# St Bernards Maths Homework Uic 210

The first written homework will be assigned on Wednesday, January 17th, in Crowdmark. A link to the assignment will be sent to your UIC email address. A pdf with questions will also be posted on the Blackboard. The solutions will be uploaded on the Blackboard in about 2 days after the deadline for submission.

Date | Sections | MyMathLab Problems (Additional HW may be assigned via BlackBoard) |
---|---|---|

Mon Jan 15 | Martin Luther King Day | |

Wed Jan 17 | 11.1 Vectors in the Plane, discussion of course policies | 11.1: #5,15,21,28,36,47,51; 11: #6,9,11,12,13,15,17 |

Fri Jan 19 | 11.2 Vectors in Three Dimensions | 11.2: #4,11,25,27,29,32,41,51; 11:#30,33,36 |

Mon Jan 22 | 11.3 Dot Products | 11.3: #11,13,19,27,37; 11:#44,53 |

Wed Jan 24 | 11.4 Cross Products | 11.4: #3,4,11,14,23,25,31,35,41; 11:#58 |

Fri Jan 26 | 11.5 Lines and Curves in Space | 11.5: #1,3,6,9,15,19,28,62,67;11: #67,73,74,77 |

Mon Jan 29 | 11.6 Calculus of Vector-Valued Functions | 11.6: #3,4,9,11,18,27,49,55,59; 11:#78,79 |

Wed Jan 31 | 11.7 Motion in Space | 11.7: #3,11,33,37,43; 11: #83,86,87,88 |

Fri Feb 2 | 11.8 Length of Curves | 11.8: #5,9,18,23,35,43; 11: #95 |

Mon Feb 5 | 12.1 Planes and Surfaces | 12.1: #1,3,13,15,19,21,25,31,38; 12:#2 |

Wed Feb 7 | 12.1 Planes and Surfaces | 12.1: #7,41,52,79; 12:#9,10,11,13,14 |

Fri Feb 9 | 12.2 Graphs and Level Curves | 12.2: #1,3,5,21,22,34; 12: #24,25,28,30,35 |

Mon Feb 12 | 12.3 Limits and Continuity | 12.3: #3,5,6,7,11,13,15,19,24,27,29; 12: #47 |

Wed Feb 14 | 12.4 Partial Derivatives; 12.5 The Chain Rule | 12.4: #6,13,17,22,30,41,49,72,74; 12.5: #3,9,11,17,21,23,28,36,58 |

Fri Feb 16 | 12.6 Directional Derivatives and the Gradient | 12.6: #3,7,11,15,21,24,27,65; 12: #63 |

Mon Feb 19 | 12.6 Directional Derivatives and the Gradient | |

Wed Feb 21 | Review of 11.1 - 11.8, 12.1 - 12.6 | |

Fri Feb 23 | 12.7 Tangent Planes and Linear Approximation | 12.7: #14,17,25,28,31,36,37; 12:#78 |

Mon Feb 26 | 12.7 Tangent Planes and Linear Approxmations, 12.8 Maximum/Minimum Problems | |

Wed Feb 28 | 12.8 Maximum/Minimum Problems | 12.8: #4,10,21,27,33,42,62; 12: #86,87 |

Fri Mar 2 | 12.8 Maximum/Minimum Problems, 12.9 Lagrange Multipliers | 12.8: #35,43,48,50,57; 12.9: #4,9,11,26 |

Mon Mar 5 | 12.9 Lagrange Multipliers | 12.9: #30,38,46,56; 12: #99,101 |

Wed Mar 7 | 13.1 Double Integrals over Rectangular Regions | 13.1: #1,7,9,17,21,23,29,50,55 |

Fri Mar 9 | 13.2 Double Integrals over General Regions | 13.2: #10,12,14,15,26,29,50,57,63,77 |

Mon Mar 12 | 13.3 Double Integrals in Polar Coordinates | 13.3: #11,21,28,31,33,37 |

Wed Mar 14 | 13.4 Triple Integrals | 13.4: #9,15,20,39; 13: #41 |

Fri Mar 16 | 13.5 Triple Integrals in Cylindrical Coordinates | 13.5: #11,13,15,18,27; 13: #52b |

Mon Mar 19 | 13.5 Triple Integrals in Cylindrical Coordinates | |

Wed Mar 21 | Review of 12.7 - 13.5 (cylindrical coordinates) | |

Fri Mar 23 | 13.5 Triple Integral in Spherical Coordinates | 13.5: #35,37,39,42,49 |

Mon Mar 26 | Spring Break | |

Wed Mar 28 | Spring Break | |

Fri Mar 30 | Spring Break | |

Mon Apr 2 | 13.5 Triple Integral in Spherical Coordinates, 13.6* Integrals for Mass Calculation | 13.6: #33, 36 |

Wed Apr 4 | 13.7 Change of Variables in Multiple Integrals | 13.7: #2,7,14,31,35 |

Fri Apr 6 | 14.1 Vector Fields | 14.1: #3,4,9,13,17,25,34,37,39,49,50 |

Mon Apr 9 | 14.2 Line Integrals | 14.2: #15,17,27,33,37,39,47,49,64 |

Wed Apr 11 | 14.2 Line Integrals | |

Fri Apr 13 | 14.3 Conservative Vector Fields | 14.3: #3,9,12,14,17,22,26,29,31,33,35,49 |

Mon Apr 16 | 14.4 Green's Theorem | 14.4: #11,14,25,26,33 |

Wed Apr 18 | 14.4 Green's Theorem | |

Fri Apr 20 | 14.5 Divergence and Curl | 14.5: #9,13,24,27,31 |

Mon Apr 23 | 14.6 Surface Integrals | 14.6: #4,15,17,21,23,32,43,48 |

Wed Apr 25 | 14.6 Surface Integrals | |

Fri Apr 27 | 14.7 Stokes' Theorem | 14.7: #4,14,41 |

Mon Apr 30 | 14.8 The Divergence Theorem | 14.8: #17,20,29 |

Wed May 2 | Review for the final | |

Fri May 4 | Review for the final |

## Presentation on theme: "1 Data Mining: Concepts and Techniques (3 rd ed.) — Chapter 3 —"— Presentation transcript:

1 1 Data Mining: Concepts and Techniques (3 rd ed.) — Chapter 3 —

2 22 Chapter 3: Data Preprocessing Data Preprocessing: An Overview Data Quality Major Tasks in Data Preprocessing Data Cleaning Data Integration Data Reduction Data Transformation and Data Discretization Summary

3 3 Data Quality: Why Preprocess the Data? Measures for data quality: A multidimensional view Accuracy: correct or wrong, accurate or not Completeness: not recorded, unavailable, … Consistency: some modified but some not, dangling, … Timeliness: timely update? Believability: how trustable the data are correct? Interpretability: how easily the data can be understood?

4 4 Major Tasks in Data Preprocessing Data cleaning Fill in missing values, smooth noisy data, identify or remove outliers, and resolve inconsistencies Data integration Integration of multiple databases, data cubes, or files Data reduction Dimensionality reduction Numerosity reduction Data compression Data transformation and data discretization Normalization Concept hierarchy generation

5 55 Chapter 3: Data Preprocessing Data Preprocessing: An Overview Data Quality Major Tasks in Data Preprocessing Data Cleaning Data Integration Data Reduction Data Transformation and Data Discretization Summary

6 6 Data Cleaning Data in the Real World Is Dirty: Lots of potentially incorrect data, e.g., instrument faulty, human or computer error, transmission error incomplete: lacking attribute values, lacking certain attributes of interest, or containing only aggregate data e.g., Occupation=“ ” (missing data) noisy: containing noise, errors, or outliers e.g., Salary=“−10” (an error) inconsistent: containing discrepancies in codes or names, e.g., Age=“42”, Birthday=“03/07/2010” Was rating “1, 2, 3”, now rating “A, B, C” discrepancy between duplicate records Intentional (e.g., disguised missing data) Jan. 1 as everyone’s birthday?

7 7 Incomplete (Missing) Data Data is not always available E.g., many tuples have no recorded value for several attributes, such as customer income in sales data Missing data may be due to equipment malfunction inconsistent with other recorded data and thus deleted data not entered due to misunderstanding certain data may not be considered important at the time of entry not register history or changes of the data Missing data may need to be inferred

8 8 How to Handle Missing Data? Ignore the tuple: usually done when class label is missing (when doing classification)—not effective when the % of missing values per attribute varies considerably Fill in the missing value manually: tedious + infeasible? Fill in it automatically with a global constant : e.g., “unknown”, a new class?! the attribute mean the attribute mean for all samples belonging to the same class: smarter the most probable value: inference-based such as Bayesian formula or decision tree

9 9 Noisy Data Noise: random error or variance in a measured variable Incorrect attribute values may be due to faulty data collection instruments data entry problems data transmission problems technology limitation inconsistency in naming convention Other data problems which require data cleaning duplicate records incomplete data inconsistent data

10 10 How to Handle Noisy Data? Binning first sort data and partition into (equal-frequency) bins then one can smooth by bin means, smooth by bin median, smooth by bin boundaries, etc.

11 11 How to Handle Noisy Data? Regression smooth by fitting the data into regression functions

12 September 9, 2015Data Mining: Concepts and Techniques12 Regression x y y = x + 1 X1 Y1 Y1’

13 13 How to Handle Noisy Data? Clustering detect and remove outliers

14 September 9, 2015Data Mining: Concepts and Techniques14 Cluster Analysis

15 15 How to Handle Noisy Data? Regression smooth by fitting the data into regression functions Clustering detect and remove outliers Combined computer and human inspection detect suspicious values and check by human (e.g., deal with possible outliers)

16 16 Data Cleaning as a Process Data discrepancy detection Use metadata (e.g., domain, range, dependency, distribution) Check field overloading Check uniqueness rule, consecutive rule and null rule Use commercial tools Data scrubbing: use simple domain knowledge (e.g., postal code, spell-check) to detect errors and make corrections Data auditing: by analyzing data to discover rules and relationship to detect violators (e.g., correlation and clustering to find outliers) Data migration and integration Data migration tools: allow transformations to be specified ETL (Extraction/Transformation/Loading) tools: allow users to specify transformations through a graphical user interface Integration of the two processes Iterative and interactive (e.g., Potter’s Wheels)

17 17 Chapter 3: Data Preprocessing Data Preprocessing: An Overview Data Quality Major Tasks in Data Preprocessing Data Cleaning Data Integration Data Reduction Data Transformation and Data Discretization Summary

18 18 Data Integration Data integration: Combines data from multiple sources into a coherent store Schema integration: e.g., A.cust-id B.cust-# Integrate metadata from different sources Entity identification problem: Identify real world entities from multiple data sources, e.g., Bill Clinton = William Clinton Detecting and resolving data value conflicts For the same real world entity, attribute values from different sources are different Possible reasons: different representations, different scales, e.g., metric vs. British units

19 19 Handling Redundancy in Data Integration Redundant data occur often when integration of multiple databases Object identification: The same attribute or object may have different names in different databases Derivable data: One attribute may be a “derived” attribute in another table, e.g., annual revenue

20 20 Handling Redundancy in Data Integration Redundant attributes may be able to be detected by correlation analysis and covariance analysis Given two attributes, correlation analysis can measure how strongly one attribute implies the other based on the available data. For nominal data, chi-square test is used. For numeric data, the correlation coefficient and covariance are used. Both access how one attribute’s values vary from those of another. Careful integration of the data from multiple sources may help reduce/avoid redundancies and inconsistencies and improve mining speed and quality

21 21 Correlation Analysis (Nominal Data) Χ 2 (chi-square) test The larger the Χ 2 value, the more likely the variables are related

22 Example Suppose that a group of 1500 people was surveyed. The gender of each person was noted. Each person was polled as to whether his or her preferred type of reading material was fiction or nonfiction. Thus, we have two attributes, gender and preferred reading. The observed frequency (or count) of each possible joint event is summarized in the contingency table 9/9/2015Data Mining: Concepts and Techniques22

23 23 Correlation Analysis (Nominal Data) Χ 2 (chi-square) test The larger the Χ 2 value, the more likely the variables are related The cells that contribute the most to the Χ 2 value are those whose actual count is very different from the expected count Correlation does not imply causality # of hospitals and # of car-theft in a city are correlated Both are causally linked to the third variable: population

24 24 Chi-Square Calculation: An Example Χ 2 (chi-square) calculation (numbers in parenthesis are expected counts calculated based on the data distribution in the two categories) It shows that like_science_fiction and play_chess are correlated in the group Play chessNot play chessSum (row) Like science fiction250(90)200(360)450 Not like science fiction50(210)1000(840)1050 Sum(col.)30012001500

25 25 Correlation Analysis (Numeric Data) Correlation coefficient (also called Pearson’s product moment coefficient) where n is the number of tuples, and are the respective means of A and B, σ A and σ B are the respective standard deviation of A and B, and Σ(a i b i ) is the sum of the AB cross-product. If r A,B > 0, A and B are positively correlated (A’s values increase as B’s). The higher, the stronger correlation. r A,B = 0: independent; r AB < 0: negatively correlated

26 26 Visually Evaluating Correlation Scatter plots showing the similarity from –1 to 1.

27 Homework 3 (2) 27

28 28 Correlation (viewed as linear relationship) Correlation measures the linear relationship between objects To compute correlation, we standardize data objects, A and B, and then take their dot product

29 29 Covariance (Numeric Data) Covariance is similar to correlation where n is the number of tuples, and are the respective mean or expected values of A and B, σ A and σ B are the respective standard deviation of A and B. Positive covariance: If Cov A,B > 0, then A and B both tend to be larger than their expected values. Negative covariance: If Cov A,B < 0 then if A is larger than its expected value, B is likely to be smaller than its expected value. Independence: Cov A,B = 0 but the converse is not true: Some pairs of random variables may have a covariance of 0 but are not independent. Only under some additional assumptions (e.g., the data follow multivariate normal distributions) does a covariance of 0 imply independence Correlation coefficient:

30 Co-Variance: An Example It can be simplified in computation as Suppose two stocks A and B have the following values in one week: (2, 5), (3, 8), (5, 10), (4, 11), (6, 14). Question: If the stocks are affected by the same industry trends, will their prices rise or fall together? E(A) = (2 + 3 + 5 + 4 + 6)/ 5 = 20/5 = 4 E(B) = (5 + 8 + 10 + 11 + 14) /5 = 48/5 = 9.6 Cov(A,B) = (2×5+3×8+5×10+4×11+6×14)/5 − 4 × 9.6 = 4 Thus, A and B rise together since Cov(A, B) > 0.

31 31 Chapter 3: Data Preprocessing Data Preprocessing: An Overview Data Quality Major Tasks in Data Preprocessing Data Cleaning Data Integration Data Reduction Data Transformation and Data Discretization Summary

32 This is all for today!

33 33 Data Reduction Strategies Data reduction: Obtain a reduced representation of the data set that is much smaller in volume but yet produces the same (or almost the same) analytical results Why data reduction? — A database/data warehouse may store terabytes of data. Complex data analysis may take a very long time to run on the complete data set. Data reduction strategies Dimensionality reduction, e.g., remove unimportant attributes Wavelet transforms Principal Components Analysis (PCA) Feature subset selection, feature creation Numerosity reduction (some simply call it: Data Reduction) Regression and Log-Linear Models Histograms, clustering, sampling Data cube aggregation Data compression

34 34 Data Reduction 1: Dimensionality Reduction Curse of dimensionality When dimensionality increases, data becomes increasingly sparse Density and distance between points, which is critical to clustering, outlier analysis, becomes less meaningful The possible combinations of subspaces will grow exponentially Dimensionality reduction Avoid the curse of dimensionality Help eliminate irrelevant features and reduce noise Reduce time and space required in data mining Allow easier visualization Dimensionality reduction techniques Wavelet transforms Principal Component Analysis Supervised and nonlinear techniques (e.g., feature selection)

35 35 Mapping Data to a New Space Two Sine Waves Two Sine Waves + NoiseFrequency Fourier transform Wavelet transform

36 36 What Is Wavelet Transform? Decomposes a signal into different frequency subbands Applicable to n- dimensional signals Data are transformed to preserve relative distance between objects at different levels of resolution Allow natural clusters to become more distinguishable Used for image compression

37 37 Wavelet Transformation Discrete wavelet transform (DWT) for linear signal processing, multi-resolution analysis Compressed approximation: store only a small fraction of the strongest of the wavelet coefficients Similar to discrete Fourier transform (DFT), but better lossy compression, localized in space Method: Length, L, must be an integer power of 2 (padding with 0’s, when necessary) Each transform has 2 functions: smoothing, difference Applies to pairs of data, resulting in two set of data of length L/2 Applies two functions recursively, until reaches the desired length Haar2 Daubechie4

38 38 Wavelet Decomposition Wavelets: A math tool for space-efficient hierarchical decomposition of functions S = [2, 2, 0, 2, 3, 5, 4, 4] can be transformed to S ^ = [2 3 / 4, -1 1 / 4, 1 / 2, 0, 0, -1, -1, 0] Compression: many small detail coefficients can be replaced by 0’s, and only the significant coefficients are retained

39 39 Haar Wavelet Coefficients Coefficient “Supports” 2 2 0 2 3 5 4 4 -1.252.750.5 0 0 0 + - + + + ++ + + - - ---- + - + + - + - + - + - - + + - 0.5 0 2.75 -1.25 0 0 Original frequency distribution Hierarchical decomposition structure (a.k.a. “error tree”)

40 40 Why Wavelet Transform? Use hat-shape filters Emphasize region where points cluster Suppress weaker information in their boundaries Effective removal of outliers Insensitive to noise, insensitive to input order Multi-resolution Detect arbitrary shaped clusters at different scales Efficient Complexity O(N) Only applicable to low dimensional data

41 41 x2x2 x1x1 e Principal Component Analysis (PCA) Find a projection that captures the largest amount of variation in data The original data are projected onto a much smaller space, resulting in dimensionality reduction. We find the eigenvectors of the covariance matrix, and these eigenvectors define the new space

42 42 Given N data vectors from n-dimensions, find k ≤ n orthogonal vectors (principal components) that can be best used to represent data Normalize input data: Each attribute falls within the same range Compute k orthonormal (unit) vectors, i.e., principal components Each input data (vector) is a linear combination of the k principal component vectors The principal components are sorted in order of decreasing “significance” or strength Since the components are sorted, the size of the data can be reduced by eliminating the weak components, i.e., those with low variance (i.e., using the strongest principal components, it is possible to reconstruct a good approximation of the original data) Works for numeric data only Principal Component Analysis (Steps)

43 43 Attribute Subset Selection Another way to reduce dimensionality of data Redundant attributes Duplicate much or all of the information contained in one or more other attributes E.g., purchase price of a product and the amount of sales tax paid Irrelevant attributes Contain no information that is useful for the data mining task at hand E.g., students' ID is often irrelevant to the task of predicting students' GPA

44 44 Heuristic Search in Attribute Selection There are 2 d possible attribute combinations of d attributes Typical heuristic attribute selection methods: Best single attribute under the attribute independence assumption: choose by significance tests Best step-wise feature selection: The best single-attribute is picked first Then next best attribute condition to the first,... Step-wise attribute elimination: Repeatedly eliminate the worst attribute Best combined attribute selection and elimination Optimal branch and bound: Use attribute elimination and backtracking

45 45 Attribute Creation (Feature Generation) Create new attributes (features) that can capture the important information in a data set more effectively than the original ones Three general methodologies Attribute extraction Domain-specific Mapping data to new space (see: data reduction) E.g., Fourier transformation, wavelet transformation, manifold approaches (not covered) Attribute construction Combining features (see: discriminative frequent patterns in Chapter 7) Data discretization

46 46 Data Reduction 2: Numerosity Reduction Reduce data volume by choosing alternative, smaller forms of data representation Parametric methods (e.g., regression) Assume the data fits some model, estimate model parameters, store only the parameters, and discard the data (except possible outliers) Ex.: Log-linear models—obtain value at a point in m- D space as the product on appropriate marginal subspaces Non-parametric methods Do not assume models Major families: histograms, clustering, sampling, …

47 47 Parametric Data Reduction: Regression and Log-Linear Models Linear regression Data modeled to fit a straight line Often uses the least-square method to fit the line Multiple regression Allows a response variable Y to be modeled as a linear function of multidimensional feature vector Log-linear model Approximates discrete multidimensional probability distributions

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