Reckon

Reckon is a library containing various algorithms for scientific applications.

Regularization

Regularization is a method used to control how closely a model fits the data, especially in ill-posed or ill-conditioned problems. One common approach is to modify the loss function by adding a penalty term that constrains the solution. This leads to the linear regularization problem, which can be expressed as:

Here, $p$ and $q$ represent the norm of the vectors. $\mathbf{x}$ is the unknown source (or the model parameters) to be estimated, $\mathbf{A}$ is the forward model matrix with high condition number or it is rank deficient. $\mathbf{L}$ is the regularization operator, which is used to adjust the properties of the output model. $\lambda$ is a regularization parameter employed to determine the weight of the regularization term in the solution. $\mathbf{y}$ is the noisy measurements that can be expressed as follows;

$\mathbf{y}= \mathbf{A}\mathbf{x} + \mathbf{n}$

Where $\mathbf{n}$ is the measurement noise vector.

• If $p = q = 2$, then the problem is called Tikhonov regularization or $L_{2}$ regularization that usually yields smooth estimation.

• If $p = 2$, $q = 1$, then the problem is called $L_{1}$ regularization, which promotes sparsity in the solution.

Kalman Filter

Kalman filtering is an algorithm that allows us to estimate the state of a system based on observations or measurements. The state of a system at time k evolved from the prior state at time k-1, expressed in the following form;

$\mathbf{x}(k)= \mathbf{A}\mathbf{x}(k-1) + \mathbf{B}\mathbf{u}(k-1) + \mathbf{w}(k-1)$

And the accompanying observation is defined as;

$\mathbf{y}(k)= \mathbf{H}\mathbf{x}(k) + \mathbf{v}(k-1)$

where

$p(\mathbf{w}) \sim \mathcal{N}(0,\mathbf{Q})$

$p(\mathbf{v}) \sim \mathcal{N}(0, \mathbf{R})$

The cycle of discrete Kalman filter.