A metric that isn't translation-invariant is a measurement tool that changes when the position or orientation of an object is altered. This means that the metric is affected by the translation, rotation, or reflection of the object.
Translation-invariance is important in metrics because it allows for consistent and accurate measurements regardless of the position or orientation of an object. This is especially important in scientific research and data analysis.
One example of a metric that isn't translation-invariant is the Euclidean distance in three-dimensional space. This distance measure changes depending on the position and orientation of the objects being measured.
Non-translation-invariance can affect data analysis by introducing bias and inaccuracies in the measurements. This can lead to incorrect conclusions and hinder the understanding of relationships between variables.
While translation-invariance is generally desirable in metrics, there are some cases where a non-translation-invariant metric may be more useful. For example, in image processing, certain features may be better captured using a metric that takes into account the position and orientation of an object.