Metadata-Version: 2.1
Name: gpar
Version: 0.2.1
Summary: Implementation of the Gaussian Process Autoregressive Regression Model
Home-page: https://github.com/wesselb/gpar
Author: Wessel Bruinsma
Author-email: wessel.p.bruinsma@gmail.com
License: MIT
Description: # [GPAR](http://github.com/wesselb/gpar)
        
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        Implementation of the Gaussian Process Autoregressive Regression Model.
        See the [paper](https://arxiv.org/abs/1802.07182), and see the [docs](https://wesselb.github.io/gpar).
        
        * [Requirements and Installation](#requirements-and-installation)
        * [Basic Usage](#basic-usage)
        * [Features](#features)
            - [Input and Output Dependencies](#input-and-output-dependencies)
            - [Output Transformation](#output-transformation)
            - [Sampling](#sampling)
            - [Logpdf Computation](#logpdf-computation)
            - [Inducing Points](#inducing-points)
         * [Example (examples/paper/synthetic.py)](#example-examples-paper-synthetic-py)
        
        ## Requirements and Installation
        
        See [the instructions here](https://gist.github.com/wesselb/4b44bf87f3789425f96e26c4308d0adc).
        Then simply
        
        ```
        pip install gpar
        ```
        
        ## Basic Usage
        A simple instance of GPAR can be created as follows:
        
        ```python
        from gpar import GPARRegressor
        
        gpar = GPARRegressor(
            replace=True,
            impute=True,
            scale=1.0,
            linear=True,
            linear_scale=100.0,
            nonlinear=True,
            nonlinear_scale=1.0,
            noise=0.1,
            normalise_y=True
        )
        ```
        
        Here the keyword arguments have the following meaning:
        
        * `replace=True`: Replace data points with the posterior mean of the previous
            layer before feeding them into the next layer. This helps the model deal 
            with noisy data, but may discard important structure in the data if  the 
            fit is bad.
            
        * `impute=True`: GPAR requires that data is _closed downwards_. If this is 
            not the case, the model will be unable to use part of the data. Setting 
            `impute` to `True` lets GPAR impute data points to ensure that the data is
            closed downwards.
            
        * `scale=1.0`: Initialisation of the length scale with respect to the inputs.
        
        * `linear=True`: Use linear dependencies between outputs.
        
        * `linear_scale=True`: Initialisation of the length scale of the linear 
            dependencies between outputs.
            
        * `nonlinear=True`: Also use nonlinear dependencies between outputs.
        
        * `nonlinear_scale=1.0`: Initialisation of the length scale of the nonlinear 
            dependencies between outputs. _Important:_ this length scale applies 
            _after_ possible normalisation of the outputs (see below), in which case 
            `nonlinear_scale=1.0` corresponds to a simple, but nonlinear relationship.
            
        * `noise=0.1`: Initialisation of the variance of the observation noise.
        
        * `normalise_y=True`: Internally, work with a normalised version of the 
            outputs by subtracting the mean and dividing by the standard deviation.
            Predictions will be transformed back appropriately.
        
        In the above, any scalar hyperparameter may be replaced by a list of values 
        to initialise each layer separately, e.g. `scale=[1.0, 2.0]`. See the 
        documentation for a full overview of the keywords that may be passed to 
        `GPARRegressor`.
        
        To fit GPAR, call `gpar.fit(x_train, y_train)` where `x_train` are the training 
        inputs and `y_train` the training outputs.
        The inputs `x_train` must have shape $n$ or $n \times m$, where $n$ is the
        number of data points and $m$ the number of input features, and the outputs
        `y_train` must have shape $n$ or $n \times p$, where $p$ is the number of
        outputs.
        To condition GPAR on data without optimising its hyperparameters, use
        `gpar.condition(x_train, y_train)` instead.
        
        Finally, to make predictions, call
        
        ```python
        means = gpar.predict(x_test, num_samples=100)
        ```
        
        to get the predictive means, or 
        
        ```python
        means, lowers, uppers = gpar.predict(
            x_test,
            num_samples=100, 
            credible_bounds=True
          )
        ```
        
        to also get lower and upper 95% central marginal credible bounds. If you wish
         to predict the underlying latent function instead of the observed values, set
        `latent=True` in the call to `GPARRegressor.predict`.
        
        ## Features
        
        ### Input and Output Dependencies
        Using keywords arguments, `GPARRegressor` can be configured to specify the 
        dependencies with respect to the inputs and between the outputs. The following
        dependencies can be specified:
        
        * **Linear input dependencies:** Set `linear_input=True` and specify the 
            length scale with `linear_input_scale`.
            
        * **Nonlinear input dependencies:** This is enabled by default. The length 
            scale can be specified using `scale`. To tie these length scales across all
            layers, set `scale_tie=True`.
            
        * **Locally periodic input dependencies:** Set `per=True` and specify the period
            with `per_period`, the length scale with `per_scale`, and the length 
            scale on which the periodicity changes with `per_decay`.
            
        * **Linear output dependencies:** Set `linear=True` and specify the length 
            scale with `linear_scale`.
            
        * **Nonlinear output dependencies:** Set `nonlinear=True` and specify the 
            length scale with `nonlinear_scale`.
            
        All nonlinear kernels are exponentiated quadratic kernels. If you wish to 
        instead use rational quadratic kernels, set `rq=True`.
        
        All parameters can be set to a list of values to initialise the value for 
        each layer separately.
        
        To let every layer depend only the `k`th previous layers, set `markov=k`.
        
        
        ### Output Transformation
        
        One may want to apply a transformation to the data before fitting the model, 
        e.g. $y\mapsto\log(y)$ in the case of positive data. Such a transformation can
        be specified by setting the `transform_y` keyword argument for `GPARRegressor`.
        The following transformations are available:
        
        * `log_transform`: $y \mapsto \log(y)$.
        
        * `squishing_transform`: $y \mapsto \operatorname{sign}(y) \log(1 + |y|)$.
        
        
        ### Sampling
        
        Sampling from the model can be done with `GPARRegressor.sample`. The keyword 
        argument `num_samples` specifies the number of samples, and `latent` 
        specifies whether to sample from the underlying latent function or the 
        observed values. Sampling from the _prior_ and _posterior_ (model must be fit
        first) can be done as follows:
         
        ```python
        sample = gpar.sample(x, p=2)  # Sample two outputs from the prior.
        
        sample = gpar.sample(x, posterior=True)  # Sample from the posterior.
        ```
        
        ### Logpdf Computation
        The logpdf of data can be computed with `GPARRegressor.logpdf`. To compute the
        logpdf under the posterior, set `posterior=True`. To sample missing data to 
        compute an unbiased estimate of the *pdf*, *not logpdf*, set 
        `sample_missing=True`.
        
        ### Inducing Points
        Inducing points can be used to scale GPAR to large data sets. Simply set `x_ind`
        to the locations of the inducing points in `GPARRegressor`.
        
        
        ## Example (`examples/paper/synthetic.py`)
        
        ![Prediction](https://raw.githubusercontent.com/wesselb/gpar/master/readme_example_prediction.png)
        
        ```python
        import matplotlib.pyplot as plt
        import numpy as np
        from gpar.regression import GPARRegressor
        from wbml.experiment import WorkingDirectory
        import wbml.plot
        
        if __name__ == "__main__":
            wd = WorkingDirectory("_experiments", "synthetic", seed=1)
        
            # Create toy data set.
            n = 200
            x = np.linspace(0, 1, n)
            noise = 0.1
        
            # Draw functions depending on each other in complicated ways.
            f1 = -np.sin(10 * np.pi * (x + 1)) / (2 * x + 1) - x ** 4
            f2 = np.cos(f1) ** 2 + np.sin(3 * x)
            f3 = f2 * f1 ** 2 + 3 * x
            f = np.stack((f1, f2, f3), axis=0).T
        
            # Add noise and subsample.
            y = f + noise * np.random.randn(n, 3)
            x_obs, y_obs = x[::8], y[::8]
        
            # Fit and predict GPAR.
            model = GPARRegressor(
                scale=0.1,
                linear=True,
                linear_scale=10.0,
                nonlinear=True,
                nonlinear_scale=0.1,
                noise=0.1,
                impute=True,
                replace=False,
                normalise_y=False,
            )
            model.fit(x_obs, y_obs)
            means, lowers, uppers = model.predict(
                x, num_samples=100, credible_bounds=True, latent=True
            )
        
            # Fit and predict independent GPs: set `markov=0` in GPAR.
            igp = GPARRegressor(
                scale=0.1,
                linear=True,
                linear_scale=10.0,
                nonlinear=True,
                nonlinear_scale=0.1,
                noise=0.1,
                markov=0,
                normalise_y=False,
            )
            igp.fit(x_obs, y_obs)
            igp_means, igp_lowers, igp_uppers = igp.predict(
                x, num_samples=100, credible_bounds=True, latent=True
            )
        
            # Plot the result.
            plt.figure(figsize=(15, 3))
        
            for i in range(3):
                plt.subplot(1, 3, i + 1)
        
                # Plot observations.
                plt.scatter(x_obs, y_obs[:, i], label="Observations", style="train")
                plt.plot(x, f[:, i], label="Truth", style="test")
        
                # Plot GPAR.
                plt.plot(x, means[:, i], label="GPAR", style="pred")
                plt.fill_between(x, lowers[:, i], uppers[:, i], style="pred")
        
                # Plot independent GPs.
                plt.plot(x, igp_means[:, i], label="IGP", style="pred2")
                plt.fill_between(x, igp_lowers[:, i], igp_uppers[:, i], style="pred2")
        
                plt.xlabel("$t$")
                plt.ylabel(f"$y_{i + 1}$")
                wbml.plot.tweak(legend=i == 2)
        
            plt.tight_layout()
            plt.savefig(wd.file("synthetic.pdf"))
        ```
        
        
        
        
        
Platform: UNKNOWN
Requires-Python: >=3.6
Description-Content-Type: text/markdown
