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Convert math into HTML.
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manu-mannattil committed Dec 25, 2017
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Expand Up @@ -4,98 +4,98 @@ Examples
General Workflow
----------------

1. Check stationarity.
1. Estimate time delay (e.g., autcorrelation, mutual information, 2D/3D
phase portraits, etc.).
1. Estimate embedding dimension (AFN, IFNN, FNN, etc.).
1. Noise reduction (if required).
1. Estimate an invariant (e.g., Lyapunov exponent, correlation
dimension/entropy, etc.).
1. Surrogate analysis (IAAFT, FT, cycle shuffled, etc.)
1. Additional tests for nonlinearity (e.g., prediction error, etc.)
1. Conclusion.
1. Check stationarity.
2. Estimate time delay (e.g., autcorrelation, mutual information, 2D/3D
phase portraits, etc.).
3. Estimate embedding dimension (AFN, IFNN, FNN, etc.).
4. Noise reduction (if required).
5. Estimate an invariant (e.g., Lyapunov exponent, correlation
dimension/entropy, etc.).
6. Surrogate analysis (IAAFT, FT, cycle shuffled, etc.)
7. Additional tests for nonlinearity (e.g., prediction error, etc.)
8. Conclusion.

Practical Demonstrations
------------------------

* Generating test data sets
+ [Mackey–Glass system](data/mackey-glass.py)

* Estimating the time delay
+ [Autocorrelation function of a *finite* sine wave](delay/sine.py)
+ [How many bins should one take while estimating the delayed mutual
information?](delay/dmibins.py)
+ [Delayed mutual information for map like data can give bad
estimates](delay/henon.py)
+ Time delay estimation for
- [Henon map](delay/henon.py)
- [Ikeda map](delay/ikeda.py)
- [Rössler oscillator](delay/roessler.py)
- [Lorenz attractor](delay/lorenz.py)
+ Average deviation from the diagonal (ADFD)
- [Mackey–Glass system](delay/adfd_mackey-glass.py)
- [Rössler oscillator](delay/adfd_roessler.py)

* Averaged false neighbors (AFN), aka Cao's test
+ Averaged false neighbors for:
- [Henon map](afn/henon.py)
- [Ikeda map](afn/ikeda.py)
- [Lorenz attractor](afn/lorenz.py)
- [Mackey–Glass system](afn/mackey-glass.py)
- [Rössler oscillator](afn/roessler.py)
- [Data from a far-infrared laser](afn/laser.py)
+ [AFN is not impervious to every stochastic data](afn/ar1.py)
+ [AFN can cause trouble with discrete data](afn/roessler-8bit.py)

* False nearest neighbors (FNN)
+ FNN for:
- [Henon map](fnn/henon.py)
- [Ikeda map](fnn/ikeda.py)
- [Mackey–Glass system](fnn/mackey-glass.py)
- [Uncorrelated noise](fnn/noise.py)
+ [FNN results depend on the metric used](fnn/metric.py)
+ [FNN can fail when temporal correlations are not removed](fnn/corrnum.py)

* Noise reduction
+ [Filtering human breath data](noise/breath.py)
+ [Cleaning the "GOAT" vowel](noise/goat.py)
+ [Simple moving average vs. nonlinear noise reduction](noise/henon_sma.py)
+ [Filtering data from a far-infrared laser](noise/laser.py)

* Correlation sum/correlation dimension
+ Computing the correlation sum for:
- [Henon map](d2/henon.py)
- [Ikeda map](d2/ikeda.py)
- [Lorenz attractor](d2/lorenz.py)
- [Rössler oscillator](d2/roessler.py)
- [Mackey–Glass system](d2/mackey-glass.py)
- [White noise](d2/white.py)
+ Subtleties
- [AR(1) process can mimic a deterministic process](d2/ar1.py)
- [Brown noise can have a saturating $D_2$](d2/brown.py)
+ Computing $D_2$ of geometrical object
- [Nonstationary spiral](d2/spiral.py)
- [Closed noisy curve](d2/curve.py)

* Maximum Lyapunov exponent (MLE)
+ Computing the MLE for:
- [Henon map](lyapunov/henon.py)
- [Lorenz attractor](lyapunov/lorenz.py)
- [Rössler oscillator](lyapunov/roessler.py)
- [Closed noisy curve](lyapunov/curve.py)

* Surrogate analysis
+ Algorithms
- [AAFT surrogates](surrogates/aaft.py)
- [IAAFT surrogates](surrogates/iaaft.py)
+ Examples of surrogate analysis of
- [Linearly correlated noise](surrogates/corrnoise.py)
- [Nonlinear time series](surrogates/lorenz.py)
+ Time reversal asymmetry:
- [Skew statistic fails for linear stochastic data](surrogates/skewnoise.py)
- [Skew statistic fails for Lorenz](surrogates/skewlorenz.py)
+ [Why end point mismatch in a time series ought to be reduced](surrogates/mismatch.py)
+ [Converting a time series to a uniform deviate is harmful](surrogates/unidev.py)
- Generating test data sets
- [Mackey–Glass system](data/mackey-glass.py)
- Estimating the time delay
- [Autocorrelation function of a <em>finite</em> sine
wave](delay/sine.py)
- [How many bins should one take while estimating the delayed
mutual information?](delay/dmibins.py)
- [Delayed mutual information for map like data can give bad
estimates](delay/henon.py)
- Time delay estimation for
- [Henon map](delay/henon.py)
- [Ikeda map](delay/ikeda.py)
- [Rössler oscillator](delay/roessler.py)
- [Lorenz attractor](delay/lorenz.py)
- Average deviation from the diagonal (ADFD)
- [Mackey–Glass system](delay/adfd_mackey-glass.py)
- [Rössler oscillator](delay/adfd_roessler.py)
- Averaged false neighbors (AFN), aka Cao’s test
- Averaged false neighbors for:
- [Henon map](afn/henon.py)
- [Ikeda map](afn/ikeda.py)
- [Lorenz attractor](afn/lorenz.py)
- [Mackey–Glass system](afn/mackey-glass.py)
- [Rössler oscillator](afn/roessler.py)
- [Data from a far-infrared laser](afn/laser.py)
- [AFN is not impervious to every stochastic data](afn/ar1.py)
- [AFN can cause trouble with discrete data](afn/roessler-8bit.py)
- False nearest neighbors (FNN)
- FNN for:
- [Henon map](fnn/henon.py)
- [Ikeda map](fnn/ikeda.py)
- [Mackey–Glass system](fnn/mackey-glass.py)
- [Uncorrelated noise](fnn/noise.py)
- [FNN results depend on the metric used](fnn/metric.py)
- [FNN can fail when temporal correlations are not
removed](fnn/corrnum.py)
- Noise reduction
- [Filtering human breath data](noise/breath.py)
- [Cleaning the “GOAT” vowel](noise/goat.py)
- [Simple moving average vs. nonlinear noise
reduction](noise/henon_sma.py)
- [Filtering data from a far-infrared laser](noise/laser.py)
- Correlation sum/correlation dimension
- Computing the correlation sum for:
- [Henon map](d2/henon.py)
- [Ikeda map](d2/ikeda.py)
- [Lorenz attractor](d2/lorenz.py)
- [Rössler oscillator](d2/roessler.py)
- [Mackey–Glass system](d2/mackey-glass.py)
- [White noise](d2/white.py)
- Subtleties
- [AR(1) process can mimic a deterministic process](d2/ar1.py)
- [Brown noise can have a saturating
<em>D</em><sub>2</sub>](d2/brown.py)
- Computing <em>D</em><sub>2</sub> of geometrical object
- [Nonstationary spiral](d2/spiral.py)
- [Closed noisy curve](d2/curve.py)
- Maximum Lyapunov exponent (MLE)
- Computing the MLE for:
- [Henon map](lyapunov/henon.py)
- [Lorenz attractor](lyapunov/lorenz.py)
- [Rössler oscillator](lyapunov/roessler.py)
- [Closed noisy curve](lyapunov/curve.py)
- Surrogate analysis
- Algorithms
- [AAFT surrogates](surrogates/aaft.py)
- [IAAFT surrogates](surrogates/iaaft.py)
- Examples of surrogate analysis of
- [Linearly correlated noise](surrogates/corrnoise.py)
- [Nonlinear time series](surrogates/lorenz.py)
- Time reversal asymmetry:
- [Skew statistic fails for linear stochastic
data](surrogates/skewnoise.py)
- [Skew statistic fails for Lorenz](surrogates/skewlorenz.py)
- [Why end point mismatch in a time series ought to be
reduced](surrogates/mismatch.py)
- [Converting a time series to a uniform deviate is
harmful](surrogates/unidev.py)

General Tips
------------
Expand All @@ -104,56 +104,60 @@ While there is no dearth of good literature on nonlinear time series
analysis, here are a few things that I found to be useful in practical
situations.

1. Picking a good time delay hinges on balancing redundance and
irrelevance between the components of the time delayed vectors and
there are no straightforward theoretical results that help us do
this. Therefore, always plot two- and three-dimensional phase
portraits of the reconstructed attractor before settling on a time
delay and verify that the attractor (or whatever structure appears)
looks unfolded. Don't blindly pick values by just looking at the
delayed mutual information or the autocorrelation function of the
time series.

1. If possible, use the Chebyshev metric while computing the correlation
sum $C(r)$. The Chebyshev metric has many advantages, especially
when we are trying to evaluate $C(r)$ after embedding the time
series.

* It's computationally faster than the cityblock and the Euclidean
metric. For larger data sets, this is an obvious advantage.

* Distances are independent of the embedding dimension $d$ and
always remain bounded as opposed to Euclidean and cityblock
distances, which crudely go as $\sqrt{d}$ and $d$ respectively.
This helps in comparing the correlation sum plots at different
embedding dimensions.

* Since the distances always remain bounded, we can evaluate $C(r)$
at the same $r$'s for all embedding dimensions. And $C(r)$ at
$r_{\text{max}} = \max_i x_i - \min_i x_i$ is $1$ regardless of
the embedding dimension. Of course, $C(r)$ could be $1$ even for
$r < r_{\text{max}}$. Nonetheless, this helps in choosing a range
of $r$ values.

1. Ensure that temporal correlations between points are removed in all
cases where it is known to result in spurious estimates of dimension
and/or determinism.

1. Avoid the skew statistic which attempts to measures asymmetry w.r.t.
time reversal for detecting nonlinearity. It tells very little about
the origin of nonlinearity and fails miserably in many cases.

1. The second FNN test tests for "boundedness" of the reconstructed
attractor.

1. When plotting the reconstructed phase space, use same scaling for all
the axes as all $d$ coordinates are sampled from the same
distribution.

1. If the series has a strong periodic component, a reasonable time
delay is the quarter of the time period.

1. Some chaotic time series such as those from Lorenz attractor may
display long-range correlations (because of its "reversing" nature).
In such cases, a delay may be determined by computing autocorrelation
function of the *square* of the original time series.
1. Picking a good time delay hinges on balancing redundance and
irrelevance between the components of the time delayed vectors and
there are no straightforward theoretical results that help us do
this. Therefore, always plot two- and three-dimensional phase
portraits of the reconstructed attractor before settling on a time
delay and verify that the attractor (or whatever structure appears)
looks unfolded. Don’t blindly pick values by just looking at the
delayed mutual information or the autocorrelation function of the
time series.

2. If possible, use the Chebyshev metric while computing the
correlation sum <em>C</em>(<em>r</em>). The Chebyshev metric
has many advantages, especially when we are trying to evaluate
<em>C</em>(<em>r</em>) after embedding the time series.

- It’s computationally faster than the cityblock and the Euclidean
metric. For larger data sets, this is an obvious advantage.

- Distances are independent of the embedding dimension
<em>d</em> and always remain bounded as opposed to Euclidean
and cityblock distances, which crudely go as <em>d<sup>1/2</sup></em> and
<em>d</em> respectively. This helps in comparing the
correlation sum plots at different embedding dimensions.

- Since the distances always remain bounded, we can evaluate
<em>C</em>(<em>r</em>) at the same <em>r</em>’s for
all embedding dimensions. And <em>C</em>(<em>r</em>) at
<em>r</em><sub>max</sub> = max<sub><em>i</em></sub><em>x</em><sub><em>i</em></sub> − min<sub><em>i</em></sub><em>x</em><sub><em>i</em></sub>
is 1 regardless of the embedding dimension. Of course,
<em>C</em>(<em>r</em>) could be 1 even for
<em>r</em> &lt; <em>r</em><sub>max</sub>.
Nonetheless, this helps in choosing a range of <em>r</em>
values.

3. Ensure that temporal correlations between points are removed in all
cases where it is known to result in spurious estimates of dimension
and/or determinism.

4. Avoid the skew statistic which attempts to measures asymmetry w.r.t.
time reversal for detecting nonlinearity. It tells very little about
the origin of nonlinearity and fails miserably in many cases.

5. The second FNN test tests for “boundedness” of the reconstructed
attractor.

6. When plotting the reconstructed phase space, use same scaling for
all the axes as all <em>d</em> coordinates are sampled from the
same distribution.

7. If the series has a strong periodic component, a reasonable time
delay is the quarter of the time period.

8. Some chaotic time series such as those from Lorenz attractor may
display long-range correlations (because of its “reversing” nature).
In such cases, a delay may be determined by computing
autocorrelation function of the <em>square</em> of the original
time series.

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