Concentrated Matrix Exponential Distributions

Abstract

We revisit earlier attempts for finding matrix exponential (ME) distributions of a given order with low coefficient of variation ($\hbox {cv}$). While there is a long standing conjecture that for the first non-trivial order, which is order 3, the $\hbox {cv}$ cannot be less than 0.200902 but the proof of this conjecture is still missing.

In previous literature ME distributions with low $\hbox {cv}$ are obtained from special subclasses of ME distributions (for odd and even orders), which are conjectured to contain the ME distribution with minimal $\hbox {cv}$. The numerical search for the extreme distribution in the special ME subclasses is easier for odd orders and previously computed for orders up to 15. The numerical treatment of the special subclass of the even orders is much harder and extreme distribution had been found only for order 4.

In this work, we further extend the numerical optimization for subclasses of odd orders (up to order 47), and also for subclasses of even order (up to order 14). We also determine the parameters of the extreme distributions, and compare the properties of the optimal ME distributions for odd and even order.

Finally, based on the numerical results we draw conclusions on both, the behavior of the odd and the even orders.

Appendices

A Various Forms of Matrix Exponential Functions

This section is dedicated to show the equivalence of the various forms of matrix exponential functions throughout the paper. Specifically, we show the equivalence of the forms (2) and (3) and also show how (4) (and also (5)) can be brought to a form consistent with (2) and (3).

As mentioned in Sect. 2, (2) can be converted directly into (3) using the Jordan-decomposition of A.

From a pdf given in the form (3), one can reconstruct a matrix-vector representation in (2) in the following manner: A will be in block-diagonal form, with each block corresponding to either a single real eigenvalue $\lambda _j$ or a pair of complex eigenvalues $\lambda _j,\lambda _{j+1}=a\pm \mathcal{I} b$, where $\mathcal{I} = \sqrt{-1}$.

If $\lambda _j$ is real, then the block in A is

$$\left[ \lambda _j\right] \quad \text {and} \quad \left[ \begin{array}{ccccc} \lambda _j &{} 1 &{} 0 &{} \dots &{} 0\\ 0 &{} \lambda _j &{} 1 &{} \dots &{} 0\\ \vdots &{} &{} \ddots \\ &{} &{} &{} \lambda _j &{} 1\\ 0 &{} &{} \dots &{} 0 &{} \lambda _j \end{array} \right] $$

for multiplicity 1 and $N_j>1$, respectively.

For a complex pair of eigenvalues $\lambda _j,\lambda _{j+1}=a\pm \mathcal{I} b$, the block is

$$\left[ \begin{array}{cc} a &{} b\\ -b &{} a \end{array} \right] \quad \text {or} \quad \left[ \begin{array}{ccccccccc} a &{} b &{} 1 &{} 0 &{} &{} &{} &{} &{} 0 \\ -b &{} a &{} 0 &{} 1 &{} 0 &{} &{} \dots &{} &{} 0\\ 0 &{} 0 &{} a &{} b &{} 1 &{} 0 &{} &{} &{} 0\\ 0 &{} 0 &{} -b &{} a &{} 0 &{} 1 &{} 0 &{} &{} 0\\ 0 \\ \vdots &{} &{} &{} &{} &{} \ddots &{} &{} &{} \vdots \\ \\ &{} &{} &{} &{} &{} &{} &{} a &{} b\\ 0 &{} &{} &{} \dots &{} &{} &{} &{} -b &{} a \end{array}\right] $$

for multiplicity 1 or $N_j>1$ respectively (the matrix on the right is size $2N_j\times 2N_j$). Once A is constructed, $\alpha$ can be obtained by solving a system of linear equations.

Finally, (4) can be represented in a form consistent with (3); the eigenvalues are $-1,(-1\pm 2\mathcal{I}\omega ),\dots ,\left( -1\pm (n-1)\mathcal{I}\omega \right)$. We demonstrate this for $n=5$; for higher odd values of n, it follows a similar structure, albeit with more terms:

$$\begin{aligned} f(t) =&\mathrm {e}^{-t} \cos ^2(\omega t -\phi _1)\cos ^2(\omega t -\phi _2)\\ =&\frac{1}{8}\mathrm {e}^{-t} \big (2+\cos (2\phi _1-2\phi _2)\big ) +\frac{1}{8}\mathrm {e}^{-t} \big (\cos (2\phi _1)\cos (2t\omega )\\&+\sin (2\phi _1)\sin (2t\omega )+ \cos (2\phi _2)\cos (2t\omega )+\sin (2\phi _2)\sin (2t\omega )\big )\\&+\frac{1}{8}\mathrm {e}^{-t}\big (\cos (2\phi _1+2\phi _2)\cos (4t\omega ) +\sin (2\phi _1+2\phi _2)\sin (4t\omega ) \big )\\ =&\frac{1}{8} \mathrm {e}^{-t}(2+\cos (2\phi _1-2\phi _2))+\\&+\frac{1}{8} (\mathrm {e}^{2i\phi _1}+\mathrm {e}^{2i\phi _2})e^{(-1-2i\omega )t}+ \frac{1}{8} (\mathrm {e}^{-2i\phi _1}-\mathrm {e}^{2i\phi _2})e^{(-1+2i\omega )t}+\\&\frac{1}{16}\mathrm {e}^{2i(\phi _1+\phi _2)} \mathrm {e}^{2t(-1+4i\omega )}+ \frac{1}{16}\mathrm {e}^{-2i(\phi _1+\phi _2)} \mathrm {e}^{2t(-1-4i\omega )}. \end{aligned}$$

Representing (5) in a form consistent with (3) is essentially the same, just with one extra real eigenvalue.

B Optimal Parameter Values

Table 4. Optimal parameter values for odd values $19\le n\le 47$