#' ---	
#' title: "Data Science and Predictive Analytics (UMich HS650)"	
#' subtitle: "<h2><u>Dimensionality Reduction</u></h2>"	
#' author: "<h3>SOCR/MIDAS (Ivo Dinov)</h3>"	
#' date: "`r format(Sys.time(), '%B %Y')`"	
#' tags: [DSPA, SOCR, MIDAS, Big Data, Predictive Analytics] 	
#' output:	
#'   html_document:	
#'     theme: spacelab	
#'     highlight: tango	
#'     includes:	
#'       before_body: SOCR_header.html	
#'     toc: true	
#'     number_sections: true	
#'     toc_depth: 2	
#'     toc_float:	
#'       collapsed: false	
#'       smooth_scroll: true	
#'     code_folding: show	
#'     self_contained: yes	
#' ---	
#' 	
#' Now that we have most of the fundamentals covered in the previous chapters, we can delve into the first data analytic method, *dimension reduction*, which reduces the number of features when dealing with a very large number of variables. Dimension reduction can help us extract a set of "uncorrelated" principal variables and reduce the complexity of the data. We are not simply picking some of the original variables. Rather, we are constructing new "uncorrelated" variables as functions of the old features.	
#' 	
#' Dimensionality reduction techniques enable exploratory data analyses by reducing the complexity of the dataset, still approximately preserving important properties, such as retaining the distances between cases or subjects. If we are able to reduce the complexity down to a few dimensions, we can then plot the data and untangle its intrinsic characteristics.	
#' 	
#' All *dimensionality reduction* techniques carry some resemblance to *variable selection*, which we will see in [Chapter 16](https://www.socr.umich.edu/people/dinov/courses/DSPA_notes/16_FeatureSelection.html). The *continuous* process of dimensionality reduction yields lower variability compared to the *discrete* feature selection process, e.g., see [Wolfgang M. Hartmann's paper](https://link.springer.com/chapter/10.1007/11558958_113).	
#' 	
#' We will (1) start with a synthetic example demonstrating the reduction of a 2D data into 1D, (2) explain the notion of rotation matrices, (3) show examples of principal component analysis (PCA), singular value decomposition (SVD), independent component analysis (ICA), factor analysis (FA), and t-distributed Stochastic Neighbor Embedding (t-SNE), Uniform Manifold Approximation and Projection (UMAP), and (4) present a Parkinson's disease case-study at the end.	
#' 	
#' # Example: Reducing 2D to 1D	
#' We consider an example looking at twin heights. Suppose we simulate 1,000 2D points that representing `normalized` individual heights, i.e., number of standard deviations from the mean height. Each 2D point represents a pair of twins. We will simulate this scenario using [Bivariate Normal Distribution](https://socr.umich.edu/HTML5/BivariateNormal/).	
#' 	
#' 	
library(MASS)	
	
set.seed(1234)	
n <- 1000	
y=t(mvrnorm(n, c(0, 0), matrix(c(1, 0.95, 0.95, 1), 2, 2)))	
#' 	
#' 	
#' $Twin1_{Height}$ | $Twin2_{Height}$	
#' -----------------|-----------------	
#' $y[1,1]$  | $y[1,2]$ 	
#' $y[2,1]$  | $y[2,2]$ 	
#' $y[3,1]$  | $y[3,2]$ 	
#' $\cdots$ | $\cdots$	
#' $y[500,1]$  | $y[500,2]$ 	
#' 	
#' $$ y^T_{2\times500} = \begin{bmatrix}	
#'     y[1, ]=Twin1_{Height} \\	
#'     y[2, ]=Twin2_{Height}	
#' \end{bmatrix} \sim BVN \left ( \mu= \begin{bmatrix}	
#'     Twin1_{Height} \\	
#'     Twin2_{Height}	
#' \end{bmatrix} , \Sigma=\begin{bmatrix}	
#'     1 & 0.95 \\	
#'     0.95 & 1	
#' \end{bmatrix}	
#' \right ) .$$	
#'  	
#' 	
# plot(y[1, ], y[2, ], xlab="Twin 1 (standardized height)", 	
#      ylab="Twin 2 (standardized height)", xlim=c(-3, 3), ylim=c(-3, 3))	
# points(y[1, 1:2], y[2, 1:2], col=2, pch=16)	
	
library(plotly)	
plot_ly() %>% 	
  add_markers(x = ~y[1, ], y = ~y[2, ], name="Data Scatter") %>% 	
  add_markers(x = y[1, 1], y = y[2, 1], marker=list(color = 'red', size = 20,	
                   line = list(color = 'yellow', width = 2)), name="Twin-pair 1") %>% 	
  add_markers(x = y[1, 2], y = y[2, 2], marker=list(color = 'green', size = 20,	
                   line = list(color = 'orange', width = 2)), name="Twin-pair 2")%>% 	
    layout(title='Scatter Plot Simulated Twin Data (Y)', 	
           xaxis = list(title="Twin 1 (standardized height)"), yaxis = list(title="Twin 2 (standardized height)"),	
           legend = list(orientation = 'h'))	
#' 	
#' 	
#' These data may represent a fraction of the information included in a high-throughput neuroimaging genetics study of twins. You can see [one example of such pediatric study here](https://wiki.socr.umich.edu/index.php/SOCR_Data_Oct2009_ID_NI). 	
#' 	
#' Tracking the distances between any two samples can be accomplished using the `dist` function. `stats::dist()` is a function computing the distance matrix (according to a user-specified  distance metric, default=Euclidean) representing the distances between the *rows of the data matrix*. For example, here is the distance between the two RED points in the figure above:	
#' 	
#' 	
d=dist(t(y))	
as.matrix(d)[1, 2]	
#' 	
#' 	
#' To reduce the 2D data to a simpler 1D plot we can transform the data to a 1D matrix (vector) preserving (approximately) the distances between the 2D points.	
#' 	
#' The 2D plot shows the Euclidean distance between the pair of RED points, the length of this line is the distance between the 2 points. In 2D, these lines tend to go along the direction of the diagonal. If we `rotate` the plot so that the diagonal aligned with the x-axis we get the following *raw* and *transformed* plots:	
#' 	
#' 	
z1 = (y[1, ]+y[2, ])/2 # the sum (actually average)	
z2 = (y[1, ]-y[2, ])   # the difference	
	
z = rbind( z1, z2) # matrix z has the same dimension as y	
	
thelim <- c(-3, 3)	
# par(mar=c(1, 2))	
# par(mfrow=c(2,1))	
# plot(y[1, ], y[2, ], xlab="Twin 1 (standardized height)", 	
#      ylab="Twin 2 (standardized height)", 	
#      xlim=thelim, ylim=thelim)	
# points(y[1, 1:2], y[2, 1:2], col=2, pch=16)	
	
# plot(z[1, ], z[2, ], xlim=thelim, ylim=thelim, xlab="Average height", ylab="Difference in height")	
# points(z[1, 1:2], z[2, 1:2], col=2, pch=16)	
# par(mfrow=c(1,1))	
	
plot_ly() %>% 	
  add_markers(x = ~z[1, ], y = ~z[2, ], name="Data Scatter") %>% 	
  add_markers(x = z[1, 1], y = z[2, 1], marker=list(color = 'red', size = 20,	
                   line = list(color = 'yellow', width = 2)), name="Twin-pair 1") %>% 	
  add_markers(x = y[1, 2], y = y[2, 2], marker=list(color = 'green', size = 20,	
                   line = list(color = 'orange', width = 2)), name="Twin-pair 2") %>% 	
  layout(title='Scatter Plot Transformed Twin Data (Z)', 	
         xaxis = list(title="Twin 1 (standardized height)", scaleanchor = "y", scaleratio = 2), 	
         yaxis = list(title="Twin 2 (standardized height)", scaleanchor = "x", scaleratio = 0.5),	
         legend = list(orientation = 'h'))	
#' 	
#' 	
#' Of course, matrix linear algebra notation can be used to represent this affine transformation of the data. Here we can see that to get the result `z`, we multiplied `y` by the matrix $A$:	
#' 	
#' $$ A = \begin{pmatrix}	
#' 1/2&1/2\\	
#' 1&-1\\	
#' \end{pmatrix}	
#' \implies z = A \times y .$$	
#' 	
#' We can invert this transform by multiplying the result by the `inverse` (rotation matrix?) $A^{-1}$ as follows:	
#' 	
#' $$ A^{-1} = \begin{pmatrix}	
#' 1&1/2\\	
#' 1&-1/2\\	
#' \end{pmatrix}	
#' \implies y = A^{-1} \times z $$	
#' 	
#' You can try this in `R`:	
#' 	
A <- matrix(c(1/2, 1, 1/2, -1), nrow=2, ncol=2); A   # define a matrix	
A_inv <- solve(A); A_inv  # inverse	
A %*% A_inv # Verify result	
#' 	
#' 	
#' Note that this matrix transformation did not preserve distances, i.e., it's not a simple rotation in 2D:	
#' 	
#' 	
d=dist(t(y)); as.matrix(d)[1, 2]     # distance between first two points in Y	
d1=dist(t(z)); as.matrix(d1)[1, 2]   # distance between first two points in Z=A*Y	
#' 	
#' 	
#' # Matrix Rotations 	
#' 	
#' One important question is how to identify transformations that preserve distances. In mathematics, transformations between metric spaces that are distance-preserving are called `isometries` (or `congruences` or congruent transformations).	
#' 	
#' First, let's test the MA transformation we used above:	
#' $$A=\frac{Y_1+Y_2}{2} \\ M=Y_1 - Y_2.$$	
#' 	
#' 	
MA <- matrix(c(1/2, 1, 1/2, -1), 2, 2)	
	
MA_z <- MA%*%y	
d <- dist(t(y))	
d_MA <- dist(t(MA_z))	
	
# plot(as.numeric(d), as.numeric(d_MA)) 	
# abline(0, 1, col=2)	
	
plot_ly() %>% 	
  add_markers(x = ~as.numeric(d)[1:5000], y = ~as.numeric(d_MA)[1:5000], name="Transformed Twin Distances") %>% 	
  add_markers(x = ~as.numeric(d)[1], y = ~as.numeric(d_MA)[1], marker=list(color = 'red', size = 20,	
                   line = list(color = 'yellow', width = 2)), name="Twin-pair 1") %>% 	
  add_markers(x = ~as.numeric(d)[2], y = ~as.numeric(d_MA)[2], marker=list(color = 'green', size = 20,	
                   line = list(color = 'orange', width = 2)), name="Twin-pair 2") %>% 	
  add_trace(x = ~c(0,8), y = ~c(0,8), mode="lines",	
        line = list(color = "red", width = 4), name="Preserved Distances") %>% 	
    layout(title='Preservation of Distances Between Twins (Transform=MA_Z)', 	
           xaxis = list(title="Original Twin Distances", range = c(0, 8)), 	
           yaxis = list(title="Transformed Twin Distances", range = c(0, 8)),	
           legend = list(orientation = 'h'))	
#' 	
#' 	
#' Observe that this MA transformation is not an isometry - the distances are not preserved. Here is one example with $v1=\begin{bmatrix} v1_x=0 \\ v1_y=1 \end{bmatrix}$, $v2=\begin{bmatrix} v2_x=1 \\ v2_y=0 \end{bmatrix}$, which are distance $\sqrt{2}$ apart in their native space, but separated further by the transformation $MA$, $d(MA(v1),MA(v2))=2$.	
#' 	
#' 	
MA; t(MA); solve(MA); t(MA) - solve(MA)	
v1 <- c(0,1); v2 <- c(1,0); rbind(v1,v2)	
euc.dist <- function(x1, x2) sqrt(sum((x1 - x2) ^ 2))	
euc.dist(v1,v2)	
v1_t <- MA %*% v1; v2_t <- MA %*% v2	
euc.dist(v1_t,v2_t)	
#' 	
#' 	
#' More generally, if	
#' $$ \begin{pmatrix}	
#' Y_1\\	
#' Y_2\\	
#' \end{pmatrix}	
#' \sim BVN \left( \begin{pmatrix}	
#' \mu_1\\	
#' \mu_2\\	
#' \end{pmatrix},	
#' \begin{pmatrix}	
#' \sigma_1^2&\sigma_{12}\\	
#' \sigma_{12}&\sigma_2^2\\	
#' \end{pmatrix}	
#' \right)$$	
#' Then,	
#' $$ Z = AY + \eta \sim BVN(\eta + A\mu,A\Sigma A^{T}).$$	
#' Where [BVN denotes bivariate normal distribution](http://www.distributome.org/V3/calc/2D_BivariateNormalCalculator.html), $A = \begin{pmatrix}a&b\\c&d\\ \end{pmatrix}$, $Y=(Y_1,Y_2)^T$, $\mu = (\mu_1,\mu_2)$, $\Sigma = \begin{pmatrix} \sigma_1^2&\sigma_{12}\\ \sigma_{12}&\sigma_2^2\\ \end{pmatrix}$.	
#' 	
#' You can verify this by using the [change of variable theorem](https://en.wikipedia.org/wiki/Change_of_variables). Thus, affine transformations preserve bivariate normality. However, in general, the linear transformation ($A$) is not guaranteed to be an isometry. 	
#' 	
#' The question now is under what additional conditions for the transformation matrix $A$, can we guarantee an isometry.	
#' 	
#' Notice that,	
#' $$ d^2(P_i,P_j) =\sum_{k=1}^{n} (P_{jk}-P_{ik})^2 = ||P||^2 = P^TP,$$	
#' 	
#' where $P = (P_{j,1}-P_{i,1},...,P_{j,n}-P_{i,n})^T$, $P_i,P_j \in R^n$ are any two points in $n$ dimensions.	
#' 	
#' Thus, the only requirement we need for $A$ to be an isometry is $(AY)^T(AY)=Y^TY$, i.e., $A^TA=I$, which implies that $A$ is an orthogonal (rotational) matrix.	
#' 	
#' Let's use a two dimension orthogonal matrix to illustrate this.	
#' 	
#' Set $A = \frac{1}{\sqrt{2}} \begin{pmatrix}1&1\\1&-1\\ \end{pmatrix}$. It's easy to verify that $A$ is an orthogonal (2D rotation) matrix.	
#' 	
#' The simplest way to test the isometry is to perform the linear transformation directly as follow.	
#' 	
#' 	
A <- 1/sqrt(2)*matrix(c(1, 1, 1, -1), 2, 2)	
z <- A%*%y	
d <- dist(t(y))	
d2 <- dist(t(z))	
	
# plot(as.numeric(d), as.numeric(d2)) 	
# abline(0, 1, col=2)	
	
plot_ly() %>% 	
  add_markers(x = ~as.numeric(d)[1:5000], y = ~as.numeric(d2)[1:5000], name="Transformed Twin Distances") %>% 	
  add_trace(x = ~c(0,8), y = ~c(0,8), mode="lines",	
        line = list(color = "red", width = 4), name="Preserved Distances") %>% 	
  add_markers(x = ~as.numeric(d)[1], y = ~as.numeric(d2)[1], marker=list(color = 'red', size = 20,	
                   line = list(color = 'yellow', width = 2)), name="Twin-pair 1") %>% 	
  add_markers(x = ~as.numeric(d)[2], y = ~as.numeric(d2)[2], marker=list(color = 'green', size = 20,	
                   line = list(color = 'orange', width = 2)), name="Twin-pair 2")  %>% 	
    layout(title='Preservation of Distances Between Twins (Transform=A)', 	
           xaxis = list(title="Original Twin Distances", range = c(0, 8)), 	
           yaxis = list(title="Transformed Twin Distances", range = c(0, 8)),	
           legend = list(orientation = 'h'))	
#' 	
#' 	
#' We can observe that the distances between points computed from original data and the transformed data are the same. Thus, the transformation $A$ is called a rotation (isometry) of $y$. 	
#' 	
#' The alternative method is to simulate from joint distribution of $Z = (Z_1,Z_2)^T$. 	
#' 	
#' As we have mentioned above: $Z = AY + \eta \sim BVN(\eta + A\mu,A\Sigma A^{T})$.	
#' 	
#' where $\eta = (0,0)^T$, $\Sigma = \begin{pmatrix} 1&0.95\\0.95&1\\ \end{pmatrix}$, $A = \frac{1}{\sqrt{2}} \begin{pmatrix}1&1\\1&-1\\ \end{pmatrix}$.	
#' 	
#' We can compute $A\Sigma A^{T}$ by hand or using matrix multiplication in `R`:	
#' 	
#' 	
sig <- matrix(c(1,0.95,0.95,1),nrow=2)	
A%*%sig%*%t(A)	
#' 	
#' 	
#' $A\Sigma A^{T}$ represents the variance-covariance matrix, $cov(z_1,z_2)$.  We can simulate $z_1$, $z_2$ independently from $z_1\sim N(0,1.95)$ and $z_2 \sim N(0,0.05)$. Note that *independence* and *uncorrelation* are equivalent for bivariate normal distribution.	
#' 	
#' 	
set.seed(2017)	
zz1 = rnorm(1000,0,sd = sqrt(1.95))	
zz2 = rnorm(1000,0,sd = sqrt(0.05))	
zz = rbind(zz1,zz2)	
d3 = dist(t(zz))	
# qqplot(d,d3)	
# abline(a = 0,b=1,col=2)	
	
# compute the quantiles	
QQ <- qqplot(d,d3, plot.it=FALSE)	
# take a smaller sample size to expedite the viz	
ind <-  sample(1:length(QQ$x), 10000, replace = FALSE)	
x1 <- QQ$x[ind]	
y1 <- QQ$y[ind]	
plot_ly() %>%	
  # add_markers(x = ~as.numeric(d)[1:1000], y = ~as.numeric(d3)[1:1000], name="Transformed Twin Distances") %>%	
  add_markers(x = ~x1, y = ~y1, name="Transformed Distances") %>%	
  add_trace(x = ~c(0,8), y = ~c(0,8), mode="lines",	
        line = list(color = "red", width = 4), name="Preserved Distances") %>%	
  add_markers(x = ~x1[1], y = ~y1[1], marker=list(color = 'red', size = 20,	
                   line = list(color = 'yellow', width = 2)), name="Pair 1") %>%	
  add_markers(x = ~x1[2], y = ~y1[2], marker=list(color = 'green', size = 20,	
                   line = list(color = 'orange', width = 2)), name="Pair 2")  %>%	
    layout(title='Preservation of Distances (Random Bivariate))',	
           xaxis = list(title="Original Distances", range = c(0, 8)),	
           yaxis = list(title="Transformed Distances", range = c(0, 8)),	
           legend = list(orientation = 'h'))	
#' 	
#' 	
#' Let's demonstrate that the rotation transform $A = \frac{1}{\sqrt{2}} \begin{pmatrix}1&1\\1&-1\\ \end{pmatrix}: y\to z\equiv Ay$ preserves twin-pair distances by plotting the raw and transformed distances and computing the pre- and post-transform distances between two pair of twins.	
#' 	
#' 	
	
# thelim <- c(-3, 3)	
# #par(mfrow=c(2,1))	
# plot(y[1, ], y[2, ], xlab="Twin 1 (standardized height)", 	
#      ylab="Twin 2 (standardized height)", 	
#      xlim=thelim, ylim=thelim)	
# points(y[1, 1:2], y[2, 1:2], col=2, pch=16)	
	
# plot(z[1, ], z[2, ], xlim=thelim, ylim=thelim, xlab="Average height", ylab="Difference in height")	
# points(z[1, 1:2], z[2, 1:2], col=2, pch=16)	
# par(mfrow=c(1,1))	
	
# Original (pre-transform)	
euc.dist <- function(x1, x2) sqrt(sum((x1 - x2) ^ 2))	
plot_ly() %>% 	
  add_markers(x = ~y[1,], y = ~y[2,], name="Twin Distances") %>% 	
  add_markers(x = ~y[1, 1], y = ~y[2, 1], marker=list(color = 'red', size = 20,	
                   line = list(color = 'yellow', width = 2)), name="Twin-pair 1") %>% 	
  add_markers(x = ~y[1, 2], y = ~y[2, 2], marker=list(color = 'green', size = 20,	
                   line = list(color = 'orange', width = 2)), name="Twin-pair 2")  %>% 	
    layout(title=paste0('(Pre-Transform) Original Twin Heights (standardized): Twin-Pair Distance = ', 	
              round(euc.dist(y[, 1], y[, 2]),3)),	
           xaxis = list(title="Twin 1"), 	
           yaxis = list(title="Twin 2"),	
           legend = list(orientation = 'h'))	
	
# Transform	
plot_ly() %>% 	
  add_markers(x = ~z[1,], y = ~z[2,], name="Transformed Twin Distances") %>% 	
  add_markers(x = ~z[1, 1], y = ~z[2, 1], marker=list(color = 'red', size = 20,	
                   line = list(color = 'yellow', width = 2)), name="Twin-pair 1") %>% 	
  add_markers(x = ~z[1, 2], y = ~z[2, 2], marker=list(color = 'green', size = 20,	
                   line = list(color = 'orange', width = 2)), name="Twin-pair 2")  %>% 	
    layout(title=paste0('(Transform) Twin Heights: Twin-Pair Distance = ', 	
              round(euc.dist(z[, 1], z[, 2]),3)),	
           xaxis = list(title="Twin 1", scaleanchor = "y", scaleratio = 2), 	
           yaxis = list(title="Twin 2", scaleanchor = "x", scaleratio = 0.5),	
           legend = list(orientation = 'h'))	
#' 	
#' 	
#' We applied this transformation and observed that the *distances between points were unchanged after the rotation* $A$. This rotation achieves the goals of:	
#' 	
#' * preserving the distances between points, and 	
#' * reducing the dimensionality of the data (see plot reducing 2D to 1D). 	
#' 	
#' *Removing the second dimension* and recomputing the distances, we get:	
#' 	
#' 	
d4 = dist(z[1, ]) ##distance computed using just first dimension	
# plot(as.numeric(d), as.numeric(d4)) 	
# abline(0, 1)	
	
# take a smaller sample size to expedite the viz	
ind <-  sample(1:length(d4), 10000, replace = FALSE)	
x1 <- d[ind]	
y1 <- d4[ind]	
plot_ly() %>%	
  add_markers(x = ~x1, y = ~y1, name="Transformed Distances") %>%	
  add_trace(x = ~c(0,8), y = ~c(0,8), mode="lines",	
        line = list(color = "red", width = 4), name="Preserved Distances") %>%	
  layout(title='Approximate Distance Preservation in 1D',	
           xaxis = list(title="(1D) Original Distances", range = c(0, 8)),	
           yaxis = list(title="(1D) A-Transformed Distances", range = c(0, 8)),	
           legend = list(orientation = 'h'))	
  # add_markers(x = ~x1[1], y = ~y1[1], marker=list(color = 'red', size = 20,	
  #                  line = list(color = 'yellow', width = 2)), name="Pair 1") %>%	
  # add_markers(x = ~x1[2], y = ~y1[2], marker=list(color = 'green', size = 20,	
  #                  line = list(color = 'orange', width = 2)), name="Pair 2")  %>%	
  # layout(title=paste0('Approximate Distance Estimated in 1D(Transform): Twin-Pair Distance = ', 	
  #             round(euc.dist(x1[1], y1[1]),3)),	
  #          xaxis = list(title="Original Distances", range = c(0, 8)),	
  #          yaxis = list(title="Transformed Distances", range = c(0, 8)),	
  #          legend = list(orientation = 'h'))	
#' 	
#' 	
#' The 1D distance provides a very good approximation to the actual 2D distance. This first dimension of the transformed data is called the first `principal component`. In general, this idea motivates the use of principal component analysis (PCA) and the singular value decomposition (SVD) to achieve dimension reduction. 	
#' 	
#' # Notation	
#' 	
#' In the notation above, the rows represent variables and columns represent cases. In general, rows represent cases and columns represent variables. Hence, in our example shown here, $Y$ would be transposed to be an $N \times 2$ matrix. This is the most common way to represent the data: individuals in the rows, features are columns. In genomics, it is more common to represent subjects/SNPs/genes in the columns. For example, genes are rows and samples are columns. The sample covariance matrix usually denoted with	
#' $\mathbf{X}^\top\mathbf{X}$ and has cells representing covariance between two units. Yet for this to be the case, we need the rows of $\mathbf{X}$ to represents units. Here, we have to compute, $\mathbf{Y}\mathbf{Y}^\top$ instead following the rescaling.	
#' 	
#' # Summary (PCA vs. ICA vs. FA)	
#' 	
#' Principle Component Analysis (PCA), Independent Component Analysis (ICA), and Factor Analysis (FA) are similar strategies, seeking to identify a new basis (vectors representing the principal directions) that the data is projected against to maximize certain (specific to each technique) objective function. These basis functions, or vectors, are just linear combinations of the original features in the data/signal. 	
#' 	
#' The singular value decomposition (SVD), discussed later in this chapter, provides a specific matrix factorization algorithm that can be employed in various techniques to decompose a data matrix $X_{m\times n}$ as ${U\Sigma V^{T}}$, where ${U}$ is an $m \times m$ real or complex unitary matrix (${U^TU=UU^T=I}$, i.e., $|\det(U)|=1$), ${\Sigma }$ is a $m\times n$ rectangular diagonal matrix of *singular values*, representing non-negative values on the diagonal, and ${V}$ is an $n\times n$ unitary matrix.	
#' 	
#' Method | Assumptions | Cost Function Optimization | Applications	
#' -------|-------------|----------------------------|-------------	
#' *PCA* |  Gaussian signals, linear bivariate relations | Aims to explain the variance in the original signal. Minimizes the covariance of the data and yields high-energy orthogonal vectors in terms of the signal variance. PCA looks for an orthogonal linear transformation that maximizes the variance of the variables   |  Relies on $1^{st}$ and $2^{nd}$ moments of  the measured data, which makes it useful when data features are close to Gaussian	
#' *ICA* |  No Gaussian signal assumptions  | Minimizes higher-order statistics (e.g., $3^{rd}$ and $4^{th}$ order skewness and kurtosis), effectively minimizing the *mutual information* of the transformed output. ICA seeks a linear transformation where the basis vectors are statistically independent, but neither Gaussian, orthogonal or ranked in order  | Applicable for non-Gaussian, very noisy, or mixture processes composed of simultaneous input from multiple sources	
#' *FA* | Approximately Gaussian data | Objective function relies on second order moments to compute likelihood ratios. FA *factors* are linear combinations that maximize the shared portion of the variance underlying *latent variables*, which may use a variety of optimization strategies (e.g., maximum likelihood)  | PCA-generalization used to test a theoretical model of latent factors causing the observed features	
#' 	
#' # Principal Component Analysis (PCA)	
#' 	
#' PCA (principal component analysis) is a mathematical procedure that transforms a number of possibly correlated variables into a smaller number of uncorrelated variables through a process known as orthogonal transformation.	
#' 	
#' **Principal Components**	
#' 	
#' Let's consider the simplest situation where we have *n* observations $\{p_1, p_2, ..., p_n\}$ with 2 features $p_i=(x_i, y_i)$. When we draw them on a plot, we use x-axis and y-axis for positioning. However, we can make our own coordinate system by principal components.	
#' 	
#' 	
ex<-data.frame(x=c(1, 3, 5, 6, 10, 16, 50), y=c(4, 6, 5, 7, 10, 13, 12))	
# reg1<-lm(y~x, data=ex)	
# plot(ex)	
# abline(reg1, col='red', lwd=4)	
# text(40, 10.5, "pc1")	
# segments(10.5, 11, 15, 7, lwd=4)	
# text(11, 7, "pc2")	
	
yLM <- lm(y ~ x, data=ex)	
perpSlope = - 1/(yLM$coefficients[2]) # slope of perpendicular line	
newX <- data.frame(x = mean(ex$x))	
newY <- predict.lm(yLM,newX)	
	
point0 <- c(x=newX, y=newY)  # (x,y) coordinates of point0 on LM line	
point1 <- c(x=newX[1]-1, y=newY-perpSlope)  # (x,y) coordinates of point1 on perpendicular line	
point2 <- c(x=newX[1]+1, y=newY+perpSlope)  # (x,y) coordinates of point2 on perpendicular line	
	
modelLabels <- c('PC 1 (LM)', 'PC 2 (Perp_LM)')	
modelLabels.x <- c(40, 20)	
modelLabels.y <- c(10, 6)	
modelLabels.color <- c("blue", "green")	
	
plot_ly(ex) %>%	
  add_lines(x = ~x, y = ~yLM$fitted, name="First PC, Linear Model lm(Y ~ X)", 	
            line = list(width = 4)) %>%	
  add_markers(x = ~x, y = ~y, name="Sample Simulated Data") %>%	
  add_lines(x = ~c(point1$x,point2$x), y = ~c(point1$y,point2$y), name="Second PC, Orthogonal to lm(Y ~ X)", 	
            line = list(width = 4))  %>% 	
  add_markers(x = ~newX[1]$x, y = ~newY, name="Center (avg(x),avg(y))", marker = list(size = 20,	
                                    color = 'green', line = list(color = 'yellow', width = 2))) %>%	
  layout(xaxis = list(title="X", scaleanchor  = "y"),  # control the y:x axes aspect ratio	
         yaxis = list(title="Y", scaleanchor  = "x"), legend = list(orientation = 'h'),	
         annotations = list(text=modelLabels,  x=modelLabels.x, y=modelLabels.y, 	
                            color = modelLabels.color, showarrow=FALSE ))	
#' 	
#' 	
#' Illustrated on the graph, the first PC, $pc_1$ is a minimum distance fit in the feature space. The second PC is a minimum distance fit to a line perpendicular to the first PC. Similarly, the third PC would be a minimum distance fit to all previous PCs. In our case of a 2D space, two PC is the most we can have. In higher dimensional spaces, we need to consider how many PCs do we need to make the best performance.	
#' 	
#' In general, the formula for the first PC is $pc_1=a_1^TX=\sum_{i=1}^N a_{i, 1}X_i$ where $X_i$ is a $n\times 1$ vector representing a column of the matrix $X$ (representing a total of n observations and N features). The weights $a_1=\{a_{1, 1}, a_{2, 1}, ..., a_{N, 1}\}$ are chosen to maximize the variance of $pc_1$. According to this rule, the $k^{th}$ PC is $pc_k=a_k^TX=\sum_{i=1}^N a_{i, k}X_i$. $a_k=\{a_{1, k}, a_{2, k}, ..., a_{N, k}\}$ has to be constrained by more conditions:	
#' 	
#' 1. Variance of $pc_k$ is maximized	
#' 2. $Cov(pc_k, pc_l)=0$, $\forall 1\leq l<k$	
#' 3. $a_k^Ta_k=1$ (the weights vectors are unitary)	
#' 	
#' Let's figure out how to find $a_1$. First we need to express the variance of our first principal component using the variance covariance matrix of $X$:	
#' $$Var(pc_1)=E(pc_1^2)-(E(pc_1))^2=$$	
#' $$\sum_{i, j=1}^N a_{i, 1} a_{j, 1} E(x_i x_j)-\sum_{i, j=1}^N a_{i, 1} a_{j, 1} E(x_i)E(x_j)=$$	
#' $$\sum_{i, j=1}^N a_{i, 1} a_{j, 1} S_{i, j}.$$	
#' 	
#' Where $S_{i, j}=E(x_i x_j)-E(x_i)E(x_j)$.	
#' 	
#' This implies $Var(pc_1)=a_1^TS a_1$ where $S=S_{i, j}$ is the covariance matrix of $X=\{X_1, ..., X_N\}$. Since $a_1$ maximized $Var(pc_1)$ and the constrain $a_1^T a_1=1$ holds, we can rewrite $a_1$ as:	
#' $$a_1=max_{a_1}(a_1^TS a_1-\lambda (a_1^T a_1-1))$$	
#' Where the second part after the minus sign should be 0. To maximize this quadratic expression, we can take the derivative of this expression w.r.t. $a_1$ and set it to 0. This yields $(S-\lambda I_N)a_1=0$.	
#' 	
#' In [Chapter 4](https://www.socr.umich.edu/people/dinov/2017/Spring/DSPA_HS650/notes/04_LinearAlgebraMatrixComputing.html) we showed that $a_1$ will correspond to the largest eigenvalue of $S$, the variance covariance matrix of $X$. Hence, $pc_1$ retains the largest amount of variation in the sample. Likewise, $a_k$ is the $k^{th}$ largest eigenvalue of $S$.	
#' 	
#' PCA requires the mean for each column in the data matrix to be zero. That is, the sample mean of each column is shifted to zero. 	
#' 	
#' Let's use a subset (N=33) of [Parkinson's Progression Markers Initiative (PPMI) data](https://wiki.socr.umich.edu/index.php/SMHS_PCA_ICA_FA#Parkinson.27s_disease_case_study) to demonstrate the relationship between $S$ and PC loadings. First, we need to import the dataset into R and delete the patient ID column.	
#' 	
#' 	
library(rvest)	
wiki_url <- read_html("https://wiki.socr.umich.edu/index.php/SMHS_PCA_ICA_FA")	
html_nodes(wiki_url, "#content")	
pd.sub <- html_table(html_nodes(wiki_url, "table")[[1]])	
summary(pd.sub)	
pd.sub<-pd.sub[, -1]	
#' 	
#' 	
#' Then, we need to center the `pdsub` by subtracting the average of all column means from each element in the column. Next, we cast `pd.sub` as a matrix and compute its variance covariance matrix, $S$. Finally, we can calculate the corresponding eigenvalues and eigenvectors of $S$.	
#' 	
#' 	
mu<-apply(pd.sub, 2, mean)	
mean(mu)	
pd.center<-as.matrix(pd.sub)-mean(mu)	
S<-cov(pd.center)	
eigen(S)	
#' 	
#' 	
#' The next step would be calculating the PCs using the `prcomp()` function in R. Note that we will use the raw (`uncentered`) version of the data and have to specify the `center=TRUE` option to ensure the column means are trivial. We can save the model information into `pca1` where `pca1$rotation` provides the loadings for each PC.	
#' 	
#' 	
pca1<-prcomp(as.matrix(pd.sub), center = T)	
summary(pca1)	
pca1$rotation	
#' 	
#' 	
#' We notice that the loadings are just the eigenvectors times `-1`. These loadings represent vectors in 6D space (we have 6 columns in the original data). The scale factor `-1`  just represents the opposite direction of the eigenvector. We can also load the `factoextra` package an compute the eigenvalues of each PC.	
#' 	
#' 	
# install.packages("factoextra")	
library("factoextra")	
eigen<-get_eigenvalue(pca1)	
eigen	
#' 	
#' 	
#' The eigenvalues correspond to the amount of the variation explained by each principal component (PC), which is the same as the eigenvalues of the $S$ matrix.	
#' 	
#' To see a detailed information about the *variances* explained by each PC, relative to the corresponding PC loadings.	
#' 	
#' In the 3D loadings interactive plot below, you need to zoom-in to see the smaller projections (original features that have lower impact after the linear PC rotation).	
#' 	
#' 	
# plot(pca1)	
# library(graphics)	
# biplot(pca1, choices = 1:2, scale = 1, pc.biplot = F)	
	
plot_ly(x = c(1:length(pca1$sdev)), y = pca1$sdev*pca1$sdev, name = "Scree", type = "bar") %>%	
  layout(title="Scree Plot", xaxis = list(title="PC's"),  yaxis = list(title="Variances (SD^2)"))	
	
	
# Scores	
scores <- pca1$x	
	
# Loadings	
loadings <- pca1$rotation	
	
# Visualization scale factor for loadings	
scaleLoad <- 10	
	
p <- plot_ly() %>%	
  add_trace(x=scores[,1], y=scores[,2], z=scores[,3], type="scatter3d", mode="markers", name="",	
            marker = list(color=scores[,2], colorscale = c("#FFE1A1", "#683531"), opacity = 0.7)) 	
for (k in 1:nrow(loadings)) {	
   x <- c(0, loadings[k,1])*scaleLoad	
   y <- c(0, loadings[k,2])*scaleLoad	
   z <- c(0, loadings[k,3])*scaleLoad	
   p <- p %>% add_trace(x=x, y=y, z=z, type="scatter3d", mode="lines", 	
  name=paste0("Loading PC ", k, " ", colnames(pd.sub)[k]), line=list(width=8), opacity=1) 	
  # %>%	
  # add_annotations( x = 0, y = 0, z = 0,	
  #           xref = "x", yref = "y",  zref = "z",	
  #           # axref = "x", ayref = "y", azref = "z",	
  #           text = "", showarrow = T,	
  #           ax = c(0, loadings[k,1])*scaleLoad, ay = c(0, loadings[k,2])*scaleLoad, az = c(0,	
  #                  loadings[k,3])*scaleLoad)	
}	
	
p <- p %>%	
  layout(legend = list(orientation = 'h'), title="3D Projection of 6D Data along First 3 PCs") 	
p	
	
#          scene = list(	
#             dragmode = "turntable",	
#             annotations = list(	
#               list(showarrow = T, x = c(0, loadings[1,1])*scaleLoad, y = c(0, loadings[1,2])*scaleLoad, 	
#                    z = c(0, loadings[1,3])*scaleLoad, text = "Point 1", xanchor = "left", xshift = 2, opacity=0.7),	
#               list(showarrow = T, x = c(0, loadings[2,1])*scaleLoad, y = c(0, loadings[2,2])*scaleLoad, 	
#                    z = c(0, loadings[2,3])*scaleLoad, text="Point 2", textangle=0, ax = 0, ay = -1, font = list(	
#                 color = "black", size = 12), arrowcolor = "black", arrowsize = 3, arrowwidth = 1, arrowhead = 1),	
#               list(showarrow = T, x = c(0, loadings[3,1])*scaleLoad, y = c(0, loadings[3,2])*scaleLoad, 	
#                    z = c(0, loadings[3,3])*scaleLoad, text = "Point 3", arrowhead = 1,	
#                    xanchor = "left", yanchor = "bottom")	
#     )))	
	
# library("factoextra")	
# # Data for the supplementary qualitative variables	
# qualit_vars <- as.factor(pd.sub$Part_IA)	
# head(qualit_vars)	
# # for plots of individuals	
# # fviz_pca_ind(pca1, habillage = qualit_vars, addEllipses = TRUE, ellipse.level = 0.68) +	
# #  theme_minimal()	
# # for Biplot of individuals and variables 	
# fviz_pca_biplot(pca1, axes = c(1, 2), geom = c("point", "text"),	
#   col.ind = "black", col.var = "steelblue", label = "all",	
#   invisible = "none", repel = T, habillage = qualit_vars, 	
#   palette = NULL, addEllipses = TRUE, title = "PCA - Biplot")	
#' 	
#' 	
#' The scree-plot has a clear "elbow" point at the second PC, suggesting that the first two PCs explain about 95% of the variation in the original dataset. Thus, we say we can use the first 2 PCs to represent the data. In this case, the dimension of the data is substantially reduced. 	
#' 	
#' The dynamic 3D `plot_ly` graph uses PC1, PC2, and PC3 as the the coordinate axes to represent the new variables and the lines radiating from the origin show the loadings on the original features. This *triplot* help us to visualize how the loadings are used to rearrange the structure of the data. 	
#' 	
#' Next, let's try to obtain a bootstrap test for the confidence interval of the explained variance.	
#' 	
#' 	
set.seed(12)	
num_boot = 1000	
bootstrap_it = function(i) {	
  data_resample = pd.sub[sample(1:nrow(pd.sub), nrow(pd.sub), replace=TRUE),] 	
  p_resample = princomp(data_resample,cor = T) 	
  return(sum(p_resample$sdev[1:3]^2)/sum(p_resample$sdev^2))	
  }	
	
pco = data.frame(per=sapply(1:num_boot, bootstrap_it)) 	
quantile(pco$per, probs = c(0.025,0.975))  # specify 95-th % Confidence Interval	
corpp = sum(pca1$sdev[1:3]^2)/sum(pca1$sdev^2)	
	
# require(ggplot2)	
# plot = ggplot(pco, aes(x=pco$per)) +	
#   geom_histogram() + geom_vline(xintercept=corpp, color='yellow')+ 	
#   labs(title = "Percent Var Explained by the first 3 PCs") +	
#   theme(plot.title = element_text(hjust = 0.5))+	
#   labs(x='perc of var')	
# show(plot)	
	
plot_ly(x = pco$per, type = "histogram", name = "Data Histogram") %>% 	
    layout(title='Histogram of a Bootstrap Simulation Showing Percent of Data Variability Captured by first 3 PCs', 	
           xaxis = list(title = "Percent of Variability"), yaxis = list(title = "Frequency Count"), bargap=0.1)	
#' 	
#' 	
#' Suppose we want to fit a linear model `Top_of_SN_Voxel_Intensity_Ratio ~ Side_of_SN_Voxel_Intensity_Ratio + Part_IA`. We can use `plot_ly` to show a 3D scatterplot and the (univariate) linear model. 	
#' 	
#' 	
library(scatterplot3d)	
	
#Fit linear model	
lm.fit <- lm(Top_of_SN_Voxel_Intensity_Ratio ~	Side_of_SN_Voxel_Intensity_Ratio + Part_IA, data = pd.sub)	
	
# Get the ranges of the variable.names	
summary(pd.sub$Side_of_SN_Voxel_Intensity_Ratio)	
summary(pd.sub$Part_IA)	
summary(pd.sub$Top_of_SN_Voxel_Intensity_Ratio)	
	
# #plot results	
# myPlot <- scatterplot3d(pd.sub$Side_of_SN_Voxel_Intensity_Ratio, pd.sub$Part_IA,	
#                 pd.sub$Top_of_SN_Voxel_Intensity_Ratio)	
# # Plot the linear model (line in 3D)	
# myCoef <- lm.fit$coefficients	
# plotX <- seq(0.93, 1.4,length.out = 100)	
# plotY <- seq(0,6,length.out = 100)	
# plotZ <- myCoef[1] + myCoef[2]*plotX + myCoef[3]*plotY # linear model	
# #Add the linear model to the 3D scatterplot	
# myPlot$points3d(plotX,plotY,plotZ, type = "l", lwd=2, col = "red")	
	
cf = lm.fit$coefficients	
pltx = seq(summary(pd.sub$Side_of_SN_Voxel_Intensity_Ratio)[1],	
           summary(pd.sub$Side_of_SN_Voxel_Intensity_Ratio)[6], 	
           length.out = length(pd.sub$Side_of_SN_Voxel_Intensity_Ratio))	
plty = seq(summary(pd.sub$Part_IA)[1], summary(pd.sub$Part_IA)[6],length.out = length(pd.sub$Part_IA))	
pltz = cf[1] + cf[2]*pltx + cf[3]*plty	
	
# Plot Scatter and add the LM line to the plot	
plot_ly() %>%	
  add_trace(x = ~pltx, y = ~plty, z = ~pltz, type="scatter3d", mode="lines",	
        line = list(color = "red", width = 4), 	
        name="lm(Top_of_SN_Voxel_Intensity_Ratio ~	Side_of_SN_Voxel_Intensity_Ratio + Part_IA)") %>% 	
  add_markers(x = ~pd.sub$Side_of_SN_Voxel_Intensity_Ratio, y = ~pd.sub$Part_IA, 	
              z = ~pd.sub$Top_of_SN_Voxel_Intensity_Ratio, color = ~pd.sub$Part_II, mode="markers") %>% 	
  layout(title="lm(Top_of_SN_Voxel_Intensity_Ratio ~	Side_of_SN_Voxel_Intensity_Ratio + Part_IA)",	
    legend=list(orientation = 'h'), showlegend = F,	
    scene = list(xaxis = list(title = 'Side_of_SN_Voxel_Intensity_Ratio'),	
                 yaxis = list(title = 'Part_IA'),	
                 zaxis = list(title = 'Top_of_SN_Voxel_Intensity_Ratio'))) %>% 	
  hide_colorbar()	
#' 	
#' 	
#' We can also plot in 3D a bivariate 2D plane model (e.g., `lm.fit', or `pca1`) for the 3D scatter (Top_of_SN_Voxel_Intensity_Ratio, Side_of_SN_Voxel_Intensity_Ratio, Part_IA). Below is an example using the linear model.	
#' 	
#' 	
myPlot <- scatterplot3d(pd.sub$Side_of_SN_Voxel_Intensity_Ratio, pd.sub$Part_IA, pd.sub$Top_of_SN_Voxel_Intensity_Ratio)	
	
# Static Plot	
myPlot$plane3d(lm.fit, lty.box = "solid")	
# planes3d(a, b, c, d, alpha = 0.5)	
# planes3d draws planes using the parametrization a*x + b*y + c*z + d = 0.	
# Multiple planes may be specified by giving multiple values for the normal	
# vector (a, b, c) and the offset parameter d	
#' 	
#' 	
#' Next are examples of plotting the 3D scatter along with the 2D PCA model using either `rgl` or `plot_ly`.	
#' 	
#' 	
pca1 <- prcomp(as.matrix(cbind(pd.sub$Side_of_SN_Voxel_Intensity_Ratio, pd.sub$Part_IA, pd.sub$Top_of_SN_Voxel_Intensity_Ratio)), center = T); summary(pca1)	
	
# Given two vectors PCA1 and PCA2, the cross product V = PCA1 x PCA2	
	
# is orthogonal to both A and to B, and a normal vector to the 	
# plane containing PCA1 and PCA2	
# If PCA1 = (a,b,c) and PCA2 = (d, e, f), then the cross product is	
# PCA1 x PCA2 =  (bf - ce, cd - af, ae - bd)	
# PCA1 = pca1$rotation[,1] and PCAS2=pca1$rotation[,2]	
# https://en.wikipedia.org/wiki/Cross_product#Names	
#normVec = c(pca1$rotation[,1][2]*pca1$rotation[,2][3]-	
#              pca1$rotation[,1][3]*pca1$rotation[,2][2],	
#            pca1$rotation[,1][3]*pca1$rotation[,2][1]-	
#              pca1$rotation[,1][1]*pca1$rotation[,2][3],	
#            pca1$rotation[,1][1]*pca1$rotation[,2][2]-	
#              pca1$rotation[,1][2]*pca1$rotation[,2][1]	
#            )	
normVec = c(pca1$rotation[2,1]*pca1$rotation[3,2]-	
              pca1$rotation[3,1]*pca1$rotation[2,2],	
            pca1$rotation[3,1]*pca1$rotation[1,2]-	
              pca1$rotation[1,1]*pca1$rotation[3,2],	
            pca1$rotation[1,1]*pca1$rotation[2,2]-	
              pca1$rotation[2,1]*pca1$rotation[1,2]	
            )	
	
# Interactive RGL 3D plot with PCA Plane	
library(rgl) 	
	
# Compute the 3D point representing the gravitational balance	
dMean <- apply( cbind(pd.sub$Side_of_SN_Voxel_Intensity_Ratio, pd.sub$Top_of_SN_Voxel_Intensity_Ratio, pd.sub$Part_IA), 2, mean)	
# then the offset plane parameter is (d):	
d <- as.numeric((-1)*normVec %*% dMean)  # force the plane to go through the mean	
	
# Plot the PCA Plane	
plot3d(pd.sub$Side_of_SN_Voxel_Intensity_Ratio, pd.sub$Part_IA, pd.sub$Top_of_SN_Voxel_Intensity_Ratio, type = "s", col = "red", size = 1)	
planes3d(normVec[1], normVec[2], normVec[3], d, alpha = 0.5)	
#' 	
#' 	
#' Another alternative is to use `plot_ly` for the interactive 3D visualization. First, we demonstrate displaying the `lm()` derived plane modeling superimposed on the 3D scatterplot (using `pd.sub$Side_of_SN_Voxel_Intensity_Ratio`, `pd.sub$Top_of_SN_Voxel_Intensity_Ratio`, and `pd.sub$Part_IA`).	
#' 	
#' 	
# Define the 3D features	
x <- pd.sub$Side_of_SN_Voxel_Intensity_Ratio	
y <- pd.sub$Top_of_SN_Voxel_Intensity_Ratio	
z <- pd.sub$Part_IA	
myDF <- data.frame(x, y, z)	
	
### Fit a (bivariate-predictor) linear regression model	
lm.fit <- lm(z ~ x+y)	
coef.lm.fit <- coef(lm.fit)	
	
### Reparameterize the 2D (x,y) grid, and define the corresponding model values z on the grid	
x.seq <- seq(min(x),max(x),length.out=100)	
y.seq <- seq(min(y),max(y),length.out=100)	
z.seq <- function(x,y) coef.lm.fit[1]+coef.lm.fit[2]*x+coef.lm.fit[3]*y	
# define the values of z = z(x.seq, y.seq), as a Matrix of dimension c(dim(x.seq), dim(y.seq))	
z <- t(outer(x.seq, y.seq, z.seq))	
	
# First draw the 2D plane embedded in 3D, and then add points with "add_trace"	
# library(plotly)	
# myPlotly <- plot_ly(x=~x.seq, y=~y.seq, z=~z,	
#     colors = c("blue", "red"),type="surface", opacity=0.7) %>%	
#   	
#   add_trace(data=myDF, x=x, y=y, z=pd.sub$Part_IA, mode="markers", 	
#             type="scatter3d", marker = list(color="green", opacity=0.9, 	
#                                             symbol=105)) %>%	
#   	
#   layout(scene = list(	
#     aspectmode = "manual", aspectratio = list(x=1, y=1, z=1),	
#     xaxis = list(title = "Side_of_SN_Voxel_Intensity_Ratio"),	
#     yaxis = list(title = "Top_of_SN_Voxel_Intensity_Ratio"),	
#     zaxis = list(title = "Part_IA"))	
#     )	
# # print(myPlotly)	
# myPlotly	
	
#Setup Axis	
axis_x <- seq(min(x), max(x), length.out=100)	
axis_y <- seq(min(y), max(y), length.out=100)	
	
#Sample points	
library(reshape2)	
lm_surface <- expand.grid(x = axis_x, y = axis_y, KEEP.OUT.ATTRS = F)	
lm_surface$z <- predict.lm(lm.fit, newdata = lm_surface)	
lm_surface <- acast(lm_surface, x ~ y, value.var = "z") # z ~ 0 + x + y	
	
plot_ly(myDF, x = ~x, y = ~y, z = ~z,	
        text = paste0("Part_II: ", pd.sub$Part_II), type="scatter3d", mode="markers", color=pd.sub$Part_II) %>%	
  add_trace(x=~axis_x, y=~axis_y, z=~lm_surface, type="surface", color="gray", name="LM model", opacity=0.3) %>%	
  layout(title="3D Plane Regression (Part_IA ~ Side_of_SN_Voxel + Top_of_SN_Voxel); Color=Part II", showlegend = F,	
         scene = list ( xaxis = list(title = "Side_of_SN_Voxel_Intensity_Ratio"),	
                        yaxis = list(title = "Top_of_SN_Voxel_Intensity_Ratio"),	
                        zaxis = list(title = "Part_IA"))) %>% 	
  hide_colorbar()	
#' 	
#' 	
#' Second, we show the PCA-derived 2D plane model superimposed on the 3D scatterplot.	
#' 	
#' 	
# define the original 3D coordinates	
x <- pd.sub$Side_of_SN_Voxel_Intensity_Ratio	
y <- pd.sub$Top_of_SN_Voxel_Intensity_Ratio	
z <- pd.sub$Part_IA	
myDF <- data.frame(x, y, z)	
	
### Fit (compute) the 2D PCA space (dimensionality reduction)	
pca1 <- prcomp(as.matrix(cbind(pd.sub$Side_of_SN_Voxel_Intensity_Ratio, pd.sub$Top_of_SN_Voxel_Intensity_Ratio, pd.sub$Part_IA)), center = T); summary(pca1)	
# Compute the Normal to the 2D PC plane	
normVec = c(pca1$rotation[2,1]*pca1$rotation[3,2]-	
              pca1$rotation[3,1]*pca1$rotation[2,2],	
            pca1$rotation[3,1]*pca1$rotation[1,2]-	
              pca1$rotation[1,1]*pca1$rotation[3,2],	
            pca1$rotation[1,1]*pca1$rotation[2,2]-	
              pca1$rotation[2,1]*pca1$rotation[1,2]	
            )	
# Compute the 3D point of gravitational balance (Plane has to go through it)	
dMean <- apply( cbind(pd.sub$Side_of_SN_Voxel_Intensity_Ratio, pd.sub$Top_of_SN_Voxel_Intensity_Ratio, pd.sub$Part_IA), 2, mean)	
	
d <- as.numeric((-1)*normVec %*% dMean)  # force the plane to go through the mean	
	
# Reparameterize the 2D (x,y) grid, and define the corresponding model values z on the grid. Recall z=-(d + ax+by)/c, where normVec=(a,b,c)	
x.seq <- seq(min(x),max(x),length.out=100)	
y.seq <- seq(min(y),max(y),length.out=100)	
z.seq <- function(x,y) -(d + normVec[1]*x + normVec[2]*y)/normVec[3]	
# define the values of z = z(x.seq, y.seq), as a Matrix of dimension c(dim(x.seq), dim(y.seq))	
#z <- t(outer(x.seq, y.seq, z.seq))/10; range(z)  # we need to check this 10 correction, to ensure the range of z is appropriate!!!	
z <- t(outer(x.seq, y.seq, z.seq)); range(z)	
	
# Draw the 2D plane embedded in 3D, and then add points with "add_trace"	
# library(plotly)	
myPlotly <- plot_ly(x=~x.seq, y=~y.seq, z=~z,	
    colors = c("blue", "red"),type="surface", opacity=0.7) %>%	
    add_trace(data=myDF, x=x, y=y, z=(pd.sub$Part_IA-mean(pd.sub$Part_IA))*10, mode="markers", 	
            showlegend=F, type="scatter3d", marker=list(color="green", opacity=0.9, symbol=105)) %>%	
    layout(scene = list(	
    aspectmode = "manual", aspectratio = list(x=1, y=1, z=1),	
    xaxis = list(title = "Side_of_SN_Voxel_Intensity_Ratio"),	
    yaxis = list(title = "Top_of_SN_Voxel_Intensity_Ratio"),	
    zaxis = list(title = "Part_IA")) ) %>%	
  hide_colorbar()	
# print(myPlotly)	
myPlotly	
#' 	
#' 	
#' As stated in the [summary table](https://www.socr.umich.edu/people/dinov/courses/DSPA_notes/05_DimensionalityReduction.html#4_summary_(pca_vs_ica_vs_fa)), classical PCA assumes that the bivariate relations are linear in nature. The non-linear PCA is a generalization that allows us to incorporate nominal and ordinal variables, as well as to handle and identify nonlinear relationships between variables in the dataset. See Chapter 2 of this textbook [Nonparametric inference in nonlinear principal components analysis: exploration and beyond](https://openaccess.leidenuniv.nl/handle/1887/12386).	
#' 	
#' Non-linear PCA assigns values to the categories representing the numeric variables, which maximize the association (e.g., correlation) between the quantified variables (i.e., optimal scaling to quantify the variables according to their analysis levels). The [Bioconductor's pcaMethods](https://www.bioconductor.org/packages/release/bioc/html/pcaMethods.html) package provides the functionality for non-linear PCA.	
#' 	
#' # Independent component analysis (ICA)	
#' ICA aims to find basis vectors representing independent components of the original data. For example, this may be achieved by maximizing the norm of the $4^{th}$ order normalized kurtosis, which iteratively projects the signal on a new basis vector, computes the objective function (e.g., the norm of the kurtosis) of the result, slightly adjusts the basis vector (e.g., by gradient ascent), and recomputes the kurtosis again. At the end, this iterative process generates a basis vector corresponding to the highest (residual) kurtosis representing the next independent component. 	
#' 	
#' The process of Independent Component Analysis is to maximize the statistical independence of the estimated components. Assume that each variable $X_i$ is generated by a sum of *n* independent components.	
#' $$X_i=a_{i, 1}s_1+...+a_{i, n}s_n$$	
#' Here, $X_i$ is generated by $s_1, ..., s_n$ and $a_{i,1}, ... a_{i,n}$ are the corresponding weights. Finally, we rewrite $X$ as	
#' $$X=As,$$	
#' where $X=(X_1, ..., X_n)^T$, $A=(a_1, ..., a_n)^T$, $a_i=(a_{i,1}, ..., a_{i,n})$ and $s=(s_1, ..., s_n)^T$. Note that $s$ is obtained by maximizing the independence of the components. This procedure is done by maximizing some *independence* objective function.	
#' 	
#' ICA does not assume that all of its components ($s_i$) are Gaussian and independent of each other.	
#' 	
#' We will utilize the `fastICA` function in R.	
#' 	
#' `fastICA(X, n.comp, alg.typ, fun, rownorm, maxit, tol)`	
#' 	
#' * **X**: data matrix	
#' * **n.comp**:number of components, 	
#' * **alg.type**: components extracted simultaneously(`alg.typ == "parallel"`) or one at a time(`alg.typ == "deflation"`)	
#' * **fun**: functional form of F to approximate to neg-entropy, 	
#' * **rownorm**: whether rows of the data matrix X should be standardized beforehand	
#' * **maxit**: maximum number of iterations 	
#' * **tol**: a positive scalar giving the tolerance at which the un-mixing matrix is considered to have converged.	
#' 	
#' Now we can create a correlated matrix $X$. 	
#' 	
#' 	
S <- matrix(runif(10000), 5000, 2)	
S[1:10, ]	
A <- matrix(c(1, 1, -1, 3), 2, 2, byrow = TRUE)	
X <- S %*% A # In R,  "*" and "%*%" indicate "scalar" and matrix multiplication, respectively!	
cor(X)	
#' 	
#' The correlation between two variables is -0.4. Then we can start to fit the ICA model.	
#' 	
# install.packages("fastICA")	
library(fastICA)	
a <- fastICA(X, 2, alg.typ = "parallel", fun = "logcosh", alpha = 1, 	
            method = "C", row.norm = FALSE, maxit = 200, tol = 0.0001)	
#' 	
#' 	
#' To visualize the correlation of the original pre-processed data ($X$) and the independence of the corresponding ICA components $S=fastICA(X)\$S$ we can draw the following composite scatter-plot.	
#' 	
#' 	
# par(mfrow = c(1, 2))	
# plot(a$X, main = "Pre-processed data")	
# plot(a$S, main = "ICA components")	
	
plot_ly() %>%  	
  add_markers(x = a$X[ , 1], y =~a$X[ , 2], name="Pre-processed data", 	
              marker = list(color="green", opacity=0.9, symbol=105)) %>%	
  add_markers(x = a$S[ , 1], y = a$S[ , 2], name="ICA components",	
              marker = list(color="blue", opacity=0.99, symbol=5))  %>% 	
  layout(title='Scatter Plots of the Original (Pre-processed) Data and the corresponding ICA Transform', 	
         xaxis = list(title="Twin 1 (standardized height)", scaleanchor = "y"), 	
         yaxis = list(title="Twin 2 (standardized height)", scaleanchor = "x"),	
         legend = list(orientation = 'h'))	
#' 	
#' 	
#' Finally we can confirm that the correlation of two components is nearly 0.	
#' 	
#' 	
cor(a$S)	
#' 	
#' 	
#' Let's look at a more interesting example, based on the `pd.sub` dataset. It has 6 variables and the correlation is relatively high. After fitting the ICA model. The components are nearly independent.	
#' 	
#' 	
cor(pd.sub)	
a1<-fastICA(pd.sub, 2, alg.typ = "parallel", fun = "logcosh", alpha = 1, 	
                 method = "C", row.norm = FALSE, maxit = 200, 	
                 tol = 0.0001)	
par(mfrow = c(1, 2))	
cor(a1$X)	
cor(a1$S)	
#' 	
#' Notice that we only have 2 components instead of 6 variables. We successfully reduced the dimension of the data.	
#' 	
#' # Factor analysis (FA)	
#' 	
#' Similar to ICA and PCA, FA tries to find special principal components in data. As a generalization of PCA, FA requires that the number of components is smaller than the original number of variables (or columns of the data matrix). FA optimization relies on iterative perturbations with full-dimensional Gaussian noise and maximum-likelihood estimation where every observation in the data represents a sample point in a higher dimensional space. Whereas PCA assumes the noise is spherical, Factor Analysis allows the noise to have an arbitrary diagonal covariance matrix and estimates the subspace as well as the noise covariance matrix.	
#' 	
#' Under FA, the centered data can be expressed in the following from:	
#' 	
#' $$x_i-\mu_i=l_{i, 1}F_1+...+l_{i, k}F_k+\epsilon_i=LF+\epsilon_i,$$	
#' 	
#' where $i\in {1, ..., p}$, $j \in{1, ..., k}$, $k<p$ and $\epsilon_i$ are independently distributed error terms with zero mean and finite variance.	
#' 	
#' Let's try FA in R using the function `factanal()`. According to the previous PCA, our `pd.sub` dataset can explain 95% of variance using only the first two principal components. This suggest that we might need 2 factors in FA. We can double check that by examining the *scree* plot.	
#' 	
#' 	
# Determine Number of Factors to Extract	
# install.packages("nFactors")	
	
library(nFactors)	
par(mfrow=c(1, 1))	
ev <- eigen(cor(pd.sub)) # get eigenvalues	
ap <- parallel(subject=nrow(pd.sub), var=ncol(pd.sub), rep=100, cent=.05)	
nS <- nScree(x=ev$values, aparallel=ap$eigen$qevpea)	
summary(nS)	
# plotnScree(nS)	
	
plot_ly() %>% 	
  add_trace(y = nS$Analysis$Eigenvalues, type="scatter", name = 'Eigenvalues', 	
              mode = 'lines+markers', marker = list(opacity=0.99, size=20, symbol=5)) %>%	
  add_trace(y = nS$Analysis$Par.Analysis, type="scatter", 	
            name = 'Parallel Analysis (centiles of random eigenvalues)', 	
            mode = 'lines+markers', marker = list(opacity=0.99, size=20, symbol=2)) %>%	
  # add_trace(y = nS$Analysis$OC, type="scatter", 	
  #           name = 'Critical Optimal Coordinates', 	
  #           mode = 'lines+markers', marker = list(opacity=0.99, size=20, symbol=3)) %>%	
  add_trace(y = nS$Analysis$Acc.factor, type="scatter", 	
            name = 'Acceleration Factor', 	
            mode = 'lines+markers', marker = list(opacity=0.99, size=20, symbol=15)) %>%	
  layout(title='Scree plot', 	
         xaxis = list(title="Components)", scaleanchor = "y"), 	
         yaxis = list(title="Eigenvalues)", scaleanchor = "x"),	
         legend = list(orientation = 'h'))	
#' 	
#' 	
#' Note that 3 out of 4 Cattell's Scree test rules summary suggest we should use 2 factors. Thus, in the function `factanal()` we can specify `factors=2`. In addition, we can use `varimax` rotation of the factor axes maximizing the variance of the squared loadings of factors (columns) on the original variables (rows), which effectively differentiates the original variables by the extracted factors. Oblique `promax` and `Procrustes rotation` (projecting the loadings to a target matrix with a simple structure) are two alternative and commonly used matrix rotations that may be specified.	
#' 	
#' 	
fit<-factanal(pd.sub, factors=2, rotation="varimax")	
# fit<-factanal(pd.sub, factors=2, rotation="promax") # the most popular oblique rotation; And fitting a simple structure	
fit	
#' 	
#' 	
#' Here the p-value 0.854 is very large, suggesting that we failed to reject the null-hypothesis that 2 factors are sufficient. We can also visualize the loadings for all the variables.	
#' 	
#' 	
load <- fit$loadings	
# plot(load, type="n") # set up plot 	
# text(load, labels=colnames(pd.sub), cex=.7) # add variable names	
	
df <- as.data.frame(load[])	
Features <- rownames(df)	
X <- df$Factor1	
Y <- df$Factor2	
df1 <- data.frame(Features, X, Y)	
cols <- palette(rainbow(6))   # as.numeric(as.factor(Features))	
cols <- cols[2:7]   # this is necessary as cols has 8 rows (not 6, as df1 does!)	
	
plot_ly(df1, x = ~X, y = ~Y, text = ~Features, color = cols) %>% 	
  add_markers(marker = list(opacity=0.99, size=20, color=cols, symbol=~as.numeric(as.factor(Features)))) %>% 	
  add_text(textfont = list(family= "Times", size= 20, color= cols), textposition="top right") %>% 	
  layout(title = '2D FA', xaxis = list(title = 'Factor 1', zeroline = TRUE,range = c(-0.5, 1)),	
         yaxis = list(title = 'Factor 2'), showlegend = FALSE)	
#' 	
#' 	
#' This plot displays *factors 1* and *2* on the x-axis and y-axis, respectively.	
#' 	
#' # Singular Value Decomposition (SVD)	
#' 	
#' SVD is a factorization of a real or complex matrix. If we have a data matrix $X$ with $n$ observation and $p$ variables it can be factorized into the following form:	
#' $$X=U D V^T,$$ 	
#' where $U$ is a $n \times p$ unitary matrix that $U^TU=I$, $D$ is a $p \times p$ diagonal matrix, and $V^T$ is a $p \times p$ unitary matrix, which is the conjugate transpose of the $n\times n$ unitary matrix, $V$. Thus, we have $V^TV=I$.	
#' 	
#' SVD is closely linked to PCA (when correlation matrix is used for calculation). $U$ represents the left singular vectors, $D$ the singular values, $U%*%D$ yields the PCA scores, and $V$  the right singular vectors - PCA loadings.	
#' 	
#' Using the `pd.sub` dataset, we can compare the outputs from the `svd()` function and the `princomp()` function (another R function for PCA). Prior to the SVD, we need to `scale` the data matrix.	
#' 	
#' 	
#SVD output	
df<-nrow(pd.sub)-1	
zvars<-scale(pd.sub)	
z.svd<-svd(zvars)	
z.svd$d/sqrt(df)	
z.svd$v	
#PCA output	
pca2<-princomp(pd.sub, cor=T)	
pca2	
loadings(pca2)	
#' 	
#' 	
#' When correlation matrix is used for calculation (`cor=T`), the $V$ matrix of SVD contains the corresponding PCA loadings.	
#' 	
#' ## SVD Summary	
#' Intuitively, the SVD approach $X= UD V^T$ represents a composition of the (centered!) data into 3 geometrical transformations: a rotation or reflection ($U$), a scaling ($D$), and a rotation or reflection ($V$). Here we assume that the data $X$ stores samples/cases in rows and variables/features in columns. If these are reversed, then the interpretations of the $U$ and $V$ matrices reverse as well.	
#' 	
#' * The columns of $V$ represent the directions of the principal axes, the columns of $UD$ are the principal components, and the singular values in $D$ are related to the eigenvalues of data variance-covariance matrix ($\Sigma$) via $\lambda_i = \frac{d_i^2}{n-1}$, where the eigenvalues $\lambda_i$ capture the magnitude of the data variance in the respective PCs.	
#' * The standardized scores are given by columns of $\sqrt{n-1}U$, the corresponding loadings are given by columns of $\frac{1}{n-1}VD$. However, these "loadings" *are not* the principal directions. The requirement for $X$ to be centered is needed to ensure that the covariance matrix $Cov(X) = \frac{1}{n-1}X^TX$.	
#' * Alternatively, to perform PCA on the *correlation matrix* (instead of the covariance matrix), the columns of $X$ need to be *scaled* (centered and standardized).	
#' * Reduce the data dimensionality from $p$ to $k<p$, we multiply the first $k$ columns of $U$ by the $k\times k$ upper-left corner of the matrix $D$ to get an $n\times k$ matrix $U_kD_k$ containing the first $k$ PCs.	
#' * Multiplying the first $k$ PCs by their corresponding principal directions $V_k^T$ reconstructs the original data from the first $k$ PCs, $X_k=U_k D_kV_k^T$, with the lowest possible reconstruction error.	
#' * Typically we have more subjects/cases ($n$) than variables/features ($p<n$). As $U_{n\times n}$ and $V_{p\times p}$, the last $n-p>0$ columns of $U$ may be trivial (zeros). It's customary to drop the zero columns of $U$ for $n>>p$ to avid dealing with unnecessarily large (trivial) matrices. 	
#' 	
#' # t-SNE (t-distributed Stochastic Neighbor Embedding)	
#' 	
#' The t-SNE technique represents a recent machine learning strategy for nonlinear dimensionality reduction that is useful for embedding (e.g., scatter-plotting) of high-dimensional data into lower-dimensional (1D, 2D, 3D) spaces. For each object (point in the high-dimensional space), the method models *similar objects* using nearby and *dissimilar objects* using remote distant objects. The two steps in t-SNE include (1) construction of a probability distribution over pairs of the original high-dimensional objects where similar objects have a high probability of being paired and correspondingly, dissimilar objects have a small probability of being selected; and (2) defining a similar probability distribution over the points in the derived low-dimensional embedding minimizing the [Kullback-Leibler divergence](https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence) between the high- and low-dimensional distributions relative to the locations of the objects in the embedding map. Either Euclidean or non-Euclidean distance measures between objects may be used as similarity metrics.	
#' 	
#' ## t-SNE Formulation	
#' Suppose we have high dimensional data ($N$D): $x_1, x_2,..., x_N$. In `step 1`, for each pair ($x_i, x_j$), t-SNE estimates the probabilities $p_{i,j}$ that are proportional to their corresponding similarities, $p_{j | i}$:	
#' 	
#' $$p_{j | i} = \frac{\exp\left (\frac{-||x_i - x_j||^2}{2\sigma_i^2} \right )}{\sum_{k \neq i} \exp\left (\frac{-||x_i - x_k||^2}{2\sigma_i^2} \right )}.$$	
#' 	
#' The similarity between $x_j$ and $x_i$ may be thought of as the conditional probability, $p_{j | i}$. That is, assuming $N$D Gaussian distributions centered at each point $x_i$, neighbors are selected based on a probability distribution (proportion of their probability density), which represents the chance that $x_i$ may select $x_j$ as its neighbor, $p_{i,j} = \frac{p_{j | i} + p_{i |j}}{2N}$.	
#' 	
#' The **perplexity** ($perp$) of a discrete probability distribution, $p$, is defined as an exponential function of the entropy, $H(p)$, over all discrete events: $perp(x)=2^{H(p)}=2^{-\sum _{x}p(x)\log_{2}p(x)}$. t-SNE performs a binary search for the value $\sigma_i$ that produces a predefined value $perp$. The simple interpretation of the perplexity at a data point $x_i$, $2^{H(p_i)}$, is as a smooth measure of the effective number of points in the $x_i$ neighborhood. The performance of t-SNE may vary with the perplexity value, which is typically specified by the user, e.g., between $5\leq perp\leq 50$.	
#' 	
#' Then, the precision (variance, $\sigma_i$) of the local Gaussian kernels may be chosen to ensure that the *perplexity* of the conditional distribution equals a specified perplexity. This allows adapting the kernel bandwidth to the sample data density -- smaller $\sigma_i$ values are fitted in denser areas of the sample data space, and correspondingly, larger $\sigma_i$ are fitted in sparser areas. A particular value of $\sigma_i$ yields a probability distribution, $p_i$, over all of other data points, which has an increasing entropy as $\sigma_i$ increases.	
#' 	
#' t-SNE learns a mapping $f: \{x_1, x_2, ..., x_N\} \longrightarrow \{y_1, y_2, ..., y_d\}$, where $x_i\in R^N$ and $y_i \in R^d$ ($N\gg d$) that resembles closely the *original similarities*, $p_{i,j}$ and represents the *derived similarities*, $q_{i,j}$ between pairs of embedded points $y_i,y_j$, defined by:	
#' 	
#' $$q_{i,j} = \frac{(1 + ||y_i - y_j||^2)^{-1}}{\sum_{k \neq i} (1 + ||y_i - y_k||^2)^{-1}}.$$	
#' 	
#' The `t-distributed` reference in t-SNE refers to the heavy-tailed *Student-t distribution* ($t_{df=1}$) which [coincides](https://wiki.socr.umich.edu/index.php/AP_Statistics_Curriculum_2007_StudentsT) with [Cauchy distribution](https://wiki.socr.umich.edu/index.php/AP_Statistics_Curriculum_2007_Cauchy), $f(z)=\frac{1}{1+z^2}$. It is used to model and measure similarities between closer points in the embedded low-dimensional space, as well as dissimilarities of objects that map far apart in the embedded space. 	
#' 	
#' The rationale for using *Student t distribution* for mapping the points is based on the fact that the volume of an $N$D ball of radius $r$, $B^N$, is proportional to $r^N$. Specifically, $V_N(r) = \frac{\pi^\frac{N}{2}}{\Gamma\left(\frac{N}{2} + 1\right)}r^N$, where $\Gamma()$ is the [Euler's gamma function](https://en.wikipedia.org/wiki/Gamma_function), which is an extension of the factorial function to non-integer arguments. For large $N$, when we select uniformly random points inside $B^N$, most points will be expected to be close to the ball surface (boundary), $S^{N-1}$, and few will be expected near the $B^N$ center, as half the volume of $B^N$ is included in the hyper-area *inside* $B^N$ and *outside* a ball of radius $r_1=\frac{1}{\sqrt[N]{2}}\times r \sim r$. You can try this with $N=2$, $\{x\in R^2 |\ ||x||\leq r\}$, representing a disk in a 2D plane. 	
#' 	
#' When reducing the dimensionality of a dataset, if we used the  Gaussian distribution for the mapping embedding into the lower dimensional space, there will be a distortion of the distribution of the distances between neighboring objects. This is simply because the *distribution* of the distances is much different between the original (high-dimensional) and a the map-transformed low-dimensional spaces. t-SNE tries to (approximately) preserve the distances in the two spaces  to avoid imbalances that may lead to biases due to excessive attraction-repulsion forces. Using Student t distribution $df=1$ (aka Cauchy distribution) for mapping the points preserves (to some extent) the distance similarity distribution, because of the heavier tails of $t$ compared to the Gaussian distribution. For a given similarity between a pair of data points, the two corresponding map points will need to be much further apart in order for their similarity to match the data similarity. 	
#' 	
#' A minimization process with respect to the objects $y_i$ using gradient descent of a (non-symmetric) objective function, *Kullback-Leibler divergence* between the distributions $Q$ and $P$ , is used to determine the object locations $y_i$ in the map, i.e.,	
#' 	
#' $$KL(P || Q) = \sum_{i \neq j} p_{i,j} \log \frac{p_{i,j}}{q_{i,j}}.$$	
#' 	
#' The minimization of the KL objective function by gradient descent may be analytically represented by:	
#' 	
#' $$\frac{\partial {KL(P||Q)}}{\partial {y_i}}= \sum_{j}{(p_{i,j}-q_{i,j})f(|x_i-x_j|) u_{i,j}},$$	
#' where $f(z)=\frac{z}{1+z^2}$ and $u_{i,j}$ is a unit vector from $y_j$ to $y_i$. This gradient represents the aggregate sum of all spring forces applied to map point $x_i$.	
#' 	
#' This optimization leads to an embedding mapping that "preserves" the object (data point) similarities of the original high-dimensional inputs into the lower dimensional space. Note that the data similarity matrix ($p_{i,j}$) is fixed, whereas its counterpart, the map similarity matrix ($q_{i,j}$) depends on the embedding map. Of course, we want these two distance matrices to be as close as possible, implying that similar data points in the original space yield similar map-points in the reduced dimension.	
#' 	
#' ## t-SNE Example: Hand-written Digit Recognition	
#' 	
#' Later, in [Chapter 10](https://www.socr.umich.edu/people/dinov/courses/DSPA_notes/10_ML_NN_SVM_Class.html) and [Chapter 22](https://www.socr.umich.edu/people/dinov/courses/DSPA_notes/22_DeepLearning.html), we will present the Optical Character Recognition (OCR) and analysis of hand-written notes (unstructured text). 	
#' 	
#' Below, we show a simple example of generating a 2D embedding of the [hand-written digits dataset](https://www.socr.umich.edu/people/dinov/2017/Spring/DSPA_HS650/data/DigitRecognizer_TrainingData.zip) using t-SNE.	
#' 	
#' 	
# install.packages("tsne"); library (tsne)	
# install.packages("Rtsne")	
library(Rtsne)	
	
# Download the hand-written digits data	
pathToZip <- tempfile()		
download.file("https://www.socr.umich.edu/people/dinov/2017/Spring/DSPA_HS650/data/DigitRecognizer_TrainingData.zip", pathToZip)		
train <- read.csv(unzip(pathToZip))		
dim(train)		
unlink(pathToZip)		
	
# identify the label-nomenclature - digits 0, 1, 2, ..., 9 - and map to diff colors	
colMap <- function(x){	
  cols <- rainbow(length(x))[order(order(x))] # reindexing by ranking the observed values	
}	
	
# Note on "order(order())": set.seed(123); x <- sample(10)	
# This experiment shows that order(order()) = rank()	
# set.seed(12345); data <- sample(6); data	
# order(data); order(order(data)); rank(data)	
# Ordering acts as its own inverse and returns a sequence starting with the index of the	
# smallest data value (1). Nested odd "order" applications yield the same vector outputs.	
# Order outputs an index vector useful for sorting the original data vector. 	
# The location of the smallest value is in the first position of the order-output.	
# The index of the second smallest data value is next, etc.	
# The last order output item represents the index of the largest data value.	
# Double-order application yields an indexing vector where the first element is the index 	
# of the smallest first-order-index, etc., which corresponds to the data vector's rank.	
	
train.labels<-train$label	
train$label<-as.factor(train$label)	
train.labels.colors <- colMap(train.labels)	
names(train.labels.colors) <- train$label # unique(train$label)	
	
# May need to check and increase the RAM allocation	
memory.limit() 	
memory.limit(50000)	
	
# Remove the labels (column 1) and Scale the image intensities to [0; 1]	
train  <- data.matrix(train[, -1]); dim(train)	
train <- t(train/255)	
	
# Visualize some of the images	
library("imager")	
# first convert the CSV data (one row per image, 42,000 rows)	
array_3D <- array(train[ , ], c(28, 28, 42000))	
mat_2D <- matrix(array_3D[,,1], nrow = 28, ncol = 28)	
plot(as.cimg(mat_2D))	
	
N <- 42000	
img_3D <- as.cimg(array_3D[,,], 28, 28, N)	
	
# plot the k-th image (1<=k<=N)	
k <- 5; plot(img_3D, k)	
k <- 6; plot(img_3D, k)	
k <- 7; plot(img_3D, k)	
	
pretitle <- function(index) bquote(bold("Image: "~.(index)))	
#layout(t(1:2))	
op <- par(mfrow = c(2,2), oma = c(5,4,0,0) + 0.1, mar = c(0,0,1,1) + 0.1)	
	
for (k in 1:4) {	
  plot(img_3D, k, xlim = c(0,28), ylim = c(28,0), axes=F, ann=T, main=pretitle(k))	
}	
	
# Run the t-SNE, tracking the execution time (artificially reducing the sample-size to get reasonable calculation time)	
execTime_tSNE <- system.time(tsne_digits <- Rtsne(t(train)[1:10000 , ], dims = 2, perplexity=30, verbose=TRUE, max_iter = 500)); execTime_tSNE	
# Full dataset(42K * 1K) execution may take over 5-mins	
# execTime_tSNE <- system.time(tsne_digits <- Rtsne(train[ , ], dims = 2, perplexity=30, verbose=TRUE, max_iter = 500)); execTime_tSNE	
	
# Plot only first 1,000 cases (to avoid clutter)	
# plot(tsne_digits$Y[1:1000, ], t='n', main="t-SNE") # don't plot the points to avoid clutter	
# text(tsne_digits$Y[1:1000, ], labels=names(train.labels.colors)[1:1000], col=train.labels.colors[1:1000])	
	
# 2D t-SNE Plot	
df <- data.frame(tsne_digits$Y[1:1000, ], train.labels.colors[1:1000])	
plot_ly(df, x = ~X1, y = ~X2, mode = 'text') %>%	
  add_text(text = names(train.labels.colors)[1:1000], textfont = list(color = df$train.labels.colors.1.1000.)) %>%	
  layout(title = "t-SNE 2D Embedding", xaxis = list(title = ""),  yaxis = list(title = ""))	
	
# 3D t_SNE plot	
execTime_tSNE <- system.time(tsne_digits3D <- Rtsne(t(train)[1:10000 , ], dims = 3, perplexity=30, verbose=TRUE, max_iter = 500)); execTime_tSNE	
	
df3D <- data.frame(tsne_digits3D$Y[1:1000, ], train.labels.colors[1:1000])	
plot_ly(df3D, x = ~df3D[, 1], y = ~df3D[, 2], z= ~df3D[, 3], mode = 'markers+text') %>%	
  add_text(text = names(train.labels.colors)[1:1000], textfont = list(color = df$train.labels.colors.1.1000.)) %>%	
  layout(title = "t-SNE 3D Embedding", 	
         scene = list(xaxis = list(title=""),  yaxis=list(title=""), zaxis=list(title="")))	
	
# Classic plot all cases as solid discs with colors corresponding to each of the 10 numbers	
# plot(tsne_digits$Y, main="t-SNE Clusters", col=train.labels.colors, pch = 19)	
# legend("topright", unique(names(train.labels.colors)), fill=unique(train.labels.colors), bg='gray90', cex=0.5)	
	
plot_ly(df3D, x = ~df3D[, 1], y = ~df3D[, 2], z= ~df3D[, 3], mode = 'markers+text') %>%	
  add_text(text = names(train.labels.colors)[1:1000], textfont = list(color = df$train.labels.colors.1.1000.)) %>%	
  layout(title = "t-SNE 3D Embedding", 	
         scene = list(xaxis = list(title=""),  yaxis=list(title=""), zaxis=list(title="")))	
	
cols <- palette(rainbow(10)) 	
	
plot_ly(df3D, x=~df3D[, 1], y=~df3D[, 2], z=~df3D[, 3], color=train.labels.colors[1:1000], 	
        colors=cols, name=names(train.labels.colors)[1:1000]) %>% 	
  add_markers() %>% 	
  layout(scene=list(xaxis=list(title=''), yaxis=list(title=''), zaxis=list(title='')), showlegend=F) %>%	
  hide_colorbar()	
#' 	
#' 	
#' The [hands-on interactive SOCR t-SNE Dimensionaltiy Reduction Activity](https://socr.umich.edu/HTML5/SOCR_TensorBoard_UKBB) provides an interactive demonstration of t-SNE utilizing TensorBoard and the UK Biobank data.	
#' 	
#' # Uniform Manifold Approximation and Projection (UMAP)	
#' 	
#' In 2018, McInnes and Healy proposed the [Uniform Manifold Approximation and Projection (UMAP)](https://arxiv.org/abs/1802.03426) (UMAP) technique for dimensional reduction.	
#' 	
#' SImilar to t-SNE, UMAP first constructs a high dimensional graph representation of initial data and employs graph-layout algorithms to project the original high-dimensional data into a lower dimensional space. The iterative process aims to preserve the *graph structure* as much as possible. The initial high-dimensional graph representation uses *simplicial complexes*, as weighted graphs where edge weights represent likelihoods of the connectivity between two graph points are connected, i.e., neighborhoods. For a given point, the UMAP connectedness metric computes the distance with other points based on the overlap of their respective neighborhoods.	
#' 	
#' The parameter controlling the neighborhood size (i.e., radius) determines the tradeoff between within and between cluster sizes. Choosing too small radius may lead to small and rather isolated clusters. Selecting too large radius may cluster all points into a single group. For local radius selection, UMAP preprocessing utilizes the distances between each point and its nth nearest neighbor. 	
#' 	
#' A "fuzzy" simplicial complex (graph) is constructed by iterative minimization of the connectivity likelihood function as the radius increases. Assuming that each point is connected to at least one of its closest neighbors, UMAP ensures that local and global graph structure is (somewhat protected and) preserved during the optimization process (e.g., based on stochastic gradient descent).	
#' 	
#' ## Mathematical formulation	
#' 	
#' UMAP relies on local approximations of patches on the manifold to construct local fuzzy simplicial complex (topological) representations of the high dimensional data.  For each low dimensional representation of the projection of the data, UMAP tries to generate an analogous equivalent simplicial complex representation. The iterative UMAP optimization process aims to preserve the topological layout in the low dimensional space by minimizing the cross-entropy between the high- and low-dimensional topological representations.	
#' 	
#' The [DSPA Appendix on Shape](http://www.socr.umich.edu/people/dinov/courses/DSPA_notes/DSPA_Appendix_03.1_Geometric_Parametric_Surface_Viz.html) includes examples of how to generate some of geometric and topological primitives, including 0, 1, and 2 simplicial complexes.	
#' 	
#' The figure below shows the first few such primitives - a point, line, triangle, and tetrahedron.	
#' 	
#' 	
library(plotly)	
	
p <- plot_ly(type = 'mesh3d',	
             # Define all (4) zero-cells (points or vertices)	
             # P_i(x,y,z), 0<=i<4	
             x = c(0,           1/sqrt(3), -1/(2*sqrt(3)), -1/(2*sqrt(3))),	
             y = c(sqrt(2/3),   0,          0,             0),	
             z = c(0,           0,          -1/2,          1/2),	
             	
             # Next define all triples (i,j,k) of vertices that form a 2-cell face. 	
             # All Tetrahedra have 4 faces	
             i = c(0, 0, 0, 1),	
             j = c(1, 2, 3, 2),	
             k = c(2, 3, 1, 3),	
             	
             # Define the appearance of the 4 faces (2-cells)	
             facecolor = toRGB(viridisLite::viridis(4)),	
             showscale = TRUE, 	
             opacity=0.8	
) 	
traceEdge <- list( 	
  x1 = c(-1/(2*sqrt(3)), -1/(2*sqrt(3))), 	
  y1 = c(0, 0), 	
  z1 = c(-1/2, 1/2), 	
  line = list(	
    color = "rgb(1, 1, 1)",  #dark color for line traces	
    width = 20 # width of line	
  ), 	
  mode = "lines", 	
  opacity = 1,	
  type = "scatter3d"	
)	
	
# emphasize one of the faces by stressing the three 1-cells (edges)	
p <- add_trace(p, x=~traceEdge$x1, y=~traceEdge$y1, z=~traceEdge$z1,	
               type="scatter3d", mode=traceEdge$mode,	
               opacity=traceEdge$opacity,	
               line = list(color=traceEdge$line$color,	
                           width=traceEdge$line$width), showlegend=F)	
# add one 0-cell (point)	
p <- add_trace(p, x=-1/(2*sqrt(3)), y=0, z=1/2, type="scatter3d",	
                mode="markers", marker = list(size =16, 	
                                      color="blue", opacity = 1.0)) %>%	
  layout(title = "Simplicial Complexes (0,1,2,3) Cells", showlegend = FALSE,	
         scene = list(	
          xaxis = list(title = "X"),	
          yaxis = list(title = "Y"),	
          zaxis = list(title = "Z")	
        ))	
p	
#' 	
#' 	
#' For a given finite set of data observations, we are trying to represent the underlying topological space the data is sampled from. This topological formulation can be approximated as patches of open covers modeled by simplicial complexes. Locally, the data may be assumed to lie in a metric space where distances between data points can be measured. This leads to neighborhood approximations that can locally be represented as $nD$ balls centered at each data point. This topological space covering may not represent a complete and open cover as all data samples are finite and small, biased, or incomplete samples may lead to poor approximations of the topology of the problem state-space. Zero-cells (0-simplexes, points) are fit for for each observed data point, 1-cells (lines) are fit for each pair of data in the same neighborhood, and so on. More information about simplex trees is available in this article [The Simplex Tree: An Efficient Data Structure for General Simplicial Complexes](https://link.springer.com/chapter/10.1007/978-3-642-33090-2_63).	
#' 	
#' Example of simplicial tree decomposition of 10 data points using the `R` package [simplextree](https://cran.r-project.org/web/packages/simplextree/).	
#' 	
#' 	
# install.packages("simplextree")	
library(simplextree)	
	
simplicialTreeExample <- simplex_tree(list(1:3, 2:5, 5:8, 7:8, c(7,9,10)))	
plot(simplicialTreeExample, color_pal = rainbow(simplicialTreeExample$dimension + 1))	
	
# Generate some data	
n <- 30 #number of points to generate	
	
#generate space of parameter	
theta = seq(length.out=n, from=0, to=pi) 	
	
a <- 0.0; b <- 0.0; r <- 5.0	
	
x = a + r*cos(theta) + rnorm(n, 0, 0.4)	
y = b + r*sin(theta) + rnorm(n, 0, 0.4)	
	
x1 = a + r*cos(theta)	
y1 = b + r*sin(theta)	
	
df <- as.data.frame(cbind(x,y))	
#code to plot the circle for visualization	
# plot(x,y)	
	
plot_ly() %>% 	
  add_trace(x=x, y=y, type="scatter", name = 'Simulated Data Sample', 	
              mode = 'markers', marker = list(opacity=0.99, size=20, symbol=1)) %>%	
  add_trace(x=x1, y = y1, type="scatter", 	
            name = '(Quadratic) Model', 	
            mode = 'lines') %>%	
  layout(title='Sample Data from Quadratic Model', 	
         xaxis = list(title="X)", scaleanchor = "y"), 	
         yaxis = list(title="Y)", scaleanchor = "x"),	
         legend = list(orientation = 'h'))	
	
	
# install.packages("reticulate")	
library(reticulate)	
# specify the path of the Python version that you want to use	
py_path = "C:/Users/Dinov/Anaconda3/"  # manual	
# py_path = Sys.which("python3")       # automated	
use_python(py_path, required = T) 	
Sys.setenv(RETICULATE_PYTHON = "C:/Users/Dinov/Anaconda3/")	
#' 	
#' 	
#' Go into python to generate the simplicial complex for the data.	
#' 	
#' 	
import matplotlib	
import matplotlib.pyplot as plt	
import numpy as np	
import os 	
os.environ['QT_QPA_PLATFORM_PLUGIN_PATH'] = 'C:/Users/Dinov/Anaconda3/Library/plugins/platforms'	
	
print(r.df[1:6])	
	
#This set of functions allows for building a Vietoris-Rips simplicial complex from point data	
	
def euclidianDist(a,b):	
  return np.linalg.norm(a - b) #Euclidean distance metric	
	
#Build neighborhood graph	
def buildGraph(raw_data, epsilon = 3.1, metric=euclidianDist): #raw_data is a numpy array	
  nodes = [x for x in range(raw_data.shape[0])] #initialize node set, reference indices from original data array	
  edges = [] #initialize empty edge array	
  weights = [] #initialize weight array, stores the weight (which in this case is the distance) for each edge	
  for i in range(raw_data.shape[0]): #iterate through each data point	
      for j in range(raw_data.shape[0]-i): #inner loop to calculate pairwise point distances	
          a = raw_data[i]	
          b = raw_data[j+i] #each simplex is a set (no order), hence [0,1] = [1,0]; so only store one	
          if (i != j+i):	
              dist = metric(a,b)	
              if dist <= epsilon:	
                  edges.append({i,j+i}) #add edge	
                  weights.append([len(edges)-1,dist]) #store index and weight	
  return nodes,edges,weights	
	
def lower_nbrs(nodeSet, edgeSet, node):	
    return {x for x in nodeSet if {x,node} in edgeSet and node > x}	
	
def rips(graph, k):	
    nodes, edges = graph[0:2]	
    VRcomplex = [{n} for n in nodes]	
    for e in edges: #add 1-simplexes (edges)	
        VRcomplex.append(e)	
    for i in range(k):	
        for simplex in [x for x in VRcomplex if len(x)==i+2]: #skip 0-simplexes	
            #for each u in simplex	
            nbrs = set.intersection(*[lower_nbrs(nodes, edges, z) for z in simplex])	
            for nbr in nbrs:	
                VRcomplex.append(set.union(simplex,{nbr}))	
    return VRcomplex	
	
def drawComplex(origData, ripsComplex, axes=[-6,8,-6,6]):	
  plt.clf()	
  plt.axis(axes)	
  plt.scatter(origData[:,0],origData[:,1]) #plotting just for clarity	
  for i, txt in enumerate(origData):	
      plt.annotate(i, (origData[i][0]+0.05, origData[i][1])) #add labels	
	
  #add lines for edges	
  for edge in [e for e in ripsComplex if len(e)==2]:	
      #print(edge)	
      pt1,pt2 = [origData[pt] for pt in [n for n in edge]]	
      #plt.gca().add_line(plt.Line2D(pt1,pt2))	
      line = plt.Polygon([pt1,pt2], closed=None, fill=None, edgecolor='r')	
      plt.gca().add_line(line)	
	
  #add triangles	
  for triangle in [t for t in ripsComplex if len(t)==3]:	
      pt1,pt2,pt3 = [origData[pt] for pt in [n for n in triangle]]	
      line = plt.Polygon([pt1,pt2,pt3], closed=False, color="blue",alpha=0.3, fill=True, edgecolor=None)	
      plt.gca().add_line(line)	
  plt.show()	
#' 	
#' 	
#' Visualize the simplicial complex.	
#' 	
#' 	
newData = np.array(r.df)	
# print(newData)	
graph = buildGraph(raw_data=newData, epsilon=1.7)	
ripsComplex = rips(graph=graph, k=3)	
drawComplex(origData=newData, ripsComplex=ripsComplex)	
#' 	
#' 	
#' The 2D simplicial complex construction above illustrated a weighted graph representation of the 2D data using 0, 1, and 2 cells. In general, similar decomposition using higher order simplexes can be obtained, with increasing computational complexity. This simplex graph help with the lower dimensional projections in the UMAP algorithm as the simplicial decomposition captures the topological structure of the data.	
#' 	
#' Our aim is to generate a low dimensional representation of the data that has similar (or homologous) topological structure as the higher-dimensional simplicial decomposition. The UMAP projection is low dimensional representation of the data into the desired embedding space (e.g., $R^2$ or $R^3$). We will measure distances on the manifolds using some metric, e.g., standard Euclidean distance. In particular, we focus on distances to the nearest neighbors, where neighborhoods are defined as balls centered at specific data point of a `min_dist` diameter, a user-specified hyper-parameter.	
#' 	
#' The lower dimensional projections are driven by iteratively solving optimization problems of finding the low dimensional simplicial graph that closely resembles the fuzzy topological structure of the previous graph (or the original simplicial decomposition. The simplexes are represented as graphs with weighted edges that can be thought of as probabilities that compare the homologous 0-simplexes, 1-simplexes, and so on. These probabilities can be modeled as Bernoulli processes as (corresponding) simplexes either exist or don't exist and the probability is the univariate parameter of the Bernoulli distribution, which can be estimated using the cross entropy measure.	
#' 	
#' For example, if $S1$ the set of all possible 1-simplexes, let's denote by $\omega(e)$ and $\omega'(e)$ the weight functions of the 1-simplex $e$ in the high dimensional space and the corresponding lower dimensional counterpart. Then, the cross entropy measure for the 1-simplexes is:	
#' 	
#' $$\sum_{e\in E} {\left [ \underbrace{\omega(e)log\left ( \frac{\omega(e)}{\omega'(e)}\right )}_\text{attractive force}+	
#' \underbrace{(1-\omega(e))log\left ( \frac{1-\omega(e)}{1-\omega'(e)}\right ) }_\text{repulsive force}\right ]} .$$	
#' 	
#' The iterative optimization process would minimize the objective function composed of all cross entropies for all simplicial complexes using a strategy like stochastic gradient descent.	
#' 	
#' The optimization process balances the push-pull between the *attractive forces* between the points favoring larger values of $\omega'(e)$ (that correspond to small distances between the points), and the *repulsive forces* between the ends of $e$ when $\omega(e)$ is small (that correspond to small values of $\omega'(e)$.	
#' 	
#' ## Hand-Written Digits Recognition	
#' 	
#' Let's go back to the [MNIST digits dataset](https://en.wikipedia.org/wiki/MNIST_database) and apply UMAP projection from the original 784D space to 2D.	
#' 	
#' The R package `umap` provides functionality to use UMAP for dimensional reduction of high-dimensional data.	
#' 	
#' Let's demonstrate UMAP projecting the the hand-written digits dataset in 2D and using the projection for prediction and forecasting using new testing/validation data.	
#' 	
#' 	
# install.packages('plotly')	
library(plotly)	
# install.packages('umap')	
library(umap)	
# 	
# # Specify the input data 	
# umapData <- iris[ , 1:4]	
# umapLabels <- iris[ , 5]	
# 	
# # UMAP projection in 2D	
# umapData.umap <- umap(umapData)   # exclude the iris 3-class labels	
# 	
# # Using Python through R-reticulate interface	
# # library(reticulate)	
# # iris.umap_learn <- umap(umapData, method="umap-learn")	
# 	
# # Plot UMAP reduction in 2D	
# plot(umapData.umap$layout, col=as.factor(umapLabels))	
	
# https://rdrr.io/cran/umap/man/umap.defaults.html	
custom.settings = umap.defaults	
custom.settings$n_neighbors = 5	
custom.settings$n_components=3	
# custom.settings	
execTime_UMAP <- system.time(umap_digits3D <- umap(t(train)[1:10000 , ], umap.config=custom.settings)); execTime_UMAP	
	
cols <- palette(rainbow(10)) 	
	
# 2D UMAP Plot	
dfUMAP <- data.frame(umap_digits3D$layout[1:1000, ], train.labels.colors[1:1000])	
plot_ly(dfUMAP, x = ~X1, y = ~X2, mode = 'text') %>%	
  add_text(text=names(train.labels.colors)[1:1000], textfont=list(color=dfUMAP$train.labels.colors.1.1000.)) %>%	
  layout(title="UMAP (748D->2D) Embedding", xaxis = list(title = ""),  yaxis = list(title = ""))	
	
# # 3D t_SNE plot	
# execTime_UMAP <- system.time(umap_digits3D <- umap(t(train)[1:10000 , ], umap.config=custom.settings)); execTime_UMAP	
# 	
# df3D <- data.frame(tsne_digits3D$Y[1:1000, ], train.labels.colors[1:1000])	
# plot_ly(df3D, x = ~df3D[, 1], y = ~df3D[, 2], z= ~df3D[, 3], mode = 'markers+text') %>%	
#   add_text(text = names(train.labels.colors)[1:1000], textfont = list(color = df$train.labels.colors.1.1000.)) %>%	
#   layout(title = "t-SNE 3D Embedding", 	
#          scene = list(xaxis = list(title=""),  yaxis=list(title=""), zaxis=list(title="")))	
#' 	
#' 	
#' 	
#' 	
# execTime_UMAP <- system.time(umapData.umap <- 	
#                                umap(t(train)[1:10000 , ])) 	
# execTime_UMAP	
# # Full dataset(42K * 1K) execution may take much longer	
# 	
# # Plot the result 2D UMAP embedding of the data with digits or solid discs	
# par(mfrow=c(1,1)) # reset the plot canvas	
# 	
# # Plot only first 1,000 cases (to avoid clutter)	
# plot(umapData.umap$layout[1:200 , ], t='n', main="UMAP (748D->2D)") # don't plot the points to avoid clutter	
# text(umapData.umap$layout[1:200,], labels=names(train.labels.colors)[1:200], col=train.labels.colors[1:200])	
# 	
# # plot(umapData.umap$layout[1:200 , ],	
# #      col=as.factor(train.labels.colors[1:200]))	
#' 	
#' 	
#' ## Apply UMAP for class-prediction using new data	
#' 	
#' Using the MNIST hand-written digits dataset, or the simpler iris dataset, we will show how we can use UMAP for predicting the image-to-digit, or flower-to-species (taxa) mappings. In the MNIST data, we use a new 1K 2d images for prediction and for the iris data, we simulate new iris flower features by introducing random noise $N(0,0.1)$ to the original Iris data.	
#' 	
#' 	
# for IRIS data	
# umapData <- iris[ , 1:4]	
# umapLabels <- iris[ , 5]	
# 	
# # UMAP projection in 2D	
# umapData.umap <- umap(umapData)	
# 	
# # Generate a new random set (from the original iris data)	
# umapDataNoise <- umapData + matrix(rnorm(150*40, 0, 0.1), ncol=4)	
# colnames(umapDataNoise) <- colnames(umapData)	
# head(umapDataNoise, 5)	
# 	
# # Prediction/Forecasting	
# umapDataNoisePred <- predict(umapData.umap, umapDataNoise)	
# 	
# # Stack the original and noise-corrupted data into the same object and plot	
# plot(umapData.umap$layout, col=umapLabels, pch = 1, 	
#      xlab='UMAP Dim 1', ylab='UMAP Dim 2')  # UMAP of Original data	
# par(new=TRUE)	
# plot(umapDataNoisePred, col=umapLabels, pch = 3, 	
#      xaxt='n', yaxt='n', ann=FALSE)  # UMAP of Predictions on Noisy data	
# Stack the original and noise-corrupted data into the same object and plot	
# plot(umap_digits3D$layout, col=umapLabels, pch = 1, 	
#      xlab='UMAP Dim 1', ylab='UMAP Dim 2')  # UMAP of Original data	
# par(new=TRUE)	
# plot(umapDataNoisePred, col=umapLabels, pch = 3, 	
#      xaxt='n', yaxt='n', ann=FALSE)  # UMAP of Predictions on Noisy data	
	
	
# Use the next 1,000 images as prospective new data to classify into 0,1, ..., 9 labels	
umapData_1000 <- train[ , 10001:11000]	
str(umapData_1000)	
	
# Prediction/Forecasting	
umapData_1000_Pred <- predict(umap_digits3D, t(umapData_1000))  # mind transpose of data (case * Feature)	
# train.labels.colors <- colMap(train.labels)	
	
# 2D UMAP Plot	
dfUMAP <- data.frame(x=umap_digits3D$layout[1:1000, 1], y=umap_digits3D$layout[1:1000, 2],	
                     col=train.labels.colors[1:1000])	
plot_ly(dfUMAP, x = ~x, y = ~y, mode = 'text') %>%	
    # Add training data UMAP projection scatter Digits	
    add_text(text=names(train.labels.colors)[1:1000], textfont=list(color=~col, size=15), showlegend=F) %>%	
    # Add 1,000 testing hand-written digit cases onto the 2D UMAP projection pane as colored scatter points	
    # add_markers(marker = list(opacity=0.99, size=10, color=train.labels.colors[10001:11000],	
    #                           symbol=~as.numeric(as.factor(train.labels.colors[10001:11000]))))	
    add_markers(x=umapData_1000_Pred[ , 1], y=umapData_1000_Pred[ , 2], 	
                name =train.labels[10001:11000], marker = list(color=train.labels.colors[10001:11000])) %>%	
    layout(title="UMAP Prediction: Projection of 1,000 New Images in 2D", 	
           xaxis = list(title = ""),  yaxis = list(title = ""))	
	
# #############################################################################	
# # plot(range(umapData.umap$layout)[ , 1], range(umapData.umap$layout[ , 2]))	
# # points(umapData.umap$layout[,1], umapData.umap$layout[,2],	
# #        col=umapLabels, cex=2, pch=1)	
# # mtext(side=3, main="UMAP")	
# # df <- as.data.frame(cbind(umapData.umap$layout[ , 1], umapData.umap$layout[ , 2]))	
# library(plotly)	
# plot_ly(x = umapData.umap$layout[ , 1] , 	
#         y = umapData.umap$layout[ , 2], z = umapData.umap$layout[ , 2], 	
#         color = ~umapLabels,	
#         opacity = .5,	
#         colors = c('darkgreen', 'red'), 	
#         type = "scatter3d", 	
#         mode = "markers",	
#         marker = list(size = 5, width=2), 	
#         text = ~umapLabels,	
#         hoverinfo = "text") %>%	
#   layout(	
#     title = "UMAP clusters"	
#   )	
# 	
# # p <-plot_ly(	
# #   x = umapData.umap$layout[ , 1] , 	
# #         y = umapData.umap$layout[ , 2], z = umapData.umap$layout[ , 2], 	
# #   type="scatter3d",	
# #   mode = "markers",	
# #   color = as.factor(umapLabels)) %>%	
# #   layout(	
# #     title = "UMAP clusters"	
# #   )	
# # p	
#' 	
#' 	
#' ## UMAP Parameters	
#' 	
#' The UMAP configuration allows specifying the 21 parameters controlling the UMAP projection embedding of the function `umap()`.	
#' 	
#'   - *n_neighbors*: integer; number of nearest neighbors	
#'   - *n_components*: integer; dimension of target (output) space	
#'   - *metric*: character or function determining how distances between data points are computed. String-labeled metrics include : *euclidean*, *manhattan*, *cosine*, *pearson*, *pearson2*. The triangle inequality may not be satisfied by some  metrics and the internal **knn** search may not be optimal 	
#'   - *n_epochs*: integer; number of iterations performed during layout optimization	
#'   - *input*: character, use either "data" or "dist"; determines whether the primary input argument to `umap()` is treated as a data matrix or as a distance matrix	
#'   - *init*: character or matrix. The default string "spectral" computes an initial embedding using eigenvectors of the connectivity graph matrix. An alternative is the string "random", which creates an initial layout based on random coordinates. This setting.can also be set to a matrix, in which case layout optimization begins from the provided coordinates.	
#'   - *min_dist*: numeric; determines how close points appear in the final layout	
#'   - *set_op_ratio_mix_ratio*: numeric in range [0,1]; determines the knn-graph  used to create a fuzzy simplicial graph	
#'   - *local_connectivity*: numeric; used during construction of fuzzy simplicial set	
#'   - *bandwidth*: numeric; used during construction of fuzzy simplicial set	
#'   - *alpha*: numeric; initial value of "learning rate" of layout optimization	
#'   - *gamma*: numeric; determines, together with alpha, the learning rate of layout optimization	
#'   - *negative_sample_rate*: integer; determines how many non-neighbor points are used per point and per iteration during layout optimization	
#'   - *a*: numeric; contributes to gradient calculations during layout optimization. When left at NA, a suitable value will be estimated automatically.	
#'   - *b*: numeric; contributes to gradient calculations during layout optimization.	
#'   - *spread*: numeric; used during automatic estimation of a/b parameters.	
#'   - *random_state*: integer; seed for random number generation used during umap()	
#'   - *transform_state*: integer; seed for random number generation used during `predict()`	
#'   - *knn.repeat*: number of times to restart knn search	
#'   - *verbose*: logical or integer; determines whether to show progress messages	
#'   - *umap_learn_args*: vector of arguments to python package umap-learn	
#' 	
#' 	
#' ## Stability, Replicability and Reproducibility	
#' 	
#' *Reproducibility* is a stronger condition that requires obtaining consistent computational results using a *fixed* protocol including input data, computational steps, scientific methods, software code, compiler, and other experimental conditions. 	
#' 	
#' *Replicability* is a weaker condition that requires obtaining homologous results across multiple studies examining the same phenomenon aiming at answering the same scientific questions, each study using its own independent (yet homologous) input data.	
#' 	
#' As most UMAP implementations involve stochastic processes that employ random number generation, there may be variations in the output results from repeated runs of the algorithm. To stabilize these outputs and ensure result reproducibility we can use seed-setting, `umapData.umap.1234 = umap(umapData, random_state=1234)`. 	
#' 	
#' ## UMAP Interpretation	
#' 	
#' Below is a summary of key UMAP interpretation points:	
#' 	
#'   - The choice of the hyperparameters will effect the output: Choosing optimal parameter values may be difficult and is always data dependent. The stability of running UMAP repeatedly with a range of hyperparameters may provide a more sensible interpretation of the lower dimensional UMAP projection.	
#'   - The sizes of the lower dimensional UMAP cluster should not be over-interpreted, since UMAP relies only on local graph neighborhood distances (not global) to construct the initial high-dimensional graph representation.	
#'   - Similarly, between-cluster distances are not representative of dissimilarities between the cluster elements.	
#'   - Initial random noise in the data will not be preserved as random variation in the UMAP projections and some spurious clustering may be present that may not be signal related.	
#' 	
#' 	
#' 	
#' # Dimensionality Reduction Case Study (Parkinson's Disease)	
#' 	
#' ## Step 1: Collecting Data	
#' 	
#' The data we will be using in this case study is the Clinical, Genetic and Imaging Data for Parkinson's Disease in the SOCR website. A detailed data explanation is on the following link [PD data](https://wiki.socr.umich.edu/index.php/SOCR_Data_PD_BiomedBigMetadata). Let's import the data into R.	
#' 	
#' 	
# Loading required package: xml2	
wiki_url <- read_html("https://wiki.socr.umich.edu/index.php/SOCR_Data_PD_BiomedBigMetadata")	
html_nodes(wiki_url, "#content")	
pd_data <- html_table(html_nodes(wiki_url, "table")[[1]])	
head(pd_data); summary(pd_data)	
#' 	
#' 	
#' ## Step 2: Exploring and preparing the data	
#' 	
#' To make sure that the data is ready for further modeling, we need to fix a few things. Firstly, the `Dx` variable or diagnosis is a factor. We need to change it to a numeric variable. Second, we don't need the patient ID and time variable in the dimension reduction procedures. 	
#' 	
#' 	
pd_data$Dx <- gsub("PD", 1, pd_data$Dx)	
pd_data$Dx <- gsub("HC", 0, pd_data$Dx)	
pd_data$Dx <- gsub("SWEDD", 0, pd_data$Dx)	
pd_data$Dx <- as.numeric(pd_data$Dx)	
attach(pd_data)	
pd_data<-pd_data[, -c(1, 33)]	
#' 	
#' 	
#' ## PCA	
#' 	
#' Now we start the process of fitting a PCA model. Here we will use the `princomp()` function and use the correlation rather than the covariance matrix for calculation. Mind that previous, we used `prcomp()` and now we are employing `princomp()`, two different functions, to compute the PCA.	
#' 	
#' 	
pca.model <- princomp(pd_data, cor=TRUE)	
summary(pca.model) # pc loadings (i.e., eigenvector columns)	
plot(pca.model)	
biplot(pca.model)	
	
fviz_pca_biplot(pca.model, axes = c(1, 2), geom = "point",	
  col.ind = "black", col.var = "steelblue", label = "all",	
  invisible = "none", repel = F, habillage = pd_data$Sex, 	
  palette = NULL, addEllipses = TRUE, title = "PCA - Biplot")	
	
	
plot_ly(x = c(1:length(pca.model$sdev)), y = pca.model$sdev*pca.model$sdev, name = "Scree", type = "bar") %>%	
  layout(title="Scree Plot", xaxis = list(title="PC's"),  yaxis = list(title="Variances (SD^2)"))	
	
# Scores	
scores <- pca.model$scores	
# Loadings	
loadings <- pca.model$loadings	
	
# Visualization scale factor for loadings	
scaleLoad <- 10	
	
p <- plot_ly() %>%	
  add_trace(x=scores[,1], y=scores[,2], z=scores[,3], type="scatter3d", mode="markers", name=pd_data$Dx,	
            marker = list(color=pd_data$Dx, colorscale = c("gray", "red"), opacity = 0.7), showlegend=F) 	
for (k in 1:nrow(loadings)) {	
   x <- c(0, loadings[k,1])*scaleLoad	
   y <- c(0, loadings[k,2])*scaleLoad	
   z <- c(0, loadings[k,3])*scaleLoad	
   p <- p %>% add_trace(x=x, y=y, z=z, type="scatter3d", mode="lines", 	
  name=paste0("Loading PC ", k, " ", colnames(pd.sub)[k]), line=list(width=8), opacity=1) 	
}	
	
p <- p %>%	
  layout(legend = list(orientation = 'h'), 	
         title=paste0("3D Projection of ", length(pca.model$sdev),"D PD Data along First 3 PCs (Colored by Dx)")) 	
p	
#' 	
#' 	
#' Albeit the two cohorts (normal controls and patients, $DX$, red and gray colored markers in the 3D scene) are slightly separated in the second principal direction, we can see in this real world example that PCs do not necessarily correspond to a definitive "elbow" plot suggesting an optimal number of components. In our PCA model, each PC explains about the same amount of variation. Thus, it is hard to tell how many PCs, or factors, we need to select. This would be an *ad hoc* decision in this case. We can understand this better after understanding the following FA model. 	
#' 	
#' ## FA	
#' 	
#' Let's set up an Cattel's Scree test to determine the number of factors first. 	
#' 	
#' 	
ev <- eigen(cor(pd_data)) # get eigenvalues	
ap <- parallel(subject=nrow(pd_data), var=ncol(pd_data), rep=100, cent=.05)	
nS <- nScree(x=ev$values, aparallel=ap$eigen$qevpea)	
summary(nS)	
#' 	
#' 	
#' Although the Cattel's Scree test suggest that we should use 14 factors, the real fit shows 14 is not enough. Previous PCA results suggest we need around 20 PCs to obtain a cumulative variance of 0.6. After a few trials we find that 19 factors can pass the chi square test for sufficient number of factors at $0.05$ level.	
#' 	
#' 	
fa.model<-factanal(pd_data, 19, rotation="varimax")	
fa.model	
#' 	
#' 	
#' This data matrix has relatively low correlation. Thus, it is not suitable for ICA. 	
#' 	
#' 	
cor(pd_data)[1:10, 1:10]	
	
# generate some random categorical labels for all N observations	
class <- pd_data$Dx	
df <- as.data.frame(pd_data[1:5], class=class)	
	
plot_ly(df) %>%	
    add_trace(type = 'splom', dimensions = list( list(label=colnames(pd_data)[1], values=~L_caudate_ComputeArea),	
                                                 list(label=colnames(pd_data)[2], values=~L_caudate_Volume),	
                                                 list(label=colnames(pd_data)[3], values=~R_caudate_ComputeArea),	
                                                 list(label=colnames(pd_data)[4], values=~R_caudate_Volume),	
                                                 list(label=colnames(pd_data)[5], values=~L_putamen_ComputeArea)),	
              text=~class, marker = list(line = list(width = 1, color = 'rgb(230,230,230)'))) %>%	
    layout(title= 'Parkinsons Disease (PD) Data Pairs Plot', hovermode='closest', dragmode= 'select',	
        plot_bgcolor='rgba(240,240,240, 0.95)')	
#' 	
#' 	
#' ## t-SNE	
#' 	
#' Next, let's try the t-Distributed Stochastic Neighbor Embedding method on the [PD data](https://wiki.socr.umich.edu/index.php/SOCR_Data_PD_BiomedBigMetadata).	
#' 	
#' 	
# install.packages("Rtsne")	
library(Rtsne)	
	
# If working with post-processed PD data above: remove duplicates (after stripping time)	
# pd_data <- unique(pd_data[,])	
	
# If working with raw PD data: reload it	
pd_data <- html_table(html_nodes(wiki_url, "table")[[1]])	
	
# Run the t-SNE, tracking the execution time (artificially reducing the sample-size to get reasonable calculation time)	
execTime_tSNE <- system.time(tsne_PD <- Rtsne(pd_data, dims = 3, perplexity=30, verbose=TRUE, max_iter = 1000)); execTime_tSNE	
	
# Plot the result 2D map embedding of the data	
# table(pd_data$Sex)	
# plot(tsne_PD$Y, main="t-SNE Clusters", col=rainbow(length(unique(pd_data$Sex))), pch = 1)	
#legend("topright", c("Male", "Female"), fill=rainbow(length(unique(pd_data$Sex))), bg='gray90', cex=0.5)	
	
table(pd_data$Dx)	
	
# Either use the DX label column to set the colors col = as.factor(pd_data$Dx) 	
#plot(tsne_PDs$Y, main="t-SNE Clusters", col=as.factor(pd_data$Dx), pch = 15)	
#legend("topright", c("HC", "PD", "SWEDD"), fill=unique(as.factor(pd_data$Dx)), bg='gray90', cex=0.5)	
	
	
# Or to set the colors explicitly	
CharToColor = function(input_char){ 	
    mapping = c("HC"="blue", "PD"="red", "SWEDD"="yellow")	
    mapping[input_char]	
}	
pd_data$Dx.col = sapply(pd_data$Dx, CharToColor)	
	
# plot(tsne_PD$Y, main="t-SNE Clusters", col=pd_data$Dx.col, pch = 15)	
# legend("topright", c("HC", "PD", "SWEDD"), fill=unique(pd_data$Dx.col), bg='gray90', cex=0.5)	
	
df3D <- data.frame(tsne_PD$Y, pd_data$Dx.col)	
plot_ly(df3D, x = ~df3D[, 1], y = ~df3D[, 2], z= ~df3D[, 3], type="scatter3d",  mode = 'markers', 	
        color = pd_data$Dx.col, name=pd_data$Dx) %>%	
  layout(title = "PD t-SNE 3D Embedding", 	
         scene = list(xaxis = list(title=""),  yaxis=list(title=""), zaxis=list(title="")))	
#' 	
#' 	
#' ## Uniform Manifold Approximation and Projection (UMAP)	
#' 	
#' Similarly, we can try the UMAP method on the PD data.	
#' 	
#' 	
execTime_UMAP <- system.time(umap_PD_3D <- umap(pd_data[, -c(27, 34)]))   # remove "Dx.col" & 	
execTime_UMAP	
	
cols <- palette(rainbow(3)) 	
	
# 2D UMAP Plot	
dfUMAP <- data.frame(umap_PD_3D$layout, df3D$pd_data.Dx.col)	
plot_ly(dfUMAP, x = ~X1, y = ~X2, mode = 'text') %>%	
  add_text(text=pd_data$Dx, textfont=list(color=dfUMAP$df3D.pd_data.Dx.col)) %>%	
  layout(title="UMAP PD (32D->2D) Embedding", xaxis = list(title = ""),  yaxis = list(title = ""))	
#' 	
#' 	
#' The results of the PCA, ICA, FA, t-SNE, and UMAP methods on the [PD data](https://wiki.socr.umich.edu/index.php/SOCR_Data_PD_BiomedBigMetadata) imply that the data is complex and intrinsically high-dimensional, which prevents explicit embeddings into a low-dimensional (e.g., 2D or 3D) space. More advanced methods to interrogate this dataset will be demonstrated later. The [SOCR publications site](https://www.socr.umich.edu/people/dinov/publications.html) provides additional examples of Parkinson's disease studies.	
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