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.

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

The ordered indices may be formally determined through the inductive formula

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

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

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

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 $$x$$):

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

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 (IODataset) – 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.

DataFormatters
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)

Predict output data from input data.

The user can specify these input data either as a NumPy array, e.g. array([1., 2., 3.]) or as a dictionary, e.g. {'a': array([1.]), 'b': array([2., 3.])}.

If the numpy arrays are of dimension 2, their i-th rows represent the input data of the i-th sample; while if the numpy arrays are of dimension 1, there is a single sample.

The type of the output data and the dimension of the output arrays will be consistent with the type of the input data and the size of the input arrays.

Parameters:

input_data (ndarray | Mapping[str, ndarray]) – The input data.

Returns:

The predicted output data.

Return type:
predict_proba(input_data, hard=True)

Predict the probability of belonging to each cluster from input data.

The user can specify these input data either as a numpy array, e.g. array([1., 2., 3.]) or as a dictionary, e.g. {'a': array([1.]), 'b': array([2., 3.])}.

If the numpy arrays are of dimension 2, their i-th rows represent the input data of the i-th sample; while if the numpy arrays are of dimension 1, there is a single sample.

The type of the output data and the dimension of the output arrays will be consistent with the type of the input data and the size of the input arrays.

Parameters:
• input_data (DataType) – The input data.

• hard (bool) –

Whether clustering should be hard (True) or soft (False).

By default it is set to True.

Returns:

The probability of belonging to each cluster.

Return type:

ndarray

to_pickle(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.transformers.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.

resampling_results: dict[str, tuple[Resampler, list[MLAlgo], list[ndarray] | ndarray]]

The resampler class names bound to the resampling results.

A resampling result is formatted as (resampler, ml_algos, predictions) where resampler is a Resampler, ml_algos is the list of the associated machine learning algorithms built during the resampling stage and predictions are the predictions obtained with the latter.

resampling_results stores only one resampling result per resampler type (e.g., "CrossValidation", "LeaveOneOut" and "Boostrap").

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