TABLE I.

Summary of all the non-TDMI based metrics used to assess homogeneity in a population (both among the graphs and the supports) used to verify the TDMI-type analysis.

non-TDMI-based quantities for characterizing a population
$H_{\overline{x}}$	difference between the population and individual element means	∼0 implies either (1) most elements have a similar number of measurements or (2) the individuals come from distributions with similar means; $\gg 0$ implies the converse
V(f(n))	variance of the PDF of the number of measurements per individual	(1) V ∼ 0, $H_{\overline{x}}~0$ imply elements were measured similarly; $\gg 0$, $H_{\overline{x}}~0$ implies elements measured at different rates; $\gg 0$, $H_{\overline{x}}\gg 0$ implies elements measured at different rates with differing source distributions
${\overline{s}}_{\mathit{min}}$	E[smin(i)]	lower support boundary mean
${\overline{s}}_{\mathit{max}}$	E[smax(i)]	upper support boundary mean
$V_{s_{\mathit{min}}}$	Var(smin)	lower support boundary variance
$V_{s_{\mathit{max}}}$	Var(smax)	upper support boundary variance
$\overline{\left|S\right|}$	${\overline{s}}_{\mathit{max}}-{\overline{s}}_{\mathit{min}}$	length of support mean.
$V_{\overline{\left|S\right|}}$	$\mathit{Var}({\overline{s}}_{\mathit{max}}-{\overline{s}}_{\mathit{min}})$	length of support variance
HRA	area between the (point-wise) least and greatest PDF graph	quantifies variance between the PDFs of the population; ∼0 implies element PDFs are homogeneous; very sensitive
VS(p)	${\int }_{S}E[{(p(x))}^2]-E{[p(x)]}^2\mathit{dx}$, variance of the PDFs relative to a specified support, S	∼0 implies homogeneity in PDFs; larger VarS(f) implies greater heterogeneity in the PDFs.
$V_{\hat{S}}(p)$	VS(p) calculated relative to the support of the aggregate population; $\hat{S}={\cup }_{i=1}^N{\hat{S}}_i$; note that there does exist an aggregate normalized support, $\hat{\mathfrak{S}}$, but we will not use this quantity here.	$V_{\hat{S}}(p)$ has the same interpretation as VS(p) in general, but has the potential to include support-based effects.
$V_{\overline{\mathfrak{S}}}(p)$	VS(p) calculated relative to the abstract support of the population, $\overline{\mathfrak{S}}$	$V_{\overline{\mathfrak{S}}}(p)$ has the same interpretation as VS(p) in general, but excludes support-based effects.