 20:11

format: plain


text:
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# Is x between 0 and 1? (1) - x < x^3 (2) x < x^2

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Is x between 0 and 1? (1) - x < x^3 (2) x < x^2 [#permalink]

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05 Jan 2011, 11:23
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Is x between 0 and 1?

(1) - x < x^3
(2) x < x^2
[Reveal] Spoiler: OA

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05 Jan 2011, 11:38
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rxs0005 wrote:
Is x between 0 and 1?

(1) - x < x^3
(2) x < x^2

Is $$0<x<1$$?

(1) -x<x^3 --> $$x^3+x>0$$ --> $$x(x^2+1)>0$$ --> $$x>0$$ (as x^2+1 is always positive). Not sufficient.

(2) x<x^2 --> $$x^2-x>0$$ --> $$x(x-1)>0$$ --> either $$x>1$$ or $$x<0$$, so the answer is NO. Sufficient.

Check this for more: x2-4x-94661.html#p731476
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05 Jan 2011, 13:48
Bunuel

thanks , I had a Q , your approach to these DS algebra has always been using algebra itself to get the answers will this ( should ) work most of the times right ?

I prefer your approach as its clear cut rather than plug in values and check more time consuming

do share your thoughts on this

thanks

rxs0005
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05 Jan 2011, 16:28
rxs0005 wrote:
Bunuel

thanks , I had a Q , your approach to these DS algebra has always been using algebra itself to get the answers will this ( should ) work most of the times right ?

I prefer your approach as its clear cut rather than plug in values and check more time consuming

do share your thoughts on this

thanks

rxs0005

Yes, algebraic approach will work most of the times, though sometimes other approaches might be faster or/and simpler, also there are certain GMAT questions which are pretty much only solvable with plug-in or trial and error methods (well at leas in 2-3 minutes).

Check this for more on this issue:
positive-integers-85437.html#p831985
gmat-prep-exponent-101358.html#p797434
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17 Mar 2011, 23:05
pick a positive integer, positive fraction. Negative integer, negative fraction approach to this problem...
statement 1 says that x is > 0 (integer or fraction doesn't matter...it satisfies the condition)
statement 2 says that x is <0 and >1 hence it never satisfies the original problem. NO is an acceptable answer.
Solved. B

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18 Mar 2011, 00:32
(1)

-x < x^3 , this is correct for any +ve fraction, but also for a number > 1, not sufficient.

(2)

x < x^2

This is correct for any number(-ve integer or -ve fraction), but also for a number > 1

x< 0 or x > 1, so answer is B
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Re: Is x between 0 and 1? (1) - x < x^3 (2) x < x^2 [#permalink]

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17 Jul 2012, 13:29
If someone could elaborate the first statement in more detail. From the statement I know that x(x^2 + 1)> 0, so x>0 and x^2 + 1> 1 isn't it?
Furthermore, I can conclude that x could be x>0 and x>1, so because I don't know the exact value of x the statement is insufficient.
Is that correct?
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Re: Is x between 0 and 1? (1) - x < x^3 (2) x < x^2 [#permalink]

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18 Jul 2012, 02:54
Stiv wrote:
If someone could elaborate the first statement in more detail. From the statement I know that x(x^2 + 1)> 0, so x>0 and x^2 + 1> 1 isn't it?
Furthermore, I can conclude that x could be x>0 and x>1, so because I don't know the exact value of x the statement is insufficient.
Is that correct?

Not quite.

The question asks: is $$0<x<1$$?

The first statement says: $$x(x^2+1)>0$$. So, we have that the product of two multiples, $$x$$ and $$x^2+1$$, is positive. Now, since the second multiple is always positive ($$x^2+1= nonnegative +positive=positive$$), then the first multiple must also be positive in order the product to be positive, therefore $$x>0$$. So, from this statement, we cannot say whether $$0<x<1$$ is true.

Hope it's clear.
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20 Jul 2012, 10:06
Bunuel wrote:
rxs0005 wrote:
Is x between 0 and 1?

(1) - x < x^3
(2) x < x^2

Is $$0<x<1$$?

(1) -x<x^3 --> $$x^3+x>0$$ --> $$x(x^2+1)>0$$ --> $$x>0$$ (as x^2+1 is always positive). Not sufficient.

(2) x<x^2 --> $$x^2-x>0$$ --> $$x(x-1)>0$$ --> either $$x>1$$ or $$x<0$$, so the answer is NO. Sufficient.

Dear Bunuel, if x is a negative integer, even then we can conclude that x < x^2

from B, how can we know that x cannot be a negative integer??

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20 Jul 2012, 11:24
asifibaju wrote:
Bunuel wrote:
rxs0005 wrote:
Is x between 0 and 1?

(1) - x < x^3
(2) x < x^2

Is $$0<x<1$$?

(1) -x<x^3 --> $$x^3+x>0$$ --> $$x(x^2+1)>0$$ --> $$x>0$$ (as x^2+1 is always positive). Not sufficient.

(2) x<x^2 --> $$x^2-x>0$$ --> $$x(x-1)>0$$ --> either $$x>1$$ or $$x<0$$, so the answer is NO. Sufficient.

Dear Bunuel, if x is a negative integer, even then we can conclude that x < x^2

from B, how can we know that x cannot be a negative integer??

Not quite sure understand your question.

We are asked: is $$0<x<1$$? From (2) we have that $$x>1$$ or $$x<0$$, which means that $$x$$ could be any negative number (including integers) as well as any number greater than 1 . Hence the answer to the question whether $$0<x<1$$ is NO.
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20 Jul 2012, 11:54
oops !! i am really sorry !! Got it now.. I was out of my mind and asked such a silly question.. effect of a hectic day.. sorry again

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Re: Is x between 0 and 1? (1) - x < x^3 (2) x < x^2 [#permalink]

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Re: Is x between 0 and 1? (1) - x < x^3 (2) x < x^2 [#permalink]

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18 Jun 2017, 23:17
dear Bunuel

could you please explain the statement 2 solution in more detail. i followed till the eq x(x-1) > 0..after it how are we moving to x<0 or X>1

Thanks

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Re: Is x between 0 and 1? (1) - x < x^3 (2) x < x^2 [#permalink]

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19 Jun 2017, 03:59

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Re: Is x between 0 and 1? (1) - x < x^3 (2) x < x^2   [#permalink] 19 Jun 2017, 03:59
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# Of all probability matrix $P$ having stationary distribution $\pi$, find the one having smallest diagonal

I am requesting your help today trying to solve a somewhat odd problem. Is there a way to find through some numerical algorithm such as Newton's method the stochastic matrix $$\boldsymbol{P}$$ having stationary distribution $$\boldsymbol{\pi}$$ (column vector, given as an input) and lowest diagonal (whose elements are closest to 0)?

I initially thought about minimizing the following loss function:

$$L(\boldsymbol{P}) = {(\mathrm{diag}(\boldsymbol{P}))}^\mathsf{T} \mathrm{diag}(\boldsymbol{P}) + \lambda_1 {||\boldsymbol{P}^\mathsf{T}\boldsymbol{\pi} - \boldsymbol{\pi}||}^2 + \lambda_2 {||\boldsymbol{P}\boldsymbol{e} - \boldsymbol{e}||}^2,$$

where $$\mathrm{diag}$$ returns the diagonal of its argument, $$\lambda_1$$ and $$\lambda_2$$ are both Lagrange multipliers, and $$\boldsymbol{e}$$ is a vector filled with 1 (same dimensions as $$\boldsymbol{\pi}$$).

As far as I know, this is a case of differentiating a scalar ($$L$$) with respect to a matrix ($$\boldsymbol{P}$$). I think it may involve something called tensors, but to be honest I have little to zero experience with this, and even less once you throw in Lagrange multipliers.

I did some calculations, and it would appear the differential is given by

$$\tfrac{\partial{}L}{\partial{}\boldsymbol{P}} = 2\mathrm{\boldsymbol{D}}(\boldsymbol{P}) - 2\lambda_1\boldsymbol{\pi}\boldsymbol{\pi}^\mathsf{T}(\boldsymbol{I} - \boldsymbol{P}) - 2\lambda_2 \boldsymbol{e} \boldsymbol{e}^\mathsf{T} (\boldsymbol{I} - \boldsymbol{P}^\mathsf{T}),$$

which at some point involved an "outer product" or "Kronecker product", but could be simplified to that. The $$D$$ function outputs a diagonal matrix having same diagonal as its argument. In turn, the Hessian matrix (matrix of second order derivatives) would be given by

$$\tfrac{\partial{}^2L}{\partial{}\boldsymbol{P}\partial{}\boldsymbol{P}^\mathsf{T}} = 2\boldsymbol{I} + 2\lambda_1\boldsymbol{\pi}\boldsymbol{\pi}^\mathsf{T} + 2\lambda_2 \boldsymbol{e} \boldsymbol{e}^\mathsf{T}.$$

I tried inputting these in a "Newton's method-like" program, but all it outputted was gibberish.

All of this is a bit out of my league, but I really tried to make it work by myself before running here. I would be so grateful if someone could help me out. I know a solution exists, because Excel's solver is able to find solutions (don't ask why I use Excel, in this case I don't have a choice).

Thanks,

RSMax

P.S. Just in case there would be multiple definitions around, by "stochastic matrix" I mean a square matrix whose elements are probabilities, and whose rows all sum to 1.

P.P.S. By stationary distribution, I am referring to "long terms odds", as given by:

$$\boldsymbol{\pi} = {(\boldsymbol{I} + \boldsymbol{E} - \boldsymbol{P}^\mathsf{T})}^{-1} \boldsymbol{e},$$

where $$\boldsymbol{E}$$ is a square matrix filled with ones (same dimensions as $$\boldsymbol{P}$$).

EDIT No1: Fixed some typos and added some clarifications regarding what is an input.

EDIT No2: Substituting the sum of the squares of the diagonal elements for the trace indeed made this a linear problem which I was then able to solve using the simplex's "Big M" method. Works like a charm, even on Excel! My VBA code:

Public Const MAXVALUE As Double = 1.79769313486231E+308 'Arbitrary large value
Public Const EPSILON As Double = 0.0001 'Tolerance parameter for the Simplex method
Public Const BIGM As Double = 100# 'Big M constant for the Simplex method

Public Function Simplex(odds As Range) As Variant
'///////////////////////////////////////////////////////////////////////
'/// :function:    Simplex
'/// :scope:       Public
'/// :description: For odds.Cells.Count > 10, there can be instability issues.
'/// :return:      Wrapper (data validation) for the Simplex_ function.
'/// :odds:        An array of long-term odds (Range).
'///////////////////////////////////////////////////////////////////////
Dim n As Long
n = odds.Cells.Count

With WorksheetFunction
If .Count(odds) <> n Then
Simplex = CVErr(xlErrNum)
ElseIf .CountIf(odds, ">1") > 0 Then
Simplex = CVErr(xlErrValue)
ElseIf .CountIf(odds, "<0") > 0 Then
Simplex = CVErr(xlErrValue)
ElseIf Abs(.Sum(odds) - 1) > EPSILON Then
Simplex = CVErr(xlErrValue)
Else
Simplex = Simplex_(odds)
End If
End With
End Function

Private Function Simplex_(pi As Range) As Double()
'///////////////////////////////////////////////////////////////////////
'/// :function:    Simplex_
'/// :scope:       Public
'/// :description: Implements the "Big M" variant of the Simplex method.
'/// :return:      The lowest trace probability matrix having stationary distribution Odds.
'/// :pi:          An array of long-term odds.
'///////////////////////////////////////////////////////////////////////
Dim StopFlag As Boolean
Dim output() As Double
Dim MinRatio As Double
Dim MaxCost As Double
Dim factor As Double
Dim Ratio As Double
Dim Pivot As Double
Dim x() As Variant
Dim Row As Long
Dim Col As Long
Dim n As Long
Dim i As Long
Dim j As Long
Dim k As Long

n = pi.Cells.Count
'As we're looking for a matrix, the number of unknowns is n^2 + others for the constraints
ReDim x(0 To 2 * n + 1, 0 To n * n + 2 * n + 2) As Variant
ReDim output(1 To n, 1 To n) As Double
x(0, 0) = "BASIS"
x(0, 1) = "Z"

For i = 1 To n * n
Row = 1 + (i - 1) \ n
Col = 1 + (i - 1) Mod n
x(0, i + 1) = "p" & Row & Col
Next

For i = 1 To n
Col = 1 + (i - 1) Mod n
x(0, i + n * n + 1) = "c.r" & Col
x(0, i + n * n + n + 1) = "c.pi" & Col
x(i, 0) = "c.r" & Col
x(i + n, 0) = "c.pi" & Col
Next

x(0, n * n + 2 * n + 2) = "RHS" 'Right-hand side
x(2 * n + 1, 0) = "COST"

For i = 1 To n
For j = 1 To n * n
Row = 1 + (j - 1) \ n
x(i, j + 1) = IIf(Row = i, 1, 0)
Next

x(i, n * n + 2 * n + 2) = 1

For j = 1 To n * n
Row = 1 + (j - 1) \ n
Col = 1 + (j - 1) Mod n
x(i + n, j + 1) = IIf(Col = i, pi(Row).Value, 0)
Next

x(i + n, n * n + 2 * n + 2) = pi(i).Value
Next

For i = 1 To n * n
Row = 1 + (i - 1) \ n
Col = 1 + (i - 1) Mod n
x(2 * n + 1, i + 1) = IIf(Row = Col, -1, 0)
Next

For i = 1 To 2 * n
For j = i + 1 To 2 * n
x(i, n * n + 1 + j) = 0
x(j, n * n + 1 + i) = 0
Next

x(i, 1) = 0
x(i, n * n + 1 + i) = 1
x(2 * n + 1, n * n + 1 + i) = -BIGM
Next

x(2 * n + 1, 1) = 1
x(2 * n + 1, n * n + 2 * n + 2) = 0

For k = 1 To 2 * n
Col = n * n + 1 + k
MinRatio = MAXVALUE

For i = 1 To 2 * n
If x(i, Col) > 0 Then
Ratio = x(i, n * n + 2 * n + 2) / x(i, Col)

If Ratio < MinRatio Then
MinRatio = Ratio
Row = i
End If
End If
Next

Pivot = x(Row, Col)

For i = 1 To 2 * n + 1
If i <> Row Then
factor = -x(i, Col) / Pivot

For j = 1 To n * n + 2 * n + 2
x(i, j) = x(i, j) + factor * x(Row, j)
Next
End If
Next

For j = 1 To n * n + 2 * n + 2
x(Row, j) = x(Row, j) / Pivot
Next
Next

Do
MaxCost = -MAXVALUE

For i = 1 To n * n
If x(2 * n + 1, i + 1) > MaxCost Then
MaxCost = x(2 * n + 1, i + 1)
Col = i + 1
End If
Next

If MaxCost <= 0 Then
Exit Do
Else
MinRatio = MAXVALUE

For i = 1 To 2 * n
If x(i, Col) > 0 Then
Ratio = x(i, n * n + 2 * n + 2) / x(i, Col)

If Ratio < MinRatio Then
MinRatio = Ratio
Row = i
End If
End If
Next

Pivot = x(Row, Col)

For i = 1 To 2 * n + 1
If i <> Row Then
factor = -x(i, Col) / Pivot

For j = 1 To n * n + 2 * n + 2
x(i, j) = x(i, j) + factor * x(Row, j)
Next
End If
Next

For j = 1 To n * n + 2 * n + 2
x(Row, j) = x(Row, j) / Pivot
Next

x(Row, 0) = x(0, Col)
End If
Loop

For i = 1 To n * n
Row = 1 + (i - 1) \ n
Col = 1 + (i - 1) Mod n

For j = 1 To 2 * n
If x(j, 0) = "p" & Row & Col Then
output(Row, Col) = x(j, n * n + 2 * n + 2)
Exit For
End If
Next
Next

Simplex_ = output
End Function

• Usually this sort of matrices is called "stochastic". Jun 11, 2020 at 7:01
• @<NAME>lka You are perfectly right, in fact I initially learned about these in a class named "Stochastic Processes", so in makes so much sense, but since I'm translating everything in my head as I write, I simply didn't make the connection right away. Jun 11, 2020 at 17:16

If you make the objective to minimize the sum of the diagonal entries (i.e. the trace), your problem becomes a linear programming problem, solvable with readily available software (I think even Excel). In many cases the optimal solution will have all diagonal entries $$0$$.

EDIT: It may help to think of the problem this way. There are $$n$$ people, numbered from $$1$$ to $$n$$. Each person $$i$$ has a nonnegative amount $$\pi_i$$ of money, and we want to move all the money around so that everyone ends up with the same amount they started with. $$P_{ij}$$ is the fraction of person $$i$$'s money that goes to person $$j$$. Of course this will not be possible if one person has more than half the total amount of money. Otherwise it will be possible, by the following algorithm. We arrange the people in order clockwise from richest to poorest around a circular table, and everyone puts their money on the table in front of them (let's say in coins of a given denomination). The richest persion (#1) takes the amount $$\pi_1$$ off the table, starting with #2's coins, which come after his own coins. Each person then continues, taking the correct number of coins starting where the previous one left off. By induction, for $$k$$ from $$2$$ to $$n$$, when #k's turn comes, all his coins have already been taken.

EDIT: Here is a more explicit version of the algorithm I alluded to above.
We may assume that $$\pi_1 \ge \pi_2 \ge \ldots \pi_n > 0$$. If necessary, we reorder the rows and columns to make them decrease. If some $$\pi_i$$ are $$0$$, we make the corresponding columns all $$0$$ and put a $$1$$ arbitrarily (off the diagonal) in each of the corresponding rows, and follow the following algorithm for the rows and columns where $$\pi > 0$$.

If $$\pi_1 > 1/2$$, we let $$P_{11} = 2 - 1/\pi_1$$, $$P_{i1}=1$$ and $$P_{1i} = \pi_i/\pi_1$$ for $$i > 1$$, and all other $$P_{ij} = 0$$. It is easy to check that this works, and no solution can have $$P_{11} < 2 - 1/\pi_1$$.

Now suppose $$\pi_1 \le 1/2$$. Let $$S_k = \sum_{i=1}^k \pi_i$$ be the partial [[sums of $$\pi$$, so $$S_0 = 0$$ and $$S_n = 1$$. Let $$r_{ij}$$ be the length of the intersection of the intervals $$[S_{i-1}, S_i]$$ and $$[S_{j-1}+\pi_1, S_j + \pi_1] \mod 1$$. Then $$P_{ij} = r_{ij}/\pi_i$$.

For example, suppose $$\pi = [0.3, 0.24, 0.24, 0.16, 0.06]$$. The partial sums are $$[0, 0.3, 0.54, 0.78, 0.94, 1]$$. The shifted partial sums $$S_i + \pi_1$$ are $$[.3, .0.6, 0.84, 1.08, 1.24, 1.30]$$. Then, for example, the intersection of $$[S_{2} + \pi_1, S_3 + \pi_1] \mod 1 = [.84, 1] \cup [0, .08]$$ and $$[S_0, S_1]=[0,0.3]$$ has length $$0.08$$, so $$P_{13} = 0.08/0.3 = 4/15$$, while the intersection of $$[S_2+\pi_1, S_3 + \pi_1] \mod 1$$ with $$[S_3, S_4] =[.78, .94]$$ has length $$.1$$ so $$P_{43} = .1/.16 = 5/8$$ and the intersection of $$[S_2+\pi_1, S_3 + \pi_1] \mod 1$$ with $$[S_4, S_5] = [.94, 1]$$ has length $$.06$$ so $$P_{53} = .06/.06 = 1$$. The resulting matrix is $$\pmatrix{ 0 & 0 & 4/15 & 8/15 & 1/5\cr 1 & 0 & 0 & 0 & 0\cr 1/4 & 3/4 & 0 & 0 & 0\cr 0 & 3/8 & 5/8 & 0 & 0\cr 0 & 0 & 1 & 0 & 0\cr}$$

• Alternatively, an objective to minimize $\sum_i P_{i,i}^2$ yields a (convex) quadratic programming problem. Jun 11, 2020 at 15:04
• I totally agree with the EDIT section, this is something I have observed in practice using Excel's solver. What remains encouraging is that through this scheme, $P_{11}$ will nonetheless be less than $\pi_1$, given that $\pi_1 \geq 0.5$, ergo the diagonal has still been minimized in some way. Jun 11, 2020 at 17:03
• Whether I minimize the trace or the sum the squares of all elements on the diagonal, my current issue (under the Newton's method approach) is that my "main" unknown is a matrix ($\boldsymbol{P}$, whereas the Lagrange multipliers are scalars, and I'm confused about how to wrap those 3 under a common structure so that I can recursively update it and converge toward a solution. Should I add rows and columns to the unknown $\boldsymbol{P}$ matrix so that it incorporates $\lambda_1$ and $\lambda_2$ on the main diagonal? Jun 11, 2020 at 17:08
• If by linear programming you mean something else than Newton's method, I am very much interested. As I need to ultimately code the result, anything easier and/or more efficient than Newton's method will be received with open arms. Would you mind elaborating a bit? Thank you sir. Jun 11, 2020 at 17:14
• Linear programming in Excel. But the algorithm I presented in the EDIT is simpler (though maybe not so easy to program in Excel). Maybe I should make it more explicit. Jun 11, 2020 at 19:49

The following is too large for a comment and definitely not an answer.

Fix $$n \in \mathbb{N}$$ and let $$\cal{P}$$ be the set of stochastic $$n \times n$$-matrices with $$\text{diag}(P) = 0$$. Let further $$\mathcal{E} := \left\{\pi \in \mathbb{R}^n \colon \pi_i \geq 0, \sum \pi_i = 1, \exists P \in \mathcal{P} \text{ with } \pi \cdot P = \pi\right\},$$ each $$\pi$$ being a row vector. Then both $$\mathcal{P}$$ and $$\mathcal{E}$$ are compact convex subsets of $$\mathbb{R}^{n \times n}$$ resp. $$\mathbb{R}^n$$ and not empty if $$n \geq 2$$ and for $$n = 2$$ we have $$\mathcal{E} = \{(0.5,0.5\}$$. The problem is that $$(1,0,\ldots,0), \ldots, (0,0,\ldots,1) \not\in \mathcal{E}$$.

But for large $$n$$ it seems (I have no proof) it seems that almost any random stochastic $$\pi$$ is in $$\mathcal{E}$$. For this case your original problem is equivalent to "Find a $$P \in \mathcal{P}$$ with $$\pi \cdot P = \pi$$". Here the explicit construction of a homogeneous Markov chain with transition probability matrix $$P$$ (appropriately chosen) may be helpful to get a "feeling" for the solution. Of course this is impossible to formalize.

It seems to me that for $$n$$ large using numerical methods may be too heavy for this sort of problem. Edit (I've just seen the answer of <NAME>) A reformulation as a linear programm with criterion $$\sum P_{ii} = \min!$$ and variables $$P_{ij}$$ is possible.

• But $\pi_i$ is given, right? Jun 11, 2020 at 15:01
• Right, I just edited my answer. Jun 11, 2020 at 15:04
• I tried with a $10\times{}10$ matrix using Excel's solver and it took about half an hour. That's much too slow for my purpose. I doubt the order of the $\boldsymbol{P}$ matrix will ever exceed $n = 10$ states though. In fact, I won't allow it too. Jun 11, 2020 at 17:11
• If you have access to a unix/linux computer you should try the glpk package on mirrors.ocf.berkeley.edu/gnu/glpk It's very much faster than excel. After installing this package you have glpsol. With glpsol you can solve among others linear programs. There are examples where you can start Jun 11, 2020 at 17:23
• @RSMax: If you have a windows computer, there is a windows version of glpk: winglpk.sourceforge.net There is glpsol.exe in the package. I don't know anything about the quality of this solver. It should be much faster than excel, but you have to invest some time. Jun 12, 2020 at 11:50


text:
Question

# Resultant of two vectors $$\vec{A}$$ and $$\vec{B}$$ is of magnitude P. If $$\vec{B}$$ is reversed, then resultant is of magnitude Q. What is the value of P$$^{2}$$ + Q$$^{2}$$ ?

A
2(A2 + B2)
B
2(A2-B2)
C
A2-B2
D
A2 + B2

Solution

## The correct option is B 2(A$$^{2}$$ + B$$^{2}$$)Let $$\theta$$ be angle between $$\vec{A}$$ and $$\vec{B}$$.$$\therefore$$ Resultant of $$\vec{A}$$ and $$\vec{B}$$ is $$P =\sqrt{A^{2}+B^{2}+2ABcos\theta}$$   ---(i)When $$\vec{B}$$ is reversed, then the angle between $$\vec{A}$$ and - $$\vec{B}$$ is (180$$^{\circ}$$ - $$\theta$$).Resultant of $$\vec{A}$$ and - $$\vec{B}$$ isQ = $$\sqrt{A^{2}+B^{2}+2ABcos(180^ {\circ} - \theta)}$$Q = $$\sqrt{A^{2}+B^{2}+2ABcos\theta}$$      --(ii)Squaring and adding (i) and (ii), we get$$P^{2} + Q^{2} = 2(A^{2} + B^{2})$$Physics

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text:
# How should I analyze this design with no replication?

I have data from an experiment where I measured the $CO_2$ from:

• 4 different dissolved organic carbon (DOC) solutions
• each at 4 different concentrations
• each in light and in dark.

Each combination of DOC source, concentration, and light occurred only once (e.g., only one jar each of DOC-A: conc-A: Dark, and one jar for DOC-B: conc-A: Dark, etc...) but a series of 4 measurements was taken from each combination (i.e., jar).

The production of $CO_2$ was linear over the 4 time points so, the response variable can be represented as a single rate from each jar.

I am interested in the effect of light, DOC source, and concentration on $CO_2$ production, plus interactions. I am not necessarily interested in the effect of the different time pointsGiven that each experimental unit was unreplicated, how can I best analyze this experiment?

I agree with your assessment that the 4 measurements of a jar are pseudo-replicates, and the easiest way of dealing with it is to combine them into one value (I think you are creating a rate, but any meaningful combination could be used). Let's concentrate on modeling one $Y$ value per jar. Consider the ANOVA table for the sources of variability:

Source       Degrees of freedom
DOC             4-1=3
Conc            4-1=3
Light           2-1=1
DOC*Conc        3*3=9
DOC*Light       3*1=3
Conc*Light      3*1=3
DOC*Conc*Light  3*3*1=9
Residual        0
-------------------------
Total           32-1=31

As you can see, if you include all 2- and 3-way interactions, then the observed value for each jar is predicted perfectly, and there are no degrees of freedom left to estimate the residual variability. The only alternative to giving up at this point is to assume that there is no 3-way interaction, and use the corresponding term as the residual. The exact implementation depends on the software, but it could be done by hand from the ANOVA table. If this assumption is not reasonable a-priori, then you have to replicate the experiment.

I think you're believing "replication" in this instance to mean sampling a unit with the exact same concentration and light which is inherently continuously valued. Obviously, this is impossible to do with any moderately sensitive equipment. It's also something that arises in real life, no two units are the same, but intuitively one can pool adjacent data to make apples-to-apples comparisons between units.

You should handle this effect using methods for continuously valued data. This sounds to me like a repeated measures ANOVA problem or a linear mixed model, accounting for random effects due to "jar". If you adjust continuously for darkness and concentration, the model naturally borrows information across groups and estimates a consistent trend, regardless of whether there were replications within any one continuous measure.

• Thanks for you input. The concentration, light, and DOC source were all experimentally controlled and thus do not vary to the degree that we can measure them. So they are continuous variables as you say be in our experiment we do not have a measure of this variation. For example, all of the low DOC concentrations were exactly 2 mg/L, all of the high were 20 mg/L etc... Since each jar is an experimental unit and is not replicated, isn't it true that there is no replication? – DQdlM Jul 2 '12 at 17:39
• No, the jar itself is not the experimental unit. I think of experimental units as rows in a dataset. Your fundamental level of observation is the measurement you take on the jar, for some combination of controlled, uncontrolled, and semi-controlled environmental variable. This is the fundamental concept of repeated measures. – AdamO Jul 2 '12 at 19:06


text:
## Non linear pseudo-bosons versus hidden Hermiticity

<NAME>, <NAME>

The increasingly popular concept of a hidden Hermiticity of operators (i.e., of their Hermiticity with respect to an {\it ad hoc} inner product in Hilbert space) is compared with the recently introduced notion of {\em non-linear pseudo-bosons}. The formal equivalence between these two notions is deduced under very general assumptions. Examples of their applicability in quantum mechanics are discussed.

http://arxiv.org/abs/1109.0605
Mathematical Physics (math-ph); Functional Analysis (math.FA); Quantum Physics (quant-ph)


text:
# What is the domain and range of y=-|x|-9?

Feb 4, 2018

Domain: x inℝ (all real numbers)

Range: $y \le - 9$

#### Explanation:

The domain of the function $y = - | x | - 9$ is all real numbers because any number plugged in for $x$ yields a valid output $y$.

Since there is a minus sign in front of the absolute value, we know that the graph "opens downward," like this:
graph{|x|*-1 [-10, 10, -5, 5]}
(This is the graph of $- | x |$.)

This means that the function has a maximum value. If we find the maximum value, we can say that the function's range is $y \le n$, where $n$ is that maximum value.

The maximum value can be found by graphing the function:

graph{|x|*-1-9 [-10, 10, -15, -5]}

The highest value that the function reaches is $- 9$, so this is the maximum value. Finally, we can say that the range of the function is $y \le - 9$.


text:
# An algebra problem by <NAME>

Algebra Level 3

In $$a{ x }^{ 2 }+bx+c=0$$ , if $$a+c= b$$ then the roots of the equation are

×

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text:
# Why does this floor() call appear to have no effect?

I'm trying to get an effect where my material forms a mosaic which is aligned to the material. So I slapped a simple quad in the scene with a new material on it and set a new shader to that material.

My inputs (at least one thing in here will be removed later since it turns out I'm not using it):

        struct Inputs
{
float4 vertex : POSITION;
float2 uv_panel : TEXCOORD0;
};
struct FragmentData
{
float4 vertex : POSITION;
float4 uv_grab : TEXCOORD1;
};

Vertex function:

        FragmentData Vertex(Inputs i)
{
FragmentData o;

o.vertex = UnityObjectToClipPos(i.vertex);

// what I actually want to do (the exact maths might be
//   wrong still):
float2 uv_panel = floor(i.uv_panel * _Pixels) / _Pixels;

// Convert that UV value for inside the panel
// to a UV value for the grab texture
float4 object_coords = float4(0, 0, 0, 1);
object_coords.xy = uv_panel - 0.5;
float4 clip_coords = UnityObjectToClipPos(object_coords);
o.uv_grab = ComputeGrabScreenPos(clip_coords);

return o;
}

And then the fragment function is pretty trivial:

        half4 Fragment(FragmentData i) : COLOR
{
return tex2Dproj(_GrabTexture, UNITY_PROJ_COORD(i.uv_grab));
}

The end result is basically equivalent to doing nothing at all. I can see the objects behind the quad as if it weren't there. Upside is that it means my transformations are correct. Downside is that it means the mosaic effect somehow isn't being applied.

To check whether the floor call was working (and because I currently know no better way to debug this crap) I tried applying it like so:

            float2 uv_panel = floor(i.uv_panel);

This has no effect either.

This seems odd to me, because I thought uv_panel values in x and y would range from 0.0 to 1.0, so almost all values should floor to 0.0.

When I do this:

            float2 uv_panel = float2(0, 0);

It acts like I expected, with the bottom of the grab texture being smeared up the panel with the panel's alignment. So it's definitely using whatever value I put in here.

At this point I have no idea how to proceed because it's like my assumptions about the values in a float2:TEXCOORD0 are completely incorrect, or the floor() function does something completely different to what I expected.

What am I doing wrong here?

You're using floor in your vertex shader, which runs only once for each vertex.

So if you're rendering a standard quad or a blit pass, it only sees 4 UV values: (0, 0), (0, 1), (1, 0), (1, 1). And for numbers like these (if _Pixels is an integer), your formula is a no-op:

   floor( (0, 1) * _Pixels ) / _Pixels
= floor( (0, _Pixels) ) / _Pixels
= (0, _Pixels) / _Pixels
= (0, 1)

For each fragment rasterized to the screen, these UV values output by the vertex shader are interpolated. The interpolation unit has no knowledge of how those numbers were calculated, so it doesn't know to perform a floor evaluation in between.

Instead, if you want to get a flooring or any other effect that varies pixel-by-pixel across a triangle, rather than only vertex-by-vertex, you'll want to put it into the fragment shader, which runs for every rasterized fragment individually.

• Ah....a........ That makes more sense now. Though I could swear I had seen some other shaders doing between-vertex things in a similar way. :/ Feb 28, 2018 at 12:34
• Some effects interpolate safely, or sometimes we're using a mesh that's subdivided densely enough that evaluating per vertex is enough. Feb 28, 2018 at 12:40