Is noise present in the differenced series (taking non-stationary data and making it stationary through first difference of a time series)?
To determine if noise is present in the differenced series, you can analyze the autocorrelation function (ACF) and partial autocorrelation function (PACF) of the differenced series.
Here are the steps to analyze the differenced series:
1. Calculate the first difference of the time series data. This can be done by subtracting the current observation from the previous observation in the series.
2. Plot the differenced series.
3. Analyze the ACF and PACF plots of the differenced series:
a. Autocorrelation Function (ACF): The ACF plot shows the correlation between the differenced series and its lagged values. If the ACF plot shows a significant correlation at lag 1 and gradually decreases to zero or becomes random, it indicates that noise is present.
b. Partial Autocorrelation Function (PACF): The PACF plot shows the correlation between the differenced series and its lagged values, while controlling for the correlations at shorter lags. If the PACF plot shows a significant spike at lag 1 and gradually decreases to zero or becomes random, it indicates that noise is present.
If both the ACF and PACF plots show a gradual decrease to zero or randomness, it suggests that noise is present in the differenced series. However, if there are significant spikes at certain lags in the ACF and PACF plots, it indicates that there may be some remaining patterns or autocorrelations in the differenced series, possibly indicating the presence of non-random noise.
Keep in mind that this is just one approach to analyze the noise in the differenced series, and it may vary depending on the specific characteristics of your data and the analysis you are performing.