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What is the expected distance between two points when one of the points has some unknown (or missing) coordinate values?

The two points are in the same finite dimensional real space. Assume that the probability density function that describes the missing coordinates varies uniformly between $[-1,1]$.

Here is an Adobe PDF file showing the solution for a point that has either one or two unknown coordinate values. I would appreciate any information leading to a solution for the general case of $m$ missing coordinates, or useful lower and upper bounds for the expected distance between these two points.

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You might want to edit your question as follows: 'point' instead of 'vector'; 'coordinate' instead of 'element'; 'expected distance' instead of 'distance'. ---------- The exact lower and upper bounds are easy to find: just minimise or maximise the distance in each coordinate separately. Or did you mean "useful lower and upper bounds for the expected distance between these two vectors"? –  TonyK May 4 '10 at 11:01
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1 Answer 1

You can rephrase your question as follows: first we subtract the known vector from both and then take care of the known coordinates. So assuming the coordinates of the two points are $(\alpha,\beta)$ and $(\gamma,X)$ where $\alpha,\gamma \in \mathbb{R}^m$ are known, and $\beta \in \mathbb{R}^n$ is known, but $X$ represent the unknown coordinates constrainted to lie inside the cube $[-1,1]^n$, the integral to evaluate becomes $$\frac{1}{2^n}\int_{[-1,1]^n} \sqrt{ |\alpha-\gamma|^2 + |\beta - X|^2 } dX$$ In the lower dimensional case this can be integrated. But an analytical expression in higher dimensions is elusive. For the case $\alpha = \gamma$ and $\beta = 0$, some bounds were obtained in an old paper of Anderssen et al. http://dx.doi.org/10.1137/0130003 For more general probability distributions there is a recent paper with some bounds by Burgstaller and Pillichshammer. http://journals.cambridge.org/action/displayAbstract?aid=6622208

Of course, one can get a fairly trivial bound by Cauchy-Schwartz $$ \int_{[-1,1]^n} f(X) dX \leq 2^{n/2} \left( \int_{[-1,1]^n} f(X)^2 dX \right)^{1/2} $$ and that $$ \int_{[-1,1]^n} R^2 + |\beta - X|^2 dX = 2^n (R^2 + \beta^2) + \int_{[-1,1]^n} X^2 dX$$ the last term is simply evaluated as $n 2^n / 3$, so putting it all together we have the upper bound for the expected value by $$ \frac{1}{2^n}\int_{[-1,1]^n} \sqrt{ |\alpha-\gamma|^2 + |\beta - X|^2 } dX \leq \sqrt{ |\alpha -\gamma|^2 + \beta^2 + \frac{n}{3}}$$ which is slight improvement over the utterly trivial upper/lower bound of $\sqrt{|\alpha-\gamma|^2 + \beta^2 \pm n}$ if you just maximize/minimize each coordinates.

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Oops, looks like my $n$ is your $m$... sorry about that. –  Willie Wong May 6 '10 at 14:21
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