week43 update

doc/src/week43/week43.do.txt

@@ -2411,17 +2411,19 @@ o "Introduction to Partial Differential Equations by A. Tveito, R. Winther":"htt
 
 The principal component analysis deals with the problem of fitting a
 low-dimensional affine subspace $S$ of dimension $d$ much smaller than
-the totaldimension $D$ of the problem at hand (our data
+the total dimension $D$ of the problem at hand (our data
 set). Mathematically it can be formulated as a statistical problem or
 a geometric problem.  In our discussion of the theorem for the
-classical PCA, we will stay with a statistical approach. This is also
-what set the scene historically which for the PCA.
+classical PCA, we will stay with a statistical approach. 
+Historically, the PCA was first formulated in a statistical setting in order to estimate the principal component of a multivariate random variable.
 
 We have a data set defined by a design/feature matrix $\bm{X}$ (see below for its definition) 
 * Each data point is determined by $p$ extrinsic (measurement) variables
 * We may want to ask the following question: Are there fewer intrinsic variables (say $d << p$) that still approximately describe the data?
 * If so, these intrinsic variables may tell us something important and finding these intrinsic variables is what dimension reduction methods do. 
 
+A good read is for example "Vidal, Ma and Sastry":"https://www.springer.com/gp/book/9780387878102".
+
 
 !split
 ===== Introducing the Covariance and Correlation functions  =====
@@ -2706,12 +2708,10 @@ print(covariance_matrix)
 
 We note here that the covariance is zero for the first rows and
 columns since all matrix elements in the design matrix were set to one
-(we are fitting the function in terms of a polynomial of degree $n$).
-
-This means that the variance for these elements will be zero and will
-cause problems when we set up the correlation matrix.  We can simply
+(we are fitting the function in terms of a polynomial of degree $n$). We would however not include the intercept
+and wee can simply
 drop these elements and construct a correlation
-matrix without these elements. 
+matrix without them. 
 
 
 !split
@@ -3013,7 +3013,7 @@ $\bm{z}_i$ with at most $l$ vectors, with $l << p$, defined by the
 orthogonal projection of the data onto the columns spanned by the
 eigenvectors of the covariance(correlations matrix).
 
-The proof which follows will be updated by mid January 2020.
+
 
 !split
 ===== Proof of the PCA Theorem  =====
@@ -3132,12 +3132,12 @@ discussion in chapter 12.2 of Murphy's text has also a nice link with
 the Singular Value Decomposition theorem. For categorical data, see
 chapter 12.4 and discussion therein.
 
-Additional part of the proof for the other eigenvectors will be added by mid January 2020.
+For more details, see for example "Vidal, Ma and Sastry, chapter 2":"https://www.springer.com/gp/book/9780387878102".
 
 !split
 ===== Geometric Interpretation and link with Singular Value Decomposition =====
 
-This material will be added by mid January 2020.
+For a detailed demonstration of the geometric interpretation, see "Vidal, Ma and Sastry, section 2.1.2":"https://www.springer.com/gp/book/9780387878102".
 
 
 !split