<BODY bgcolor="#aaeebb"> <H3><CENTER>Universal Alignment Probability Surface<BR> P(|G&#171;S|&gt;k; sigma<sup>2</sup>, g, s, N)</CENTER> </H3> <UL> <LI><i>sigma</i><sup>2</sup>: large , medium, small (range of noise/range of performance)<BR> <BR> <LI><i>N</i>: size of sample <BR> <BR> <LI><i>g</i>: size of <B>G</B>, <i>s</i>: size of <B>S</B> (usually we let <i>s</i>=<i>g</i>)<BR> <BR> <LI><i>k</i>: size of overlap of <B>G</B>&#171;<B>S</B> </UL> <BR> <BR> <tt> <A HREF="Slide03.html" target="_parent">To Next Slide</A> / <A HREF="Slide01.html" target="_parent">To Previous Slide</A> / <A HREF="../OOTOC.html" target="_parent">Table of Content</A> </tt> <BR> <hr> <hr> <P> Let us give a simple example. Suppose we use the crudest selection method, namely we blindly pick |<B>S</B>| members of the <i>N</i> samples and ask the probability that among these |<B>S</B>| selected designs there are at least <i>k</i> designs that will rank in the top-<i>g</i> of the performance order (recall the engineering design example in the earlier transparency). Blind Pick (BP) is equivalent to a selection method where the error variance of selection is infinity. It turns out this probability can be calculated in closed form. There are <sub>g</sub>C<sub>i</sub> ways of picking <i>i</i> elements out of the top-<i>g</i> designs and <sub>N-g</sub>C<sub>g-i</sub> ways picking the remaining (<i>g</i>-<i>i</i>) designs out the <i>N</i>-</i>g</i> samples; this is divided by the total number of ways of selecting <i>g</i> designs out of <i>N</i> samples <sub>N</sub>C<sub>g</sub> to arrive at the alignment probability. The curves shown are for <i>N</i> = 1000. As we can see for <i>k</i>=1, <i>P</i> ~ 1 for size of |<B>G</B>|=<i>g</i>=50, i.e., without any knowledge we can effect a 20:1 reduction and still be sure that at least one of the top-50 design is in the blindly picked subset.<BR> </BODY> </HTML>