We turn to the **tall and skinny** case associated with **Dynamic Mode Decomposition**, the case in which $ m >>n $.

We turn to the $ m >>n $ **tall and skinny** case associated with **Dynamic Mode Decomposition**.

Here an $ m \times n $ data matrix $ \tilde X $ contains many more attributes $ m $ than individuals $ n $.

Here an $ m \times n+1 $ data matrix $ \tilde X $ contains many more attributes (or variables) $ m $ than time periods $ n+1 $.

Dynamic mode decomposition was introduced by {cite}`schmid2010`,

$$ (eq:Ahatwithtildes)

Paralleling a step used to construct Representation 1, define a transition matrix for a rotated $p \times 1$ state $\tilde b_t$ by

**Computing Dominant Eigenvectors of $\hat A$**

We begin by paralleling a step used to construct Representation 1, define a transition matrix for a rotated $p \times 1$ state $\tilde b_t$ by

$$

\tilde A =\tilde U^\top \hat A \tilde U

= \tilde U^\top X' \tilde V \tilde \Sigma^{-1} \tilde U^\top

$$ (eq:tildeAverify)

Next, we'll just compute the regression coefficients in a projection of $\hat A$ on $\tilde U$ using a standard least-squares formula

\tilde U^\top X' \tilde V \tilde \Sigma^{-1} \tilde U^\top = \tilde A .

$$

Thus, we have verified that $\tilde A$ is a least-squares projection of $\hat A$ onto $\tilde U$.

**An Inverse Challenge**

Note that because we are using a reduced SVD, $\tilde U \tilde U^\top \neq I$.

Because we are using a reduced SVD, $\tilde U \tilde U^\top \neq I$.

Consequently,

so we can't simply recover $\hat A$ from $\tilde A$ and $\tilde U$.

**A Blind Alley**

Nevertheless, we hope for the best and proceed to construct an eigendecomposition of the

$p \times p$ matrix $\tilde A$:

We can start by hoping for the best and proceeding to construct an eigendecomposition of the $p \times p$ matrix $\tilde A$:

$$

\tilde A = \tilde W \Lambda \tilde W^{-1} .

\tilde A = \tilde W \Lambda \tilde W^{-1}

$$ (eq:tildeAeigenred)

where $\Lambda$ is a diagonal matrix of $p$ eigenvalues and the columns of $\tilde W$

are corresponding eigenvectors.

Mimicking our procedure in Representation 2, we cross our fingers and compute an $m \times p$ matrix

$ \hat A \tilde \Phi_s \neq \tilde \Phi_s \Lambda $ means that, unlike the corresponding situation in Representation 2, columns of $\tilde \Phi_s = \tilde U \tilde W$

are **not** eigenvectors of $\hat A$ corresponding to eigenvalues on the diagonal of matix $\Lambda$.

But in a quest for eigenvectors of $\hat A$ that we **can** compute with a reduced SVD, let's define the $m \times p$ matrix

**An Approach That Works**

Continuing our quest for eigenvectors of $\hat A$ that we **can** compute with a reduced SVD, let's define an $m \times p$ matrix

$\Phi$ as

$$

It turns out that columns of $\Phi$ **are** eigenvectors of $\hat A$.

This is a consequence of a result established by Tu et al. {cite}`tu_Rowley`, which we now present.

This is a consequence of a result established by Tu et al. {cite}`tu_Rowley` that we now present.

\end{aligned}

$$

Thus, we have deduced that

so that

$$

\hat A \Phi = \Phi \Lambda

\hat A \Phi = \Phi \Lambda .

$$ (eq:APhiLambda)

Let $\phi_i$ be the $i$th column of $\Phi$ and $\lambda_i$ be the corresponding $i$ eigenvalue of $\tilde A$ from decomposition {eq}`eq:tildeAeigenred`.

Equating the $m \times 1$ vectors that appear on the two sides of equation {eq}`eq:APhiLambda` gives

Also see {cite}`DDSE_book` (p. 238)

### Decoder of $X$ as a linear projection

### Decoder of $\check b$ as a linear projection

$$ (eq:Aform12)

From formula {eq}`eq:Aform12` we can deduce the reduced dimension dynamics

From formula {eq}`eq:Aform12` we can deduce dynamics of the $p \times 1$ vector $\check b_t$:

$$

\check b_{t+1} = \Lambda \check b_t

is an $m \times n$ matrix of least squares projections of $X$ on $\Phi$.

**Variance Decomposition of $X$**

By virtue of the least-squares projection theory discussed in this quantecon lecture <https://python-advanced.quantecon.org/orth_proj.html>, we can represent $X$ as the sum of the projection $\check X$ of $X$ on $\Phi$ plus a matrix of errors.

### A useful approximation

### An Approximation

There is a useful way to approximate the $p \times 1$ vector $\check b_t$ instead of using formula {eq}`eq:decoder102`.

We now describe a way to approximate the $p \times 1$ vector $\check b_t$ instead of using formula {eq}`eq:decoder102`.

In particular, the following argument adapted from {cite}`DDSE_book` (page 240) provides a computationally efficient way to approximate $\check b_t$.

For convenience, we'll do this first for time $t=1$.

For convenience, we'll apply the method at time $t=1$.

where $\check b_1$ is a $p \times 1$ vector.

Recall from representation 1 above that $X_1 = U \tilde b_1$, where $\tilde b_1$ is a time $1$ basis vector for representation 1 and $U$ is from a full SVD of $X$.

Recall from representation 1 above that $X_1 = U \tilde b_1$, where $\tilde b_1$ is a time $1$ basis vector for representation 1 and $U$ is from the full SVD $X = U \Sigma V^\top$.

It then follows from equation {eq}`eq:Xbcheck` that

$$

Replacing the error term $U^\top \epsilon_1$ by zero, and replacing $U$ from a full SVD of $X$ with $\tilde U$ from a reduced SVD, we obtain an approximation $\hat b_1$ to $\tilde b_1$:

Replacing the error term $U^\top \epsilon_1$ by zero, and replacing $U$ from a **full** SVD of $X$ with $\tilde U$ from a **reduced** SVD, we obtain an approximation $\hat b_1$ to $\tilde b_1$:

$$ (eq:bphieqn)

(To highlight that {eq}`eq:beqnsmall` is an approximation, users of DMD sometimes call components of the basis vector $\check b_t = \Phi^+ X_t $ the **exact** DMD modes.)

(To highlight that {eq}`eq:beqnsmall` is an approximation, users of DMD sometimes call components of basis vector $\check b_t = \Phi^+ X_t $ the **exact** DMD modes and components of $\hat b_t = ( \tilde W \Lambda)^{-1} \tilde U^\top X_t$ the **approximate** modes.)

Conditional on $X_t$, we can compute our decoded $\check X_{t+j}, j = 1, 2, \ldots $ from either

Conditional on $X_t$, we can compute a decoded $\check X_{t+j}, j = 1, 2, \ldots $ from the exact modes via

$$

\check X_{t+j} = \Phi \Lambda^j \Phi^{+} X_t

$$ (eq:checkXevoln)

or use the approximation

or use compute a decoded $\hat X_{t+j}$ from approximate modes via

$$

\hat X_{t+j} = \Phi \Lambda^j (\tilde W \Lambda)^{-1} \tilde U^\top X_t .

$$ (eq:checkXevoln2)

We can then use $\check X_{t+j}$ or $\hat X_{t+j}$ to forecast $X_{t+j}$.

We can then use a decoded $\check X_{t+j}$ or $\hat X_{t+j}$ to forecast $X_{t+j}$.