gemseo / mlearning / classification

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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)

Bases: MLClassificationAlgo

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.

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.

input_names: list[str]

The names of the input variables.

input_space_center: dict[str, ndarray]

The center of the input space.

learning_set: IODataset

The learning dataset.

n_classes: int

The number of classes.

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