# Tag Info

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PCA(Principal Component Analysis) is the most interesting topic in QF. PCA is at the heart of quantitative data analysis. It is used in factor analysis, factor loadings, finding principal component of interest rate term structure for derivative and option pricing, data compression, eigenfaces( find the best match from a set of pictures with a , say, fuzzy ...

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PCA gives you a decomposition of the covariance matrix of the form $$\Sigma = V \Lambda V^T$$ where $\Lambda$ is diagonal with the eigenvalues in the diagonal. Your portfolio variance is $$w^T \Sigma w = (V^T w )^T \Lambda (V^T w)$$ On the other hand if you take your return matrix $R$ and define $$F = V^T R$$ then the covariance matrix of these so ...

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Following @silencer's comment, your formula for variance is wrong. I would suggest that instead of trying to re-invent the wheel, you just use the formula that everyone else uses. So I'd replace your first indented line with $$w^{*}\equiv argmin\left\{ \frac{1}{2}w'\varSigma w-\lambda\left(w'\mathbf{1}-1\right)\right\}$$ which will give you $$... 2 When I use PCA, I follow a few typical steps. First, I would apply PCA to the covariance matrix, I would then designate certain eigenvalues as dominant or significant (such as by those that contribute up to x\% of variance or by RMT), and then I would identify the eigenvectors that match up with those significant eigenvalues. I think you're with me at ... 2 In the chapter that deals with NMF of the book "Programming collective intelligence" , the author did NMF on several stock trading volumes and found some comovement. I googled a little. This did NMF on 40 chinese stock close prices. This developed A variant of nonnegative matrix factorization for Stock Trend Extraction. Another google found this also did ... 2 Perhaps this paper by Hyun Woo Byun and coauthors is what you're looking for: Using a Principal Component Analysis to develop Multi-Currency Trading algorithms in the FX market They apply principal component analysis to a currency basket of 9 pairs with a 2 month rolling window. In a second step, various techniques (logistic regression, decision trees, ... 2 If you look at changes of the points on the yield curve, then you probably find something stationary - right? Applying PCA on the covariance of these changes makes sense. E.g. you will find out that on PC describes a parallel shift (a change in the yield curve). Look at this question too: What do eigenvalues/eigenvectors of the yield/forward rates ... 1 The choice of normalization depends on your data set: Without normalization : variable with high variance will have more impact on the PCA. You will have size effects. For exemple if you have one variable in meters and the other one in kilometers the one in meters will have way more impact. To avoid that you can normalize but now every variable will have ... 1 The derivation is correct and given the formula you should get w^{*'} \vec{1} = 1. My guess is that the inversion of the \Omega matrix is numerically badly conditioned. Instead of implementing the formula as it is, have you tried to calculate \vec{1}^{'} \Omega^{-1} and e^{'} \Omega^{-1} only once and rewrite:$$ w^{*\prime} = \frac{1}{2}\left[ ...

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The paper alternatives between using eigenportfolios and sector/industry ETFs for statistical arbitrage. For instance, sections 2.1-2 vs. 2.3. The trade in Section 4.1 is long some stock and short an appropriate amount of sector/industry ETFs. That being said Sections 5.3 and 5.4 discuss PCA strategies in a backtest, with relatively little additional ...

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Q #1: I'm not sure if you have the answer quite right. The signs for the loadings are arbitrary, but you cannot take the absolute value. You can multiply by -1. Q #2: It might be helpful to think about what PCA is actually doing. This paper might be helpful: http://arxiv.org/pdf/1404.1100v1.pdf (A Tutorial on Principal Component Analysis by Jonathon ...

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Trying to answer: in the blog post that you mention the author looks at three equity funds and one REIT fund. One could say that these markets are different to FX markets (for various reasons but let's start with the question whether there is a risk premium in FX markets). what he does is the usual PCA analysis on the data. You find various questions in ...

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