# Testing if a vector is in a population of vectors

Suppose that I am concerned with objects that are described by N 2D points. These points all have equal importance to me. I begin with a large number R of such objects, each with its corresponding N-element list of 2D points. Let's call this the population.

I'm now given a new object B, with the N locations of its 2D points. I'd like to know how confident I can be that B belongs to the population.

I'd also be happy with a probability, or any other number with a simple interpretation. I'm willing to assume that everything is uncorrelated, point $i$ in each object is uniformly distributed with respect to point $i$ in the other objects, and other such typical assumptions.

When there's just one variable, Student's t-test seems the right tool. That's about the limit of my statistical depth. I've read about multidimensional t-tests, Hotelling tests, and related topics, but understanding how to use these tools requires substantially more sophistication and depth than I have. Frankly, I can't turn the references I've found (such as the Wikipedia page on "Multivariate t-distribution"), into a practical solution. I'm encouraged that these tools are built into Mathematica (the system with which I'm familiar - I don't know R), and that they seem very close to what I need, but figuring out how to use them for this question seems just out of reach. The example uses I've read involve comparing things like the means of two populations and other comparisons, but I don't see how to phrase my question in terms of those sorts of tests.

Ideally I'd like to spend the time to learn enough statistical theory to understand these tools well enough to answer this question for myself. But at the moment I'm faced with a practical situation for which I need to get an answer.

My goal is a little Mathematica routine that takes two inputs: a matrix of R vectors, each made up of N 2D points, and a second vector of N 2D points, and it returns to me a number expressing the confidence that the vector belongs to the population.

Or, equally useful, some kind of measure that tells me how "far" the new vector is from belonging to the population. If anyone can help me get to that goal in concrete, practical terms I would be very grateful.

• Does your 2-D data follow a certain pattern e.g. linear correlations (fitting more or less on a straight line)? if so, you can calculate the confidence band for the data and find where you need to place your boundaries to exactly go through your new data point. Otherwise, depending on how simple or complex the structure of the cloud of your points is, the solution may be simple or complex. You would need to provide more details (e.g., post a scatter plot of your data). – Lucozade Jul 2 '13 at 15:45
• Even in the one-dimensional case the t-test is not correct: please see stats.stackexchange.com/questions/62634/…. You are asking for a multivariate prediction region. For certain kinds of multivariate distribution, the Mahalanobis distance satisfies your requirements for a distance metric. A related problem (but not quite the same) is to find multivariate outliers. – whuber Jul 2 '13 at 15:56
• Thank you both for your replies. Lucozade, my data represents a simple (but not convex) shape. Think of the data as describing the locations of a few points on a stick figure of a person. I can easily normalize the data, so for example the head will be at (0,0) and the height of the figure is exactly 1. I have around a half-dozen points for each stick figure, so that's about 12 real numbers (x and y for each point). whuber, thanks for the pointer and a new search term! – user2541222 Jul 2 '13 at 16:08