数学统计知识

Posted 2023-02-22 银灯玉箫

Coefficient of determination

https://en.wikipedia.org/wiki/Coefficient_of_determination

In the best case, the modeled values exactly match the observed values, which results in \\displaystyle SS_\\textres=0\\displaystyle SS_\\textres=0 and \\displaystyle R^2=1\\displaystyle R^2=1. A baseline model, which always predicts \\displaystyle \\bar y\\bar y, will have \\displaystyle R^2=0\\displaystyle R^2=0. Models that have worse predictions than this baseline will have a negative \\displaystyle R^2R2.

pdf

The probability density function of a continuous random variable X, if it exists, expresses the complete probability model of a random as the following mathematical function:
$$f_X(x)=\frac{d{F}_x(x)}{dx}$$
This derivative, when it exists that is when $F_x(x)$ is differentiable at x, is non-negative, as the cdf is non-decreasing function. The pdf has larger values in regions of high probability and smaller values in regions of low probability.

pmf

The probability mass function of a discrete random variable X expresses the complete probability model of a random experiment as the following mathematical function:
$$p_X(x)=P(X=x),x\in S_x$$
Note that $p_X(x)$ is a function ranging over real numbers x and that $p_X(x)$ can be nonzero at the values $x\in S_x$.

expected value of a random variable

In summary, the expected value is obtained by multiplying each possible value by its respective probability and the summing these products over all the values that have nonzero probability.

Note that the statistical expectation is a linear operation, that is the expected value of a linear combination of n random functions, where n is a positive integer, is equal to the same linear combination of n expected values of the random functions.
$${\sigma }_X^2=E[X^2]-(E[X]{)}^2=E[X^2]-{\mu }_x^2$$

standard deviation

THe square root of the variance of X, denoted by ${\sigma }_X$, is called the standard deviation of the random variable X, and is a positive quantity with the same unit as X.

GLM Generalized Linear Models

GLM models allow us to build a linear relationship between the response and predictiors, even though their underlying relationship is not linear. This is made possible by using a link function, which links the response variable to a linear model. Unlike Linear Regression models, the error distribution of the response variable need not to be normally distributed. The errors in the response varialbe are assumed to follow an exponential family of distribution (i.e., normal, binomial, Poission, or gamma distribution).