How to Estimate the Support of a High-Dimensional Distribution

The support of a probability distribution is the set of values the random variable corresponding to the probability distribution can take on. For instance, a uniform distribution U(0,1) has the support of all real numbers between 0 and 1. In many real-world situations dealing with high-dimensional distributions, it is not possible to perform the complicated mathematical analysis needed to find the true support of the distribution. In such a case, it is preferable to estimate the distribution's support through the plotting of data from a study or simulation.

Instructions

    • 1

      Marginalize the distribution for each dimension. Choose a single dimension in the probability distribution. Integrate over all other dimensions in the distribution, leaving a function of a single dimension. Do this for all dimensions in the probability distribution. You will be left with a set of single dimension functions, one for each dimension in the distribution.

    • 2

      Estimate the end points of every single dimension function. Plot the data for each function. For each function, find where the plot seems to begin (i.e., the first point from the left in which the function does not evaluate to zero) and ends (the farthest point on the right in which the function is not zero). Write the ranges for each function as [a, b], where "a" is the value of the variable's left end point and "b" is the value of the variable's right end point.

    • 3

      Observe the functions, looking for gaps. Sometimes a probability distribution will have a gap or series of gaps. Pour over the function plots, looking for gaps, which are places between the end points in which the plot stays at zero for an extended range. For each single dimension plot, write down these gaps in the same notation as you did the end points (e.g., [c, d]).

    • 4

      Remove the gaps from the ranges found earlier. For each variable's single-dimension function, remove the gaps from the range by rewriting the intervals. For example, if the dimension "x" originally had range [2, 4] and you found a gap at [3, 3.5], rewrite the range in a way that excludes the gap, namely as [2, 3]U[3.5, 5], where "U" represents the union function, which combines intervals.

    • 5

      Write the set of gap-excluded ranges in mathematical form, as a support. Basically, you need to specify what gap-excluded range corresponds to a given variable. Mathematically, you can write this as (using the example above) "x is an element of the set [2, 3]U[3.5, 5]." The full statement that includes every dimension is the support of the original probability distribution.

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