Linear Algebra Theory of Evolution

 

Linear Algebra Theory of Evolution

1. Genetic State Representation

• Represent the genetic state of an organism as a vector in a high-dimensional space. Each dimension corresponds to a specific gene or trait.

$$\mathbf{g} = \begin{pmatrix} g_1 \\ g_2 \\ \vdots \\ g_n \end{pmatrix}$$

Where $g_i$ represents the value or expression level of the $i$-th gene.

2. Population State

• Represent the state of a population as a matrix, where each column represents the genetic state of an individual.

$$\mathbf{P} = \begin{pmatrix} \mathbf{g}_1 & \mathbf{g}_2 & \cdots & \mathbf{g}_m \end{pmatrix}$$

Where $\mathbf{g}_j$ is the genetic state vector of the $j$-th individual in a population of size $m$.

3. Selection Matrix

• Define a selection matrix $\mathbf{S}$ that models natural selection. This matrix modifies the population state based on fitness, favoring certain genetic states over others.

$$\mathbf{P}' = \mathbf{S} \mathbf{P}$$

Where $\mathbf{P}'$ is the new population state after selection.

4. Mutation Matrix

• Define a mutation matrix $\mathbf{M}$ that models random genetic mutations. This matrix adds variability to the genetic states.

$$\mathbf{P}'' = \mathbf{M} \mathbf{P}'$$

Where $\mathbf{P}''$ is the population state after mutation.

5. Reproduction and Genetic Crossover

• Model reproduction and genetic crossover using matrix operations. For instance, a crossover matrix $\mathbf{C}$ could combine genetic material from two parent vectors to produce offspring.

$$\mathbf{o} = \mathbf{C} \begin{pmatrix} \mathbf{g}_{\text{parent1}} \\ \mathbf{g}_{\text{parent2}} \end{pmatrix}$$

Where $\mathbf{o}$ is the genetic state vector of the offspring.

6. Iterative Process

• The evolutionary process can be modeled as an iterative application of selection, mutation, and reproduction matrices over multiple generations.

$$\mathbf{P}_{t+1} = \mathbf{M} \mathbf{S} \mathbf{C} (\mathbf{P}_t)$$

Where $\mathbf{P}_{t+1}$ is the population state at generation $t+1$, and $\mathbf{P}_t$ is the population state at generation $t$.

7. Fitness Landscape

• The fitness landscape can be represented as a function $f(\mathbf{g})$ that maps genetic states to fitness values. This function guides the construction of the selection matrix $\mathbf{S}$.

$$f: \mathbb{R}^n \rightarrow \mathbb{R}$$

Where $f(\mathbf{g})$ gives the fitness of the genetic state $\mathbf{g}$.

Application of the Theory

1. Simulation of Evolutionary Dynamics

   • Use the matrices and vectors to simulate the evolution of a population over time, observing how genetic states change and adapt to their environment.

2. Analyzing Genetic Diversity

   • Analyze the diversity within the population by examining the eigenvalues and eigenvectors of the population state matrix $\mathbf{P}$.

3. Predicting Evolutionary Outcomes

   • Predict potential evolutionary outcomes by studying the stability and convergence properties of the iterative process.

Example Scenario

• Population size: $m = 3$
• Number of genes: $n = 2$
• Selection pressures: favor higher values of the first gene

1. Genetic State Representation

Each individual’s genetic state is represented by a vector in $\mathbb{R}^2$.

$$\mathbf{g}_1 = \begin{pmatrix} 2 \\ 3 \end{pmatrix}, \quad \mathbf{g}_2 = \begin{pmatrix} 4 \\ 1 \end{pmatrix}, \quad \mathbf{g}_3 = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$$

2. Population State Matrix

The population state matrix $\mathbf{P}$ includes all individuals’ genetic states as columns.

$$\mathbf{P} = \begin{pmatrix} 2 & 4 & 3 \\ 3 & 1 & 2 \end{pmatrix}$$

3. Selection Matrix

The selection matrix $\mathbf{S}$ models selection pressures. Suppose higher values of the first gene are favored, and individuals with low values have reduced fitness.

$$\mathbf{S} = \begin{pmatrix} 1.2 & 0 & 0 \\ 0 & 0.8 & 0 \\ 0 & 0 & 1.0 \end{pmatrix}$$

4. Mutation Matrix

The mutation matrix $\mathbf{M}$ introduces small random changes to the genes. Assume a small mutation rate.

$$\mathbf{M} = \begin{pmatrix} 0.99 & 0.01 \\ 0.01 & 0.99 \end{pmatrix}$$

5. Reproduction and Genetic Crossover

To model reproduction, we can create a crossover matrix $\mathbf{C}$. Assume a simple average crossover for two parents.

$$\mathbf{C} = \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{pmatrix}$$

6. Iterative Evolution Process

Let's illustrate one iteration of the evolutionary process:

1. Initial Population State:

$$\mathbf{P} = \begin{pmatrix} 2 & 4 & 3 \\ 3 & 1 & 2 \end{pmatrix}$$

2. Apply Selection:

$$\mathbf{P}' = \mathbf{S} \mathbf{P} = \begin{pmatrix} 1.2 & 0 & 0 \\ 0 & 0.8 & 0 \\ 0 & 0 & 1.0 \end{pmatrix} \begin{pmatrix} 2 & 4 & 3 \\ 3 & 1 & 2 \end{pmatrix} = \begin{pmatrix} 2.4 & 0 & 0 \\ 2.4 & 0.8 & 0 \end{pmatrix}$$

3. Apply Mutation:

$$\mathbf{P}'' = \mathbf{M} \mathbf{P}' = \begin{pmatrix} 0.99 & 0.01 \\ 0.01 & 0.99 \end{pmatrix} \begin{pmatrix} 2.4 & 0 & 0 \\ 2.4 & 0.8 & 0 \end{pmatrix} = \begin{pmatrix} 2.376 & 0.008 & 0 \\ 2.376 & 0.792 & 0 \end{pmatrix}$$

4. Reproduction and Crossover (for one pair of parents):

$$\mathbf{g}_{\text{offspring}} = \mathbf{C} \begin{pmatrix} \mathbf{g}_1 \\ \mathbf{g}_2 \end{pmatrix} = \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{pmatrix} \begin{pmatrix} 2 \\ 3 \\ 4 \\ 1 \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$$

7. Iterative Process

Repeat the process over multiple generations to observe the changes in the population state.

Extended Example Scenario

• Population size: $m = 3$
• Number of genes: $n = 2$
• Selection pressures: favor higher values of the first gene

Initial Genetic States

Each individual’s genetic state is represented by a vector in $\mathbb{R}^2$.

$$\mathbf{g}_1 = \begin{pmatrix} 2 \\ 3 \end{pmatrix}, \quad \mathbf{g}_2 = \begin{pmatrix} 4 \\ 1 \end{pmatrix}, \quad \mathbf{g}_3 = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$$

Population State Matrix

The population state matrix $\mathbf{P}$ includes all individuals’ genetic states as columns.

$$\mathbf{P} = \begin{pmatrix} 2 & 4 & 3 \\ 3 & 1 & 2 \end{pmatrix}$$

Selection Matrix

The selection matrix $\mathbf{S}$ models selection pressures. Suppose higher values of the first gene are favored, and individuals with low values have reduced fitness.

$$\mathbf{S} = \begin{pmatrix} 1.2 & 0 & 0 \\ 0 & 0.8 & 0 \\ 0 & 0 & 1.0 \end{pmatrix}$$

Mutation Matrix

The mutation matrix $\mathbf{M}$ introduces small random changes to the genes. Assume a small mutation rate.

$$\mathbf{M} = \begin{pmatrix} 0.99 & 0.01 \\ 0.01 & 0.99 \end{pmatrix}$$

Reproduction and Genetic Crossover

To model reproduction, we can create a crossover matrix $\mathbf{C}$. Assume a simple average crossover for two parents.

$$\mathbf{C} = \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{pmatrix}$$

Iterative Evolution Process

Initial Population State:

$$\mathbf{P} = \begin{pmatrix} 2 & 4 & 3 \\ 3 & 1 & 2 \end{pmatrix}$$

Apply Selection:

$$\mathbf{P}' = \mathbf{S} \mathbf{P} = \begin{pmatrix} 1.2 & 0 & 0 \\ 0 & 0.8 & 0 \\ 0 & 0 & 1.0 \end{pmatrix} \begin{pmatrix} 2 & 4 & 3 \\ 3 & 1 & 2 \end{pmatrix} = \begin{pmatrix} 2.4 & 0 & 0 \\ 2.4 & 0.8 & 0 \end{pmatrix}$$

Apply Mutation:

$$\mathbf{P}'' = \mathbf{M} \mathbf{P}' = \begin{pmatrix} 0.99 & 0.01 \\ 0.01 & 0.99 \end{pmatrix} \begin{pmatrix} 2.4 & 0 & 0 \\ 2.4 & 0.8 & 0 \end{pmatrix} = \begin{pmatrix} 2.376 & 0.008 & 0 \\ 2.376 & 0.792 & 0 \end{pmatrix}$$

Reproduction and Crossover (for one pair of parents):

$$\mathbf{g}_{\text{offspring}} = \mathbf{C} \begin{pmatrix} \mathbf{g}_1 \\ \mathbf{g}_2 \end{pmatrix} = \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{pmatrix} \begin{pmatrix} 2 \\ 3 \\ 4 \\ 1 \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$$

Fitness Landscape

Define a fitness function $f(\mathbf{g})$ that evaluates the fitness of a genetic state.

$$f(\mathbf{g}) = 2g_1 + g_2$$

Evaluate the fitness of each individual in the population:

$$f(\mathbf{g}_1) = 2(2) + 3 = 7, \quad f(\mathbf{g}_2) = 2(4) + 1 = 9, \quad f(\mathbf{g}_3) = 2(3) + 2 = 8$$

Multiple Generations

Let's iterate this process over three generations.

Generation 1:

1. Initial Population:

   $$\mathbf{P}_1 = \begin{pmatrix} 2 & 4 & 3 \\ 3 & 1 & 2 \end{pmatrix}$$

2. Selection:

   $$\mathbf{P}_1' = \mathbf{S} \mathbf{P}_1 = \begin{pmatrix} 2.4 & 0 & 0 \\ 2.4 & 0.8 & 0 \end{pmatrix}$$

3. Mutation:

   $$\mathbf{P}_1'' = \mathbf{M} \mathbf{P}_1' = \begin{pmatrix} 2.376 & 0.008 & 0 \\ 2.376 & 0.792 & 0 \end{pmatrix}$$

Generation 2:

1. Initial Population:

   $$\mathbf{P}_2 = \mathbf{P}_1''$$

2. Selection:

   $$\mathbf{P}_2' = \mathbf{S} \mathbf{P}_2 = \begin{pmatrix} 2.8512 & 0 & 0 \\ 2.8512 & 0.6336 & 0 \end{pmatrix}$$

3. Mutation:

   $$\mathbf{P}_2'' = \mathbf{M} \mathbf{P}_2' = \begin{pmatrix} 2.822688 & 0.006336 & 0 \\ 2.822688 & 0.633264 & 0 \end{pmatrix}$$

Generation 3:

1. Initial Population:

   $$\mathbf{P}_3 = \mathbf{P}_2''$$

2. Selection:

   $$\mathbf{P}_3' = \mathbf{S} \mathbf{P}_3 = \begin{pmatrix} 3.3872256 & 0 & 0 \\ 3.3872256 & 0.5066112 & 0 \end{pmatrix}$$

3. Mutation:

   $$\mathbf{P}_3'' = \mathbf{M} \mathbf{P}_3' = \begin{pmatrix} 3.3533533 & 0.005066 & 0 \\ 3.3533533 & 0.5065545 & 0 \end{pmatrix}$$

Extended Linear Algebra Theory of Evolution

Elements to Add:

1. Genetic Drift: Random changes in allele frequencies due to chance.
2. Migration: Movement of individuals between populations.
3. Complex Crossover: More realistic genetic recombination methods.
4. Population Diversity Analysis: Measure genetic diversity within the population over time.

1. Genetic Drift

Genetic drift can be modeled by adding a random perturbation matrix to the population state matrix.

$$\mathbf{D} = \begin{pmatrix} \delta_{11} & \delta_{12} & \delta_{13} \\ \delta_{21} & \delta_{22} & \delta_{23} \end{pmatrix}$$

Where $\delta_{ij}$ are small random values representing random genetic changes.

2. Migration

Migration can be modeled by adding a migration matrix $\mathbf{R}$ that represents the exchange of individuals between populations.

$$\mathbf{R} = \begin{pmatrix} r_{11} & r_{12} & r_{13} \\ r_{21} & r_{22} & r_{23} \end{pmatrix}$$

3. Complex Crossover

We can define a more complex crossover operation by using matrices that represent different crossover strategies.

$$\mathbf{C}_1 = \begin{pmatrix} 0.7 & 0.3 \\ 0.3 & 0.7 \end{pmatrix}, \quad \mathbf{C}_2 = \begin{pmatrix} 0.6 & 0.4 \\ 0.4 & 0.6 \end{pmatrix}$$

4. Population Diversity Analysis

We measure genetic diversity using the variance-covariance matrix of the population state matrix $\mathbf{P}$.

$$\mathbf{\Sigma} = \frac{1}{m} \mathbf{P} \mathbf{P}^\top - \mathbf{\mu} \mathbf{\mu}^\top$$

Where $\mathbf{\mu}$ is the mean genetic state vector.

Example with New Elements

Initial Population State:

$$\mathbf{P} = \begin{pmatrix} 2 & 4 & 3 \\ 3 & 1 & 2 \end{pmatrix}$$

Selection Matrix:

$$\mathbf{S} = \begin{pmatrix} 1.2 & 0 & 0 \\ 0 & 0.8 & 0 \\ 0 & 0 & 1.0 \end{pmatrix}$$

Mutation Matrix:

$$\mathbf{M} = \begin{pmatrix} 0.99 & 0.01 \\ 0.01 & 0.99 \end{pmatrix}$$

Genetic Drift Matrix:

$$\mathbf{D} = \begin{pmatrix} 0.01 & -0.02 & 0.01 \\ -0.01 & 0.02 & -0.01 \end{pmatrix}$$

Migration Matrix:

$$\mathbf{R} = \begin{pmatrix} 0.95 & 0.05 & 0 \\ 0.05 & 0.90 & 0.05 \end{pmatrix}$$

Complex Crossover Matrices:

$$\mathbf{C}_1 = \begin{pmatrix} 0.7 & 0.3 \\ 0.3 & 0.7 \end{pmatrix}, \quad \mathbf{C}_2 = \begin{pmatrix} 0.6 & 0.4 \\ 0.4 & 0.6 \end{pmatrix}$$

Iterative Evolution Process with New Elements

Initial Population State:

$$\mathbf{P} = \begin{pmatrix} 2 & 4 & 3 \\ 3 & 1 & 2 \end{pmatrix}$$

Apply Selection:

$$\mathbf{P}' = \mathbf{S} \mathbf{P} = \begin{pmatrix} 2.4 & 0 & 0 \\ 2.4 & 0.8 & 0 \end{pmatrix}$$

Apply Mutation:

$$\mathbf{P}'' = \mathbf{M} \mathbf{P}' = \begin{pmatrix} 2.376 & 0.008 & 0 \\ 2.376 & 0.792 & 0 \end{pmatrix}$$

Apply Genetic Drift:

$$\mathbf{P}_{\text{drift}} = \mathbf{P}'' + \mathbf{D} = \begin{pmatrix} 2.386 & -0.012 & 0.01 \\ 2.366 & 0.812 & -0.01 \end{pmatrix}$$

Apply Migration:

$$\mathbf{P}_{\text{migr}} = \mathbf{R} \mathbf{P}_{\text{drift}} = \begin{pmatrix} 2.2727 & -0.0174 & 0.001 \\ 2.2707 & 0.7307 & 0.005 \end{pmatrix}$$

Reproduction and Complex Crossover (for one pair of parents):

$$\mathbf{g}_{\text{offspring1}} = \mathbf{C}_1 \begin{pmatrix} 2 \\ 3 \\ 4 \\ 1 \end{pmatrix} = \begin{pmatrix} 2.4 \\ 1.9 \end{pmatrix}$$ $$\mathbf{g}_{\text{offspring2}} = \mathbf{C}_2 \begin{pmatrix} 3 \\ 2 \\ 4 \\ 1 \end{pmatrix} = \begin{pmatrix} 3.4 \\ 2.4 \end{pmatrix}$$

Diversity Analysis

Calculate Mean Genetic State:

$$\mathbf{\mu} = \frac{1}{3} \sum_{i=1}^3 \mathbf{g}_i = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$$

Calculate Variance-Covariance Matrix:

$$\mathbf{\Sigma} = \frac{1}{3} \begin{pmatrix} 2 & 4 & 3 \\ 3 & 1 & 2 \end{pmatrix} \begin{pmatrix} 2 & 3 \\ 4 & 1 \\ 3 & 2 \end{pmatrix} - \mathbf{\mu} \mathbf{\mu}^\top = \begin{pmatrix} 2.67 & -0.33 \\ -0.33 & 0.67 \end{pmatrix}$$

Multiple Generations

Generation 1:

1. Initial Population:

   $$\mathbf{P}_1 = \begin{pmatrix} 2 & 4 & 3 \\ 3 & 1 & 2 \end{pmatrix}$$

2. Selection:

   $$\mathbf{P}_1' = \mathbf{S} \mathbf{P}_1 = \begin{pmatrix} 2.4 & 0 & 0 \\ 2.4 & 0.8 & 0 \end{pmatrix}$$

3. Mutation:

   $$\mathbf{P}_1'' = \mathbf{M} \mathbf{P}_1' = \begin{pmatrix} 2.376 & 0.008 & 0 \\ 2.376 & 0.792 & 0 \end{pmatrix}$$

4. Genetic Drift:

   $$\mathbf{P}_{1, \text{drift}} = \mathbf{P}_1'' + \mathbf{D} = \begin{pmatrix} 2.386 & -0.012 & 0.01 \\ 2.366 & 0.812 & -0.01 \end{pmatrix}$$

5. Migration:

   $$\mathbf{P}_{1, \text{migr}} = \mathbf{R} \mathbf{P}_{1, \text{drift}} = \begin{pmatrix} 2.2727 & -0.0174 & 0.001 \\ 2.2707 & 0.7307 & 0.005 \end{pmatrix}$$

6. Reproduction and Crossover:

   $$\mathbf{g}_{\text{offspring1}} = \mathbf{C}_1 \begin{pmatrix} 2 \\ 3 \\ 4 \\ 1 \end{pmatrix} = \begin{pmatrix} 2.4 \\ 1.9 \end{pmatrix}$$ $$\mathbf{g}_{\text{offspring2}} = \mathbf{C}_2 \begin{pmatrix} 3 \\ 2 \\ 4 \\ 1 \end{pmatrix} = \begin{pmatrix} 3.4 \\ 2.4 \end{pmatrix}$$

Diversity Analysis

Calculate Mean Genetic State:

$$\mathbf{\mu} = \frac{1}{3} \sum_{i=1}^3 \mathbf{g}_i = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$$

Calculate Variance-Covariance Matrix:

$$\mathbf{\Sigma} = \frac{1}{3} \begin{pmatrix} 2 & 4 & 3 \\ 3 & 1 & 2 \end{pmatrix} \begin{pmatrix} 2 & 3 \\ 4 & 1 \\ 3 & 2 \end{pmatrix} - \mathbf{\mu} \mathbf{\mu}^\top = \begin{pmatrix} 2.67 & -0.33 \\ -0.33 & 0.67 \end{pmatrix}$$

Summary

This extended example demonstrates how genetic states evolve over multiple generations through the iterative application of selection, mutation, genetic drift, migration, and reproduction matrices. Each step in the process refines the population state, driving the genetic makeup of the population toward higher fitness as defined by the fitness landscape. By incorporating additional elements like genetic drift and migration, the model becomes more realistic and better captures the complexities of natural evolution.

General Equations

1. Genetic State Representation

Each individual's genetic state is represented as a vector in $\mathbb{R}^n$.

$$\mathbf{g}_i = \begin{pmatrix} g_{i1} \\ g_{i2} \\ \vdots \\ g_{in} \end{pmatrix} \quad \text{for } i = 1, 2, \ldots, m$$

2. Population State Matrix

The population state matrix $\mathbf{P}$ contains all individuals' genetic states as columns.

$$\mathbf{P} = \begin{pmatrix} g_{11} & g_{12} & \cdots & g_{1m} \\ g_{21} & g_{22} & \cdots & g_{2m} \\ \vdots & \vdots & \ddots & \vdots \\ g_{n1} & g_{n2} & \cdots & g_{nm} \end{pmatrix}$$

3. Selection Matrix

The selection matrix $\mathbf{S}$ models the selection pressures on each genetic state.

$$\mathbf{P}' = \mathbf{S} \mathbf{P}$$

Where $\mathbf{S}$ is a diagonal matrix:

$$\mathbf{S} = \begin{pmatrix} s_{11} & 0 & \cdots & 0 \\ 0 & s_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & s_{nn} \end{pmatrix}$$

4. Mutation Matrix

The mutation matrix $\mathbf{M}$ introduces random changes to the genetic states.

$$\mathbf{P}'' = \mathbf{M} \mathbf{P}'$$

Where $\mathbf{M}$ is a matrix that models mutation rates:

$$\mathbf{M} = \begin{pmatrix} m_{11} & m_{12} & \cdots & m_{1n} \\ m_{21} & m_{22} & \cdots & m_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ m_{n1} & m_{n2} & \cdots & m_{nn} \end{pmatrix}$$

5. Genetic Drift

Genetic drift is represented by adding a random perturbation matrix $\mathbf{D}$.

$$\mathbf{P}_{\text{drift}} = \mathbf{P}'' + \mathbf{D}$$

Where $\mathbf{D}$ is a matrix of small random values:

$$\mathbf{D} = \begin{pmatrix} \delta_{11} & \delta_{12} & \cdots & \delta_{1m} \\ \delta_{21} & \delta_{22} & \cdots & \delta_{2m} \\ \vdots & \vdots & \ddots & \vdots \\ \delta_{n1} & \delta_{n2} & \cdots & \delta_{nm} \end{pmatrix}$$

6. Migration

Migration is modeled by applying a migration matrix $\mathbf{R}$ that represents the movement of individuals between populations.

$$\mathbf{P}_{\text{migr}} = \mathbf{R} \mathbf{P}_{\text{drift}}$$

Where $\mathbf{R}$ is a migration matrix:

$$\mathbf{R} = \begin{pmatrix} r_{11} & r_{12} & \cdots & r_{1m} \\ r_{21} & r_{22} & \cdots & r_{2m} \\ \vdots & \vdots & \ddots & \vdots \\ r_{m1} & r_{m2} & \cdots & r_{mm} \end{pmatrix}$$

7. Reproduction and Crossover

Reproduction and crossover are modeled using crossover matrices $\mathbf{C}$.

$$\mathbf{g}_{\text{offspring}} = \mathbf{C} \begin{pmatrix} \mathbf{g}_{\text{parent1}} \\ \mathbf{g}_{\text{parent2}} \end{pmatrix}$$

Where $\mathbf{C}$ is a crossover matrix:

$$\mathbf{C} = \begin{pmatrix} c_{11} & c_{12} & \cdots & c_{1n} \\ c_{21} & c_{22} & \cdots & c_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ c_{n1} & c_{n2} & \cdots & c_{nn} \end{pmatrix}$$

Iterative Process

The evolutionary process over multiple generations can be represented as:

$$\mathbf{P}_{t+1} = \mathbf{R} (\mathbf{P}_{\text{drift}} + \mathbf{D}) = \mathbf{R} (\mathbf{M} \mathbf{S} \mathbf{P}_t + \mathbf{D})$$

This equation can be iterated over multiple generations to model the evolutionary dynamics.

Population Diversity Analysis

To measure genetic diversity within the population, we calculate the variance-covariance matrix $\mathbf{\Sigma}$.

Calculate Mean Genetic State Vector:

$$\mathbf{\mu} = \frac{1}{m} \sum_{i=1}^m \mathbf{g}_i$$

Calculate Variance-Covariance Matrix:

$$\mathbf{\Sigma} = \frac{1}{m} \mathbf{P} \mathbf{P}^\top - \mathbf{\mu} \mathbf{\mu}^\top$$