Deep-Learning-Based COVID-19 Time Series Prediction

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Code & Notebooks: Google Drive Folder

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Project Overview

A deep-learning and statistical approach to forecast active COVID-19 cases (as a percentage of total population) for the ten countries with the highest case counts as of May 4, 2020. The goal is to help governments plan public‐health interventions by accurately predicting when the outbreak will peak.

• ✔️ Use ARIMA, Holt–Winters Additive (HWAAS), TBAT, Prophet, DeepAR, and N-Beats on univariate time series.
• ✔️ Compare classical statistical models (ARIMA, HWAAS, TBAT) with modern deep‐learning approaches (Prophet, DeepAR, N-Beats).
• ✔️ Train on daily active case percentages (active cases ÷ total population) for USA, UK, Italy, Spain, Russia, France, Turkey, Germany, Iran, and Brazil.
• ✔️ Evaluate forecasting accuracy via Root Mean Square Error (RMSE) over a 7-day forecast horizon.

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Table of Contents

1. Abstract
2. Introduction
3. Related Work
4. Time Series Models
   • ARIMA
   • Holt–Winters Additive (HWAAS)
   • TBAT
   • Prophet
   • DeepAR
   • N-Beats
5. Data Description
6. Experiments & Results
   • Model Training & Evaluation
   • Performance Comparison (RMSE)
   • Statistical Ranking (Friedman Test)
   • Holm’s Post‐hoc Analysis (TBAT vs All)
   • Forecasting Examples
7. Conclusions & Future Work
8. References

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Abstract

The ongoing COVID-19 pandemic has caused worldwide socioeconomic unrest, forcing governments to introduce extreme measures to reduce its spread. Accurately forecasting outbreak peaks would significantly diminish the disease’s impact, enabling proactive policy changes (public health messaging, resource allocation, etc.). This study investigates the accuracy of six time‐series modeling approaches—ARIMA, HWAAS, TBAT, Prophet, DeepAR, and N-Beats—on daily active COVID-19 case percentages for the ten countries with the highest confirmed cases as of May 4, 2020. Using two publicly available datasets (daily case counts & population), we demonstrate that classical statistical models (ARIMA, TBAT) often outperform deep‐learning methods when data are limited. ARIMA and TBAT achieve the lowest RMSE in 7 out of 10 countries, indicating that data‐scarce forecasting scenarios still favor simpler, interpretable models.

Keywords: COVID-19, Time Series Forecasting, ARIMA, TBAT, DeepAR, N-Beats, RMSE, Pandemic Modeling

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1. Introduction

Since its emergence in December 2019 (Wuhan, China), COVID-19 has spread to 187 countries, causing over 3.5 million confirmed cases and 248,000 deaths (as of May 4, 2020). Governments worldwide have implemented self‐isolation and social‐distancing measures to “flatten the curve” and preserve healthcare capacity. However, large‐scale testing remains inconsistent: by April 23, 2020, no country had tested more than 13.4% of its population; the global average was only 1.3%.

Accurate estimation of active case counts (active cases ÷ total population) is crucial for:

• Public‐health planning & resource allocation
• Timing of lockdowns and social‐distancing policies
• Anticipating healthcare system loads (hospital beds, ICU, ventilators)

This project compares six time‐series methods—three classical (ARIMA, HWAAS, TBAT) and three deep‐learning (Prophet, DeepAR, N-Beats)—to forecast active‐case percentages for: A 7-day forecast horizon is used; performance is measured in RMSE (Root Mean Square Error).

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2. Related Work

2.1. Classical Time Series in Infectious Diseases

• Leptospirosis & Climate: Models linking monthly leptospirosis incidence to rainfall and temperature (Chadsuthi et al., 2012).
• Malaria & ENSO: Time‐series correlations between Plasmodium falciparum cases and El Niño Southern Oscillation (Hanf et al., 2011).
• Influenza Forecasting:
  • ARIMA to predict monthly influenza incidence in China (Song et al., 2016).
  • SARIMA + Internet search data to forecast US flu (Zhang et al., 2019).
  • TBAT for complex seasonal influenza cycles (De Livera et al., 2011).
  • Twitter‐based real‐time flu‐spread estimation (Lee et al., 2017).

2.2. COVID-19 Forecasting

• China‐focused Models:
  • Phenomenological models for province-level cumulative cases (Roosa et al., 2020).
  • SEIR + AI hybrid using migration data (Yang et al., 2020).
  • Stacked autoencoder for Chinese confirmed case forecasts (Zheng et al., 2020).
  • ANFIS + metaheuristic for 10-day confirmed case forecasts (Al-qaness et al., 2020).
• Global Predictions:
  • Simple exponential smoothing for global case counts (Petropoulos & Makridakis, 2020).
  • IHME: Statistical modeling for US healthcare resource utilization (beds, ICU, ventilators).
  • R₀ Estimation: MCMC (Wu et al., 2020), SIRD models (Anastassopoulou et al., 2020).
  • Diamond Princess outbreak analysis (Zhang et al., 2020).

Our work expands upon these by:

1. Focusing on active-case percentage (active cases ÷ population)
2. Comparing six diverse methods on the same 10-country dataset
3. Emphasizing simple statistical models vs. cutting-edge deep-learning approaches

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3. Time Series Models

Below is a concise overview of each method used in this study. References to original papers appear in References.

3.1. ARIMA (Auto-Regressive Integrated Moving Average)

• Core Idea: Model current value as a linear combination of past values (AR), differencing (I), and past forecast errors (MA).
• Strengths:
  • High interpretability (parameters have clear statistical meaning).
  • Box–Jenkins methodology automates order selection (p, d, q).
  • Handles non-stationary data via differencing.
• Limitations:
  • Unable to model nonlinear dependencies.
  • Assumes linear relationships only.

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3.2. Holt–Winters Additive Model (HWAAS)

• Core Idea: Exponential smoothing with additive trend and additive seasonality.
• Components:
  1. Level (ℓ)
  2. Trend (b)
  3. Seasonality (s)
• Formula (Additive):
  1. Level:
     [ \ell_t = \alpha,\bigl(y_t - s_{t - m}\bigr);+;(1 - \alpha),(\ell_{t-1} + b_{t-1}) ]
  2. Trend:
     [ b_t = \beta,\bigl(\ell_t - \ell_{t-1}\bigr);+;(1 - \beta),b_{t-1} ]
  3. Seasonality:
     [ s_t = \gamma,\bigl(y_t - \ell_t\bigr);+;(1 - \gamma),s_{t - m} ]
  4. Forecast (h steps ahead):
     [ \hat{y}{t+h} = \ell_t + h,b_t + s{t + h - m,\lfloor (h-1)/m \rfloor} ]
  • α, β, γ ∈ (0, 1): smoothing parameters
  • m: seasonal period (e.g., m=7 for weekly seasonality)
• Strengths:
  • Handles both trend and seasonality additively.
  • Simpler than ARIMA; fewer parameters to tune.
• Limitations:
  • Less accurate on average than Box–Jenkins ARIMA for some datasets.
  • Sensitive to initial values and outliers.

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3.3. TBAT (Trigonometric Seasonal, Box–Cox, ARMA, Trend)

• Core Idea: Decompose a time series into trend + multiple seasonalities using a trigonometric formulation, apply Box–Cox transform, and model residuals with ARMA.
• Components:
  1. Box–Cox Transform: Stabilize variance →
     [ y_t^{(\lambda)} = \begin{cases} \frac{y_t^\lambda - 1}{\lambda}, & \lambda \neq 0, \ \ln(y_t), & \lambda = 0. \end{cases} ]
  2. Trend: Local linear or segment-wise linear trend.
  3. Seasonality: Trigonometric representation for any (including non-integer) seasonal frequency:
     [ S_{t} = \sum_{k=1}^{K} \bigl(a_{k}\cos\bigl(\tfrac{2\pi k,t}{m}\bigr) ;+; b_{k}\sin\bigl(\tfrac{2\pi k,t}{m}\bigr)\bigr) ]
     • K: number of Fourier terms; m: seasonal period
  4. ARMA Residuals: Auto-Regressive + Moving Average model on residual errors
• Strengths:
  • Captures complex/multiple seasonalities (e.g., weekly, yearly, sub-daily).
  • Handles nonlinearity via Box–Cox.
  • Identifies hidden seasonal components not obvious in raw data.
• Limitations:
  • Larger parameter space → more computationally expensive (but optimized for ML).
  • Requires tuning of Fourier terms (K) and Box–Cox parameter (λ).

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3.4. Prophet

• Core Idea: Generalized additive model (GAM) with decomposable trend, seasonality, and holiday effects; designed for business time series.
• Model Definition:
  [ y(t) = g(t) + s(t) + h(t) + \epsilon_t
  ]
  1. Trend (g):
     • Piecewise linear with change points, or
     • Logistic growth (saturating)
  2. Seasonality (s): Fourier series up to order N:
     [ s(t) = \sum_{n=1}^{N} \bigl(a_n\cos\bigl(\tfrac{2\pi n,t}{P}\bigr) + b_n\sin\bigl(\tfrac{2\pi n,t}{P}\bigr)\bigr) ]
     • P: seasonal period (e.g., 365 days, 7 days)
  3. Holiday Effects (h):
     • User-provided list of dates with additive indicators
  4. Error (ε): Gaussian noise
• Strengths:
  • User-friendly: works well “out of the box” with default parameters.
  • Explicitly handles missing data and trend change points.
  • Incorporates known holidays/events easily.
• Limitations:
  • Designed for business/seasonal data; may underperform on epidemiological data without strong seasonality.
  • Less granular control over model internals compared to ARIMA/TBAT.

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3.5. DeepAR

• Core Idea: Probabilistic forecasting using an auto-regressive RNN (LSTM) to model future distribution, trained on many time series simultaneously.
• Architecture:
  1. Input: Previous target values (t−1, t−2, …) and covariates (optional).
  2. LSTM: Encodes historical context.
  3. Output Layer: Parameterizes a chosen likelihood (e.g., Gaussian, Negative Binomial) at each time step.
  4. Training Objective: Maximize log-likelihood of observed data under predicted distribution.
• Strengths:
  • Produces full probabilistic forecasts (quantiles, prediction intervals).
  • Can leverage cross-series learning: trains on multiple related time series to improve accuracy.
  • Flexible: supports arbitrary likelihood functions (e.g., Student-T, Log-Norm, Poisson).
• Limitations:
  • Requires substantial data to train effectively; may overfit on small datasets.
  • Less interpretable than classical models; “black box” behavior.
  • Hyperparameter tuning (layers, cells, learning rate) is nontrivial.

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3.6. N-Beats

• Core Idea: Deep fully-connected residual architecture that explicitly separates trend and seasonality via basis expansions, enabling interpretability and state-of-the-art accuracy.
• Architecture Overview (Block):
  1. Input window of length L.
  2. Backcast: Model reconstructs part of input (removes its contribution).
  3. Forecast: Model outputs future horizon of length H.
  4. Blocks are stacked in two “stacks”:
     • Trend Stack: Models a polynomial basis to capture trend.
     • Seasonality Stack: Models seasonality via Fourier basis.
  5. Residual Connections (Backcast − forecast): Each block removes its backcast from the input before passing to the next block (hierarchical residual).
• Strengths:
  • Interpretable: separate trend vs. seasonality outputs.
  • State-of-the-art performance on M4/M3 forecasting competitions.
  • Fast to train; uses simple fully connected layers and ReLU.
• Limitations:
  • Requires careful choice of stack depth and width.
  • Still a “black box” to some extent (deep fully connected nets).

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4. Data Description

4.1. Datasets Used

1. Novel Corona Virus 2019 Dataset (Kaggle)
   • Daily time series of
     • Confirmed cases
     • Recovered cases
     • Deaths
   • Source: Kaggle: Novel Corona Virus 2019 Dataset
2. Population by Country (Kaggle)
   • 2019 population estimates for all countries
   • Source: Kaggle: World Population by Country

4.2. Active Case Percentage Calculation

• Active Cases (daily):
  [ \text{Active}_t = \text{Confirmed}_t ;-; \text{Recovered}_t ;-; \text{Deaths}_t ]
• Active Case Percentage (daily):
  [ \text{PctActive}_t = \frac{\text{Active}t}{\text{Population}{\text{country}}} ]
• Ten Countries Selected (highest total confirmed as of May 4, 2020):
  1. USA
  2. Spain
  3. Italy
  4. UK
  5. France
  6. Russia
  7. Germany
  8. Turkey
  9. Brazil
  10. Iran

4.3. Data Preprocessing & Splits

• Total Instances per Country:
  • From first reported case → May 4, 2020
  • 104 daily observations (approx.)
• Train / Validation / Test Splits:
  • Training: First 72 days (percentage series)
  • Validation: Next 25 days
  • Test (Forecast Horizon): Last 7 days (used only for final RMSE calculation)
• Scaling: Each country’s series is already a fraction (active cases ÷ population), so no further scaling was necessary.

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5. Experiments & Results

5.1. Model Training & Evaluation

1. Training
   • Fit each model on the 72 training days of PctActive_t.
   • Hyperparameters:
     • ARIMA: Automatic order selection via AICc (p, d, q).
     • HWAAS: Seasonal period m = 7; smoothers α, β, γ tuned via validation RMSE.
     • TBAT: Fourier terms up to order K = 2 (weekly seasonality), Box–Cox λ tuned on validation set, ARMA orders selected via BIC.
     • Prophet: Default growth (“logistic”), seasonalities (yearly, weekly), no holidays.
     • DeepAR:
       • 2 LSTM layers, 64 cells each
       • Gaussian likelihood
       • Learning rate: 1e–3, batch = 32, epochs = 100
     • N-Beats:
       • 3 blocks per stack (trend & seasonality), width = 256
       • ReLU activations, Adam optimizer (lr = 1e–3), epochs = 100
2. Validation
   • Evaluate on 25 validation days; tune hyperparameters (where applicable) to minimize RMSE.
3. Test (7-Day Forecast)
   • Generate 7-day forecasts, compute RMSE against actual PctActive_t for Days (98 – 104).
   • Report RMSE per model per country.

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5.2. Performance Comparison (RMSE)

Country	Statistical Models				Deep Learning Models
	ARIMA	Prophet	HWAAS	TBAT	N-Beats	DeepAR (GluonTS)
USA	0.007421	0.013877	0.172957	0.009873	0.036958	0.044805
Spain	0.080094	0.065433	0.031497	0.029295	0.050492	0.108842
Italy	0.005628	0.019217	0.006616	0.005810	0.008645	0.043551
UK	0.005484	0.007634	0.004366	0.004310	0.037623	0.046134
France	0.060824	0.044482	0.011007	0.007003	0.004220	0.010549
Germany	0.006431	0.037139	0.004586	0.003389	0.013192	0.057523
Russia	0.001536	0.014681	0.002295	0.002193	0.027078	0.034479
Turkey	0.004442	0.044595	0.000887	0.001946	0.018265	0.093839
Brazil	0.004194	0.009279	0.005717	0.005621	0.010870	0.002836
Iran	0.002628	0.016281	0.001046	0.000425	0.003745	0.002277

Key Observations:

• Best Performing Model (lowest RMSE) in each country (bold):
  • USA: TBAT (0.009873)
  • Spain: TBAT (0.029295)
  • Italy: TBAT (0.005810)
  • UK: TBAT (0.004310)
  • France: N-Beats (0.004220)
  • Germany: TBAT (0.003389)
  • Russia: TBAT (0.002193)
  • Turkey: HWAAS (0.000887)
  • Brazil: DeepAR (0.002836)
  • Iran: TBAT (0.000425)

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5.3. Statistical Ranking (Friedman Test)

To compare multiple models across all ten countries, we applied the non-parametric Friedman test (significance α = 0.02) on the per-country RMSE rankings. Lower rank = better average performance:

Rank	Algorithm
1.700	TBAT
2.900	ARIMA
2.900	HWAAS
4.100	N-Beats
4.600	Prophet
4.800	DeepAR

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5.4. Holm’s Post-hoc Analysis (TBAT vs All)

After the Friedman test, we used Holm’s post-hoc to compare TBAT against all other algorithms (α = 0.02).

• Null Hypothesis (H₀): There is no significant difference between TBAT and the compared algorithm.

Comparison	Test Statistic	Adjusted p-Value	Result (Reject H₀?)
TBAT vs DeepAR	3.70521	0.00106	Reject H₀
TBAT vs Prophet	3.46616	0.00211	Reject H₀
TBAT vs N-Beats	2.86855	0.01237	Reject H₀
TBAT vs ARIMA	1.43427	0.30299	Accept H₀
TBAT vs HWAAS	1.43427	0.30299	Accept H₀

Conclusion:

• TBAT is significantly better (p < 0.02) than DeepAR, Prophet, and N-Beats.
• No significant difference between TBAT vs. ARIMA or TBAT vs. HWAAS at α = 0.02.

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5.5. Forecasting Examples

Below are sample plots of actual vs. predicted active-case percentages over the final 7-day test window for selected countries. Replace placeholder images with actual plots in your ./figures/ folder.

• Figure 1. USA Forecast (Test Horizon):
  

• Figure 2. UK Forecast (Test Horizon):
  

• Figure 3. Russia Forecast (Test Horizon):
  

• Figure 4. France Forecast (Test Horizon):
  

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6. Conclusions & Future Work

• Key Findings:

  • TBAT and ARIMA are the top performers for 7-day forecasting of active-case percentages in most countries.
  • HWAAS wins for Turkey, while N-Beats wins for France, and DeepAR (GluonTS) wins for Brazil.
  • Deep-learning methods (DeepAR, N-Beats) underperform statistical methods when data volume is limited.
  • Prophet (designed for business seasonality) did not rank among top models for any country.

• Possible Explanatory Factors for Cross-Country Differences:

  1. Climate & Geography: Differences in temperature/humidity may affect virus spread.
  2. Population Density & Demographics: High density can accelerate transmission.
  3. Testing & Reporting Variability: Inconsistent testing rates/data collection → noisy time series.
  4. Intervention Policies: Timing, severity, and duration of lockdowns/social distancing differ by country.

• Future Improvements:

  1. Model Ensembles: Combine ARIMA, TBAT, HWAAS, etc., to reduce overall forecast error.
  2. Multivariate Time Series: Incorporate external covariates:
     • Climate Data: Temperature, humidity, air quality.
     • Mobility Data: Google/Apple mobility reports.
     • Policy Indices: Stringency of lockdown measures.
  3. Transfer Learning: Use learnings from data-rich countries to improve forecasts in data-scarce regions.
  4. Longer Forecast Horizons: Extend to 14 or 21 days and evaluate model stability.
  5. Real-Time Adaptation: Incorporate an online-learning setting where models retrain daily as new data arrive.

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