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| --- | ||
| title: "Covid-19 Data-Rich Dynamic Factor Model" | ||
| author: "" | ||
| format: | ||
| revealjs: | ||
| theme: blood | ||
| include-in-header: | ||
| text: | | ||
| <style> | ||
| .center-xy { | ||
| margin: 0; | ||
| position: absolute; | ||
| top: 50%; | ||
| left: 50%; | ||
| -ms-transform: translateY(-50%), translateX(-50%); | ||
| transform: translateY(-50%), translateX(-50%); | ||
| } | ||
| </style> | ||
| --- | ||
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| # Dynamic Factor Models | ||
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| ## Overview | ||
| > Dynamic Factor Models are statistical models that capture the evolving relationships among observed variables through latent factors and time dynamics. | ||
| <br> | ||
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| ::: {.incremental} | ||
| - Provides a framework to understand complex interactions in time-series data.<br><br> | ||
| - Relevant for tracking the economic impacts of COVID-19, where relationships between variables may change dynamically over time<br><br> | ||
| - Enables extraction of underlying trends and patterns from noisy and interrelated economic data | ||
| ::: | ||
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| ## Model | ||
| > Basic Dynamic Factor Model | ||
| $$y_t = \Lambda f_t + \epsilon_t$$ | ||
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| <br> | ||
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| ::: {.incremental} | ||
| - $y_t$: Observed variables at time $t$<br><br> | ||
| - $\Lambda$: Loading matrix<br><br> | ||
| - $f_t$: Latent factors<br><br> | ||
| - $\epsilon_t$: Error term | ||
| ::: | ||
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| ## Latent Factors and Loading Matrix | ||
| > Relationship between latent factors and loading matrix | ||
| <br> | ||
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| :::{.incremental} | ||
| - The loading matrix, often denoted as $\Lambda$>, establishes the linear relationship between the latent factors and the observed variables<br><br> | ||
| - The loading matrix is a bridge that connects the latent factors, which are unobservable, to the observed variables, providing a mathematical representation of how the latent factors influence the observed data | ||
| ::: | ||
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| ## Latent Factors and Loading Matrix | ||
| > Relationship between latent factors and loading matrix | ||
| ```{python} | ||
| import numpy as np | ||
| import matplotlib.pyplot as plt | ||
| # Set seed for reproducibility | ||
| np.random.seed(42) | ||
| # Generate dummy data | ||
| num_observed_variables = 4 | ||
| num_time_points = 100 | ||
| loading_matrix = np.array([[0.5, 0.3, 0.8, 0.2], | ||
| [0.7, 0.2, 0.5, 0.1]]) | ||
| latent_factors = np.random.randn(num_time_points, 2) | ||
| observed_variables = np.dot(latent_factors, loading_matrix) + np.random.randn(num_time_points, num_observed_variables) | ||
| # Plotting | ||
| plt.figure(figsize=(10, 6)) | ||
| # Plot latent factors | ||
| plt.subplot(2, 1, 1) | ||
| plt.plot(latent_factors[:, 0], label='Latent Factor 1', linestyle='--') | ||
| plt.plot(latent_factors[:, 1], label='Latent Factor 2', linestyle='--') | ||
| plt.title('Latent Factors Over Time') | ||
| plt.legend() | ||
| # Plot observed variables | ||
| plt.subplot(2, 1, 2) | ||
| for i in range(num_observed_variables): | ||
| plt.plot(observed_variables[:, i], label=f'Observed Variable {i+1}') | ||
| plt.title('Observed Variables Over Time') | ||
| plt.legend() | ||
| plt.tight_layout() | ||
| plt.show() | ||
| ``` | ||
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| ## Extending the model {.smaller} | ||
| > Incorporating time dynamics | ||
| The DFM is typically generalized to include autoregressive components<br><br> | ||
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| . . . | ||
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| $$ | ||
| \begin{split}\begin{align} | ||
| y_t & = \Lambda f_t + B x_t + u_t \\ | ||
| f_t & = A_1 f_{t-1} + \dots + A_p f_{t-p} + \eta_t \qquad \eta_t \sim N(0, I)\\ | ||
| u_t & = C_1 u_{t-1} + \dots + C_q u_{t-q} + \varepsilon_t \qquad \varepsilon_t \sim N(0, \Sigma) | ||
| \end{align}\end{split} | ||
| $$ | ||
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| . . . | ||
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| <br><br> | ||
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| Where $y_t$ is observed, $f_t$ are unobserved latent factors, $x_t$ are optional (unused for our case) exogenous variables, and the dynamic evolution of latent factors is expressed using the transition matrix $A$ with $\eta_t$ representing new information or random shocks. $u_t$ is the error or "idiosyncratic" process | ||
|
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| . . . | ||
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| <br><br> | ||
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| This model is then cast into state space form and the unobserved factors estimated via the Kalman filter. The likelihood can be evaluated as a byproduct of the filtering recursions with maximum likelihood estimation used to estimate the parameters. | ||
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| # Time Dynamics | ||
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| ## Autoregressive component | ||
| $$f_t = A f_{t-1} + \eta_t$$ | ||
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| $A$: Transition matrix<br> | ||
| $\eta_t$: Innovation term | ||
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| <br> | ||
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| :::{.incremental} | ||
| - The transition matrix, often denoted as $A$, is a square matrix that governs the temporal evolution of the latent factors | ||
| - Each element of the matrix represents the influence of one latent factor at the current time on the corresponding latent factor at the next time point | ||
| - The elements of the transition matrix $A$ determine how each latent factor at the previous time point contributes to the latent factors at the current time point | ||
| - Values in the diagonal of $A$ represent the persistence of each latent factor over time | ||
| - Off-diagonal elements indicate the influence of one latent factor on another | ||
| ::: | ||
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| ## Interpreting Transition Matrices {.smaller} | ||
| > Transition Matrix 1 | ||
| ```{python} | ||
| import numpy as np | ||
| import seaborn as sns | ||
| import matplotlib.pyplot as plt | ||
| # Set seed for reproducibility | ||
| np.random.seed(42) | ||
| # Generate two different transition matrices | ||
| transition_matrix_1 = np.array([[0.8, 0.2], | ||
| [0.3, 0.7]]) | ||
| transition_matrix_2 = np.array([[0.5, 0.5], | ||
| [0.6, 0.4]]) | ||
| # Create a figure with subplots | ||
| fig, axs = plt.subplots(1, 2, figsize=(12, 5)) | ||
| # Plot heatmap for Transition Matrix 1 | ||
| sns.heatmap(transition_matrix_1, annot=True, cmap="Reds", linewidths=.5, ax=axs[0]) | ||
| axs[0].set_title('Transition Matrix 1') | ||
| # Plot heatmap for Transition Matrix 2 | ||
| sns.heatmap(transition_matrix_2, annot=True, cmap="Reds", linewidths=.5, ax=axs[1]) | ||
| axs[1].set_title('Transition Matrix 2') | ||
| # Adjust layout | ||
| plt.tight_layout() | ||
| plt.show() | ||
| ``` | ||
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| - The diagonal elements (0.8 and 0.7) are relatively high, indicating a strong persistence of each latent factor over time. | ||
| - The off-diagonal elements (0.2 and 0.3) suggest moderate influence of one latent factor on the other, allowing for some interaction between the two factors. | ||
| - Summary: latent factors have a tendency to persist, with some interdependence. | ||
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| ## Interpreting Transition Matrices {.smaller} | ||
| > Transition Matrix 2 | ||
| ```{python} | ||
| import numpy as np | ||
| import seaborn as sns | ||
| import matplotlib.pyplot as plt | ||
| # Set seed for reproducibility | ||
| np.random.seed(42) | ||
| # Generate two different transition matrices | ||
| transition_matrix_1 = np.array([[0.8, 0.2], | ||
| [0.3, 0.7]]) | ||
| transition_matrix_2 = np.array([[0.5, 0.5], | ||
| [0.6, 0.4]]) | ||
| # Create a figure with subplots | ||
| fig, axs = plt.subplots(1, 2, figsize=(12, 5)) | ||
| # Plot heatmap for Transition Matrix 1 | ||
| sns.heatmap(transition_matrix_1, annot=True, cmap="Reds", linewidths=.5, ax=axs[0]) | ||
| axs[0].set_title('Transition Matrix 1') | ||
| # Plot heatmap for Transition Matrix 2 | ||
| sns.heatmap(transition_matrix_2, annot=True, cmap="Reds", linewidths=.5, ax=axs[1]) | ||
| axs[1].set_title('Transition Matrix 2') | ||
| # Adjust layout | ||
| plt.tight_layout() | ||
| plt.show() | ||
| ``` | ||
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| - The diagonal elements (0.5 and 0.4) are lower compared to Transition Matrix 1, suggesting less persistence of each latent factor over time. | ||
| - The off-diagonal elements (0.5 and 0.6) indicate a relatively stronger influence of one latent factor on the other compared to Transition Matrix 1. | ||
| - Summary: latent factors are less likely to persist and may be influenced more by each other, allowing for a more dynamic and responsive behavior. | ||
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| ## Factor Constraints | ||
| > Factor constraints are restrictions imposed on the parameters of the model, particularly on the loading matrix or transition matrix. | ||
| <br> | ||
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| Factor constraints can enhance the interpretability of the model by imposing structure on the relationships between latent factors and observed variables. <br><br> | ||
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| For example, setting certain elements of the loading matrix to zero might suggest that specific observed variables are not influenced by particular latent factors. | ||
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| ## Factor Constraints {.smaller} | ||
| > Factor loading constraint example | ||
| | Dep. variable | Global.1 | Pandemic | Employment | Consumption | Inflation | | ||
| |-----------------|----------|----------|------------|-------------|-----------| | ||
| | Supply_1 | X | | | | | | ||
| | Supply_7 | X | | | | | | ||
| | Monetary_5 | X | | | | | | ||
| | Monetary_9 | X | | | | | | ||
| | Supply_2 | X | | X | | | | ||
| | Supply_3 | X | | X | | | | ||
| | Demand_7 | X | | X | | | | ||
| | Demand_3 | X | | | X | | | ||
| | Demand_5 | X | | | X | | | ||
| | Monetary_2 | X | | | | X | | ||
| | Monetary_1 | X | | | | X | | ||
| | Pandemic_2 | X | X | | | | | ||
| | Pandemic_9 | X | X | | | | | ||
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| . . . | ||
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| <br> | ||
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| - Can reduce model complexity by reducing the number of free parameters<br><br> | ||
| - Incorporates domain knowledge about variable relationships | ||
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| # Python Package | ||
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| ## Implementation | ||
| > A Data-Rich Dynamic Factor Model for Covid-19 Response | ||
| We have created a self-contained Python package that is also available as a Docker container, a platform-agnostic container environment, which can run in any host environment that has Docker installed deterministically | ||
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| :::{.incremental} | ||
| - Poetry for dependency management | ||
| - CI with GitHub Actions | ||
| - Pre-commit hooks with pre-commit | ||
| - Code quality with black & ruff | ||
| - Testing and coverage with pytest and codecov | ||
| - Documentation with MkDocs | ||
| - Compatibility testing for multiple versions of Python with Tox | ||
| - Containerization with Docker | ||
| ::: | ||
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| ## Dashboard | ||
| > Included in the package is a GUI-dashboard for running the model | ||
| :::{.column-page} | ||
|  | ||
| ::: | ||
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| # Conclusion | ||
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| TBD |