Classification models We are interested in classifying a data set among many classes. Each point in the data set has many attributes. Those can be either discrete or continuous, but the classes can only be discrete. If a continuous class is required, then we should use a regression model. It is also worth noting that the classes have no order relation (we cannot say that class 5 is greater than class 2). ...

Consider a statistical model with unknown parameter $\theta$. We want to develop some methods to find $\theta$. $\DeclareMathOperator*{\argmax}{arg \,max \,} \DeclareMathOperator*{\argmin}{arg \,min \,}$ Rao-Blackwell theorem Tip For any two random variables $X$ and $Y$, $$ \begin{align*} \mathbb{E}[\mathbb{E}[X \mid Y]] &= \mathbb{E}[X] \\ \mathbb{E}[\mathbb{E}[X \mid Y]^2] &\leq \mathbb{E}[\mathbb{E}[X^2 \mid Y]] \\ \end{align*} $$ Theorem: Rao-Blackwell theorem Let $T$ be a sufficient statistic, and let $\delta$ be an unbiased estimator of $\theta$. The estimator $\hat{\theta}$, defined as follows, is unbiased and has a lower quadratic risk than $\delta$. ...

There are inherit defects when an image is captured. Noise The noise is an error of measurement in each pixel. There are essentially 2 sources: Read noise (uniform); Photonic noise (depends on the luminosity): each photon has a probability $p$ to be measured ($p < 1$ to read only the color we want) in a Poisson distribution, then, $\mathbb{E}[Y] = N p (1 - p)$ and $\mathrm{Var}(Y) = N p (1 - p)$. The relative photonic noise is, however, ...

Interpolation Our goal is to calculate the value of a pixel that is outside the grid of the image. To this end, we will interpolate the image. Depending on our hypothesis, we can arrive to different interpolation methods. Constant by parts: nearest neighbor; Continue: bi-linear; Polynomial of degree 3: bi-cubic; Limited band: Shannon interpolation. Method Initialization Operations per pixel Nearest neighbor $0$ $1$ Bi-linear $0$ $4$ Bi-cubic $4 N ^2$ $16$ Shannon $0$ $N^2$ We will call our discrete image $I_d$ and the continuous image after interpolation $I_c$. ...

Our goal is to characterize efficient estimators for $\theta$ in terms of mean squared error using the notion of Fisher information. Estimator Let $P_\theta$ be a probability distribution where $\theta \in \Theta \subset \mathbb{R}^d$, $d \in \mathbb{N}$. Definition: Estimator An estimator of $\theta$ is any statistic $\hat{\theta}$ taking values in $\Theta$. Bias We want $\hat{\theta}(X)$ to be close to $\theta$. Since the estimator is a random variable, we can calculate its expectation. ...