Making Sense of NMMI

The Normalized Mass Moment of Inertia proposed by Li et al (2013) is deceptively hard to express. Given some shape and a distribution of attributes around the shape, they state the NMMI ratio in Fan et al (forthcoming) as:

$$NMMI = \frac{I_O^’} {I^{G_i}_{i_mass}}$$

where $I$ represents the area-standardized mass moment of inertia. The $O$ subscript denotes that the numerator is the mass moment of inertia of some null reference shape. I will discuss the denominator later. Here, that reference shape is defined as an “equiperimeter” circle. And, I say “area standardized” because it is important to note that their explanation of what the mass moment of inertia for a circle may be slightly different from what is expected. For a disk of uniform mass distribution, the typical equation expressing the moment of inertia around the shape centroid is:

$$I_O = \frac{1}{2} Mr^2$$

where $M$ is the total mass of the circle and $r$ is its radius. But, the mass moment of inertia expressed in this literature on the subject expresses it in terms of circle area, $A$:

$$I_O^’ = \frac{MA_c}{2\pi} = \frac{M(\pi r^2}{2\pi} = \frac{1}{2}Mr^2$$

Not a mathematically significant difference, but an atypical expression of the concept nonetheless. Then, given this quantity, they also define the mass moment of inertia of some arbitrary region $r$ with respect to its mass distribution as:

$$I^{Gi}{r} = \frac{A_r \int{da}d^2 \rho^m_i{da_i} da} {\int_{da} \rho^m_i(da_i) da_i}$$

This is also phrased in terms of shape area, with $\rho^m_i(da_i)$ being the function for the density of mass type $m$ over the infinitesimal shape area $da_i$. The $d^2$ term is constructed using the distance from the infinitesimal shape area and the center of mass of the region with respect to the mass being studied, here denoted $Gi$.

Again, this is consistent with stating these functions with respect to shape areas rather than how the formulation is sometimes given, which is in terms of the squared torque on the center of mass, weighted by the density as a function of radii. Both are correct and equivalent, as will be shown.

However, I am interested in discrete applications of the moment of inertia. In fact, most of the times this NMMI measure has been applied have been in discrete-space cases. This is unsurprising, as geographers are very concerned with the compactness and mass distribution of things over lattices, or collections of polygons (states, congressional districts, census tracts), as that gives some indicator for how a shape “contains” its contents. That is, large moments of inertia would indicate that there is a large squared distance from that shapes’ component elements and the shape’s center. In the case of a mass-moment of inertia measure, this shows how the centrality of the shape relates to the distribution of the mass in question (raw population, racial population, votes) is distributed.

To re-express the continuous form expression of $I^{Gi}_{i}$ as a discrete-space expression, we must realize that $\int_da …$ is simply the sum of arguments *over* the discrete spaces being studied. Therefore, replacing the integral formulation with a sum over all of the polygons in the lattice (or sublattice) in question is sufficient to solve convert the integral into a discrete-space summation.

But, what does $\rho_i^{m}(da_i)$ become? It becomes the mass of the shape being studied! As the density of mass type $m$ over shape $i$, $\rho_i^m$ times the area of the shape ($da_i$, in continuous terms) is equal to the mass of that shape. In the continuous formulation, this area is infinitely small. However, in the discrete formulation, we enumerate this, and the density function $\rho^m_i$ for shape $a_i$ becomes exactly $\frac{m_i}{a_i}$. So, replacing the density function times the differential area, or $\rho_i^m(da_i)da_i$ for each shape $i$ becomes its mass!

Thus, we can restate the continuous expression of some region $r$’s moment of inertia as:

$$I^{Gi}_{r} = \frac{A_r \sum_i m_i d^2} {\sum m}$$

for all $i$ in the region $r$. Given this, the discrete-form expression of the NMMI can be stated:

$$NMMI_r = \frac{I_O^’} {I^{Gi}_r} = \frac {\frac{\sum_i m_i A_c} {2\pi} } {\frac{A_r \sum_i(m_i * d^2)} {\sum_i m_i} }$$

Just to be clear, the $d$ term is calculated from the difference between the center of mass of subarea $i$ and the region center of mass $Gi$ for some mass type $m$. This could be denoted $d^{2_m}_{i,r}$, but that would be quite silly. Thus, just remember that the $d$ term is with respect to each sub-area $i$ in shape $r$ for some mass type $m$.

But, for the sake of simplicity, let us manipulate this expression a bit. First, consider rearranging the function in terms of the things that it relates. Namely, I plan on isolating the normalizing constant $k$, which is likely some power of $2\pi$, some purely-geometric indicator derived from the region information ($G(r}$) relating the geometric properties of the shapes in question, and some mass-based indicator, relating the masses in question and their distribution around the shape ($f(m_i, d)$). So, following this logic:

\begin{align*} NMMI_r =&~ \frac{I_O^’} {I^{Gi}_r} = \frac {\frac{\sum_i m_i A_c} {2\pi} } {\frac{A_r \sum_i(m_i * d^2)} {\sum_i m_i} } \ =&~ \frac{\sum_i(m_i)A_c}{2\pi} \cdot \frac{\frac{1}{A_r \cdot \sum_i(m_id_^2}}{\sum_i m_i} \ =&~ \frac{\sum_i(m_i)A_c}{2\pi} \cdot \frac{\sum_m_i}{A_r\sum_i{m_i d^2}}\end{aligned}

Then, we can let the sum of all masses $\sum_i m_i$ be denoted $M$ again. Continuing:

\begin{align*} =&~ \frac{1}{2\pi} \cdot \frac{A_c}{A_r} \cdot \frac{M^2}{\sum_i (m_id^2)}\end{aligned}

is our first real restatement with a cogent representation in the three components mentioned above. Taking the equation from the right end first, we can see that the right-most term is the ratio of total mass to the total mass, weighted by the squared distance of that mass from the center. This term fully captures the effects of the distribution of the masses around the region, and it will not change in subsequent symbolic adjustments. The middle term is the ratio between the equiperimeter circle to the region being studied. Intuitively, this relates the area of a shape to its own perimeter, long a classic measure used in the pure analysis of compactness. The first term is simply a geometric scaling constant.

Further symbolic adjustment can refine the center area ratio to be an explicit function of the region being studied, rather than some relationship between the region and a hypothetical circle. First, though, it is important to realize that the equiperimeter circle’s radius can be expressed as $\frac{p}{2\pi}$, where $p$ is the perimeter of the region under study. This is simple algebraic adjustment of the equation relating the two quantities:

$$P_r = 2 \pi r$$

Therefore, given this property, we can substitute this in for the circle area term $A_c$ in the equation above. Keep in mind that $r$ will be used as the index of the region under study, not as an index over radii.

\begin{align*} =&~ \frac{1}{2\pi} \cdot \frac{\pi\left(\frac{P_r}{2\pi}\right)^2 }{A_r} \cdot \frac{M^2}{\sum_i (m_id^2)} \ =&~ \frac{1}{2\pi} \cdot \frac{\frac{P_r^2}{4\pi}}{A_r} \cdot \frac{M^2}{\sum_i (m_id^2) \ =&~ = \frac{1}{2\pi \cdot 4\pi} \cdot \frac{P_r^2}{A_r} \cdot \frac{M^2}{\sum_i (m_id^2)}\end{align*}

And, after this manipulation, we finish with:

\begin{align} =&~ \frac{1}{8\pi^2} \cdot \frac{P_r^2}{A_r} \cdot \frac{M^2}{\sum_i (m_i d^2}\end{align}

Thus, the $NMMI$ can be restated as only a function of properties expressed by the shape being studied itself, without reference to any constructed equiperimeteric circle. In addition, the fact that these formulations are expressed as areas can be modified if the area of the shape is known and is related to its dimensions, but is irreducible if the areas are not closed-form functions.

But, put simply, this formulation follows the desire to use only functions of the data and separation of the index into shape and mass compactness components, scaled by some factor. In addition, this expression is more efficient for database construction, as only the masses and information directly contained within the geometries of the shapes needs to be queried. Lastly, recall that this formulation exactly divides the NMMI into the three components mentioned above, but these components are not statistically independent! For instance, area is usually proportional to the distance from a region’s centroid and its constituent subareas. Thus, the terms of the denominators are not independent. In addition, the area and perimeter are usually considered related, but are not necessarily strictly so.

When expressed this way, the purpose and the logic of the statistic becomes more clear.

imported from: yetanothergeographer