Time Series Analysis

The assignment deals with characterizing time series data by eliminating the trend and seasonal component followed by the mathematical modeling of the incidental component. The verification of the model is done by analyzing the residual. A white noise residual implies that the model has sufficient accuracy. The assignment deals with performing the above mentioned procedure using R.

For more details, please refer to the complete project report.

Dow Jones Index Analysis

The increase in variation over time in the Dow Jones Index data implies an underlying multiplicative model. The model is then reduced to an additive time series and analysed further.

The Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) plots help us analyse the data correlations and estimate the order of ARMA process. The ACF plot displays weak geometric decay (can be seen by plotting the ACF till 500) whereas the PACF cuts off after 1-lag (0-lag has not been plotted). This suggests an AR(1) model or an ARMA(1,0) model. For the arima() function it implies an ARIMA(1,0,0) Model.

This can further be verified using the Akaike Information Criterion (AIC) presented for different models.

Thus, the best estimate for a model is

$$ X_t = 0.9966 X_{t-1} + \epsilon_t $$

with $\epsilon_t$ being a zero-mean white noise with variance $0.0007594$

The residual analysis shows that the residuals can be considered as white noise. The mean is 4.297635e-05, and can be considered to be negligible. The ACF Plot confirms that the residuals are indeed white noise. The ACF Plot deplays only 1 peak, at 0-lag, while the others can be considered insignificant, since they lie below the significance interval.

ECG Analysis

The ACF plot is decaying, hence AR component is present. The PACF is not abruptly reduced to zero and a slight decay is visible in it as well, hence the model is expected to be ARMA.

Calculating the AIC values for all models upto the ARMA(5,5), it can be seen that the ARMA(4,5) has the least AIC value and hence is the best model for the given dataset. ARMA(5,4) is a similarly good model. This complies with the observation made from the ACF and PACF plots.

$$ X_t = 1.1436 X_{t-1} + 0.4225 X_{t-2} − 0.943 X_{t-3} + 0.318 X_{t-4} + 0.57 \epsilon_{t-1} − 0.4815 \epsilon_{t-2} − 0.4158 \epsilon_{t-3} − 0.3354 \epsilon_{t-4} − 0.0905 \epsilon_{t-5} + \epsilon_t $$

The residuals can be considered as white noise. The mean is -6.724934e-05, and can be considered to be negligible. The ACF Plot confirms that the residuals are indeed white noise. The ACF Plot deplays only 1 peak, at 0-lag, while the others can be considered insignificant, since they lie below the significance interval.