Page 189 - Contributed Paper Session (CPS) - Volume 2

CPS1820 Shuichi S.
feasible region as RIP. For LSD, all MNMs corresponding to the feasible region
become zero. If MNM is over one, it is QP-solver does not work correctly. It
can find a unique solution to minimize the objective function that is the
quadratic function.
Thus, it cannot find SMs [12-14].
MIN = ||b|| /2; yi × ( xib + b0) >= 1; (2)
t
2
IP-OLDF based on MNM criterion found two facts:
1) The relation of NMs and all LDFs (Fact1)
If we set b0=1 and M=0 in (1), the Eq. (3) becomes the constraints of
IP-OLDF.
yi* ( xib + 1) >= 1 ; (3)
t
If we consider linear hyperplane Hi of (4), the hyperplanes divide p-
discriminant coefficients space into a finite convex polyhedron. All interior
points have the unique NMi, and all corresponding LDFs misclassify the same
NMi cases. If LDF corresponds to the vertex or edge of some convex, its NM
may increase. Thus, all NMs of LDFs, except for RIP, may not be correct. RIP
find the interior point of the optimal convex polyhedron having MNM.
t
Hi: yi* ( xib + 1) = 1 (4)
2) MNM monotonic decrease (Fact2)
MNM decreases monotonously (MNMk >= MNM(k+1)). If MNMk =0, all
models including this k- variables are LSD (MNM=0). Swiss banknote data
consists of 100 genuine and 100 counterfeit bills having six variables. Because
two variable model (X4, X6) is MNM=0, 16 models including (X4, X6) are LSD,
and other 47 models are not LSD. Six-variables model is a big Matryoshka that
includes smaller 15 Matryoshkas. We call (X4, X6) is a Basic Gene Set (BGS).
Thus, because microarrays are LSD and big Matryoshka, those have the
Matryoshka structure. Because microarray is a big data, we developed
Method2 by LINGO. LINGO Program3 finds all SMs and Program4 finds all
BGSs.

2.3 Method2 and Cancer Gene Analysis
We explain Method2 by Japanese car data that consist of the 29 regular
cars and 15 small cars. Because the emission rate (X1) and capacity (X3) of
small cars are less than those of regular cars, two MNMs of these one-variable
models are zero, and those are BGSs and SMs. Thus, 48 MNMs including (X1)
or (X3) are zero. Other 15 MNMs are not zero. When LINGO Program3
discriminates this data, we obtain the result in Table 1. “SM (big loop)” column
is the sequential number of SM found by Program3. “IT (small loop)” column
shows the iteration.

178 | I S I W S C 2 0 1 9