# 统计计算|MTH3045 Statistical Computing代写

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A variation on least squares scaling is the so-called Sammon mapping which minimizes
$$\sum_{i \neq i^{\prime}} \frac{\left(d_{i i^{\prime}}-\left|z_{i}-z_{i^{\prime}}\right|\right)^{2}}{d_{i i^{\prime}}} .$$
Here more emphasis is put on preserving smaller pairwise distances.
In classical scaling, we instead start with similarities $s_{i i^{r}}:$ often we use the centered inner product $s_{i i^{\prime}}=\left\langle x_{i}-\bar{x}, x_{i^{\prime}}-\bar{x}\right\rangle$. The problem then is to minimize
$$\sum_{i, i^{\prime}}\left(s_{i i^{\prime}}-\left\langle z_{i}-\bar{z}, z_{i^{\prime}}-\bar{z}\right\rangle\right)^{2}$$

## MTH3045 COURSE NOTES ：

We approximate each point by an affine mixture of the points in its neighborhood:
$$\min {W{i k}}\left|x_{i}-\sum_{k \in \mathcal{N}(i)} w_{i k} x_{k}\right|^{2}$$
over weights $w_{i k}$ satisfying $w_{i k}=0, k \notin \mathcal{N}(i), \sum_{k=1}^{N} w_{i k}=1 . w_{i k}$ is the contribution of point $k$ to the reconstruction of point $i$. Note that for a hope of a unique solution, we must have $K<p$.

Finally, we find points $y_{i}$ in a space of dimension $d<p$ to minimize
$$\sum_{i=1}^{N}\left|y_{i}-\sum_{k=1}^{N} w_{i k} y_{k}\right|^{2}$$
with $w_{i k}$ fixed.