# Semantically Controlled Adaptive Equalisation in Reduced Dimensionality Parameter Space +
## 4 Model
### Parameter Space Mapping  
Once the filters have been weighted by the audio signal, the mapping from 13 equaliser variables to a two-dimensional subspace can be accomplished using a range of dimensionality reduction techniques. In this study, we expand on [26] and evaluate the performance of six dimensionality  
Figure 3: An example spectrum taken from an input example, weighted by the biquad coefficients, where the red line represents a peaking filter, the black line represents the biquad-filtered spectrum, and the blue line represents the spectral energy in the rectangular window (\(R_{k}\)). reduction techniques. Here, the algorithms that were used for the dimensionality reduction are available as part of the dimensionality reduction toolbox in [31]. In addition to this, parameter space mapping is evaluated by measuring the quality of reduction with rank-based measures and nearest neighbour classification algorithms. In dimensionality reduction, the reconstruction process is often less common due to the nature of the task (e.g., feature optimisation, data reduction). We evaluate the efficacy of two regression-based techniques and three interpolation techniques at mapping two-dimensional interface variables to a vector of equaliser parameters. This is done by approximating functions using the weighted parameter data and measuring the reconstruction error. Finally, we evaluate an sAE model of data reduction, in which the parameter space is both reduced and reconstructed in the same algorithm; we are then able to isolate the reconstruction (decoder) stage for the implementation process. Dimensionality reduction is implemented using the following techniques: PCA, a widely used method of embedding data into a linear subspace of reduced dimensionality by finding the eigenvectors of the covariance matrix, originally proposed by [32]; _Kernel PCA_ (kPCA), a non-linear manifold mapping technique in which the eigenvectors are computed from a kernel matrix as opposed to the covariance matrix, as defined by [33]; _probabilistic PCA_ (pPCA), a method that considers standard PCA as a latent variable model and makes use of an Expectation Maximisation (EM) algorithm, a method for finding the maximum-likelihood estimate of the parameters in an underlying distribution from a given data set, depending on unobserved latent variables [34] as described in [35]; _Factor Analysis_ (FA), a statistical analysis technique that identifies the relationship between different variables of a dataset and groups those variables by the correlation of the underlying factors [36]; _Diffusion Maps_ (DM), a technique inspired by the field of dynamical systems, reducing the dimensionality of data by embedding the original dataset in a low-dimensional space by retrieving the eigenvectors of Markov random walks [37]; _Linear Discriminant Analysis_ (LDA), a supervised projection technique that maps to a linear subspace while maximising the separability between data points that belong to different classes (see [38]). As LDA projects the data-points onto the dimensions that maximise inter-class variance for C classes, the dimensionality of the subspace is set to \(C-1\). This means that in a binary classification problem such as ours, we need to reconstruct the second dimension arbitrarily. For each of the other algorithms, we select the first two variables for mapping, and for the kPCA algorithm, the feature distances are computed using a Gaussian kernel.