Chapter 2 Random Vectors 與他們之間的性質 (Random vectors and their properties)

1.2 如何設計一個 Classifier (Process of Classifier Design) 特徵資料收集 (Data gathering) 正規化資料 與 Registration (Normalization and Registration) Registration 需要一個範例 簡單 Bayes 分類 / 錯誤評估 (Error Estimation, non-parametric) 若 錯誤比例 (Bayeserror)> 要求, 表示特徵資料代表性不夠 若 錯誤比例 (Bayeserror)> 要求, 表示特徵資料代表性不夠 資料結構分析, 研究資料的特性 (Data Structure Analysis) Feature Extraction Techniques clustering, Statistical tests, modeling and so on. 檢驗有多少的分類資訊在前面階段遺失 設計 Classifier

Linear Classifier Quadratic Classifier Piecewise Classifier Non-Parametric classifier M1 ∑1 M2 ∑2 Quadratic decision boundary for normal distributions Quadratic decision boundary for normal distributions Error Estimation (Parametric)

Outline

Distribution and Density Functions How to characterized a random vector X How to characterized a random vector X Using the distribution or density function Using the distribution or density function But these functions cannot easily determined But these functions cannot easily determined age Probability Density Function 年齡在 [a,b] 之間 很高的比例, 接受打針. ( 單一年齡的資訊 ) ab 打針人數 Cumulative Distribution Function (CDF) age ab 打針人數比例 100% 意思是 : 年齡超過 b 以後, 大約有百分之 95 的人已經打過針了. ( 比較多統計資訊 )

Distribution function A ramdom vector may be characterized by a probability distribution function, which is defined by A ramdom vector may be characterized by a probability distribution function, which is defined by For convenience, we often write (2.2) as (2.2)

Density function

An adoptive characterization An adoptive characterization Expected vector (mean) of a random vector Expected vector (mean) of a random vector the most important parameters the most important parameters Covariance matrix Covariance matrix Normal Distributions Normal Distributions

Expected Vector (Mean) The expected vector of a random vector X is defined by The expected vector of a random vector X is defined by vector

The marginal density function Let X and Y be two random variables defined on the same sample space. Let X and Y be two random variables defined on the same sample space. Sample space Sample space all undergraduate at Queue’s all undergraduate at Queue’s X amount of outside study time put in by student (per day) amount of outside study time put in by student (per day) Y overall average marks of the student overall average marks of the student Appendix

Characterizing a random vector Distribution Function Distribution Function Density Function Density Function

Distribution function Distribution function Joint Cumulative Distribution of X and Y is defined as Joint Cumulative Distribution of X and Y is defined as Ex: Ex: P(24,100)=1 P(24,100)=1 All students study at less 24 hours a day and average less than or equal to 100 All students study at less 24 hours a day and average less than or equal to 100 P(0,0)=0 P(0,0)=0 No students study less 0 hours and have average less than or equal to 0 No students study less 0 hours and have average less than or equal to 0 大 P大 P 大 P大 P 對 random vector X,Y 而言, X<x 與 Y < y 的機率 為何 ? 對 random vector X,Y 而言, X<x 與 Y < y 的機率 為何 ?

Density function Density function Joint density of X and Y is defined as Joint density of X and Y is defined as Marginal density function for X is Marginal density function for X is X and Y are (statistically) Independent iff X and Y are (statistically) Independent iff 小 P小 P 小 P小 P 大 P大 P 大 P大 P 若只考慮 x 方面符合條件的機率, 則只要出現 x 的部分都要加起來

In pattern recognition, we deal with random vector drawn from different classes. In pattern recognition, we deal with random vector drawn from different classes. 這些 classes is characterized by it’s own density function w1 w2 w3 w4 對 random vector X 是 w1 的機率是多少 ? X Mixture density function of X is given by Mixture density function of X is given by Wi 本身的機率 * X 是 Wi 的機率

A posteriori probability A posteriori probability of ω i given X, A posteriori probability of ω i given X, can be computed by Bayes theorem, as follows: can be computed by Bayes theorem, as follows: 12-> W1 15-> W2 16-> W1

Parameters of distributions A random vector X is fully characterized by its distribution or density function. A random vector X is fully characterized by its distribution or density function. However, these function cannot be easily determined. However, these function cannot be easily determined. We preferable to adopt a less complete, but more computable, characterization We preferable to adopt a less complete, but more computable, characterization

Expected vector Expected Vector of a random vector X is defined by Expected Vector of a random vector X is defined by X 的數值 * X 本身出線的機率

Index Distribute function Distribute function The Joint Cumulative Distribution The Joint Cumulative Distribution The Density function The Density function The Joint density The Joint density marginal density function marginal density function