1. Finding the Popular Frequency!
Time Series Data with the Fourier Transformation
Olivia B. Hoff
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2. → Mathematical description:
Logistic Map
→ Mathematical description:
Xn+1= rXn(1-Xn)
→Polynomial mapping of degree 2.
→Complex and chaotic behavior can occur with the parameter, r.
→Gives us the various time series by changing r through its simplicity.
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3. Logistic Map cont...
→ A property of the logistic map is that it becomes chaotic around the value 3.57 and reaches complete chaos at 4.
→The stretching and folding of the mapping results in gradual divergence of data points and exponential divergence.
→Shown by the complexity and unpredictability of the time-related data.
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4. Logistic Map cont... Examples:
Some of regular (non-chaotic) series with
Logistic map.
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5. Logistic Map cont...
Cases for multiple periods
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6. Logistic Map cont...
Chaotic time series, r = 3.7
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7. Fourier Transformation
→ Transforms one valued function into another.
→ The domain of the original function is usually time and the domain of the new function is frequency.
→ Describes which frequencies are present in the original function.
→ The Fourier transformation measures how much of an individual frequency is present in a function f(t).
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8. Some of examples with regular time series
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9. Some of examples with regular time series 2
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10. Some of examples with chaotic time series
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11. Completely Chaotic Cases
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12. Other Cases
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13. Conclusions
The logistic map is expressed with a simple iteration, yet the time series are complicated by only changing the parameter.
The Fourier Transformation gives a qualitative index to distinguish chaotic from non-chaotic time series.
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