gemseo / mlearning / classification

# knn module¶

The k-nearest neighbors for classification.

The k-nearest neighbor classification algorithm is an approach to predict the output class of a new input point by selecting the majority class among the k nearest neighbors in a training set through voting. The algorithm may also predict the probabilities of belonging to each class by counting the number of occurrences of the class withing the k nearest neighbors.

Let $$(x_i)_{i=1,\cdots,n_{\text{samples}}}\in \mathbb{R}^{n_{\text{samples}}\times n_{\text{inputs}}}$$ and $$(y_i)_{i=1,\cdots,n_{\text{samples}}}\in \{1,\cdots,n_{\text{classes}}\}^{n_{\text{samples}}}$$ denote the input and output training data respectively.

The procedure for predicting the class of a new input point $$x\in \mathbb{R}^{n_{\text{inputs}}}$$ is the following:

Let $$i_1(x), \cdots, i_{n_{\text{samples}}}(x)$$ be the indices of the input training points sorted by distance to the prediction point $$x$$, i.e.

$\|x-x_{i_1(x)}\| \leq \cdots \leq \|x-x_{i_{n_{\text{samples}}}(x)}\|.$

The ordered indices may be formally determined through the inductive formula

$i_p(x) = \underset{i\in I_p(x)}{\operatorname{argmin}}\|x-x_i\|,\quad p=1,\cdots,n_{\text{samples}}$

where

$\begin{split}I_1(x) = \{1,\cdots,n_{\text{samples}}\}\\ I_{p+1} = I_p(x)\setminus \{i_p(x)\},\quad p=1,\cdots,n_{\text{samples}}-1,\end{split}$

that is

$I_p(x) = \{1,\cdots,n_{\text{samples}}\}\setminus \{i_1(x),\cdots,i_{p-1}(x)\}.$

Then, by denoting $$\operatorname{mode}(\cdot)$$ the mode operator, i.e. the operator that extracts the element with the highest occurrence, we may define the prediction operator as the mode of the set of output classes associated to the $$k$$ first indices (classes of the $$k$$-nearest neighbors of $$x$$):

$f(x) = \operatorname{mode}(y_{i_1(x)}, \cdots, y_{i_k(x)})$

This concept is implemented through the KNNClassifier class which inherits from the MLClassificationAlgo class.

## Dependence¶

The classifier relies on the KNeighborsClassifier class of the scikit-learn library.

class gemseo.mlearning.classification.knn.KNNClassifier(data, transformer=mappingproxy({}), input_names=None, output_names=None, n_neighbors=5, **parameters)[source]

The k-nearest neighbors classification algorithm.

Parameters:
• data (Dataset) – The learning dataset.

• transformer (TransformerType) –

The strategies to transform the variables. The values are instances of Transformer while the keys are the names of either the variables or the groups of variables, e.g. "inputs" or "outputs" in the case of the regression algorithms. If a group is specified, the Transformer will be applied to all the variables of this group. If IDENTITY, do not transform the variables.

By default it is set to {}.

• input_names (Iterable[str] | None) – The names of the input variables. If None, consider all the input variables of the learning dataset.

• output_names (Iterable[str] | None) – The names of the output variables. If None, consider all the output variables of the learning dataset.

• n_neighbors (int) –

The number of neighbors.

By default it is set to 5.

• **parameters (int | str) – The parameters of the machine learning algorithm.

Raises:

ValueError – When both the variable and the group it belongs to have a transformer.

class DataFormatters

Bases: DataFormatters

Decorators for supervised algorithms.

classmethod format_dict(predict)

Make an array-based function be called with a dictionary of NumPy arrays.

Parameters:

predict (Callable[[ndarray], ndarray]) – The function to be called; it takes a NumPy array in input and returns a NumPy array.

Returns:

A function making the function ‘predict’ work with either a NumPy data array or a dictionary of NumPy data arrays indexed by variables names. The evaluation will have the same type as the input data.

Return type:

Callable[[Union[ndarray, Mapping[str, ndarray]]], Union[ndarray, Mapping[str, ndarray]]]

classmethod format_input_output(predict)

Make a function robust to type, array shape and data transformation.

Parameters:

predict (Callable[[ndarray], ndarray]) – The function of interest to be called.

Returns:

A function calling the function of interest ‘predict’, while guaranteeing consistency in terms of data type and array shape, and applying input and/or output data transformation if required.

Return type:

Callable[[Union[ndarray, Mapping[str, ndarray]]], Union[ndarray, Mapping[str, ndarray]]]

classmethod format_samples(predict)

Make a 2D NumPy array-based function work with 1D NumPy array.

Parameters:

predict (Callable[[ndarray], ndarray]) – The function to be called; it takes a 2D NumPy array in input and returns a 2D NumPy array. The first dimension represents the samples while the second one represents the components of the variables.

Returns:

A function making the function ‘predict’ work with either a 1D NumPy array or a 2D NumPy array. The evaluation will have the same dimension as the input data.

Return type:
classmethod format_transform(transform_inputs=True, transform_outputs=True)

Force a function to transform its input and/or output variables.

Parameters:
• transform_inputs (bool) –

Whether to transform the input variables.

By default it is set to True.

• transform_outputs (bool) –

Whether to transform the output variables.

By default it is set to True.

Returns:

A function evaluating a function of interest, after transforming its input data and/or before transforming its output data.

Return type:
learn(samples=None, fit_transformers=True)

Train the machine learning algorithm from the learning dataset.

Parameters:
• samples (Sequence[int] | None) – The indices of the learning samples. If None, use the whole learning dataset.

• fit_transformers (bool) –

Whether to fit the variable transformers.

By default it is set to True.

Return type:

None

Load a machine learning algorithm from a directory.

Parameters:

directory (str | Path) – The path to the directory where the machine learning algorithm is saved.

Return type:

None

predict(input_data, *args, **kwargs)

Evaluate ‘predict’ with either array or dictionary-based input data.

Firstly, the pre-processing stage converts the input data to a NumPy data array, if these data are expressed as a dictionary of NumPy data arrays.

Then, the processing evaluates the function ‘predict’ from this NumPy input data array.

Lastly, the post-processing transforms the output data to a dictionary of output NumPy data array if the input data were passed as a dictionary of NumPy data arrays.

Parameters:
• input_data (Union[ndarray, Mapping[str, ndarray]]) – The input data.

• *args – The positional arguments of the function ‘predict’.

• **kwargs – The keyword arguments of the function ‘predict’.

Returns:

The output data with the same type as the input one.

Return type:
predict_proba(input_data, *args, **kwargs)

Evaluate ‘predict’ with either array or dictionary-based input data.

Firstly, the pre-processing stage converts the input data to a NumPy data array, if these data are expressed as a dictionary of NumPy data arrays.

Then, the processing evaluates the function ‘predict’ from this NumPy input data array.

Lastly, the post-processing transforms the output data to a dictionary of output NumPy data array if the input data were passed as a dictionary of NumPy data arrays.

Parameters:
• input_data (Union[ndarray, Mapping[str, ndarray]]) – The input data.

• *args – The positional arguments of the function ‘predict’.

• **kwargs – The keyword arguments of the function ‘predict’.

Returns:

The output data with the same type as the input one.

Return type:
save(directory=None, path='.', save_learning_set=False)

Save the machine learning algorithm.

Parameters:
• directory (str | None) – The name of the directory to save the algorithm.

• path (str | Path) –

The path to parent directory where to create the directory.

By default it is set to “.”.

• save_learning_set (bool) –

Whether to save the learning set or get rid of it to lighten the saved files.

By default it is set to False.

Returns:

The path to the directory where the algorithm is saved.

Return type:

str

DEFAULT_TRANSFORMER: DefaultTransformerType = mappingproxy({'inputs': <gemseo.mlearning.transform.scaler.min_max_scaler.MinMaxScaler object>})

The default transformer for the input and output data, if any.

FILENAME: ClassVar[str] = 'ml_algo.pkl'
IDENTITY: Final[DefaultTransformerType] = mappingproxy({})

A transformer leaving the input and output variables as they are.

LIBRARY: Final[str] = 'scikit-learn'

The name of the library of the wrapped machine learning algorithm.

SHORT_ALGO_NAME: ClassVar[str] = 'KNN'

The short name of the machine learning algorithm, often an acronym.

Typically used for composite names, e.g. f"{algo.SHORT_ALGO_NAME}_{dataset.name}" or f"{algo.SHORT_ALGO_NAME}_{discipline.name}".

algo: Any

The interfaced machine learning algorithm.

property input_data: ndarray

The input data matrix.

property input_dimension: int

The input space dimension.

input_names: list[str]

The names of the input variables.

input_space_center: dict[str, ndarray]

The center of the input space.

property is_trained: bool

Return whether the algorithm is trained.

property learning_samples_indices: Sequence[int]

The indices of the learning samples used for the training.

learning_set: Dataset

The learning dataset.

n_classes: int

The number of classes.

property output_data: ndarray

The output data matrix.

property output_dimension: int

The output space dimension.

output_names: list[str]

The names of the output variables.

parameters: dict[str, MLAlgoParameterType]

The parameters of the machine learning algorithm.

transformer: dict[str, Transformer]

The strategies to transform the variables, if any.

The values are instances of Transformer while the keys are the names of either the variables or the groups of variables, e.g. “inputs” or “outputs” in the case of the regression algorithms. If a group is specified, the Transformer will be applied to all the variables of this group.

## Examples using KNNClassifier¶

Classification API

Classification API

K nearest neighbors classification

K nearest neighbors classification