# Case study: Lorenz attractor¶

This notebook contains a full TDA pipeline to analyse the transitions of the Lorenz system to a chaotic regime from the stable one and viceversa.

If you are looking at a static version of this notebook and would like to run its contents, head over to github.

**License: AGPLv3**

## Import libraries¶

The first step consists in importing relevant *gtda* components and
other useful libraries or modules.

```
# Import the gtda modules
from gtda.time_series import Resampler, TakensEmbedding, SlidingWindow, PermutationEntropy
from gtda.homology import VietorisRipsPersistence
from gtda.diagrams import Scaler, Filtering, PersistenceEntropy, BettiCurve, PairwiseDistance
from gtda.graphs import KNeighborsGraph, GraphGeodesicDistance
from gtda.pipeline import Pipeline
import numpy as np
from sklearn.metrics import pairwise_distances
import plotly.express as px
from plotly.offline import init_notebook_mode, iplot
init_notebook_mode(connected=True)
# gtda plotting functions
from gtda.plotting import plot_heatmap
# Import data from openml
import openml
```

```
# Plotting functions
from gtda.plotting import plot_diagram, plot_betti_surfaces
from gtda.plotting import plot_point_cloud
```

## Setting up the Lorenz attractor simulation¶

In the next block we set up all the parameters of the Lorenz system and we define also the instants at which the regime (stable VS chaotic) changes.

```
# Plotting the trajectories of the Lorenz system
from openml.datasets.functions import get_dataset
point_cloud = get_dataset(42182).get_data(dataset_format='array')[0]
plot_point_cloud(point_cloud)
```

```
# Selecting the z-axis and the label rho
X = point_cloud[:,2]
y = point_cloud[:,3]
```

```
fig = px.line(title='Trajectory of the Lorenz solution, projected along the z-axis')
fig.add_scatter(y=X, name='X')
fig.add_scatter(y=y, name='y')
fig.show()
```

## Resampling the time series¶

It is important to find the correct time scale at which key signals take
place. Here we propose one possible resampling period: *10h*. Recall
that the unit time is *1h*. The resampler method is used to perform the
resampling.

```
period = 10
periodicSampler = Resampler(period=period)
X_sampled, y_sampled = periodicSampler.fit_transform_resample(X, y)
```

```
fig = px.line(title='Trajectory of the Lorenz solution, projected along the z-axis and resampled every 10h')
fig.add_scatter(y=X_sampled.flatten(), name='X_sampled')
fig.add_scatter(y=y_sampled, name='y_sampled')
fig.show()
```

## Takens Embedding¶

In order to obtain meaningful topological features from a time series,
we use a delayed-time embedding technique, invented by F. Takens in the
late 1960s. The idea is simple: given a time series \(X(t)\), one
can extract a sequence of vectors of the form
\(X_i := [(X(t_i)), X(t_i + 2 \tau), ..., X(t_i + M \tau)]\). The
difference between \(t_i\) and \(t_{i-1}\) is called *stride*.

\(M\) and \(\tau\) are optimized automatically in this example (they can be set by the user if needed).

```
embedding_dimension = 10
time_delay = 3
TE = TakensEmbedding(
parameters_type='search', dimension=embedding_dimension, time_delay=time_delay)
```

```
TE.fit(X_sampled)
time_delay_ = TE.time_delay_
embedding_dimension_ = TE.dimension_
print('Optimal embedding time delay based on mutual information: ', time_delay_)
print('Optimal embedding dimension based on false nearest neighbors: ', embedding_dimension_)
```

```
Optimal embedding time delay based on mutual information: 3
Optimal embedding dimension based on false nearest neighbors: 10
```

```
X_embedded, y_embedded = TE.transform_resample(X_sampled, y_sampled)
```

We can also “localise” our Takens embedding procedure to multiple sliding windows over the data, rather than over the whole time series as we just did. The result is therefore a “time series of point clouds” with possibly interesting topologies, which we will be able to feed directly to our homology transformers.

```
window_width = 40
window_stride = 5
SW = SlidingWindow(width=window_width, stride=window_stride)
X_windows, y_windows = SW.fit_transform_resample(X_embedded, y_embedded)
```

We can plot the Takens embedding of a specific window either by using
`plot_point_cloud`

, or by using the `plot`

method of
`SlidingWindow`

, as follows (*note*: when `embedding_dimension > 3`

,
only the first three coordinates are plotted!):

```
window_number = 3
SW.plot(X_windows, sample=window_number)
```

For comparison, here is the portion of time series containing the data which originates this point cloud. Notice the quasi-periodicity, corresponding to the loop in the point cloud.

```
embedded_begin, embedded_end = SW._slice_windows(X_embedded)[window_number]
window_indices = np.arange(embedded_begin, embedded_end + time_delay_ * (embedding_dimension_ - 1))
fig = px.line(title=f'Resampled Lorenz solution over sliding window {window_number}')
fig.add_scatter(x=window_indices, y=X_sampled[window_indices], name='X_sampled')
fig.show()
```

## Persistence diagram¶

The topological information in the embedding is synthesised via the persistence diagram. The horizontal axis corresponds to the moment in which a homological generator is born, while the vertical axis corresponds to the moments in which a homological generator dies. The generators of the homology groups (at given rank) are colored differently.

```
homology_dimensions = (0, 1, 2)
VR = VietorisRipsPersistence(
metric='euclidean', max_edge_length=100, homology_dimensions=homology_dimensions)
X_diagrams = VR.fit_transform(X_windows)
```

We can plot the persistence diagram for the embedding of the same
sliding window as before. One way is using the `plot_diagram`

function.

```
plot_diagram(X_diagrams[window_number])
```

Alternatively, we could have used the `plot`

method of
`VietorisRipsPersistence`

as follows:

```
VR.plot(X_diagrams, sample=window_number)
```

## Scikit-learn–style pipeline¶

One of the advantages of `giotto-tda`

is the compatibility with
`scikit-learn`

. It is possible to set up and run a full pipeline such
as the one above in a few lines:

```
# Steps of the Pipeline
steps = [
('sampling', periodicSampler),
('embedding', TE),
('window', SW),
('diagrams', VR)
]
# Define the Pipeline
pipeline = Pipeline(steps)
# Run the pipeline
X_diagrams = pipeline.fit_transform(X)
```

The final result is the same as before:

```
plot_diagram(X_diagrams[window_number])
```

## Rescaling the diagram¶

Rescaling a diagram means normalizing points such that the maximum
“bottleneck distance” from the *empty diagram* (by default, across all
homology dimensions) is equal to one. Notice that this means the birth
and death scales are modified. We can use `Scaler`

as follows:

```
diagramScaler = Scaler()
X_scaled = diagramScaler.fit_transform(X_diagrams)
diagramScaler.plot(X_scaled, sample=window_number)
```

## Filtering diagrams¶

Filtering a diagram means eliminating the homology generators whose
lifespan is considererd too short to be significant. We can use
`Filtering`

as follows:

```
diagramFiltering = Filtering(epsilon=0.1, homology_dimensions=(1, 2))
X_filtered = diagramFiltering.fit_transform(X_scaled)
diagramFiltering.plot(X_filtered, sample=window_number)
```

We can add the steps above to our pipeline:

```
steps_new = [
('scaler', diagramScaler),
('filtering', diagramFiltering)
]
pipeline_filter = Pipeline(steps + steps_new)
X_filtered = pipeline_filter.fit_transform(X)
```

```
plot_diagram(X_filtered[window_number])
```

## Persistence entropy¶

In this section we show how to compute the *entropy* of persistence
diagrams.

```
PE = PersistenceEntropy()
X_persistence_entropy = PE.fit_transform(X_scaled)
```

```
fig = px.line(title='Persistence entropies, indexed by sliding window number')
for dim in range(X_persistence_entropy.shape[1]):
fig.add_scatter(y=X_persistence_entropy[:, dim], name=f'PE in homology dimension {dim}')
fig.show()
```

## Betti Curves¶

In this section we show how to compute the Betti curves of a persistence diagram. We also show the plot of the Betti surface, i.e. the time-stack of the Betti curves.

```
BC = BettiCurve()
X_betti_curves = BC.fit_transform(X_scaled)
```

```
BC.plot(X_betti_curves, sample=window_number)
```

## Distances among diagrams¶

In this section we show how to compute several notions of distances among persistence diagrams.

In each case, we will obtain distance matrices whose i-th row encodes the distance of the i-th diagram from all the others.

We start with the so-called “landscape \(L^2\) distance”: when the
parameter `order`

is `None`

, the output is one distance matrix per
sample and homology dimension.

```
p_L = 2
n_layers = 5
PD = PairwiseDistance(
metric='landscape', metric_params={'p': p_L, 'n_layers': n_layers, 'n_bins': 1000}, order=None)
X_distance_L = PD.fit_transform(X_diagrams)
print(X_distance_L.shape)
```

```
(86, 86, 3)
```

This is what distances in homology dimension 0 look like:

```
plot_heatmap(X_distance_L[:, :, 0], colorscale='blues')
```

We now change metric and compute the “\(2\)-Wasserstein distances” between the diagrams. This one takes longer to compute!

```
p_W = 2
PD = PairwiseDistance(
metric='wasserstein', metric_params={'p': p_W, 'delta': 0.1}, order=None)
X_distance_W = PD.fit_transform(X_diagrams)
```

And again this is what distances in homology dimension 0 look like:

```
plot_heatmap(X_distance_W[:, :, 0], colorscale='blues')
```

Notice that how dramatically things can change when the metrics are modified.

## New distances in the embedding space: kNN graphs and geodesic distances¶

We propose here a new way to compute distances between points in the embedding space. Instead of considering the Euclidean distance in the Takens space, we propose to build a \(k\)-nearest neighbors graph and then use the geodesic distance on such graph.

```
n_neighbors = 3
kNN = KNeighborsGraph(n_neighbors=n_neighbors)
X_kNN = kNN.fit_transform(X_windows)
```

Given the graph embedding, the natural notion of distance between
vertices corresponds to the lengths of the shortest path connecting two
vertices. This is also known as *graph geodesic distance*. We compute it
and plot it for our chosen window number.

```
GGD = GraphGeodesicDistance()
GGD.fit_transform_plot(X_kNN, sample=window_number);
```

For comparison, this is what the ordinary pairwise Euclidean distance matrix looks like for the same window:

```
plot_heatmap(pairwise_distances(X_windows[window_number]), colorscale='blues')
```