Tom's second Feb 21 edits of svd_intro masterpiece

lectures/svd_intro.md

@@ -678,7 +678,8 @@ $$
 
 ## Reduced-order VAR
 
-Consider a **vector autoregression**
+DMD  is a natural tool for estimating a **reduced order vector autoregression**,
+an object that we define in terms of the populations regression equation
 
 $$
 X_{t+1} = \check A X_t + C \epsilon_{t+1}
@@ -687,7 +688,7 @@ $$ (eq:VARred)
 where 
 
 * $X_t$ is an $m \times 1$ vector
-* $\check A$ is an $m \times m$ matrix of rank $r$
+* $\check A$ is an $m \times m$ matrix of rank $r$ whose eigenvalues are all less than $1$ in modulus
 * $\epsilon_{t+1} \sim {\mathcal N}(0, I)$ is an $m \times 1$ vector of i.i.d. shocks
 * $E \epsilon_{t+1} X_t = 0$, so that the shocks are orthogonal to the regressors
 
@@ -705,14 +706,14 @@ so that according to  model {eq}`eq:VARred`
 
 
 $$
-X' = [\check A X_1 + C \epsilon_2  \mid \check A X_2 + C \epsilon_3 \mid \cdots \mid \check A X_{n-1}  C 
-\epsilon_n ]
+X' = \begin{bmatrix} \check A X_1 + C \epsilon_2  \mid \check A X_2 + C \epsilon_3 \mid \cdots \mid \check A X_{n-1} +  C 
+\epsilon_n \end{bmatrix}
 $$
 
 To illustrate some useful calculations, assume that $n =3 $ and form
 
 $$
-X' X^T = [\check A X_1 + C \epsilon_2  \mid \check A X_2 + C \epsilon_3 ] 
+X' X^T = \begin{bmatrix} \check A X_1 + C \epsilon_2  &  \check A X_2 + C \epsilon_3 \end{bmatrix} 
    \begin{bmatrix} X_1^T \cr X_2^T \end{bmatrix} 
 $$
 
@@ -725,7 +726,7 @@ $$
 but because 
 
 $$
-E C( \epsilon_2 X_1^T + \epsilon_3 X_2^T)  = 0 
+E ( \epsilon_2 X_1^T + \epsilon_3 X_2^T)  = 0 
 $$
 
 we have
@@ -768,25 +769,28 @@ $$
 C \epsilon_{t+1} = X_{t+1} - \check A X_t , \quad t =1, \ldots, n-1
 $$
 
-and check whether they are serially uncorrelated as assumed.
+and check whether they are serially uncorrelated as assumed in {eq}`eq:VARred`.
 
 For example, we can compute spectra and cross-spectra of components of $C \epsilon_{t+1}$
+and check for serial-uncorrelatedness in the usual ways.
 
 We can also estimate the covariance matrix of $C \epsilon_{t+1}$
 from
 
 $$
-\frac{1}{n} \sum_{j=1}^{n-1} (C \epsilon_{t+1} )( C \epsilon_{t+1})^T 
+\frac{1}{n-1} \sum_{t=1}^{n-1} (C \epsilon_{t+1} )( C \epsilon_{t+1})^T 
 $$
 
-It can be useful to transform variables in our reduced order VAR
-
+It can be enlightening to diagonize  our reduced order VAR {eq}`eq:VARred` by noting that it can 
+be written
+ 
 
 $$
 X_{t+1} = \Phi \Lambda \Phi^{+} X_t + C \epsilon_{t+1}
 $$
 
-according to
+
+and then writing it as 
 
 $$
 \Phi^+ X_{t+1} = \Lambda  \Phi^{+} X_t +  \Phi^+ C \epsilon_{t+1}
@@ -796,13 +800,19 @@ or
 
 $$
 \tilde X_{t+1} = \Lambda \tilde X_t + \tilde \epsilon_{t+1} 
-$$
+$$ (eq:VARmodes)
 
 where $\tilde X_t $ is an $r \times 1$ **mode** and $\tilde \epsilon_{t+1}$ is an $r \times 1$
 shock.
 
+The $r$ modes $\tilde X_t$ obey the  first-order VAR {eq}`eq:VARmodes` in which $\Lambda$ is an $r \times r$ diagonal matrix.  
+
+Note that while $\Lambda$ is diagonal, the contemporaneous covariance matrix of $\tilde \epsilon_{t+1}$ need not be.
+
+
+**Remark:** It is permissible for $X_t$ to contain lagged values of  observables.
 
-**Remark:** It is permissible for $X_t$ to contain lagged values of  observables. For example:
+ For example, we might have a setting in which 
 
 $$
 X_t = \begin{bmatrix}
@@ -816,12 +826,6 @@ y_{2, t-1} \cr
 \end{bmatrix}
 $$
 
-
-
-
-
-
-
 +++
 
 ## Source for Some Python Code