What is the meaning of noise in a dataset with no dependant variable? My understanding of noise & signal comes from the context of bias-variance tradeoff in supervised methods.
But given a dataset with no dependant variable, how do you define noise? & how do you decompose data into signal and noise? ( there used to be the function  stats.signaltonoise in scipy which would return ratio of mean/standard_deviation given an array of numbers. What is the rationale behind this?)
 A: Noise is relative to the model.
The term noise doesn't really make sense without a dependent variable.  The signal in a matrix of independent measurements is the information that is useful in predicting the dependent variable; the rest is noise (residual). The same independent dataset may be useful in predicting different dependent variables, or the same dependent variable with different equations, where noise is different for each model.
Even in the colloquial sense, where noise is like the static degrading your TV signal, both the TV show and the noise still have a source you may care about (caring is equivalent to defining your dependent variable). If you're trying to measure cosmic background radiation in the universe, rather than watch TV, whatever that damned television broadcaster is pumping out is noise (or, perhaps a removable confound).
For that reason, @whuber is correct in the comments to the question, that noise is defined as the residual variance not related to the dependent variable(s). Without a dependent variable, it's all noise.
