`giotto-tda`

¶`giotto-tda`

includes a set of plotting functions and class methods, powered by `plotly`

. The library's plotting API is designed to facilitate the exploration of intermediate results in pipelines by harnessing the highly visual nature of topological signatures.

This notebook is a quick tutorial on how to use `giotto-tda`

's plotting functionalities and unified plotting API. The plotting functions in `gtda.mapper`

are not covered here as they are somewhat tailored to the Mapper algorithm, see the dedicated tutorial.

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

**License: AGPLv3**

`plot`

methods¶The computational building blocks of `giotto-tda`

are `scikit-learn`

–style estimators. Typically, they are also transformers, i.e. they possess a `transform`

and/or a `fit-transform`

method which:

- act on an array-like object
`X`

which collects a certain number of "samples" of a given kind; - return a transformed array-like object
`Xt`

which collects a (potentially different) number of "samples" of a potentially different kind.

The basic philosophy of `giotto-tda`

's class-level plotting API is to equip relevant transformers with `plot`

methods taking two main arguments:

- an object such as
`Xt`

above (i.e. consistent with the*outputs*of`transform`

or`fit-transform`

); - an integer index passed via the
`sample`

keyword and indicating which sample in`Xt`

should be plotted.

In other words, `<transformer>.plot(Xt, sample=i)`

will produce a plot of `Xt[i]`

which is tailored to the nature of the samples in `Xt`

.

Several `plot`

methods in `giotto-tda`

actually fall back to specialised functions which can be found in the plotting subpackage and which can be used directly instead. However, unless the additional degree of control is necessary, `plot`

methods should be preferred as they often exploit class parameters and/or attributes (e.g. those computed during `fit`

) to automatically fill some parameters in the corresponding functions.

`VietorisRipsPersistence`

¶Let's take the example of `VietorisRipsPersistence`

– a transformer also covered in another notebook. Let's create the input collection `X`

for this transformer as a collection of randomly generated point clouds, each containing 100 points positioned along two circles.

In [ ]:

```
import numpy as np
np.random.seed(seed=42)
from gtda.homology import VietorisRipsPersistence
from sklearn.datasets import make_circles
X = np.asarray([
make_circles(100, factor=np.random.random())[0]
for i in range(10)
])
```

Incidentally, samples in `X`

can be plotted using `gtda.plotting.plot_point_cloud`

.

In [ ]:

```
from gtda.plotting import plot_point_cloud
i = 0
plot_point_cloud(X[i])
```

Let us instantiate a `VietorisRipsTransformer`

object, and call the `fit-transform`

method on `X`

to obtain the transformed object `Xt`

.

In [ ]:

```
VR = VietorisRipsPersistence()
Xt = VR.fit_transform(X)
```

For any sample index i, `Xt[i]`

is a two-dimensional array encoding the multi-scale topological information which can be extracted from the i-th point cloud `X[i]`

.

It is typically too difficult to get a quick idea of the interesting information contained in `Xt[i]`

by looking at the array directly. This information is best displayed as a so-called "persistence diagram" in 2D. The `plot`

method of our `VietorisRipsPersistence`

instance achieves precisely this:

In [ ]:

```
VR.plot(Xt, sample=i)
```

In the case of `VietorisRipsPersistence`

, `plot`

is a thin wrapper around the function `gtda.plotting.plot_diagram`

, so the same result could have been achieved by importing that function and calling `plot_diagram(Xt[i])`

.

In the diagram, each point indicates a topological feature in the data which appears at a certain "birth" scale and remains present all the way up to a later "death" scale. A point's distance from the diagonal is directly proportional to the difference between the point's "death" and its "birth". Hence, this distance visually communicates how "persistent" the associated topological feature is. Topological features are partitioned by dimension using colors: above, features in dimension 0 are red while those in dimension 1 are green. In dimension 0, the diagram describes connectivity structure in the data in a very similar way to linkage clustering: we see three points along the vertical axis, which are in one-to-one correspondence with "merge" events in the sense of hierarchical clustering. In dimension 1, the diagram describes the presence of "independent" one-dimensional holes in the data: as expected, there are only two significant points, corresponding to the two "persistent" circles.

`transform_plot`

and `fit_transform_plot`

¶Where appropriate, `giotto-tda`

transformers which have a `plot`

method can also implement the two derived methods `transform_plot`

and `fit_transform_plot`

.

`transform_plot`

¶This method takes two main arguments:

- an object such as
`X`

above (i.e. consistent with the*inputs*of`transform`

or`fit-transform`

); - an integer index i passed via the
`sample`

keyword.

The logic of `transform_plot`

can be roughly described as follows: first, the sample `X[i]`

is transformed; then, the result is plotted using `plot`

and returned. [More technically: we first create a trivial collection `X_sing = [X[i]]`

, which contains a single sample from `X`

. Then, we compute `Xt_sing = <transformer>.transform(X_sing)`

. Assuming `Xt_sing`

contains a single transformed sample, we call `<transformer>.plot(Xt_sing, sample=0)`

, and also return `Xt_sing`

.]

In the example of Section 1.2, we would do:

In [ ]:

```
VR = VietorisRipsPersistence()
VR.fit(X)
VR.transform_plot(X, sample=i);
```

`fit_transform_plot`

¶This method is equivalent to first fitting the transformer using `X`

(and, optionally, a target variable `y`

), and then calling `transform_plot`

on `X`

and a given sample index.

The workflow in the example of Section 1.2 can be simplified even further, turning the entire process into a simple one-liner:

In [ ]:

```
VR = VietorisRipsPersistence()
VR.fit_transform_plot(X, sample=i);
```