This example clearly establishes our previous claim that the information dimension provides a useful upper bound for the true signal dimension (whereas other traditional methods such as Velicer's underestimate the true dimension).
In the sequel, we investigate another derivation of an upper bound regarding a bounded measurement error.
Given an arbitrary cut (L, R), we propose that r will be the least upper bound of L. The Least Upper Bound Axiom of the usual theory of the reals tells us that L has a least upper bound (L is nonempty and any element of R (which is also nonempty) is an upper bound of L, so L has a least upper bound).
SVD serves as a useful upper bound on the dimensionality of data for non-linear systems, but the difference between SVD dimensionality and true dimensionality may grow to be unacceptable for large systems.
To account for the uncertainty in fossil calibrations and the likelihood of the true age of a node being older than the first fossil record, the age constraint was set with a hard upper bound of 228 Mya and soft lower bound of 239.4 Mya.
One feasible expectation of the magnitude of ρ 1 is that ρ 1 in Eq. (5) would be larger than that in Eq. (3) if there is a positive correlation between fathers' socio-demographic variables and sons' economic status variable; Björklund and Jäntti (1997) and Ueda (2013) used it as an upper bound on the true estimates.
For technical reasons, in the bootstrap and individual subjects overlap analyses there was no extent threshold applied, whereas in the group analyses used an extent threshold of k>10; as a result the bootstrap analysis was more likely to find overlapping voxels, and therefore provides an upper bound on the true percentage overlap.
This likely represents an upper bound on the true frequency within the population, since inclusion in our study depends on coming to medical attention.
Note that, since the Gaussian distribution extends below γ i = 0, the p-value given by the Gaussian distribution is in fact an upper bound on the true p-value for small (i.e. the most significant) γ i, which means that the true p-value lies even lower.
We now discuss a useful upper bound.
