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SVD Singular-value decomposition is at the root of the three kindred techniques. Let $\bf X$ be $r \times c$ table of real values. SVD is $\bf X = U_{r\times r}S_{r\times c}V_{c\times c}'$. We may use just $m$ $[m \le\min(r,c)]$ first latent vectors and roots to obtain $\bf X_{(m)}$ as the best $m$-rank approximation of $\bf X$: $\bf X_{(m)} = U_{r\times m}... 20 As @amoeba mentioned in the comments, PCA will only look at one set of data and it will show you the major (linear) patterns of variation in those variables, the correlations or covariances between those variables, and the relationships between samples (the rows) in your data set. What one normally does with a species data set and a suite of potential ... 20 Q1 Ecologists talk of gradients all the time. There are lots of kinds of gradients, but it may be best to think of them as some combination of whatever variable(s) you want or are important for the response. So a gradient could be time, or space, or soil acidity, or nutrients, or something more complex such as a linear combination of a range of variables ... 15 I would like to suggest you a relatively recent technique for automatic structure extraction from categorical variable data (this includes binary). The method is called CorEx from Greg van Steeg from University of Southern California. The idea is to use the notion of Total Correlation based on the entropy measures. It is appealing due to its simplicity and ... 14 You can also use Multiple Correspondence Analysis (MCA), which is an extension of principal component analysis when the variables to be analyzed are categorical instead of quantitative (which is the case here with your binary variables). See for instance Husson et al. (2010), or Abdi and Valentin (2007). An excellent R package to perform MCA (and ... 11 I would suggest having a look at Linting & Kooij, 2012 "Non linear principal component analysis with CATPCA: a tutorial", Journal of Personality Assessment; 94(1). Abstract This article is set up as a tutorial for nonlinear principal components analysis (NLPCA), systematically guiding the reader through the process of analyzing actual data on ... 11 If you think of PCA as an exploratory technique to give you a way to visualise the relationships between variables (and in my opinion this is the only way to think about it) then yes, there is no reason why you can't put in binary variables. For example, here is a biplot of your data It seems reasonably useful. For example, you can see that Doc and Bashful ... 10 PCA works on the values where as CA works on the relative values. Both are fine for relative abundance data of the sort you mention (with one major caveat, see later). With % data you already have a relative measure, but there will still be differences. Ask yourself do you want to emphasise the pattern in the abundant species/taxa (i.e. the ones with large %... 8 @Silverfish asked for an expansion of the answer by PolatAlemdar, which was not given, so I will try to expand on it here. Why the name chisquare distance? The chisquare test for contingency tables is based on $$\chi^2 = \sum_{\text{cells}} \frac{(O_i-E_i)^2}{E_i}$$ so the idea is to keep this form and use it as a distance measure. This gives the ... 6 Correspondence Analysis is a method to visualize a contingency table, such as frequency cross-table. I presume that the table in your case would be 5 Brands X 11 Attributes and the entries are frequencies (counts) of 1s: attribute is characteristic of a brand. And you want the analysis to produce a biplot wherein points-brands are acommpanied by points-... 6 This is ultimately just a scatterplot. I don't think there is a special name for it. I don't see this as meaningfully related to correspondence analysis except in that you can make a scatterplot of the results from a correspondence analysis, and this is also a scatterplot. Notably, this does not have much to do with a biplot. I do see a couple features ... 5 Continuing on what @Martin F commented, recently I came across with the nonlinear PCAs. I was looking into Nonlinear PCAs as a possible alternative when a continuous variable approaches distribution of an ordinal variable as the data gets sparser (it happens in genetics a lot of times when the minor allele frequency of the variable gets lower and lower and ... 5 I'm not certain of your exact data, or the process you're using to analyze it, but what you describe makes me think of a correlation matrix. In R, generating the matrix, as well as the corresponding heat map (with dendrogram) is easy. The example below used example data to show correlations between usage rates of different IT applications, and generates the ... 4 The NMDS vegan performs is of the common or garden form of NMDS. If metaMDS() is passed the original data, then we can position the species points (shown in the plot) at the weighted average of site scores (sample points in the plot) for the NMDS dimensions retained/drawn. The weights are given by the abundances of the species. This is one way to think of ... 4 Although PCA is often used for binary data, it is argued that PCA assumptions are not appropriate for binary or count data (see e.g. Collins 2002 for an explanation) and generalizations exists: the strategy is similar in spirit to the development of generalized linear models to perform regression analysis for data belonging to the exponential family. An ... 3 A simple approach is to fit a mixture of "Naive Bayes" models using EM. The structure of the mixture model is$P(x_{i1},\ldots,x_{in}) = \sum_k P(y_i=k) \prod_j P(x_{ij}|y_i=k)$. Here,$i$indexes the data points, each of which is a vector of$n$binary features.$y_i$is the index of the cluster to which data point$i$belongs.$P(y_i=k)$is the (learned) ... 3 To learn about this you should study the website of amazon.com: http://www.amazon.com/ They obviously are using a lot of statistics! In our time, storage is cheap (almost free), so you should store just about everything, what matters is with what structure you store it. Let us have a look at that website and see what statistics they obviously used. ... 3 Why are there negative values in the range of the plot? Because the input contingency table of positive values (e.g. frequencies) was standardized by the analysis in a special way relative the central point. How is the center of the plot (i.e., the 0.0 point) found? It is the weighted mean row profile and the weighted mean column profile. It is thus the ... 3 Look into Latent Class Analysis. http://en.wikipedia.org/wiki/Latent_class_model In summary, a latent class model explains the joint distribution of some set of dichotomous variables by assuming there are sub-groups within your population, and that the observed variables are independent, given sub-group membership. The method effectively allows you to ... 3 While I don't have a proof for this, I doubt that PCA is a good method to use on binary data. It is really meant for continuous variables as far as I can tell. And actually, most clustering methods are meant so, too! But given that there can be at most$2^5=32$different values in your data set, why don't you just use the most frequent groups, then assign ... 3 It is normally considered that three is the minimum number of variables to conduct factor analysis; amongst elsewhere this is maintained in the Wikipedia article (which has a reference) and in some (most? all?) statistical software. There is no reason however that you can't do principal components analysis (which is not the same as factor analysis, although ... 3 I found this link to be quite useful: http://docs.opencv.org/2.4/doc/tutorials/imgproc/histograms/histogram_comparison/histogram_comparison.html I am not quite sure why, but OpenCV uses the 3rd formula you list for Chi-Square histogram comparison. In terms of meaning, I am not sure any measurement algorithm is going to give you a bounded range, like 0% to ... 3 The scaling argument scales one set of score to the other. You're just fiddling with the relative positions of one set versus another. The ordisurf() surface is inherently constrained to the convex hull (or very slightly beyond it) of the site scores — in whatever scaling you choose to apply — because the model it is fitting is:$$\hat{y}_i = f(... 3 Using ggvegan and ggrepel library("vegan") library("ggvegan") library("cowplot") library("ggrepel") data(varespec, varechem) vare.cca <- cca(varespec, varechem) obj <- fortify(vare.cca) want <- obj$Score %in% c("species", "sites") autoplot(vare.cca) + geom_text(data = obj[want, , drop = FALSE ], aes(x = Dim1, y = ...

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So you have two vectors of counts (length 1000) and wants to compare their shape, irrespective of the absolute counts. You can put the two vectors together as a contingency table, the individual vectors $A, B$ as rows, in R you could do mytable <- rbind(A,B). Comparing the shape or profile of the rows is just what is done by correspondence analysis, ...

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PCAmixdata #Rstats package: Implements principal component analysis, orthogonal rotation and multiple factor analysis for a mixture of quantitative and qualitative variables. Example from vignette shows results for both continuous and categorical output

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There is a recently developed approach to such problems: Generalized Low Rank Models. One of papers that use this technique is even called PCA on a Data Frame. PCA can be posed like this: For $n$ x $m$ matrix $M$ find $n$ x $k$ matrix $\hat{X}$ and $k$ x $m$ matrix $\hat{Y}$ (this encodes rank $k$ e constraint implicitly) such that $\hat{X}, \hat{Y}$ = ...

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You might be interested in correspondence analysis, which is supposed to be a categorical version of PCA. In R, these are implemented in, for example, the packages ade4 and FactoMineR. Have you tried making a dendogram of your data? This might give you a way to eyeball the number of clusters.

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Take a look at the Stata documentation for CA (even if you're not a Stata user): Correspondence analysis offers a geometric representation of the rows and columns of a two-way frequency table that is helpful in understanding the similarities between the categories of variables and the association between the variables. There are lots of examples, ...

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I'm new to CA too and have found this article to be a great resource: http://marketing-bulletin.massey.ac.nz/V14/MB_V14_T2_Bendixen.pdf Take a look at figure 5. I wonder if you could do the same with your plot. I would suggest you plot the individuals (1-39) and label the axes with the answer categories- the paper takes you through step-by-step on how to ...

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