4. Imaging time series

Imaging time series, that is transforming time series into images, is another popular transformation. One important upside of this transformation is retrieving information for any pair of time points (x_i, x_j) given a time series (x_1, \ldots, x_n). Deep neural networks, especially convolutional neural networks, have been used to classify these imaged time series. While pyts does not provide deep neural networks, it provides algorithms to transform time series into images in the pyts.image module.

4.1. Recurrence Plot

RecurrencePlot extracts trajectories from time series and computes the pairwise distances between these trajectories. The trajectories are defined as:

\vec{x}_i = (x_i, x_{i + \tau}, \ldots, x_{i + (m - 1)\tau}), \quad \forall i \in \{1, \ldots, n - (m - 1)\tau \}

where m is the dimension of the trajectories and \tau is the time_delay. The recurrence plot, denoted R, is the binarized pairwise distance matrix between the trajectories:

R_{i, j} = \Theta(\varepsilon - \| \vec{x}_i - \vec{x}_j \|), \quad \forall i,j \in \{1, \ldots, n - (m - 1)\tau \}

where \Theta is the Heaviside function and \varepsilon is the threshold. Different strategies can be used to choose the threshold, such as a given float or a quantile of the distances.

../_images/sphx_glr_plot_rp_0011.png 
>>> from pyts.datasets import load_gunpoint
>>> from pyts.image import RecurrencePlot
>>> X, _, _, _ = load_gunpoint(return_X_y=True)
>>> transformer = RecurrencePlot()
>>> X_new = transformer.transform(X)
>>> X_new.shape
(50, 150, 150)

Images can be flattened by setting the flatten parameter to True, so that classification can be directly performed:

>>> from pyts.image import RecurrencePlot
>>> from pyts.datasets import load_gunpoint
>>> from sklearn.pipeline import make_pipeline
>>> from sklearn.linear_model import LogisticRegression
>>> X_train, X_test, y_train, y_test = load_gunpoint(return_X_y=True)
>>> recurrence = RecurrencePlot(dimension=15, time_delay=3, flatten=True)
>>> logistic = LogisticRegression(solver='liblinear')
>>> clf = make_pipeline(recurrence, logistic)
>>> clf.fit(X_train, y_train)
Pipeline(...)
>>> clf.score(X_test, y_test)
0.933...

References

  • J.-P Eckmann, S. Oliffson Kamphorst and D Ruelle, “Recurrence Plots of Dynamical Systems”. Europhysics Letters (1987).
  • N. Hatami, Y. Gavet and J. Debayle, “Classification of Time-Series Images Using Deep Convolutional Neural Networks”. https://arxiv.org/abs/1710.00886

4.2. Gramian Angular Field

GramianAngularField creates a matrix of temporal correlations for each (x_i, x_j). First it rescales the time series in a range [a, b] where -1 \leq a < b \leq 1. Then it computes the polar coordinates of the scaled time series by taking the arccos. Finally it computes the cosine of the sum of the angles for the Gramian Angular Summation Field (GASF) or the sine of the difference of the angles for the Gramian Angular Difference Field (GADF).

\tilde{x}_i = a + (b - a) \times \frac{x_i - \min(x)}{\max(x) - \min(x)}, \quad \forall i \in \{1, \ldots, n\} \phi_i = \arccos(\tilde{x}_i), \quad \forall i \in \{1, \ldots, n\} GASF_{i, j} = \cos(\phi_i + \phi_j), \quad \forall i, j \in \{1, \ldots, n\} GADF_{i, j} = \sin(\phi_i - \phi_j), \quad \forall i, j \in \{1, \ldots, n\}

The method parameter controls which type of Gramian angular fields are computed.

../_images/sphx_glr_plot_gaf_0011.png 
>>> from pyts.datasets import load_gunpoint
>>> from pyts.image import GramianAngularField
>>> X, _, _, _ = load_gunpoint(return_X_y=True)
>>> transformer = GramianAngularField()
>>> X_new = transformer.transform(X)
>>> X_new.shape
(50, 150, 150)

Images can be flattened by setting the flatten parameter to True, so that classification can be directly performed:

>>> from pyts.image import GramianAngularField
>>> from pyts.datasets import load_gunpoint
>>> from sklearn.pipeline import make_pipeline
>>> from sklearn.linear_model import LogisticRegression
>>> X_train, X_test, y_train, y_test = load_gunpoint(return_X_y=True)
>>> gaf = GramianAngularField(flatten=True)
>>> logistic = LogisticRegression(solver='liblinear')
>>> clf = make_pipeline(gaf, logistic)
>>> clf.fit(X_train, y_train)
Pipeline(...)
>>> clf.score(X_test, y_test)
0.973...

References

  • Z. Wang and T. Oates, “Encoding Time Series as Images for Visual Inspection and Classification Using Tiled Convolutional Neural Networks.” AAAI Workshop (2015).

4.3. Markov Transition Field

MarkovTransitionField discretizes a time series into bins. It then computes the Markov Transition Matrix of the discretized time series. Finally it spreads out the transition matrix to a field in order to reduce the loss of temporal information.

../_images/sphx_glr_plot_mtf_0011.png 
>>> from pyts.datasets import load_gunpoint
>>> from pyts.image import MarkovTransitionField
>>> X, _, _, _ = load_gunpoint(return_X_y=True)
>>> transformer = MarkovTransitionField()
>>> X_new = transformer.transform(X)
>>> X_new.shape
(50, 150, 150)

Images can be flattened by setting the flatten parameter to True, so that classification can be directly performed:

>>> from pyts.image import MarkovTransitionField
>>> from pyts.datasets import load_gunpoint
>>> from sklearn.pipeline import make_pipeline
>>> from sklearn.linear_model import LogisticRegression
>>> X_train, X_test, y_train, y_test = load_gunpoint(return_X_y=True)
>>> mtf = MarkovTransitionField(image_size=0.1, n_bins=3, flatten=True)
>>> logistic = LogisticRegression(solver='liblinear')
>>> clf = make_pipeline(mtf, logistic)
>>> clf.fit(X_train, y_train)
Pipeline(...)
>>> clf.score(X_test, y_test)
0.92

References

  • Z. Wang and T. Oates, “Encoding Time Series as Images for Visual Inspection and Classification Using Tiled Convolutional Neural Networks.” AAAI Workshop (2015).