## Design Matrix:

* Imputations

* Design Matrix:
$$
\begin{align}
t\ | \ y_t \xrightarrow{f} v_t, | \ \ x^1_t, \ x^2_t, \ \ldots \ x^k_t \xrightarrow{g} \ g_1x^1_t, \ g_2x^2_t, \ldots, g_kx^k_t
\end{align}
$$

$$
\text{DM}_t = \text{DM}_t(y_t, f, f^{-1}, \{x^i_t\}_{i=1,k}, \{g^i_t\}_{i=1,k})
$$

### Exploratory analysis

* Autocorrelation:
    * $ \text{ACF}(x_t) $
    * $ \text{PACF}(x_t) $


* Scatter Plots:
    * $y_t$ next to  $v_t$  and $x_t^{k}$ next to $g_kx^k_t$
    * $y_t \ \text{vs.} \ x^j_{t} $ for $j=1,k$ with LOWESS for dependency shape analysis
    * $x_t \ \text{vs.} \ x_{t-h}$ with LOWESS for autocorrelation analysis
    * $y_t \ \text{vs.} \ x^k_{t-h}$ for given $k$ with LOWESS for lagged-leading relationship



## Calibrator:

$$
 \mathbb{C} ( {\mathcal{H}yperParams}) \rightarrow \mathbb{C}
$$

## Model:
$$
\text{M} = \mathbb{M}(\text{C}, \ \text{DM} ) \to  \{\hat{\theta}_l\}_{l=1,m} 
$$

### Model Specification

$$
\{\hat{\theta}_l\} \xrightarrow[\text{GridSearch}]{I(\theta): \ AIC, \ AICc, \ BIC }  \{\hat{\theta^*_l}\}
$$

### Residuals Diagnostics:
$$
\hat{\varepsilon}_t = v_{{t}}-\hat{v}_t; \ \hat{\varepsilon}_t^{std}; \ \hat{\varepsilon}_t^{stu}
$$

$$
\text{RD} = \mathcal{RD}(\hat{\varepsilon}_t, \hat{\varepsilon}_t^{std}, \hat{\varepsilon}_t^{stu}  )
$$

### Model Selection: 

#### Cross Validation 

$$
\text{CV} = \mathbb{CV}(\text{M}, \text{Partitioning}, \text{ Performance Metric}) 
$$

$$
\mathbb{C} \xrightarrow{\varepsilon_{CV}} \mathbb{C}^*
$$


## Forecast:

$$
\text{C}^*(DM_{t+1}=g_i({x^i_{t+1}})_{i=1,k} \ | \ \{\hat{\theta^*}_l\}_{l=1,m}) = \hat{v}_{t+1} \xrightarrow{f^{-1} , \  y_t} \hat{y}_{t+1}
$$


```python

```

1eb0d076af929139aa8d366f31578e178157d797
ipynb
Jupyter Notebook
jupyter/LIB_Framework.ipynb
ashubertt/statslib
5a35c0d10c3ca44c2d48f329c4f3790c91c385ac
Apache-2.0
jupyter/LIB_Framework.ipynb
ashubertt/statslib
5a35c0d10c3ca44c2d48f329c4f3790c91c385ac
Apache-2.0
2021-04-06T10:55:34.000Z
2021-04-06T10:55:34.000Z
jupyter/LIB_Framework.ipynb
ashubertt/statslib
5a35c0d10c3ca44c2d48f329c4f3790c91c385ac
Apache-2.0
Qwen/Qwen-72B
1. YES
2. YES