Author: haroonkhan
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Different types of variables
In standard inferential statistics one typically assumes that data consist of real or integer numbers. However, data may be qualitative as well, and the more dimensions we have, the more likely the joint presence of quantitative and qualitative variables will be. In some cases, dealing with qualitative variables is not that difficult. For instance, if…
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Complexity and redundancy
Visualization is not the only reason why we need data reduction methods. Quite often, multivariate data stem from the administration of a questionnaire to a sample of respondents; each question corresponds to a single variable, and a set of answers by a single respondent is a multivariate observation. It is customary to ask respondents many…
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Visualization
The first and most obvious difficulty we face with multivariate data is visualization. If we want to explore the association between variables, one possibility is to draw scatterplots for each pair of them; for instance, if we have 4 variables, we may draw a matrix of scatterplots, like the one illustrated in Fig. 15.1. The matrix…
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ISSUES IN MULTIVARIATE ANALYSIS
In the next sections we briefly outline the main complication factors that arise when dealing with multidimensional data. Some of them are to be expected, but some are a bit surprising. Getting aware of these difficulties provides the motivation for studying the wide array of sometimes quite complex methods that have been developed. Fig. 15.1 A matrix…
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Introduction
Multivariate analysis is the more-or-less natural extension of elementary inferential statistics to the case of multidimensional data. The first difficulty we encounter is the representation of data. How can we visualize data in multiple dimensions, on the basis of our limited ability to plot bidimensional and tridimensional diagrams? In Section 15.1 we show that this is just…
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A financial application: The Black–Litterman model
We considered portfolio optimization in Example 12.5 and in Section 13.2.2. For the sake of convenience, let us reconsider the problem here. We must allocate our wealth among n risky assets and a risk-free one. The returns of the risky assets are a vector of random variables with expected value μ and covariance matrix Σ; let rf be the return of the risk-free asset. Let w0 be…
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Bayesian estimation
Consider the problem of estimating a parameter θ, characterizing the probability distribution of a random available X. We have some prior information about θ, that we would like to express in a sensible way. We might assume that the unknown parameter lies anywhere in the unit interval [0, 1], or we might assume that it is close to…
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SUBJECTIVE PROBABILITY: THE BAYESIAN VIEW
We took a rather standard view. On the one hand, we have introduced events and probabilities according to an axiomatic approach. On the other hand, when dealing with inferential statistics, we have followed the orthodox approach: Parameters are unknown numbers, that we try to estimate by squeezing information out of a random sample, in the…
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DYNAMIC FEEDBACK EFFECTS AND HERDING BEHAVIOR
Game theory in its simplest form does not consider dynamics, as it revolves around a static equilibrium concept: It posits a situation such that no player has an incentive to deviate. But how is that equilibrium reached dynamically? And what about the disorderly interaction of many stakeholders, maybe stockholders in financial markets? Addressing such issues is beyond…
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BRAESS’ PARADOX FOR TRAFFIC NETWORKS
The result of the collective interaction of noncooperative players may be occasionally quite surprising. We illustrate here a little example of the Braess’ paradox for traffic networks.18 Imagine a traffic network consisting of links such as road segments, bridges, and whatnot. Most of us had some pretty bad experiences with traffic jams. Intuition would suggest that…