How does PCA and clustering treat different type of predictors where order of their values signify the result differently?

Question

Assume we have 3 variables for 3 countries as shown below:

|  Country |  GDP  |  Child Mortality  |  Health Expenditure  |
|  X       |    5  |                8  |                   4  |
|  Y       |    7  |                3  |                   1  |
|  Z       |   12  |                4  |                   9  |

Now if GDP and health expenditure increase, it is a better country. While if child mortality increases it is a worse country.

Q1. How does PCA and clustering take care of these type of predictors appropriately without having the intelligence of knowing which set of predictors makes it a better country as value increases and which set of predictors make it a worse country as value increases?

Q2. Do we need to translate some predictors like child mortality with some inverse values (eg: divide by 1 or something else on scaled data), so that even for that column an increase in value signifies a better country, so the behavior is similar to columns like GDP / Health Spend etc, so our PCA and clustering algorithms are better able to model and provide good clusters ? Note, that there is no y in the model, we just have a list of countries which we need to cluster.

Anything of this sort is done in modelling and any techniques for these ?

OR what I am seeing here is invalid, and we just dont need to care about any translation (other than simple feature scaling) and directly run PCA/clustering algorithms ? And if we do this how does it take care to create appropriate clusters mathematically ?

Answer 1

Similar combinations of values will be clustered together regardless of their meaning in your model. The meaning of the clusters is for the model builder to determine.

You could engineer another feature (call it ‘goodness’) that takes the GDP and divides by child mortality (for instance) in order to add more variables to determine clusters, by dividing GDP by mortality counties with lower GDP and higher mortality rates would have a smaller “goodness” value while higher GDP and lower mortality would have higher “goodness” values and then would give the model a better chance to cluster on this factor.