1. The white noise (WN) model
2. White noise
The white noise or W-N model is the simplest example of a stationary process.
It has:
A fixed, constant mean,
A fixed, constant variance,
And no correlation over time.
3. White noise
Let's look at some plots of white noise time series.
In figure A you note that there is no pattern or correlation in the data across time. In B, the series has shifted up, indicating a larger mean, but there is still no patterns in the data. In C, the series has more vertical variability, meaning a larger variance, but there is still no patterns in the data. Finally, in D, the series has both a larger mean and a larger variance, but once again, no clear pattern or trend over time. All four figures are examples of white noise time series.
4. White noise
Now let's look at four new figures. Which one is white noise? Figure A has an upward time trend, it's mean is not constant over time, so it is not white noise. Figure B has a clear periodic pattern with a fixed cycle length that repeats about every 12 observations. It's mean is also not constant over time, so it is not white noise. Figure C shows grater variability on the right than the left, that is, the variance seems to be increasing over time. Since the variance is not constant, it is also not white noise. That leaves D, which appears to have both a constant mean and variance, and no clear patterns or correlation over time, so it satisfies the white noise model assumptions.
5. Simulating white noise - I
The arima.sim() function can be used to simulate a WN time series.
ARIMA stands for autoregressive, integrated moving average. It is a very broad class of time series models that includes WN as a special case. To specify the white noise model we use order equal to zero, zero, zero, as shown. A series with n equals fifty observations is simulated. Some of the actual values are displayed and the full series is shown in the figure.The default mean and standard deviation for the series are zero and one, respectively.
6. Simulating white noise - II
You can redo the simulation with a mean equal to four and a standard deviation equal to 2 by adding these as additional arguments as shown.
7. Estimating white noise
Finally, for a given time series, we can estimate the white noise model. For this we will use the arima() function, first providing the data, then with order equal to zero, zero, zero, as shown. This function returns an estimate of the mean or intercept, which is about 4-point-07 for this series, an approximate standard error for this mean estimate, 0-point-2698 in this case, and an estimate of the variance or sigma squared, which was 3-point-639. The estimated standard deviation is, of course, the square-root of this variance estimate.
Alternatively, you can just use the mean() and var() functions to directly estimate the white noise model's mean and variance parameters.
8. Let's practice!
Great! Now let's practice simulating and estimating white noise models!