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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}... 19 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 ... 17 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 ... 16 French style data analysis is usually identified as work based on Correspondence Analysis and other spectrally-oriented work, but is actually more deeply grounded. Tim's reference to the Holmes piece is particularly helpful here. A slightly broad picture would be to say that French style takes an axiomatic, geometrical, and mathematical approach to data ... 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 ... 13 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 ... 10 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 ... 9 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 %... 9 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 ... 8 I feel, from your description of the task, that it can be solved by means of quantifying (turning ordinal-level into scale-level) variables the way that the predictions or associatiations are maximized. This is known as optimal scaling and is implemented in SPSS (and, of course, in R, I believe). You might choose between 3 procedures, all adopting optimal ... 7 Canonical correspondence analysis is a technique developed, I believe, by the community ecology people. A founding paper is Canonical correspondence analysis: a new eigenvector technique for multivariate direct gradient analysis by Cajo J.F. Ter Braak (1986). The method involves a canonical correlation analysis and a direct gradient analysis. The idea is to ... 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 I would go for co-inertia analysis, which is an unspoken variant of canonical analysis. This would give you a linear combination of the 21 variables that has the highest co-inertia with a linear combination of childcare data (or with child care if it is a single quantitative variable). The trick of working with co-inertia instead of correlation is that you ... 5 As Peter Ellis' comment/answer suggests you are talking about dimensionality reduction and not data reduction. You have changed the number of data points just the size of the space of covariates. Now Peter Flom is right that the PCA and FA methods can be tried with small sample sizes but it is not only the correlations that are likely to be poorly estimated ... 4 I am yet to get privilege for commenting on someone's post so I am adding my comment as a separate answer, so please bear with me. 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 ... 4 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 ... 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 Just addressing the coding part of the question, building on the comments. Most stats packages are set up to deal with a coding like What operating system are you using? 1 = Mac 2 = PC 3 = Linux with a flag on that variable that it is a categorical factor and hence the 1, 2, 3 should not be interpreted as a continuous variable but just as the form of ... 3 With the keyword "cluster" and "0/1 data", my knee-jerk reaction would be to put everything into a cluster analysis machine using a measure of "distance" between observations that only have binary variables. See e.g. Stata help file describing about a dozen such measures. I would run all possible analyses (hierarchical linkage/dendrogram) to see if there ... 3 Maybe "correspondence analysis"? : http://en.wikipedia.org/wiki/Correspondence_analysis because it was primarily developed by a French researcher Jean-Paul Benzecri ? 3 assuming your data are in a data.frame called dt, you can turn it into a table object using xtabs tbl <- xtabs(Freq ~ Species + Eco_region4, data=dt) tbl Eco_region4 Species A1 A2 A3 B1 B2 B3 C1 C2 C3 S1 10 10 2 13 9 15 12 12 18 S2 12 6 9 15 8 12 18 0 10 S3 8 11 13 7 13 13 20 11 16 You can get a three ... 3 I take it you mean "is it OK to use correspondence analysis for a table of other 'contingency' data, not just frequences?" Yes, it is OK. Your entries may me means or other values somehow reflecting affinity between rows and columns. Entries should be positive values (if negative values are there, add a constant to make them all positive). Standard CA of ... 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 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 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(...