Multivariate refers to analyses where there is >1 response / dependent variable of interest in the statistical analysis. This can be contrasted w/ *multivariable* analyses, which typically implies >1 predictor / independent variable.

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Multivariate distributions, k-margin's, Bayesian Networks, and dependency

Suppose that we have a multivariate probability distribution with four variables, X, Y, Z, and W. Let us assume that the joint distribution can be factored as follows: $$ f_{X,Y,Z,W}(x,y,z,w) = ...
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Studying complex systems (complexity)

Complex socio-technical systems is one my research interests. Since I plan to further study such systems and related phenomena, I've done a bit of reading and ran across various books, such as Bar-Yam ...
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How to account for repeated measurements and unequal sample sizes in RDA

I have a question going in the same direction as this one: Restricted Permutations However, I have a dataset of multiple Locations from which I have samples of sometimes 5 sometimes only 3 years. Not ...
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Hotelling $T^2$ distribution

Suppose $d \sim N_p(k\mu_0,\Sigma)$ and $M\sim W_p(\Sigma,m)$ are independent. Here $\mu_0\neq 0$ is a known $p$-vector. $k$ (scalar), $\Sigma$ are unknown. Find the distribution of ...
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Entropy of multivariate gaussian mixture random variable

Short: ${\bf X} \sim N({\bf 0},{\bf I}+{\bf I}_j)$; ${\bf I}_j\in S=\{I_j: I_j$ is diagonal and $ I_j \succeq 0\}, |S|=K$, and $j\sim U(1,K)$. What is $h({\bf X})$? What happens when ...
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15 views

log multivariate normal differentiation (MLE)

I've come across a lot of explanations of how to differentiate the multivariate normal, but they all appear to skip the step that I'm stuck on. Here's what I've got so far. By logging and removing ...
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29 views

How to generate multidimensional data with specific clustering properties?

In section 5.A of a research paper the researcher used the following synthetic datasets: GAUSS consisted of six Gaussian clusters with identity covariance, each with 500 points in five dimensions. ...
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How would you set up this multilevel model?

I have data at multiple levels and am trying to figure out how best to structure and analyze the data. Participants completed five measurement occasions in each of four conditions (that is, it was a ...
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22 views

On FIML assumptions

In Hayashi's Econometrics, page 529, he states one of the assumptions we need for the FIML estimator. My doubt is in the third line of point 1). He says that the vector ...
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13 views

Selecting best dependent variable in Repeated measures and multivariate data

I have a number of outcome variables both continuous, scores and categorical. I am trying to find the best methodology to select the best variable(s) among the many dependent variables that gives me ...
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Interdependence and Significance of Cluster Analysis Results

This question concerns ionic concentration data for in-stream water samples. The samples were taken longitudinally along several different streams; each of these streams has a variety of pollutant ...
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2answers
108 views

Ax = b. How can I estimate A, given multiple data vectors of x and b?

I have a problem and I believe there must be a machine learning technique to solve it, but I am new to machine learning and I have no idea where to start. So, I have multiple multivariate parameter ...
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Can we use Principal component analysis for identifying important Independent variables (X's) [duplicate]

I have read principal component analysis is mainly used for variable reduction. I have the concept that we are converting the variables in to new principal components using orthogonal transformation ...
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Which clustering method should I use and other useful statistical tools for grouping

I have 37 plant species (rows) and up to 18 corresponding traits (col). I would like to see if there are consistent species-wide attributes that are repeated in different species and can therefore be ...
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7 views

How to evaluate enrichment of features in groups of samples?

I have a matrix composed by features on rows and samples on columns. Each value in the matrix correspond to a value of activation of the feature $i$ in the sample $j$. I want to evaluate enrichment ...
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44 views

Identification problems with a structural equation model of experimental data

I have performed an experiment in which I manipulated three factors and I would like to model latent variables that those factors affect and then estimate the effects of the latents on response ...
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slope of correlation coefficient affected by independent variable?

I am interested in testing if the correlation coefficient between 2 dependent variables is significantly affected by an independent variable (which has two levels). I don't think calculating a partial ...
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46 views

R: how to do statistical inference on multiple dependent and independent variables?

In the past I've run separate multiple regression models for many correlated independent variables and one dependent variable. For this I've been using the R package multtest ...
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How to calculate Focused Information Criterion in R for Cox proportional hazards models? [migrated]

I am utilising R to perform a multivariate Cox survival regression for a research project. As I have many possible interchangeable variables in the model, I was wondering how to apply the Focused ...
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Inference in bivariate continuous distributions

We have two nodes in different positions, which are represented by two random variables X,Y, with two prior bivariate continuous distributions, p_X , p_Y. f(X,Y,U,V) is a constraint on both ...
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26 views

Identities in a VAR model

I am working on a VAR model where one of the equations is an identity. For example: $$ \begin{cases} A_t = \alpha_{11} + \alpha_{12} A_{t-1} + \alpha_{13} B_{t-1} + \alpha_{14} C_t + ...
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109 views

A Book for Multiple Regression and Multivariate analysis

I have done a course in Simple Linear Regression and I am aware of linear statistical models (I follow the book by C.R. Rao). Keeping this background in mind, please suggest some good book(s) for ...
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Clustering MVT response data by audience

We are doing multivariate testing 4 possible values for each of 3 variables on a landing page: (a) offer presented, (b) product claim message displayed and (c) image used. So, 12 dummy variables, 64 ...
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Analyzing Social Behavior in Bats

I had a question about what might be the best way to approach one of my datasets. I had been using one method, but a recent talk with one of my professors made me think that perhaps it wasn't the best ...
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Can I compare Mahalanobis distances from different distributions?

I have a multivariate dataset representing multiple locations, each of which has a set of reference observations and a single test observation. For each location, I would like to measure how anomalous ...
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Adjusting Monte Carlo estimates to generate correct even moments - improving on antitthetic draws

If I want to generate a matrix 10,000 (row) samples of 3 uniform (uncorrelated) variables it is trivial to use antithetic draws to ensure the odd moments such as the mean equal their "true" value. ...
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Chi square test of independence with multivariate nominal data

I used a list-type open-ended question to collect responses, in which respondents can list up to 6 items. After coding the items, I have the multivariate nominal data. I want to check if my IV ...
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Using Non-Euclidean Distance Matrix in Multi-Dimensional Scaling

I'm trying to use a non-euclidean distance matrix in MDS. Specifically, I'm using cmdscale in R. The resulting eigen vector appears to have huge negative values. From what I've read it appears that ...
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VARMA with t-student innovations

I'm wondering if there is a possibility to estimate VARMA model with t-student innovations in R. I found package MTS, but all models here seem to be estimated assuming multivariate normal ...
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63 views

Whitening transformation for skewness?

Let $X$ be an $(m,n)$-matrix interpreted as a two dimensional array with each column representing $m$ samples from a random variable, with known covariance matrx $M$ and mean equal to $0$, it is ...
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30 views

Goodness of fit for multi dimensional scaling

What is considered to be an acceptable value for goodness-of-fit in multi-dimensional scaling (MDS). I tried to run an MDS analysis on my data with four dimensions in R. The goodness of fit comes ...
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Multivariate Gaussian, rearranging means

Looking through the the matrix cookbook, a collection of matrix identities, I came across this one called "rearranging means" in the multivariate Normal distribution (Sec. 8.1.5 or Eq. #356, also ...
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7 views

Generate Correlated Outcomes Between Groups

For a simulation study, I want to generate an outcome (Y) for 3 groups (Control, Treatment1, and Treatment1). The outcomes for Control and Treatment1 are correlated by .50; the outcomes for Treatment1 ...
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Non parametric version of Hotelling's $T^2$

Is there a non parametric version for Hotelling's $T^2$ test? Namely, the one group test for location (not the two group).
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evaluate the similarity between two time series

I have two time series, $\mathcal{T}_1$ and $\mathcal{T}_2$, each time series is of two dimensional. One time series is collected from two sensors (SA, SB), and the other is collected from other ...
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Volatility, Spillover and Shocked

What are major difference between Volatility, Spillover and Shocked in context of Multivariate GARCH Financial modeling? or What is difference between Multivariate GARCH models of volatility, ...
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Test of Ordered, Multivariate Correlation

This should be a simple question: I have an ordered series for which I have measured three variables. Some appear to have positive correlations and some have negative correlations. Here's an example ...
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Tweedie distribution GLM for manyany() in {mvabund} package

My data follow a Tweedie distribution, and I'm working with multivariate abundances. So I'm trying to use the manyany() function in the ...
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1answer
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How to interpret this PCA biplot?

I am approaching PCA analysis for the first time, and have difficulties on interpreting the results. This is my biplot (produced by Matlab's functions pca and ...
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Binary outcome prediction based on multivariate time series in R

everyone! I have some monitoring dataset for 90 patients. It consists of about 10 parameters (continuous variables) that were recorded each 1 minute 3-4 days for every patient. I know the binary ...
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What models allow the study of the relation between a set of response variables and a set of covariates?

A first technique that comes to mind is Canonical Correlation Analysis. Bayesian Networks and other graphical models, I guess, can also be used to analyse such things. Any else that I should be aware ...
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Find an unbiased estimator of $\Sigma^{-1}$

Suppose $ X_1,\dots, X_n$ be a random sample from $N_p(\mu, \Sigma), \Sigma > 0$. Find an unbiased estimator of $\Sigma^{-1}$. I know the unbiased estimator of $\Sigma$ is $\dfrac{1}{n-1} ...
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Proving the given quadratic form is chi-squared $k$

Suppose $\underline{X}$ is an $m$-dimensional vector following multivariate Normal distribution i.e. $\underline{X}$~$N_m(\underline{\mu},\Sigma)$ where $\Sigma$ is positive definite. Let $B$ be a ...
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Can I run a multivariate quasi-binomial analysis of proportion data?

I am trying to best analyse a set of foraging ecology data with >10 behaviour categories (DVs) and 3 levels of IV (season, sex, age). The time which an animal spent engaged in a behaviour was recorded ...
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Multivariate Mahalanobis Distance Vector Normalization

I have a vector including different variables with different scales. For instance ''a'' presents ''dollar value'' in billions.''b'' is a ratio, presents value divided by quantity and it ranges 0 to 1 ...
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71 views

LDA, PCA and k-means: how are they related?

I am trying to understand how linear discriminant analysis (LDA) is related to principal component analysis (PCA) and k-means clustering method. As an example, here is a comparison between PCA and ...
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33 views

Statistical meanings of some matrix vector operations

These questions may be too elementary, but I'm new to statistics and just trying to grasp new concepts, so please be patient with me. Suppose I have data on a single variable, expressed as ${\bf{y}} ...
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Multivariate discrete distribution

Let $\mathbf{x}=(x_{1}, \dots, x_{m})$ be a vector of discrete variable; $x_{j} \in \mathbb{Z}^+$. Which distribution can I use over $\mathbf{x}$ that allows to have dependence between each couple ...
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A Conditional Probability Question

Assume we know $$p(w) \sim N(0, \Sigma)$$ $$p(e) \sim N(0, 0.01)$$ $$y=w^\intercal x$$ $$o = y + e$$ Where w/x/o/y/e are all vector How do we calculate the distribution of $$p(y|o,x)$$
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Variance estimation in a one-factor linear model

I was given a dataset (a mat file) of $100\: 000$ observations, each with $50$ dimensions (coordinates). Denote matrix $X$ a $50\times 100\:000$ matrix in which each column was generated according to: ...