STATISTICAL METHODS IN DATA SCIENCE AND LABORATORY I

From Data Science - Sapienza
Jump to navigation Jump to search
File:Perone approved.png
Prof. Marco Perone Pacifico

Professor Perone

Expectation and Variance[edit | edit source]

The variance Var(x) of a random variable X is the number: <math>Var(x)=E[(X - E[X])^2]</math>. So the variance is the expectation of the Mean Squared of the expectation. Another useful parameter is the Standard Deviation: <math>\sqrt{(Var(X))}</math>

Exploratory Data Analysis[edit | edit source]

It's important to make a distinction between probability theory and statistics. In the former the distribution function is given and we want to analyze the outcomes, in statistics we have the outcomes and we want to infer knowledge by modeling those outcomes

Graphical Summaries[edit | edit source]

Graphical Summaries Comparison
Sample Statistic Formula Distribution Feature Formula
Empirical Distribution Function <math>F_{n}</math> Distribution Function <math>F</math>
Kernel Density Estimate (Histogram) <math>f_{n,h}</math> Probability Density <math>f</math>
(Number of Probability Mass Function <math>p(a)</math>

Numerical Summaries[edit | edit source]

The numerical summaries provided in probability theory have their analogous in data analysis as reported in the table

Numerical Summaries Comparison
Sample Statistic Formula Distribution Feature Formula
Sample Mean <math>X_{n}</math> Expectation <math>\mu</math>
Sample Median <math>Med(X_{1},X_{2},...,X_{n})</math> Median <math>q_{0.5} = F^{inv}(0.5)</math>
pth Empirical Quantile <math>q_{n}(p)</math> 100pth percentile <math>q_{p} = F^{inv}(p)</math>
Sample Variance <math>S^{2}_{n}</math> Variance <math>Var(x)=E[(X - E[X])^2]</math>
Sample Standard Deviation <math>S_{n}</math> Standard Deviation <math>\sqrt{(Var(X))}</math>
Median of absolute deviation (MAD) <math>MAD(X_{1},X_{2},...X_{n})</math> Median of absolute deviation (MAD) <math>F^{inv}(0.75)-F^{inv}(0.5) for symmetric F</math>
  • Sample Mean
  • Sample Variance
  • Sample Standard Deviation
  • Median of absolute deviation (MAD)

In this chapter

Statistical Models and their Features' Estimation[edit | edit source]

Basic Statistical Models[edit | edit source]

Bootstrap[edit | edit source]

File:Bootstrap.jpg
Bootstrap Methods are named in honor of the Baron von Munchhausen Tale

Estimators[edit | edit source]

Biased and Unbiased Estimators[edit | edit source]

Estimators Efficiency Comparison: Variance and Mean Squared Error[edit | edit source]

When we have two unbiased estimators and we would like to know which one is the best to estimate parameters we use variance, instead for unbiased estimators we use Mean Squared Errors

Estimators with no analogue: Maximum Likelihood Estimators[edit | edit source]

How to Estimate Maximum Likelihood Estimators: Least Square Method[edit | edit source]