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I have a data set of 480 samples with 7-dimensions and I want to implement a Sammon mapping into 3-dimensions. In Principal Component Analysis to my understanding we need to normalize the data in order to prevent features with large scales dominating the component selection.

In Sammon mapping however I'm not sure should normalization (e.g. z-score) be applied or not. Should data be normalized in Sammon mapping? If so, why?

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As Sammon-mapping is form of multidimensional scaling, the 'general' rules apply considering normalisation and unsupervised learning.

Sammon-mapping tries to minimize the following error-function: $E = \sum_{i<j}\frac{1}{d^{*}_{ij}}\sum_{i<j}\frac{(d^{*}_{ij}-d_{ij})^2}{d^{*}_{ij}}$, with $d^{*}_{ij}$ being the distance between points $i$ and $j$ in the original space and $d_{ij}$ the distance between points $i$ and $j$ in the projected space.

If your features are not normalized, $d^{*}_{ij}$ will be more influenced by features on larger scales, if they are normalized, all features get an equal weight, resulting in a different $d^{*}_{ij}$. Of course, if it makes sense not to normalize your data, that is, it makes sense to compare the data on the scales they are represented, then you shouldn't normalize. This is application-dependent.

A number of general answers concerning the application of normalisation in an unsupervised setting can be found, see for example here or here.

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