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I faced this problem when trying to do KDE using gaussian_kde in Python where the random variables were the 784 pixels of some images. In my case the reason was that many pixels were always (in all the images) zero, so no random at all. To solve this problem I just added some small gaussian noise to the images and voilà, now the covariance matrix is ...


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R Implementation of 2-D KS test The Fasano-Franceschini test (1987) -- a 2-D Kolmogorov-Smirnov (KS) two-sample test shown to be a less computationally expensive version of the Peacock test (1983) -- has recently been implemented as R code. The package should be on CRAN shortly. In the mean time, you can check it out for yourself. R code Implementation ...


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I don't think that's the best way to go. This seems more exploratory in nature. With really large contingency tables, where you can't just look at the numbers and see the pattern, I would run a correspondence analysis and look at the plot. A correspondence analysis looks at how similar the rows are to each other, and how similar the columns are to each ...


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There are whole books written about this. Harrell's resources on Regression Modeling Strategies are a good source. His course notes are particularly helpful. Chapter 4 of the course notes, on Multivariable Modeling Strategies, deals with ways to handle multiple predictors that go beyond simply including or removing them; sometimes combining related ...


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Time-series clustering requires sample size remaining the same but the features changes over time, otherwise it makes little sense. In the question though, inferring from the description sample size increases over time. In that case, to see significant reduction on certain clusters, one should use a fixed sample-size. Then choose fixed sample from the ...


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I will try to simplify your question, and consider these simplifications. If we ignore the group (and only consider group 1). You question is if there is any relation between var1 and var2. As "hplieninger" comments above there is no way to compare the variables directly if you dont have measurements for both var1 and var2 for the same individual. ...


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The derivation relies on a result known as the matrix inversion lemma, or Woodbury matrix identity. From wikipedia: $$(A + UCV)^{-1} = A^{-1} - A^{-1}U(C^{-1} + VA^{-1}U)^{-1}VA^{-1}$$ Identifying $A = \mathbf{I}$, $U = \boldsymbol{\Lambda}^T$, $C = \boldsymbol{\Psi}^{-1}$ and $V = \boldsymbol{\Lambda}$ yields the expression you seek.


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I encountered the same problem when trying to order the nodes in a complete asymmetric graph. My objective is similar: maximize the difference between the sum of the lower entries and the sum of the upper entries. Here is a greedy solution in pytorch. I have not looked into any guarantee that it converges nor that it is optimal. However, when comparing with ...


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Doing a joint optimization for two metrics is an interesting problem, but unfortunately a bit out my expertise. However, making a decision based on a single metric is straight forward enough. Here is a simulated example of how this might be done using a Bayesian lense. Let's first simulate some data library(tidyverse) set.seed(0) N = 10000 groups = sample(...


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You asked for books not regurgitating old material ... the following is not cheap, and looks more like an article collection with examples of new applications Advances in Principal Component Analysis: Research and Development but might still be interesting. An edited volume with many authors.


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Very interesting-looking is Generalized Principal Component Analysis by René Vidal. It might be more specialized than the Izenman tome, but certainly introduces new mathematics to the game! I really want to study it, bit it will require some time to learning such things as some algebraic geometry, which was not in my studies ...


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What first comes to mind for me is Modern Multivariate Statistical Techniques: Regression, Classification, and Manifold Learning (Springer Texts in Statistics) by Izenman, which has its focus entirely on newer ideas not covered at all in most more traditional books. At Amazon.com it gets very positive reviews. I did not read it yet but ...


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The adjusted odds ratio is the result of a logistic regression analysis. Very little short of having the entire dataset would enable you to calculate these values.


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An answer to this question can be found here. In case you cannot get access to the book that is linked, I will repeat the result here: Suppose $\mathcal{C}$ is a VC class of $\mathcal{X}$. Define, $$ D_n(\mathcal{C})\equiv ||P_n - P||_\mathcal{C}$$ Then for all $\epsilon>0$, $$\sup_P \Pr[\max_{m\geq n} D_m(\mathcal{C})\geq\epsilon]=\sup_P \Pr[\max_{m\geq ...


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You are correct that we cannot interpret the main effects of variables that are interacted with another variable, or at least not in the usual way. When a variable is interacted with another variable, the interpretation is conditional on the other variable that it is being interacted with, being held at zero (or in the case of a categorical variable, at it's ...


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Note that a hazard ratio (HR) estimate of 8 with a top 95% confidence limit of 50* isn't quite as bad as it might seem. Cox regression coefficients are estimated in the log-hazard scale. A quick calculation suggests that corresponds to a regression coefficient of 2.08 with a standard error of 0.93. On that scale, things don't seem so extreme. Your model, ...


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If $X$ follows a multivariate t-distribution, then any linear combination of $X$ also follows a multivariate t-distribution with the same degrees of freedom: $$ X \sim t(\mu, \Sigma, \nu) \quad \Rightarrow \quad Y = AX + b \sim t(A\mu + b, A\Sigma A^\mathrm{T}, \nu) \; . $$ Thus, the intended combination only works, if the two multivariate t-distributions ...


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RDA and CCA are asymmetrical analysis techniques: you aren't just seeing if they are related (i.e. correlation) but are more specifically seeing how much/if a significant amount of the variation in Y can be explained in X. If you think the data sets are linearly related, RDA is a good choice (if you don't think they are but still want to use RDA instead of ...


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According to the Multivariate Normal Central Limit theorem, it will be close to Multivariate Normal DIstribution. Check section 2 of the paper by Anderson.


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