Extraction

Mathematical Framework for Cognitive Signature Extraction

This section provides the computational and mathematical underpinnings of the cognitive signature extraction framework, focusing on feature extraction, latent space representation, and cognitive trait modeling.

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1. Linguistic Feature Extraction

The first stage involves transforming linguistic data into high-dimensional numerical representations. Let X = {x₁, x₂, ..., xₙ} represent a sequence of tokens derived from the linguistic data, where each token xᵢ is embedded into a vector space ℝᵈ using a pre-trained language model such as BERT or GPT:

scssCopy codehᵢ = f_transformer(xᵢ), hᵢ ∈ ℝᵈ

Key linguistic features are then extracted, which include:

1. Syntactic Complexity The syntactic complexity is measured by analyzing the depth of dependency trees for each sentence. The average tree depth D(X) for a given set of tokens X is:

   scssCopy codeD(X) = (1/n) Σᵢ=₁ⁿ Depth(Tree(xᵢ))

2. Semantic Creativity Creativity is assessed through the density of metaphorical expressions. These are identified based on semantic similarity, with M(X) representing the number of creative elements in the linguistic output. The number of metaphorical expressions M(X) is quantified by the deviation in similarity:

   scssCopy codeM(X) = (1/n) Σᵢ=₁ⁿ 1(Sim(hᵢ, hⱼ) < ε),  i ≠ j

   where 1 is the indicator function, and ε is a threshold that helps identify novel expressions (based on semantic similarity).

3. Temporal Variability Temporal variability captures how language evolves over time. This is evaluated by measuring cosine similarity between embeddings for consecutive tokens Xₜ and Xₜ₊₁:

   scssCopy codeVariability(X) = 1 - CosSim(hₜ, hₜ₊₁)

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2. Latent Space Representation

The extracted linguistic features hᵢ are mapped into a latent cognitive space Z through dimensionality reduction methods. Below are two common approaches:

1. Variational Autoencoder (VAE) A VAE maps high-dimensional linguistic embeddings H ∈ ℝⁿˣᵈ to a latent space Z ∈ ℝᵏ. The encoder computes the posterior distribution over the latent variables z, where μφ and σφ represent the mean and variance of the learned distribution:

   scssCopy codeqφ(z | H) ~ ℕ(μφ(H), σφ(H)²)

   The VAE's loss function combines reconstruction error and a regularization term (KL divergence):

   scssCopy codeℒ_VAE = E[qφ(z | H)] [log pθ(H | z)] - D_KL(qφ(z | H) || p(z))

   where pθ is the likelihood of the data given the latent variables, and D_KL is the Kullback-Leibler divergence between the learned posterior and the prior.

2. Principal Component Analysis (PCA) PCA reduces the dimensionality of the feature set by selecting the top k principal components. The linear transformation is given by:

   makefileCopy codeZ = H Vk,  Vk ∈ ℝᵈˣᵏ

   where Vk represents the matrix of eigenvectors corresponding to the largest eigenvalues of the covariance matrix of H.

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3. Cognitive Trait Modeling

Once the features are mapped to the latent space Z, cognitive traits are modeled. These traits are predicted through either linear regression or neural network models:

1. Linear Trait Mapping In the linear case, each cognitive trait Tᵢ is modeled as a weighted sum of latent features zⱼ:

   cssCopy codeTᵢ = Σⱼ wᵢⱼ zⱼ + bᵢ,   wᵢⱼ, bᵢ ∈ ℝ

   where wᵢⱼ represents the weight matrix, and bᵢ is the bias term.

2. Nonlinear Trait Mapping A more complex model can use a neural network to predict cognitive traits. The output T is generated by applying a nonlinear activation function to the weighted sum of latent variables Z:

   scssCopy codeT = g(W Z + b),   g(x) = ReLU(x)

   where W is the weight matrix and b is the bias vector, and g(x) is the ReLU activation function which introduces nonlinearity.

3. Temporal Dynamics If cognitive traits evolve over time, Long Short-Term Memory (LSTM) networks can capture the temporal dependencies. The LSTM cell outputs a hidden state hₜ for each time step t, which is used to predict the cognitive trait Tₜ at each time step:

   scssCopy codehₜ = LSTM(hₜ₋₁, Xₜ),   Tₜ = W hₜ + b

   where hₜ₋₁ is the previous hidden state, and Tₜ is the output trait at time step t.

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4. Optimization and Training

The cognitive signature extraction framework is trained using an end-to-end approach. A composite loss function is defined to balance the trait prediction and the reconstruction loss:

rustCopy codeℒ = α ℒ_trait + β ℒ_reconstruction + γ ℒ_variability

where:

• ℒ_trait measures the accuracy of cognitive trait predictions.

• ℒ_reconstruction represents the reconstruction loss, which penalizes the difference between the original and reconstructed linguistic data.

• ℒ_variability captures the temporal and structural variability of language.

The hyperparameters α, β, and γ are tuned to optimize the trade-off between these objectives.

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