where [/(•) is the unit step function, and || • || is the vector norm. Note that the first and the last (m — l)/2 data points will not be filtered by this averaging.
3. Select the average magnitude of correction as the new neighborhood radius and the filtered data as the raw data, and go to Step 2 if deemed necessary after inspecting the filtered data. If the result looks satisfactory, or if the modified neighborhood radius drops below the distance between nonidentical nearest neighbors, iterations stop.
Example 8 Filtering the blood oxygen concentration signal
Again consider the time series data of Example 6 (Figure 5.16.a). Applying the local averaging noise filtering method to the signal with an embedding dimension of to = 5, a time delay of r = 9 sec., and a neighborhood radius of r = 400, we obtain the filtered signal shown in Figure 5.16.b. Note how the orbits became crisp, and how the conic shape of the attractor became visible in the filtered signal. □
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