Différences
Ci-dessous, les différences entre deux révisions de la page.
| Les deux révisions précédentes Révision précédente Prochaine révision | Révision précédente | ||
| en:cs:k-nn_multiple_imputation [2024/04/27 09:41] – [Definitions] fraggle | en:cs:k-nn_multiple_imputation [2024/05/04 22:33] (Version actuelle) – [Definitions] fraggle | ||
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| Ligne 2: | Ligne 2: | ||
| * $A$ and $B$ two sets, $g: A \longrightarrow B$ a function: | * $A$ and $B$ two sets, $g: A \longrightarrow B$ a function: | ||
| + | * $g$ is injective $\iff \forall x, y \in A, f(x) =_{B} f(y) \implies x =_{A} y$ | ||
| + | * $g$ is surjective $\iff \forall y \in B, \exists x \in A, y =_{B} f(x)$ | ||
| + | * $g$ is bijective $\iff g$ is injective and surjective $\iff \forall y \in B, \exists! x \in A, y =_{B} f(x) \iff \exists g^{-1}: B \longrightarrow A, g \circ g^{-1} = id_A \land g^{-1} \circ g = id_B$ | ||
| * Subset $g$ image : $A^{\prime} \subset A, g(A^{\prime}) = \{g(x) \in B| \, x \in A^{\prime}\} \subset B$ | * Subset $g$ image : $A^{\prime} \subset A, g(A^{\prime}) = \{g(x) \in B| \, x \in A^{\prime}\} \subset B$ | ||
| * Subset inverse $g$ image: $B^{\prime} \subset B, g^{-1}(B^{\prime}) = \{x \in A| \, g(x) \in B^{\prime}\} \subset A$ | * Subset inverse $g$ image: $B^{\prime} \subset B, g^{-1}(B^{\prime}) = \{x \in A| \, g(x) \in B^{\prime}\} \subset A$ | ||
| - | * $g$ is injective $\iff \forall x, y \in A, f(x) =_{B} f(y) \implies x =_{A} y$ | + | |
| - | * $g$ is surjective $\iff \forall y \in B, \exists x \in A, y =_{B} f(x)$ | + | |
| - | * $g$ is bijective $\iff g$ is injective and surjective $\iff \forall y \in B, \exists! x \in A, y =_{B} f(x)$ | + | |
| * For a given $d \in \mathbb{N}$, | * For a given $d \in \mathbb{N}$, | ||
| - | |||
| $$ \begin{array}{lrcl} | $$ \begin{array}{lrcl} | ||
| f: & \mathbb{R}^{d} & \longrightarrow & \mathbb{R}^{d} \\ | f: & \mathbb{R}^{d} & \longrightarrow & \mathbb{R}^{d} \\ | ||
| Ligne 18: | Ligne 18: | ||
| $f$ will be called the prediction function in subsequent sections. | $f$ will be called the prediction function in subsequent sections. | ||
| - | * For a given normed space vector on corpse $K$ $(E, \|~\|_{E})$ and $X \in E$, let' define the binary relation $\le_{X}$: | + | * For a given normed space vector on corpse $K$ $(E, \|~\|_{E})$ and $X \in E$, let' define the binary relation $\le_{X}$ |
| - | $$\forall X_{1} \in E \land \forall X_{2} \in E, X_{1} \le_{X} X_{2} \iff \|X - X_{1}\|_{E} \le_{K} \|X - X_{2}\|_{E}$$ | + | $$\forall X_{1} \in E \land \forall X_{2} \in E, X_{1} \le_{X} X_{2} \iff \|X - X_{1}\|_{E} \le_{K} \|X - X_{2}\|_{E}$$ |
| - | * For a given normed space vector on corpse $K$ $(E, \|~\|_{E})$ and $X \in E$, let' define the binary relation $=_{X}$: | + | * For a given normed space vector on corpse $K$ $(E, \|~\|_{E})$ and $X \in E$, let' define the binary relation $=_{X}$ |
| - | $$\forall X_{1} \in E \land \forall X_{2} \in E, X_{1} =_{X} X_{2} \iff \|X - X_{1}\|_{E} =_{K} \|X - X_{2}\|_{E}$$ | + | $$\forall X_{1} \in E \land \forall X_{2} \in E, X_{1} =_{X} X_{2} \iff \|X - X_{1}\|_{E} =_{K} \|X - X_{2}\|_{E}$$ |
| ====== k-NN multiple imputation ====== | ====== k-NN multiple imputation ====== | ||
| Ligne 43: | Ligne 43: | ||
| ===== k-NN ===== | ===== k-NN ===== | ||
| - | For $X \in \mathbb{R}^{d}$, | + | For $X \in \mathbb{R}^{d}$, |
| For $k \in \{1, | For $k \in \{1, | ||
| Ligne 64: | Ligne 64: | ||
| * Impute with the mean: | * Impute with the mean: | ||
| \[ | \[ | ||
| - | Y^* = \frac{1}{k} | + | Y^* = \frac{1}{k} |
| \] | \] | ||