cutcutcodec.core.signal.predict

Sequence prediction tools.

Terminology

• \(n_o\) is the number of observations used by the predictor.

• \(o_i\) are the past observation, with \(i \in [\![1,n_o]\!]\).

• \(\hat{o}_{n+1}\) is the prediction of the next observation.

Classes

LinearPredictor(*args[, n_alpha])	Prediction by linear combination of observations.
Predictor([memory])	Basic structuring class for predicting a sequence of reels.

Details

class cutcutcodec.core.signal.predict.LinearPredictor(*args, n_alpha: Integral | None = None, **kwargs)

Prediction by linear combination of observations.

The prediction of the next observation is defined as follow:

\[\hat{o}_{n+1} = \sum_{k=1}^{n_\alpha} \alpha_k o_{n+1-k}\]

Notation:

\[\begin{split}\begin{align} \begin{pmatrix} \hat{o}_n \\ \vdots \\ \hat{o}_{n-h} \end{pmatrix} &= \begin{pmatrix} o_{n-1} & \ldots & o_{n-1-n_\alpha} \\ \vdots & \ddots & \vdots \\ o_{n-1-h} & \ldots & o_{n-1-h-n_\alpha} \\ \end{pmatrix} \begin{pmatrix} \alpha_1 \\ \vdots \\ \alpha_{n_\alpha} \end{pmatrix} \\ \Leftrightarrow \boldsymbol{\hat{o}} &= \boldsymbol{X} \boldsymbol{\alpha} \end{align}\end{split}\]

The \(\alpha_k\) scalars are founded to minimise the mse:

\[\begin{split}\begin{align} &\boldsymbol{\alpha} = \underset{\boldsymbol{\alpha}}{\operatorname{argmin}}\left( \left\| \begin{pmatrix} \hat{o}_n \\ \vdots \\ \hat{o}_{n-h} \end{pmatrix} - \begin{pmatrix} o_n \\ \vdots \\ o_{n-h} \end{pmatrix} \right\|^2 \right) \\ \Leftrightarrow &\boldsymbol{\alpha} = \underset{\boldsymbol{\alpha}}{\operatorname{argmin}}\left( \left\| \boldsymbol{\hat{o}} - \boldsymbol{o} \right\|^2 \right) \\ \Leftrightarrow &\boldsymbol{\alpha} = \underset{\boldsymbol{\alpha}}{\operatorname{argmin}}\left( \left( \boldsymbol{X} \boldsymbol{\alpha} - \boldsymbol{o} \right)^\intercal \left( \boldsymbol{X} \boldsymbol{\alpha} - \boldsymbol{o} \right) \right) \\ \Leftrightarrow &\frac{ \partial \left( \left( \boldsymbol{X} \boldsymbol{\alpha} - \boldsymbol{o} \right)^\intercal \left( \boldsymbol{X} \boldsymbol{\alpha} - \boldsymbol{o} \right) \right) }{\partial \boldsymbol{\alpha}} = \boldsymbol{0} \\ \Leftrightarrow &\boldsymbol{\alpha} = \left( \boldsymbol{X}^\intercal \boldsymbol{X} \right)^{-1} \boldsymbol{X}^\intercal \boldsymbol{o} \\ \end{align}\end{split}\]

Examples

>>> from cutcutcodec.core.signal.predict import LinearPredictor
>>> # simple case
>>> predictor = LinearPredictor(memory=4)
>>> predictor.update([4.0, 3.0, 2.0, 1.0])  # arithmetic sequence (more recent first)
>>> round(predictor.predict_next(), 6)
5.0
>>> # complicated case
>>> predictor = LinearPredictor(memory=6)
>>> predictor.update([4.0, 2.0, 3.0, 1.0, 2.0, 0.0])  # less trivial sequence
>>> [round(p, 6) for p in predictor.predict(4)]
[6.0, 4.0, 5.0, 3.0]
>>>

Initialise the predictor.

Parameters

*args, **kwargs

Transmitted to Predictor

n_alphaint, optional

The number of internal coefficients \(n_\alpha\). By default \(n_\alpha = \lfloor \frac{n_{obs}+1}{2} \rfloor\).

class cutcutcodec.core.signal.predict.Predictor(memory: Integral | float = inf)

Basic structuring class for predicting a sequence of reels.

Attributes

memoryint

The number of memorized samples (readonly).

obslist[float]

Past observations, more recent first (readonly).

Initialize the memory.

Parameters

memoryint, default=inf

The number of samples to be memorised.