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Heart Rate Prediction using ARIMA Modelling

Detailed code + Output shown in notebook

Pretext

Given Data

In the given .xlsx sheet

  • Column 1 provides a time period over which data is collected
  • Column 2 provides the heart rate of a patient recorded over time.
Time Heart Rate
0 12:00:00 84.2500
1 13:00:00 84.2500
2 14:00:00 84.0625
3 15:00:00 85.6250
4 16:00:00 87.1875
... ... ...
1795 1900-03-15 07:00:00 103.8125
1796 1900-03-15 08:00:00 101.6250
1797 1900-03-15 09:00:00 99.5625
1798 1900-03-15 10:00:00 99.1875
1799 1900-03-15 11:00:00 98.8750

Pre-Processing

I performed some pre-processing to fix some errors associated with the time column format in .xlsx sheet, to end up with this.

Time Heart Rate
0 2022-01-01 12:00:00 84.2500
1 2022-01-01 13:00:00 84.2500
2 2022-01-01 14:00:00 84.0625
3 2022-01-01 15:00:00 85.6250
4 2022-01-01 16:00:00 87.1875
... ... ...
1795 2022-03-17 07:00:00 103.8125
1796 2022-03-17 08:00:00 101.6250
1797 2022-03-17 09:00:00 99.5625
1798 2022-03-17 10:00:00 99.1875
1799 2022-03-17 11:00:00 98.8750

Visualization

Given_Data

Question 1

Formulate a relevant null and alternative hypothesis to estimate the impact of the lagged variables. Carry out a unit root test for this series and examine if it is stationary?

Using Augmented Dicky-Fuller test,

  • $H_0: \gamma=1$ (Non-Stationary)
  • $H_1: \gamma \ne 1$ (Stationary)

If p value $\le 0.05$

  • we reject null hypothesis and accept alternate hypothesis
  • process is stationary
Statistic Value
p-value 0.00003
t value -4.96788
1% Critical Region -3.43402
5% Critical Region -2.86316
10% Critical Region -2.56763

In my result, I got p-value $<< 0.05 \implies$ process is stationary

Question 2

Identify the appropriate ARIMA model which can be used for forecasting the actual future heart rate? (Use t-test). Interpret these coefficients.

I used $z$ test, instead of $t$ test, and found that lags 1, 2, and 3 are statistically-signficant. This is also visible from the correlogram.

series_tac

series_pac
Dep. Variable: Heart Rate No. Observations: 1800
Model: ARIMA(3, 0, 0) Log Likelihood -3041.628
Date: Fri, 23 Dec 2022 AIC 6093.255
Time: 22:27:56 BIC 6120.733
Sample: 0 HQIC 6103.398
- 1800
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
const 92.5923 0.814 113.723 0.000 90.997 94.188
ar.L1 1.3264 0.008 161.655 0.000 1.310 1.342
ar.L2 -0.4632 0.028 -16.532 0.000 -0.518 -0.408
ar.L3 0.0973 0.027 3.627 0.000 0.045 0.150
sigma2 1.7161 0.014 119.613 0.000 1.688 1.744
Ljung-Box (L1) (Q): 0.04 Jarque-Bera (JB): 248894.27
Prob(Q): 0.84 Prob(JB): 0.00
Heteroskedasticity (H): 1.13 Skew: -1.43
Prob(H) (two-sided): 0.12 Kurtosis: 60.54

Hence, I identified $AR(3)$ as the most appropriate model.

$$ \begin{align} y_t &= \beta_0 + \beta_1 y_{t-1} + \beta_2 y_{t-2} + \beta_2 y_{t-3} + u_t \\ &= \beta_0 + \sum_{i=1}^3 \beta_i y_{t-i} + u_t \end{align} $$

Question 3

Compare a simple basic ARIMA (1,0,0) model with the model identified and proposed by you; and suggest a model which has better forecasting accuracy. (use RMSE estimates)

The model which has the lowest rmse for both insample and outsample prediction is ARIMA(3, 0, 0); hence it is the proposed model.

Model RMSE
Insample Prediction Heart Rate using ARIMA(3,0,0) 1.309519
Insample Prediction Heart Rate using ARIMA(1,0,0) 1.397753
Outsample Prediction Heart Rate ARIMA(3,0,0) 2.081262
Outsample Prediction Heart Rate ARIMA(1,0,0) 2.081938

Question 4

Also, Compare the adjusted r-square obtained from the two competing models.

Model Adjusted R2
Insample Prediction Heart Rate using ARIMA(3,0,0) 0.939552
Outsample Prediction Heart Rate ARIMA(1,0,0) 0.936579
Outsample Prediction Heart Rate ARIMA(3,0,0) 0.935641
Insample Prediction Heart Rate using ARIMA(1,0,0) 0.930654

Question 5

Conduct a unit root test for both the residual series obtained from the two competing models. Are they white noise?

The p value is 0, which implies that the residual series is a stationary process. From the graph in [Question 6](#Question 6), we can see that there is no autocorrelation, and hence the residual series is a white noise series.

Error p value t value
Insample Prediction Residual Series using ARIMA(1,0,0) 0 -12.07
Insample Prediction Residual Series using ARIMA(3,0,0) 0 -10.39
Outsample Prediction Residual Series using ARIMA(3,0,0) 0 -9.61
Outsample Prediction Residual Series using ARIMA(1,0,0) 0 -8.35

Question 6

Run a test for presence of autocorrelation in the error term (Use correlogram and interpret it)

errors_tac

errors_pac

Both the graphs clearly shows that the autocorrelation lines are within the shaded bands. Hence, the total and partial autocorrelation coefficient are statistically 0, ie

$$ \begin{align} \text{PAC}(u_t, u_{t-k}) &= 0 \\ \text{TAC}(u_t, u_{t-k}) &= 0 \\ (\forall k & \ne 0) \end{align} $$

Question 7

Plot the predicted and actual values obtained from the most appropriate model. What do you infer?

The predicted value almost perfectly overlaps with true value, hence we can conclude that our model is appropriate to modelling heart rate.

Prediction

The errors are also random, and mostly quite minimal.

Error

Question 8

Based on evidence from data, what will be your advice?

Firstly, despite the near-perfect fit, the model could be improved by incorporating seasonality, besides just the basic ARIMA modelling.

Secondly, the model could be improved by introducing other variables which have an effect on heart rate, such as mood, glucose level, etc.

Furthermore, the current model only predicts one step ahead, using a static approach. A more useful result would be to build a model for predicting the next 24 hours using a dynamic approach.