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$X=\{x_{i}\}$ and $Y=\{y_{i}\}$ are numeric samples: $y_i \ge 0, x_i \ge 0, i \in [0..N]$. I need to find the mapping $F(X)=\{F(x_i)\}$ with fairly simple formula such that:

  1. Euclidean distance $\rho(F(X),Y)$ between samples is minimal: $\sum_{i=1}^n(F(x_i) - y_i)^2 \rightarrow min$;
  2. Expectations of the samples F(X) and Y are equal: $E[F(X)] = E[Y]$, i.e. $\sum_{i=1}^nF(x_i) = \sum_{i=1}^ny_i$;
  3. Standard deviation of the samples F(X) and Y are close: $\sum_{i=1}^n(F(x_i) - E[F(X)])^2 - \sum_{i=1}^n(y_i - E[Y])^2 \le \varepsilon$

In addition:

  1. $F(x_i) \ge 0$;
  2. $\rho(F(X),Y) \le \rho(X,Y)$;
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1 Answer

For

$$F(x)=\sum_i\prod_{j\neq i}\frac{x-x_j}{x_i-x_j}y_i\;,$$

all three conditions are satisfied identically.

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Lagrange interpolation –  yasmar May 5 '11 at 18:20
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