# Sparse Robust Principal Components based on Projection Pursuit (PP): GRID search Algorithm Computes an approximation of the PP-estimators for sparse and robust PCA using the grid search algorithm in the plane. ```r SPcaGrid(x, ...) ## Default S3 method: SPcaGrid(x, k = 0, kmax = ncol(x), method = c ("mad", "sd", "qn", "Qn"), lambda = 1, scale=FALSE, na.action = na.fail, trace=FALSE, ...) ## S3 method for class 'formula' SPcaGrid(formula, data = NULL, subset, na.action, ...) ``` ## Arguments - `formula`: a formula with no response variable, referring only to numeric variables. - `data`: an optional data frame (or similar: see `model.frame`) containing the variables in the formula `formula`. - `subset`: an optional vector used to select rows (observations) of the data matrix `x`. - `na.action`: a function which indicates what should happen when the data contain `NA`s. The default is set by the `na.action` setting of `options`, and is `na.fail` if that is unset. The default is `na.omit`. - ``...``: arguments passed to or from other methods. - `x`: a numeric matrix (or data frame) which provides the data for the principal components analysis. - `k`: number of principal components to compute. If `k` is missing, or `k = 0`, the algorithm itself will determine the number of components by finding such `k` that $l_k/l_1 >= 10.E-3$ and $\Sigma_{j=1}^k l_j/\Sigma_{j=1}^r l_j >= 0.8$. It is preferable to investigate the scree plot in order to choose the number of components and then run again. Default is `k=0`. - `kmax`: maximal number of principal components to compute. Default is `kmax=10`. If `k` is provided, `kmax` does not need to be specified, unless `k` is larger than 10. - `method`: the scale estimator used to detect the direction with the largest variance. Possible values are `"sd"`, `"mad"` and `"Qn"`. `"mad"` is the default value. - `lambda`: the sparseness constraint's strength(`sPCAgrid` only). A single value for all components, or a vector of length `k` with different values for each component can be specified. See `opt.TPO` for the choice of this argument. - `scale`: a value indicating whether and how the variables should be scaled. If `scale = FALSE` (default) or `scale = NULL` no scaling is performed (a vector of 1s is returned in the `scale` slot). If `scale = TRUE` the data are scaled to have unit variance. Alternatively it can be a function like `sd` or `mad` or a vector of length equal the number of columns of `x`. The value is passed to the underlying function and the result returned is stored in the `scale` slot. Default is `scale = FALSE` - `trace`: whether to print intermediate results. Default is `trace = FALSE` ## Details `SPcaGrid`, serving as a constructor for objects of class `SPcaGrid-class` is a generic function with "formula" and "default" methods. For details see `sPCAgrid` and the relevant references. ## Returns An S4 object of class `SPcaGrid-class` which is a subclass of `PcaGrid-class` which in turn is a subclass of the virtual class `PcaRobust-class`. ## References C. Croux, P. Filzmoser, M. Oliveira, (2007). Algorithms for Projection-Pursuit Robust Principal Component Analysis, **Chemometrics and Intelligent Laboratory Systems**, Vol. 87, pp. 218-225. C. Croux, P. Filzmoser, H. Fritz (2013). Robust Sparse Principal Component Analysis, **Technometrics** 55 (2), pp. 202--2014, tools:::Rd_expr_doi("10.1080/00401706.2012.727746") . V. Todorov, P. Filzmoser (2013). Comparing classical and robust sparse PCA. In R Kruse, M Berthold, C Moewes, M Gil, P Grzegorzewski, O Hryniewicz (eds.), **Synergies of Soft Computing and Statistics for Intelligent Data Analysis**, volume 190 of **Advances in Intelligent Systems and Computing**, pp. 283--291. Springer, Berlin; New York. ISBN 978-3-642-33041-4, tools:::Rd_expr_doi("10.1007/978-3-642-33042-1_31") . ## Author(s) Valentin Todorov valentin.todorov@chello.at ## Examples ```r data(bus) bus <- as.matrix(bus) ## calculate MADN for each variable xmad <- apply(bus, 2, mad) cat("\nMin, Max of MADN: ", min(xmad), max(xmad), "\n") ## calculate MADN for each variable xqn <- apply(bus, 2, Qn) cat("\nMin, Max of Qn: ", min(xqn), max(xqn), "\n") ## MADN vary between 0 (for variable 9) and 34. Therefore exclude ## variable 9 and divide the remaining variables by their MADNs. bus1 <- bus[, -c(9)] p <- ncol(bus1) madbus <- apply(bus1, 2, mad) bus2 <- sweep(bus1, 2, madbus, "/", check.margin = FALSE) xsd <- apply(bus1, 2, sd) bus.sd <- sweep(bus1, 2, xsd, "/", check.margin = FALSE) xqn <- apply(bus1, 2, Qn) bus.qn <- sweep(bus1, 2, xqn, "/", check.margin = FALSE) ## Not run: spc <- SPcaGrid(bus2, lambda=0, method="sd", k=p, kmax=p) rspc <- SPcaGrid(bus2, lambda=0, method="Qn", k=p, kmax=p) summary(spc) summary(rspc) screeplot(spc, type="line", main="Classical PCA", sub="PC", cex.main=2) screeplot(rspc, type="line", main="Robust PCA", sub="PC", cex.main=2) ## find lambda K <- 4 lambda.sd <- 1.64 to.sd <- .tradeoff(bus2, k=K, lambda.max=2.5, lambda.n=100, method="sd") plot(to.sd, type="b", xlab="lambda", ylab="Explained Variance (percent)") abline(v=lambda.sd, lty="dotted") spc.sd.p <- SPcaGrid(bus2, lambda=lambda.sd, method="sd", k=p) .CPEV(spc.sd.p, k=K) spc.sd <- SPcaGrid(bus2, lambda=lambda.sd, method="sd", k=K) getLoadings(spc.sd)[,1:K] plot(spc.sd) lambda.qn <- 2.06 to.qn <- .tradeoff(bus2, k=K, lambda.max=2.5, lambda.n=100, method="Qn") plot(to.qn, type="b", xlab="lambda", ylab="Explained Variance (percent)") abline(v=lambda.qn, lty="dotted") spc.qn.p <- SPcaGrid(bus2, lambda=lambda.qn, method="Qn", k=p) .CPEV(spc.qn.p, k=K) spc.qn <- SPcaGrid(bus2, lambda=lambda.qn, method="Qn", k=K) getLoadings(spc.qn)[,1:K] plot(spc.qn) ## End(Not run) ## DD-plots ## ## Not run: ## Not run: usr <- par(mfrow=c(2,2)) plot(SPcaGrid(bus2, lambda=0, method="sd", k=4), id.n.sd=0, main="Standard PCA") plot(SPcaGrid(bus2, lambda=0, method="Qn", k=4), id.n.sd=0, ylim=c(0,20)) plot(SPcaGrid(bus2, lambda=1.64, method="sd", k=4), id.n.sd=0, main="Stdandard sparse PCA") plot(SPcaGrid(bus2, lambda=3.07, method="Qn", k=4), id.n.sd=0, main="Robust sparse PCA") par(usr) ## End (Not run) ## End(Not run) ```

SPcaGrid function