Lecture Notes for Chapter 2

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1 Lecture Notes for Chapter 2
Data Mining: Data Lecture Notes for Chapter 2

2 2.1 Types of Data - What is Data?
Collection of data objects and their attributes An attribute is a property or characteristic of an object Examples: eye color of a person, temperature, etc. Attribute is also known as variable, field, characteristic, or feature A collection of attributes describe an object Object is also known as record, point, case, sample, entity, or instance Attributes Objects

3 Attribute Values Attribute values are numbers or symbols assigned to an attribute Distinction between attributes and attribute values Same attribute can be mapped to different attribute values Example: height can be measured in feet or meters Different attributes can be mapped to the same set of values Example: Attribute values for ID and age are integers But properties of attribute values can be different ID has no limit but age has a maximum and minimum value

4 Measurement of Length The way you measure an attribute is somewhat may not match the attributes properties. 두번째 선분은 첫번째 선분 2개 합쳐진것, 세번째 선분은 첫번째 선분 3개가 합쳐진 것… 오른쪽은 이와 같은 2배, 3배,4배 등의 배수 특성이 반영되어 있지만, 왼쪽은 길이 속성의 순서 속성만 반영되어 있음

5 Types of Attributes There are different types of attributes
Nominal(명목형) Examples: ID numbers, eye color, zip codes Ordinal(서열형) Examples: rankings (e.g., taste of potato chips on a scale from 1-10), grades, height in {tall, medium, short} Interval(구간형) Examples: calendar dates, temperatures in Celsius or Fahrenheit. Ratio(비율형) Examples: temperature in Kelvin, length, time, counts

6 Properties of Attribute Values
The type of an attribute depends on which of the following properties it possesses: Distinctness(유일성): =  Order(순서): < > Addition(덧셈): + - Multiplication(곱셈): * / Nominal attribute(명목형): distinctness Ordinal attribute(서열형): distinctness & order Interval attribute(구간형): distinctness, order & addition Ratio attribute(비율형): all 4 properties

7 Attribute Type Description Examples Operations Nominal
The values of a nominal attribute are just different names, i.e., nominal attributes provide only enough information to distinguish one object from another. (=, ) (명목형은 단순히 서로 다른 이름뿐  서로 구별 정보만 제공) zip codes, employee ID numbers, eye color, sex: {male, female} mode, entropy, contingency correlation, 2 test Ordinal The values of an ordinal attribute provide enough information to order objects. (<, >) (서열형은 서로 순서를 정하는데 필요한 정보 제공) hardness of minerals, {good, better, best}, grades, street numbers Median(중간값), percentiles, rank correlation, run tests, sign tests Interval For interval attributes, the differences between values are meaningful, i.e., a unit of measurement exists. (+, - ) (구간형은 값들 사이의 차이가 의미가 있음) calendar dates, temperature in Celsius or Fahrenheit mean, standard deviation, Pearson's correlation, t and F tests Ratio For ratio variables, both differences and ratios are meaningful. (*, /) (비율형은 차이와 비율이 모두 의미가 있음) temperature in Kelvin, monetary quantities, counts, age, mass, length, electrical current geometric mean, harmonic mean, percent variation(퍼센트 편차)

8 Any permutation of values (순서 변환: 순열)
< 속성 수준을 정의한 변환예 > Attribute Level Transformation Comments Nominal Any permutation of values (순서 변환: 순열) If all employee ID numbers were reassigned, would it make any difference? Ordinal An order preserving change of values, i.e., new_value = f(old_value) where f is a monotonic function. (값의 순서 보존형 변환) An attribute encompassing the notion of good, better best can be represented equally well by the values {1, 2, 3} or by { 0.5, 1, 10}. Interval new_value =a * old_value + b where a and b are constants (새값 = a*예전값 + b) Thus, the Fahrenheit and Celsius temperature scales differ in terms of where their zero value is and the size of a unit (degree). Ratio new_value = a * old_value (새값 = a * 예전값) Length can be measured in meters or feet.

9 Discrete and Continuous Attributes
Discrete Attribute Has only a finite or countably infinite set of values Examples: zip codes, counts, or the set of words in a collection of documents Often represented as integer variables. Note: binary attributes are a special case of discrete attributes Continuous Attribute Has real numbers as attribute values Examples: temperature, height, or weight. Practically, real values can only be measured and represented using a finite number of digits. Continuous attributes are typically represented as floating-point variables.

10 Types of data sets Record Graph Ordered Data Matrix Document Data
Transaction Data Graph World Wide Web Molecular Structures Ordered Spatial Data Temporal Data Sequential Data Genetic Sequence Data

11 Important Characteristics of Structured Data
Dimensionality 데이터 집합의 객체들이 갖는 속성의 수/변수의 수 Curse of Dimensionality(차원의 저주) 문제 존재 Dimensionality reduction(차원 감소) 필요 Sparsity 객체 대부분의 속성이 0이며, 일부(1% 이내) 객체만 속성이 0이 아닌 경우  0이 아닌 값만 처리하면 됨(이 때, 희소성은 장점이 됨) Only presence counts Resolution 데이터는 상이한 수준의 해상도로 얻는 경우 많음 Patterns depend on the measurement scale(척도) 예) 지구 표면을 km단위로 보는 경우(평평함) vs. m 단위로 보는 경우

12 Record Data Data that consists of a collection of records, each of which consists of a fixed set of attributes

13 Data Matrix If data objects have the same fixed set of numeric attributes, then the data objects can be thought of as points in a multi-dimensional space, where each dimension represents a distinct attribute Such data set can be represented by an m by n matrix, where there are m rows, one for each object, and n columns, one for each attribute

14 Document Data Each document becomes a `term' vector,
each term is a component (attribute) of the vector, the value of each component is the number of times the corresponding term occurs in the document.

15 Transaction Data A special type of record data, where
each record (transaction) involves a set of items. For example, consider a grocery store. The set of products purchased by a customer during one shopping trip constitute a transaction, while the individual products that were purchased are the items.

16 Graph Data (그래프 기반 데이터) 그래프 데이터:
데이터 객체간의 관계를 나타내는 그래프 데이터 객체 자체가 그래프로 표현될 수 있음 Examples: Generic graph and HTML Links

17 Graph Data - Chemical Data
Examples: Benzene Molecule: C6H6

18 Ordered Data (서열형 데이터) 서열형 데이터 데이터 속성이 시간/공간 순서와 관련된 관계 가짐
데이터 속성이 시간/공간 순서와 관련된 관계 가짐 Temporal data(시간 데이터) Sequence data(서열 데이터) Time series data(시계열 데이터) Spatial data(공간 데이터)

19 2.2 Data Quality (데이터 품질) What kinds of data quality problems?
How can we detect problems with the data? What can we do about these problems? Examples of data quality problems: Noise and outliers missing values duplicate data

20 Noise Noise is the random component of a measurement error
It may involve the distortion of a value or the addition of spurious objects Examples: distortion of a person’s voice when talking on a poor phone and “snow” on television screen Two Sine Waves Two Sine Waves + random Noise

21 Outliers Outliers are data objects with characteristics that are considerably different than most of the other data objects in the data set

22 Missing Values It is usual for an object to be missing one or more attribute values Reasons for missing values Information is not collected (e.g., people decline to give their age and weight) Attributes may not be applicable to all cases (e.g., annual income is not applicable to children) Handling missing values Estimate Missing Values (누락값 추정) Eliminate Data Objects (누락값 제거) Ignore the Missing Value During Analysis (분석 과정에서 누락값 무시) Replace with all possible values (weighted by their probabilities) (누락값을 가능성있는 값으로 대체)

23 Duplicate Data (중복 데이터)
Data set may include data objects that are duplicates, or almost duplicates of one another Major issue when merging data from heterogeous sources Examples: Same person with multiple addresses Data cleaning (데이터 정제) Process of dealing with duplicate data issues

24 2.3 Data Preprocessing(데이터 전처리)
Aggregation(집합/집계/총계) Sampling(표본 추출) Dimensionality Reduction(차원 축소) Feature subset selection(특징 부분집합 선택) Feature creation(특징 생성) Discretization and Binarization(이산화와 이진화) Attribute Transformation(속성 변환)

25 Aggregation Combining two or more attributes (or objects) into a single attribute (or object) Purpose Data reduction Reduce the number of attributes or objects Change of scale Cities aggregated into regions, states, countries, etc More “stable” data Aggregated data tends to have less variability

26 Aggregation Variation of Precipitation in Australia (평균 월별 강수량의 표준편차)
(평균 연별 강수량의 표준 편차) 평균 연별 강수량(b)이 더 적은 가변성을 가지고 있음

27 Sampling Sampling is the main technique employed for data selection.
It is often used for both the preliminary investigation of the data and the final data analysis. Statisticians sample because obtaining the entire set of data of interest is too expensive or time consuming. Sampling is used in data mining because processing the entire set of data of interest is too expensive or time consuming.

28 Sampling … The key principle for effective sampling is the following:
using a sample will work almost as well as using the entire data sets, if the sample is representative A sample is representative if it has approximately the same property (of interest) as the original set of data

29 Types of Sampling Simple Random Sampling(단순 임의 표본추출)
There is an equal probability of selecting any particular item Sampling without replacement(무대체 표본추출) As each item is selected, it is removed from the population Sampling with replacement(대체 표본추출) Objects are not removed from the population as they are selected for the sample. In sampling with replacement, the same object can be picked up more than once Stratified sampling(층화 표본추출) Split the data into several partitions; then draw random samples from each partition (모집단을 층으로 나눈 후, 각 층에서 샘플링) 층내에서는 동질적, 층간은 이질적 특성을 가지도록 하면 적은 비용으로 더 정확한 추정 가능

30 Sample Size 표본 추출과 정보 손실 표본 크기가 커지면 표본이 전체 데이터의 대표성을 가질 확률이 높아짐(하지만, 표본추출의 장점이 퇴색됨) 표본 크기가 작으면 패턴이 누락되거나 잘못된 패컨이 감지될 수 있음 예: 8000 points Points Points

31 Sample Size 적절한 표본 크기 결정 What sample size is necessary to get at least one object from each of 10 groups. (b)는 표본 크기가 10에서 60까지 변할때, (a)이 10개 그룹에서 각각 하나의 객체를 선택할 확률임 샘플 크기가 커질 수록 10개의 모든 그룹에서 객체를 얻을 확률이 높아짐

32 Curse of Dimensionality
차원이 증가하면 데이터는 그 차원이 차지하는 공간상에서 점점더 sparse하게 됨  분석이 어려워짐 Classification에선 충분한 데이터 객체가 존재하지 않아 모델 생성이 어려워짐 Clustering에선 clustering의 핵심인 density와 두점간의 distance 정보가 작아서 군집화가 어려워짐 Randomly generate 500 points Compute difference between max and min distance between any pair of points

33 Dimensionality Reduction 기법
Purpose: Avoid curse of dimensionality Reduce amount of time and memory required by data mining algorithms Allow data to be more easily visualized May help to eliminate irrelevant features or reduce noise Techniques Principle Component Analysis (주성분 분석) Singular Value Decomposition(특이값 분해) Others: supervised and non-linear techniques

34 Dimensionality Reduction: PCA
최대한의 데이터 변이를 얻기 위함 The Goal is to find a projection that captures the largest amount of variation in data x2 e x1

35 Dimensionality Reduction: PCA
Find the eigenvectors of the covariance matrix One of the most intuitive explanations of eigenvectors of a covariance matrix is that they are the directions in which the data varies the most.  Example) Each data sample is a 2 dimensional point with coordinates x, y. The eigenvectors of the covariance matrix of these data samples are the vectors u and v; u : first eigenvector, v, the shorter arrow, is the second eigenvector The first eigenvector is the direction in which the data varies the most, the second eigenvector is the direction of greatest variance among those that are orthogonal (perpendicular) to the first eigenvector. The eigenvalues are the length of the arrows.

36 Feature Subset Selection(특징 부분집합 선택)
Another way to reduce dimensionality of data Redundant features(중복 특징) duplicate much or all of the information contained in one or more other attributes Example: purchase price of a product and the amount of sales tax paid Irrelevant features(비관련 특징) contain no information that is useful for the data mining task at hand Example: students' ID is often irrelevant to the task of predicting students' GPA

37 Feature Subset Selection
특징 부분집합 선택 기법: Brute-force approach: Try all possible feature subsets as input to data mining algorithm Embedded approaches: Feature selection occurs naturally as part of the data mining algorithm Filter approaches: Features are selected before data mining algorithm is run Wrapper approaches: Use the data mining algorithm as a black box to find best subset of attributes

38 Feature Creation Create new attributes that can capture the important information in a data set much more efficiently than the original attributes Three general methodologies: Feature Extraction(특징 추출) domain-specific Mapping Data to New Space(새로운 공간으로 데이터 매핑) Feature Construction(특징 구축) combining features

39 Mapping Data to a New Space
Fourier transform : sin/cos 함수로 time domain 신호 frequency domain으로 변환 Wavelet transform : sin/cos함수뿐만 아니라, 다양한 wavelet 모함수를 사용하여 변환함 Two Sine Waves Two Sine Waves + Noise Frequency

40 Discretization Using Class Labels (이산화)
Discretization of Two Attributes (a) x와 y 속성을 세개의 구간으로 이산화함. (b)는 5개의 구간으로 이산화함  (a),(b) 모두, 2차원에서는 잘 분리가 됨. 1차원 관점에서는 그러지 못함 (b)의 5개 구간으로 분리한 것이 더욱 잘 분리했음 (a) 3 categories for both x and y (b) 5 categories for both x and y

41 Discretization Without Using Class Labels
(d) K-means 기반 이산화가 최상임. 그 다음은 (c) 동등 주파수 기반 이산화, 그 다음이 (b) 동등폭 이산화 수행 결과 (a) 원 데이터 (b) 동등폭 이산화 수행 (c) 동등 주파수 이산화 수행 (d) K-means 이산화 수행

42 Attribute Transformation
It alters the data by replacing a selected attribute by one or more new attributes A function that maps the entire set of values of a given attribute to a new set of replacement values such that each old value can be identified with one of the new values Simple functions: xk, log(x), ex, |x| Standardization and Normalization, etc.

43 2.4 Similarity and Dissimilarity(유사도와 비유사도)
Numerical measure of how alike two data objects are. Is higher when objects are more alike. Often falls in the range [0,1] Dissimilarity Numerical measure of how different are two data objects Lower when objects are more alike Minimum dissimilarity is often 0 Upper limit varies Proximity refers to a similarity or dissimilarity

44 Similarity/Dissimilarity for Simple Attributes
p and q are the attribute values for two data objects.

45 Euclidean Distance Euclidean Distance
Where n is the number of dimensions (attributes) and pk and qk are, respectively, the kth attributes (components) or data objects p and q. Standardization is necessary, if scales differ.

46 Euclidean Distance Distance Matrix

47 Minkowski Distance Minkowski Distance is a generalization of Euclidean Distance Where r is a parameter, n is the number of dimensions (attributes) and pk and qk are, respectively, the kth attributes (components) or data objects p and q.

48 Minkowski Distance: Examples
r = 1. City block :Manhattan norm, taxicab norm, L1 norm A common example of this is the Hamming distance, which is just the number of bits that are different between two binary vectors r = 2. Euclidean distance r  . “supremum” (Lmax norm, L norm) distance. This is the maximum difference between any component of the vectors Do not confuse r with n, i.e., all these distances are defined for all numbers of dimensions. Euclidean distance L2 norm

49 Minkowski Distance Distance Matrix

50 Mahalanobis Distance This is a measure of the distance between a point P and a distribution D P가 분포 D의 평균으로부터 표준편차의 몇 배 떨어져 있는지를 알 수 있음 For red points, the Euclidean distance is 14.7, Mahalanobis distance is 6.

51 Mahalanobis Distance (example)
If, in our single observation, X = 410 and Y = 400 (other data is not shown in this example), we would calculate the Mahalanobis distance for that single value as: < Calculation of Mahalanobis distance> < covariance matrix > Var(X)=std_dev(X)^2, Cov(X,Y)=E((X-u)(Y-v)) Cov(Y,X), Var(Y)

52 Mahalanobis Distance Covariance Matrix: C A: (0.5, 0.5) B B: (0, 1)
Mahal(A,B) = 5 Mahal(A,C) = 4 B A

53 Common Properties of a Distance
Distances, such as the Euclidean distance, have some well known properties. d(p, q)  0 for all p and q, and d(p, q) = 0 only if p = q. (Positive definiteness) d(p, q) = d(q, p) for all p and q. (Symmetry) d(p, r)  d(p, q) + d(q, r) for all points p, q, and r. (Triangle Inequality) where d(p, q) is the distance (dissimilarity) between points (data objects), p and q. A distance that satisfies these properties is a metric

54 Common Properties of a Similarity
Similarities, also have some well known properties. s(p, q) = 1 (or maximum similarity) only if p = q. s(p, q) = s(q, p) for all p and q. (Symmetry) where s(p, q) is the similarity between points (data objects), p and q.

55 Similarity Between Binary Vectors
Common situation is that objects, p and q, have only binary attributes Compute similarities using the following quantities M01 = the number of attributes where p was 0 and q was 1 M10 = the number of attributes where p was 1 and q was 0 M00 = the number of attributes where p was 0 and q was 0 M11 = the number of attributes where p was 1 and q was 1 Simple Matching Coefficients (단순 매칭 계수) SMC = 매칭 속성의 수/속성의 수 = (M11 + M00) / (M01 + M10 + M11 + M00) SMC는 p도 없고 q도 없는 속성까지 고려를 하고 있음 예를 들어, 마켓에서 판매하는 1000개의 물건에서, 두 고객(p,q)의 장바구니(구매 물품)의 SMC 유사도를 본다면, p = (salt, pepper), q= (salt, sugar) M11 = 1개(salt), M00=(1000개 – 3), M01=1(sugar), M10=1(pepper) SMC = ( )/( ) =  의미없음

56 Similarity Between Binary Vectors
Jaccard Coefficients (자카드 계수) SMC와 달리 M00을 사용하지 않음 비대칭 이진 속성으로 구성된 객체의 유사도 처리에 유용 J = number of 11 matches / number of not-both-zero attributes values = (M11) / (M01 + M10 + M11) 예를 들어, 마켓에서 판매하는 1000개의 물건에서, 두 고객의 장바구니(구매 물품)의 Jaccard 유사도를 본다면, p = (salt, pepper), q= (salt, sugar) M11 = 1개(salt), M01=1(sugar), M10=1(pepper) Jaccard 유사도 = (1 )/( ) = 1/3  두 장바구니의 Jaccard 유사도는 의미있음

57 SMC versus Jaccard: Example
q = M01 = 2 (the number of attributes where p was 0 and q was 1) M10 = 1 (the number of attributes where p was 1 and q was 0) M00 = 7 (the number of attributes where p was 0 and q was 0) M11 = 0 (the number of attributes where p was 1 and q was 1) SMC = (M11 + M00)/(M01 + M10 + M11 + M00) = (0+7) / ( ) = 0.7 J = (M11) / (M01 + M10 + M11) = 0 / ( ) = 0

58 Cosine Similarity If d1 and d2 are two document vectors, then
cos( d1, d2 ) = (d1  d2) / ||d1|| ||d2|| , where  indicates vector dot product and || d || is the length of vector d. Example: d1 = d2 = d1  d2= 3*1 + 2*0 + 0*0 + 5*0 + 0*0 + 0*0 + 0*0 + 2*1 + 0*0 + 0*2 = 5 ||d1|| = (3*3+2*2+0*0+5*5+0*0+0*0+0*0+2*2+0*0+0*0)0.5 = (42) 0.5 = 6.481 ||d2|| = (1*1+0*0+0*0+0*0+0*0+0*0+0*0+1*1+0*0+2*2) 0.5 = (6) 0.5 = 2.245 cos( d1, d2 ) = .3150 Cosine 유사도가 0 이라는 것은 아무런 관련성 없음을 의미

59 Extended Jaccard Coefficient (Tanimoto)
Variation of Jaccard for continuous or count attributes Reduces to Jaccard for binary attributes

60 Correlation Correlation measures the linear relationship between objects To compute correlation, we standardize data objects, p and q, and then take their dot product 즉, 𝑪𝒐𝒓𝒓𝒆𝒍𝒂𝒕𝒊𝒐𝒏= 𝑪𝒐𝒗(𝒑,𝒒) 𝒔𝒕𝒅𝑫𝒆𝒗(𝒑)∗𝒔𝒕𝒅𝑫𝒆𝒗(𝒒)

61 Visually Evaluating Correlation
-1에서 1의 상관도를 보여주는 산포도(scatter plots)

62 General Approach for Combining Similarities
Sometimes attributes are of many different types, but an overall similarity is needed.

63 Using Weights to Combine Similarities
May not want to treat all attributes the same. Use weights wk which are between 0 and 1 and sum to 1.

64 Density Density-based clustering require a notion of density Examples:
Euclidean density Euclidean density = number of points per unit volume Probability density Graph-based density

65 Euclidean Density – Cell-based
Simplest approach is to divide region into a number of rectangular cells of equal volume and define density as # of points the cell contains

66 Euclidean Density – Center-based
Euclidean density is the number of points within a specified radius of the point