An interpretable hybrid predictive model of COVID-19 cases using autoregressive model and LSTM

The results include four sections: Model evaluations, Prediction, Interpretability, and Comparative study on the WHO datasets. In Model evaluations, we introduce the metrics we use to evaluate the models and on which we compare the models’ performances. In section Prediction, we exhibit the visualizations of several interesting trials and compare the numerical predictions and evaluations of the three models. In Interpretability, we compare the AR component of the hybrid model with the AR model. This is to examine how we may interpret the hybrid model. We leave other training details in Supplementary Material. In Comparative study on the WHO datasets, we further examine the performance of the proposed hybrid model by applying it to data of 7 different countries around the world and comparing its performance with that of its component models and 3 additional models.

Data description and statistical analysis

We utilize two primary data sources. The first data source is a dataset specific to California counties, which is available in the CHHS Open Data repository under the title COVID-19 Time-Series Metrics by County and State. This dataset includes information on populations, positive and total tests, number of deaths, and positive cases. We conducted a preliminary statistical analysis to examine correlations between these variables and the number of daily cases. The results of this analysis can be found in Supplementary Fig. 3 in Supplementary Material, and we anticipate that they will provide valuable insights for future research.

The second data source, used for comparative analysis, can be found in the WHO repository at the WHO Coronavirus (COVID-19) Dashboard. This resource presents official daily counts of COVID-19 cases, deaths, and vaccine utilization, as reported by countries, territories, and areas. In this study, we use 7 countries: Japan, Canada, Brazil, Argentina, Singapore, Italy, and the United Kingdom.

All datasets generated and analysed during the current study are also available in the author’s Github repository²⁴.

Model evaluations

We use a quantitative measure to evaluate and compare the performance of models: the Mean Absolute Percentage Error (MAPE), defined as:

A model with small values of MAPE is preferred.

We examine the performance of the three models (hybrid, AR, and LSTM) on different time periods within the available range. This is essential in our research, since the performance of a model is not constant on different trends; by intuition, a model performs better on smooth curves than it does on steep curves. By repeating our evaluation process on different time periods thus different trends, we wish to understand what trends do the model give the best performance. Such understanding will help us decide to what degrees we may trust the performance of the models. We evaluate the models repeatedly to reduce the influence brought by the instability of model training. Specifically, we leave 7 days between the first date of any two consecutive training data points. Although a larger number of repetitions seems desirable, increasing the repetition number is at the cost of making neighboring training points closer to each other. However, the difference in performance between two neighboring training points, that are too close to each other, would be attributed more to the instability of model training than to the difference in trend. Such results give us little information about the model performance over trend. In the end, we let the step number be the same as our lag number. By doing so, we suppose the concept of a week is important in forecasting.

Additional evaluation metrics

In the Supplementary Material, we additionally evaluate and compare above models using Root Mean Square Error (RMSE) and Mean Absolute Error (MAE). The evaluation is done on the same dataset across different comparing methods.

The left panels show the training and testing data. The right panels show the ground truth versus forecasts of the AR, LSTM, and hybrid model, respectively. We display the average prediction (solid line) with 2 times standard error (shaded region). The standard error across 100 runs are reported for LSTM and hybrid. The hybrid model is more stable than the LSTM.

Prediction

In this section, we present the numerical results for all three models. We perform a comprehensive comparison of the performance for the three models in multiple counties, showing the advantage of the hybrid model. All predictions are transformed back to the original scale.

Visualization

We compare the three models’ performance on COVID-19 case prediction in California 8 counties. For each county, we test the models’ performance on several different situations: for example, when the training data has an up trend and the testing data has a down trend. From all trials we practiced, we choose the following trials, presented in Figs. 3 and 4, as representatives of different combinations of training and testing data, since they reflect the general model performances well.

Figure 3a shows models being trained on curved data and being tested on down trend data, as shown on the left and right panel, respectively. Figure 3b shows models being trained on up trend data and being tested on down trend data. Figure 3c shows models being trained on up trend data and being tested on up trend data. Figure 3d shows models being trained on down trend data and being tested on down trend data. Figure 4a,b show models being trained on down trend data and being tested on up trend data, while Fig. 4a has gentle upward testing data and Fig. 4b has sharp upward testing data. Figure 4c show models being trained and tested on jagged data.

To ensure the results above are representative, we run each selected trial 100 times, visualize the mean and standard error of these trials, and present averaged MAPE. While AR outperforms LSTM on some cases, the hybrid model outperforms both in most cases, except that in Fig. 3b and in Fig. 4c. The MAPE, averaged on the 100 trials, shows that LSTM (4.469%) outperforms hybrid (4.993%) slightly in Fig. 3b. However, as shown in the right panel of Fig. 3b, the hybrid model captures the general trend of ground truth better than LSTM does. Similarly, in Fig. 4c, AR (3.675%) outperforms hybrid (3.718%) slightly. Yet, as shown in the right panel of Fig. 4c, the hybrid model captures the general trend of ground truth better than AR does.

Beside, interestingly enough, the hybrid model always seems to capture the ground truth’s trend. Actually, the shape of hybrid ’s forecasts resembles either that of the AR model or that of the LSTM model, or it resembles a combination of both. When AR model captures the trend better than the LSTM does, the hybrid model resembles the AR model in forecast shape: for example, in Fig. 3b, San Francisco 2020-02-17 to 2020-05-14, and in Fig. 4a, Santa Barbara 2022-01-17 to 2022-04-14. When LSTM model captures the trend better than the AR does, the hybrid model resembles the LSTM model in forecast shape: for example, in Fig. 3d, San Francisco 2022-06-10 to 2022-09-05, and in Fig. 4b, Riverside 2022-02-16 to 2022-12-20. On jagged testing data, where AR performs better on some part and LSTM better on the other, the hybrid model presents advantages of both models: for example, in Fig. 4c, the hybrid model resembles AR on the two ends, where AR performs better, and it resembles LSTM in shape between day 5 to day 15, where LSTM seems to capture the trend better.

General performance

We evaluated the model performances numerically, in the 8 California counties across multiple trials. The results are given in Table 2. We observe that the hybrid model outperforms the AR model and the LSTM models almost uniformly: it generally yields the smallest average MAPE. To be specific, the general MAPE of each model (AR, LSTM, LSTM with 2 layers, and hybrid), averaged on the results for all 8 counties, is 5.629%, 4.934%, 6.804%, and 4.173% in order. In general, the hybrid model has the best general performance, and it outperforms the AR model by approximately 1.5%. The LSTM model suffers from overfitting when a second LSTM layer is added. As seen in the Supplementary Material, the proposed hybrid model also yields the lowest RMSE and MAE values.

The left panels show the training and testing data. The right panels show the ground truth versus forecasts of the AR, LSTM, and hybrid model, respectively. We display the average prediction (solid line) with 2 times standard error (shaded region). The standard error across 100 runs are reported for LSTM and hybrid. The hybrid model is more stable than the LSTM.

Interpretability

Interpretability of hybrid models can be defined as the ability to provide insight into the relationships they have learned, as introduced by Murdoch et al.²³. The hybrid model proposed a decomposition approach to decipher the learned model underlying the data-generating mechanism, where the estimated AR model provides the easy-to-understand linear trend. On the other hand, the LSTM is able to capture the long-term and nonlinear trend in the time series data. Our hybrid model aims to strike a balance between interpretability and accuracy, enabling us to gain insights into the underlying data while still achieving high predictive performance.

In this section, we study how AR and LSTM components contribute to the hybrid model when fitting the data. Our purpose is to seek the insights into explaining why the hybrid model enjoys the better performance in general. And more importantly, we seek to use the interpretation from the fitted hybrid model to provide practical guidance to the public health policy making process.

Note that all models are trained on the normalized data as described in section “Training” (Supplementary Material). Consequently all figures below report predictions on the normalized scales.

In Fig.  5, we present three settings with different signal strength ratio (represented by the value of \(\alpha\)) of the AR components and LSTM components in the prediction of the hybrid model. Specifically, the larger value of \(\alpha\) indicates the AR component dominates the LSTM component in prediction, and the smaller value of \(\alpha\) indicates otherwise. We found that the component that has stronger signal characterizes the general trend in the data while the other helps to stabilize the variance. This observation sheds light into why the hybrid model provides better predictive performance in general than a single model.

Moreover, the fitted value of \(\alpha\) provides a characterization of the intrinsic nonlinearity of the data, and consequently the difficulty of exploiting interpretation in the linear components of the fitted hybrid model. The smaller the value of \(\alpha\), the higher weight the nonlinear fit using LSTM has in the final prediction. In such a setting, coefficients in the AR components should be given less weight into generating interpretation for policy making. Equivalently, for larger value of \(\alpha\), it is more trustworthy to derive coefficients interpretation from the important AR part. This observation is helpful for public policy maker to distinguish among different virus transmission stages.

Finally, we observe interesting patterns of the coefficients estimates in the AR components of the hybrid model compared with the coefficients in the pure AR model. As shown in Table 3, across the three settings of different values of \(\alpha\), the pure AR model tends to put heavier weight in coefficients of larger lags, say \(Y_{t-7}\). In contrast, the AR component in the hybrid model tends to focus on capturing the short history, i.e., the coefficients associated with smaller lags (e.g., \(Y_{t-1}\)) tend to have larger estimates. This indicates that the short history pattern in the data could be well approximated by a simple (say, linear) model, while the longer history in the data possesses more complicated nonlinear structure that requires a LSTM component to fit.

The forecasts of a hybrid model versus the ground truth, and the contribution of its AR and its LSTM component.

Comparative study on the WHO datasets

In this section, we compare our proposed hybrid model for COVID-19 prediction with its two component models, the ARIMA and LSTM models, as well as three other commonly used models: Support Vector Machines⁵³ (SVM), Random Forest⁵⁴ (RF), and eXtreme Gradient Boosting⁵⁵ (XGBoost). To ensure the effectiveness of our model in different application settings, we use a country-level data for this comparative study, focusing on datasets from seven different countries collected by the World Health Organization.

We provide a brief overview of the three additional comparing methods. Support Vector Machines (SVM)^42,47 is a machine learning model that identifies the optimal hyperplane in a high-dimensional space that maximally separates data points into different classes. An SVM applies to both classification and regression problems. SVM is know to not perform well on noisy or unbalanced data^56,57.

Random Forest^43,44,45 is an ensemble learning method that constructs a multitude of decision trees. A Random Forest is very flexible and can handle complex data types. On the other hand, the Random Forests are known for their reduced interpretability, sensitivity to noise, the need for hyperparameter tuning, and potential issues with imbalanced data. These factors may impact their performance in the context of COVID-19 predictions^58,59,60.

Extreme Gradient Boosting (XGBoost)^44,46,48 has shown exceptional performance in various tasks. XGBoost is an ensemble learning method based on gradient boosting trees. It is known for its efficiency, scalability, and accuracy. However, like other tree-based ensemble methods, it can be more challenging to interpret. This may make it difficult to understand the driving factors behind predictions. In addition, XGBoost can be prone to overfitting, especially with small datasets or when the hyperparameters are not tuned properly^61,62.

We present the numerical results of the comparative study, which are visualized in Fig. 6. The comparative study is done on data collected by the World Health Organization⁶³ in Japan (JPN), Canada (CAN), Brazil (BRA), Argentina (ARG), Singapore (SGP), Italy (ITA), and the United Kingdom (GBR).

Overall, the proposed hybrid model performs better than the other models in most cases, as evidenced by its lower MAPE. This suggests that our model is effective in various situations and outperforms other commonly used models for COVID-19 prediction.

A heatmap exhibiting the performance, measured by MAPE in percentage, of the 7 models from this study and from previous work: AR, Single LSTM(LSTM), Double LSTM(DLSTM), hybrid, SVM, Random Forest(RF), XGBoost(XGB). The assessment has been done on data collected by World Health Organization, from 7 different countries around the world: Japan(JPN), Canada(CAN), Brazil(BRA), Argentina(ARG), Singapore(SGP), Italy(ITA), and The United Kingdom(GBR).