LinearModel

class LinearModel

Bases: Model

baypy.model.model.LinearModel object.

Attributes

baypy.model.linear_model.LinearModel.datapandas.DataFrame

Data for the linear regression model, is a pandas.DataFrame containing all regressor variables \(X\) and the response variable \(y\).

baypy.model.linear_model.LinearModel.response_variablestring

Response variable \(y\) of the linear model.

baypy.model.linear_model.LinearModel.priorsdict

Priors for the regressors’ and variance parameters.

baypy.model.linear_model.LinearModel.variable_nameslist

The list of all model variables: the regressors \(X\), including the intercept and the variance \(\sigma^2\).

baypy.model.linear_model.LinearModel.posteriorsdict

Posterior samples. Posteriors and relative samples are key-value pairs. Each sample is a numpy.ndarray with a number of rows equals to the number of iterations and a number of columns equal to the number of Markov chains.

Methods

baypy.model.linear_model.LinearModel.posteriors_to_frame()

Organizes the posteriors in a pandas.DataFrame.

baypy.model.linear_model.LinearModel.residuals()

Compute the residuals \(\epsilon\) with respect to predicted values \(\hat{y}\).

baypy.model.linear_model.LinearModel.predict_distribution()

Predicts a posterior distribution for an unobserved values.

baypy.model.linear_model.LinearModel.likelihood()

Computes the likelihood of observations model.response_variable given a model 'mean' and 'variance'.

baypy.model.linear_model.LinearModel.log_likelihood()

Computes the log likelihood of observations model.response_variable given a model 'mean' and 'variance'.

property data: DataFrame

Data for the linear regression model, is a pandas.DataFrame containing all regressor variables \(X\) and the response variable \(y\).

Returns

pandas.DataFrame

Observed data of the model. It cannot be empty. It must contain regressor variables \(X\) and the response variable \(y\).

Raises

TypeError

If data is not an instance of pandas.DataFrame.

ValueError

If data is an empty pandas.DataFrame.

likelihood(data: DataFrame) → ndarray

Computes the likelihood of observations response_variable given a model 'mean' and 'variance'.

Parameters

data: pandas.DataFrame

Data to use for likelihood computation. It cannot be empty. It must contain columns response_variable, 'mean' and 'variance'.

Returns

numpy.ndarray

Array of computed likelihood. It has the same length of data. Each element is a likelihood computation of each row of data.

Raises

TypeError

If data is not an instance of pandas.DataFrame.

ValueError

  • If data is an empty pandas.DataFrame,

  • if response_variable is not a column of data,

  • if 'mean' is not a column of data,

  • if 'variance' is not a column of data.

Notes

The likelihood is computed with the normal distribution probability density function:

\[L(y) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp{- \frac{\left(y - \mu \right)^2}{2 \sigma^2}}\]

where \(\mu\) is the 'mean' column and \(\sigma^2\) is the 'variance' column.

log_likelihood(data: DataFrame) → ndarray

Computes the log likelihood of observations response_variable given a model 'mean' and 'variance'.

Parameters

data: pandas.DataFrame

Data to use for log likelihood computation. It cannot be empty. It must contain columns response_variable, 'mean' and 'variance'.

Returns

numpy.ndarray

Array of computed log likelihood. It has the same length of data. Each element is a log likelihood computation of each row of data.

Raises

TypeError

If data is not an instance of pandas.DataFrame.

ValueError

  • If data is an empty pandas.DataFrame,

  • if response_variable is not a column of data,

  • if 'mean' is not a column of data,

  • if 'variance' is not a column of data.

Notes

The log likelihood is computed as the log of the normal distribution probability density function:

\[l(y) = - \frac{1}{2} \log{2 \pi \sigma^2} - \frac{1}{2} \frac{\left(y - \mu \right)^2}{\sigma^2}\]

where \(\mu\) is the 'mean' column and \(\sigma^2\) is the 'variance' column.

property posteriors: dict

Posteriors of the regressors’ and variance parameters. Posteriors and relative samples are key-value pairs. Each sample is a numpy.ndarray with a number of rows equals to the number of iterations and a number of columns equal to the number of Markov chains.

Returns

dict

Posterior samples. Posteriors and relative samples are key-value pairs. Each sample is a numpy.ndarray with a number of rows equals to the number of iterations and a number of columns equal to the number of Markov chains.

Raises

TypeError

  • If posteriors is not a dict,

  • if a posterior sample is not a numpy.ndarray.

KeyError

If posteriors does not contain both intercept and variance keys.

ValueError

If a posterior sample is an empty numpy.ndarray.

posteriors_to_frame() → DataFrame

Organizes the posteriors in a pandas.DataFrame. Each posterior is a frame column. The length of the frame is the number of sampling iterations times the number of sampling chains.

Returns

pandas.DataFrame

Returns posterior samples. Posteriors are organized in a pandas.DataFrame, one for each column. The length of the frame is the number of sampling iterations times the number of sampling chains.

Raises

ValueError

If posteriors are not available because the method baypy.regression.LinearRegression.sample() has not been called yet.

predict_distribution(predictors: dict) → ndarray
Predicts a posterior distribution for an unobserved values. For each posterior sample, it draws a sample from

the likelihood.

Parameters

predictorsdict

Values of predictors \(X\) at which compute the posterior distribution. Each predictor has to be set as a key-value pair.

Returns

numpy.ndarray

Array of the predicted posterior distribution. It contains a number of element equal to the number of regression iterations times the number of model Markov chains.

Raises

TypeError

If predictors is not a dict.

KeyError

If a predictors key is not a key of posteriors.

ValueError

If predictors is an empty dict.

See Also

baypy.regression.linear_regression.LinearRegression

property priors: dict

Priors for the regressors’ and variance parameters. Each prior is a key-value pair, where the value is a dict with:

  • hyperparameter names as keys

  • hyperparameter values as values.

Returns

dict

Priors for each random variable. It must contain an intercept and a variance keys. Each value must be a dict with hyperparameter names as key and hyperparameter values as values.

Raises

TypeError

  • If priors is not a dict,

  • if a priors’ value is not a dict.

ValueError

  • If priors is an empty dict,

  • if a priors’ value is an empty dict,

  • if a variance value is not positive,

  • if a shape value is not positive,

  • if a scale value is not positive.

KeyError

  • If priors does not contain both intercept and variance keys,

  • if a prior’s hyperparameters are not:

    • mean and variance for a regression parameter \(\beta_j\) or

    • shape and scale for variance \(\sigma^2\).

Notes

To each random variables is assigned a prior distribution:

  • to each regressor parameter \(\beta_j\) is assigned a normal prior distribution with hyperparameters mean \(\beta_j^0\) and variance \(\Sigma_{\beta_j}^0\):

    \[\beta_j \sim N(\beta_j^0 , \Sigma_{\beta_j}^0)\]

  • to variance \(\sigma^2\) is assigned an inverse gamma distribution with hyperparameters shape \(\kappa^0\) and scale \(\theta^0\):

    \[\sigma^2 \sim \text{Inv-}\Gamma(\kappa^0, \theta^0)\]

Examples

Consider a linear regression of the response variable \(y\) with respect to regressors \(x_1\), \(x_2\) and \(x_3\), according to the following model:

\[y \sim N(\mu, \sigma^2)\]
\[\mu = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3\]
then the sampler would require priors for:

  • parameter \(\beta_0\) of variable intercept, with mean \(\beta_0^0\) and variance \(\Sigma_{\beta_0}^0\)

  • parameter \(\beta_1\) of variable \(x_1\), with mean \(\beta_1^0\) and variance \(\Sigma_{\beta_1}^0\)

  • parameter \(\beta_2\) of variable \(x_2\), with mean \(\beta_2^0\) and variance \(\Sigma_{\beta_2}^0\)

  • parameter \(\beta_3\) of variable \(x_3\), with mean \(\beta_3^0\) and variance \(\Sigma_{\beta_3}^0\)

  • variable \(\sigma^2\), with shape \(\kappa^0\) and scale \(\theta^0\)

>>> model = baypy.model.LinearModel()
>>> model.set_priors({'intercept': {'mean': 0, 'variance': 1e6},
...                   'x_1': {'mean': 0, 'variance': 1e6},
...                   'x_2': {'mean': 0, 'variance': 1e6},
...                   'x_3': {'mean': 0, 'variance': 1e6},
...                   'variance': {'shape': 1, 'scale': 1e-6}})
residuals() → DataFrame

Compute the residuals \(\epsilon\) with respect to predicted values \(\hat{y}\).

Returns

pandas.DataFrame

Returns a copy of data with 3 more columns: intercept, predicted and residuals.

Raises

ValueError

Notes

Predicted values are computed at data points \(X\) using the posteriors means for each regressor’s parameter:

\[\hat{y_i} = \beta_0 + \sum_{j = 1}^{m} \beta_j x_{i,j}\]

while residuals are the difference between the observed values and the predicted values of the response_variable:

\[\epsilon_i = y_i - \hat{y_i}\]
property response_variable: str

Response variable \(y\) of the linear model.

Returns

string

Name of the response variable \(y\). In must be one of the columns of data.

Raises

TypeError

If response_variable is not a str.

property variable_names: list

Variables of the linear model.

Returns

list

The list of all model variables: the regressors \(X\), including the intercept and the variance \(\sigma^2\).