Inertia

The origin of the concept on inertia is to be found in Mechanics : just as mass is the measure of the resistance of a body to changing its translation speed under the influence of a force, the "moment of inertia" is a measure of the resistance of a body to changing its rotation speed around a point O under the influence of a torque.

The moment of inertia M_O of a point P with respect to O is just :

M_O = m.d²

where :

    * m is the mass of the point, and

    * d is the distance between O and P.

If a body is a rigid assembly of discrete points, its moment of inertia with respect to O is the sum of the individual inertias of the points (with respect to O). If the body is rigid but continuous, the same definition applies, with "sum" being replaced by "integral", and "mass" by "density".

This density is not normalized (integral equal to 1) as it is the case for a probability density.

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The main properties of the moment of inertia are :

• Smallest value
  It is minimal when O coincides with G, the center of gravity (or "barycenter") of the body  (see animation   ).
   

• The Inertia Projection theorem 
  Project a 2-dimensional cloud on two orthogonal axes x and y that intersect in O. You then obtain two "linear clouds", with respective moments of inertia with respect to O M_x and M_y. Then :

M_O = M_x + M_y 

This point is illustrated in the interactive animation about the Covariance Matrix (  ) and is a straightforward application of the Pythagorean theorem. It generalizes to p dimensions, p > 2.
 

• Huyghens' theorem

          It goes in two steps :

• First, calculate the Moment of Inertia of the body with respect to its own center of gravity G and call it M_G .
• Denote M the total mass of the body. Then, collapse the body onto G, so that  G is now a point with mass M. It has a moment of inertia with respect to O that we denote M_OG . Then Huyghens' theorem states that :

 M_O = M_G + M_OG 

In words, the moment of inertia of a body with respect to an arbitrary point O is the sum of two terms :

    * The moment of inertia of the body with respect to its own barycenter G, and

    * The moment of inertia with respect to O of a fictitious point G whose mass is that of the body.

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All this sounds more like a course on Mechanics than on Data Modeling. But Data Modeling deals frequently with "clouds of points", where "point" usually means an "observation described by numerical attributes".

These points may have different masses, either because :

    * Several observations may have exactly the same set of attributes,

    * Nearly identical observations have been clumped together into a single, ponderated fictitious observation (clustering).

    * Some observations are more or less heavily "ponderated" to reflect their individual importances.

In Correspondance Analysis, points are modalities of nominal variables, and the "mass" of a modality is the number of observations sharing this modality.

The concept of inertia of a cloud of points ("Moment of" is usually dropped in Data Modeling) appears then quite naturally in many techniques such as :

• ANOVA, that relies on Huyghens' theorem (which is usually called "Decomposition of a Sum of Squares" in Data Modeling).
• So do some important calculations relative to the variance of residuals in Linear Regression.
• Principal Components Analysis (PCA) and Correspondance Analysis both heavily use the Inertia Projection theorem.
• Discriminant Factor Analysis may be interpreted as a PCA on the the set of class barycenters weighted by their respective class populations.

In the context of Data Modeling, the term "mass" is usually replaced by the (improper) term "weight".

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It is important to note that Inertia is not a geometric property of the cloud, as is, for example, its variance in a given direction. This is because :

    * For a given set of points, the value of the inertia depends on the values of the weights.

    * A geometric property (e.g. the position of the center of gravity, or the variance in a given direction) should change only very little when new points are added to the cloud in a way that respects the general repartition of the points in the cloud (see animation on Covariance Matrix  ). For example, the position of the center of gravity is an average calculated on a set of points, and changes little when new points are added in a way that  respects the general repartition of the points.

But Inertia is not an average, it is a cumulative quantity. It always increases when new points are added to the cloud (see animation).

A genuine (although rather crude) description of the purely geometric properties of the cloud is provided by its Covariance Matrix, and more specifically by its Diagonalized Covariance Matrix.

When weights can be ignored (that is, all set to the same value "1"), there is nevertheless a link between Inertia and Covariance Matrix : the Inertia (with respect to the barycenter) is then equal to n times the sum of the diagonal elements of the Covariance Matrix, where n is the number of points in the cloud (for more details, see Covariance Matrix).

But in the case of Correspondence Analysis, weights cannot be tempered with, and the very concept of Covariance Matrix is meaningless.

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