Statistics and Data Mining


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Statistics and Data Mining

  1. 1. Statistics & Data Mining R. Akerkar TMRF, Kolhapur, India Data Mining - R. Akerkar 1
  2. 2. Why Data Preprocessing? Data in the real world is dirty y  incomplete: lacking attribute values, lacking certain attributes of interest, or containing only aggregate data  e.g., occupation= e g occupation=“”  noisy: containing errors or outliers  e.g., Salary=“-10”  inconsistent: containing discrepancies in codes or names  e.g., Age=“42” Birthday=“03/07/1997”  e.g., Was rating “1,2,3”, now rating “A, B, C 1,2,3 , A, C”  e.g., discrepancy between duplicate records  This is the Thi i th real world… l ld Data Mining - R. Akerkar 2
  3. 3. Why Is Data Dirty? y y Incomplete data comes from  n/a data value when collected /  different consideration between the time when the data was collected and when it is analyzed.  human/hardware/software problems Noisy data comes from the process of data  Collection instrument’s f lt C ll ti i t t’ fault  Data entry  transmission Inconsistent data comes from  Different data sources  Functional d F ti l dependency violation d i l ti Data Mining - R. Akerkar 3
  4. 4. Why Is Data Preprocessing Important? No quality data no quality mining results! data,  Quality decisions must be based on quality data  e.g., duplicate or missing data may cause incorrect or even g, p g y misleading statistics.  Data warehouse needs consistent integration of quality data Data extraction, cleaning, and transformation comprises the majority of the work of building a data warehouse. Bill warehouse —Bill Inmon Data Mining - R. Akerkar 4
  5. 5. Major Tasks in Data Preprocessing Data cleaning  Fill in missing values, smooth noisy data, and resolve inconsistencies Data integration  Integration of multiple databases, data cubes, or files Data transformation  Normalization and aggregation ( a distance based mining algorithms provide better results if data is normalized and scaled to range.) Data reduction  Obtains reduced representation in volume but p p produces the same or similar analytical results (correlation analysis). Data discretization  Part of data reduction but with particular importance, especially for p p p y numerical data. Data Mining - R. Akerkar 5
  6. 6. Forms of data preprocessing Data Mining - R. Akerkar 6
  7. 7. Data Cleaning g Importance  “Data l “D t cleaning i one of th th i is f the three bi biggest problems i t bl in data warehousing”—Ralph Kimball  “Data cleaning is the number one problem in data warehousing”—DCI survey h i ” DCI Data cleaning tasks  Fill in missing values (time consuming)  Identify outliers and smooth out noisy data  Correct inconsistent data  Resolve redundancy caused by data integration y y g Data Mining - R. Akerkar 7
  8. 8. Missing Data g Data is not always available  E.g., E g many tuples have no recorded value for several attributes 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. Data Mining - R. Akerkar 8
  9. 9. How to Handle Missing Data? Ignore the tuple: usually done when class label is missing (assuming the tasks in classification—not effective when the percentage of missing values per attribute varies considerably. Fill in the missing value manually: t di i th i i l ll tedious + i f infeasible? ibl ? Fill in it automatically with  a global constant : e g “unknown”, a new class?! e.g., unknown  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 or regression. Data Mining - R. Akerkar 9
  10. 10. Noisy Data Noise: random error or variance in a measured variable. For example, for a numeric attribute “price” how can we smooth out the data to remove the noise. Incorrect attribute values may due to  faulty data collection instruments  data entry problems  data transmission problems  technology limitation gy  inconsistency in naming convention Data Mining - R. Akerkar 10
  11. 11. How to Handle Noisy Data? Binning method:  first sort data and partition into (equi-depth) bins  then one can smooth by bin means, smooth by bin median, smooth by bin boundaries etc boundaries, etc. Regression g  smooth by fitting the data into regression functions Data Mining - R. Akerkar 11
  12. 12. Binning Methods for Data Smoothing* Binning methods smooth a sorted data by consulting its neighborhood. Then sorted values are distributed in number of buckets.* Sorted data for price (in dollars): 4, 8, 9, 15, 21, 21, 24, 25, 26, 28, 29, 34* Partition into (equi-depth) bins of depth 4: - Bin 1: 4, 8, 9, 15 - Bin 2: 21, 21, 24, 25 - Bin 3: 26, 28, 29, 34* Smoothing by bin means: - Bin 1: 9, 9, 9, 9 - Bin 2: 23, 23, 23, 23 - Bin 3: 29, 29, 29, 29* S Smoothing by bin boundaries: thi b bi b d i - Bin 1: 4, 4, 4, 15 Similarly, smoothing - Bin 2: 21, 21, 25, 25 by bin median can - Bin 3: 26, 26, 26, 34 , , , be employed. Data Mining - R. Akerkar 12
  13. 13. Simple Discretization Methods: Binning Equal-width (distance) partitioning:  Divides the Di id th range i t N i t into intervals of equal size: l f l i uniform grid  if A and B are the lowest and highest values of the g attribute, the width of intervals will be: W = (B –A)/N.  The most straightforward, but outliers may dominate presentation  Skewed data is not handled well.  Binning is Bi i i applied t each i di id l f t li d to h individual feature ( (or attribute). It does not use the class information. Data Mining - R. Akerkar 13
  14. 14.  Equal depth Equal-depth (frequency) partitioning:  Divides the range into N intervals, each containing approximately same number of samples  Good data scaling  Managing categorical attributes can be tricky. Data Mining - R. Akerkar 14
  15. 15. Exercise 1 Suppose the data for analysis includes the attribute Age. The age values for the data tuples (instances) are (in increasing order): 13, 15, 16, 16, 19, 20, 20, 21, 22, 22, 25, 25, 25, 25,30, 33, 33, 35, , , , , , , , , , , , , , , , , , , 35, 35, 35, 36, 40, 45, 46, 52, 70. Use binning (by bin means) to smooth the above data using a bin data, depth of 3. Illustrate your steps, and comment on the effect of this technique for the given data data. Data Mining - R. Akerkar 15
  16. 16. Data Integration g Data integration:  combines data from multiple sources into a coherent store Schema integration  Entity identification E tit id tifi ti problem: identify real world entities f bl id tif l ld titi from multiple data sources, e.g., A.cust-id  B.cust-#  integrate metadata from different sources Detecting and resolving data value conflicts  for the same real world entity, attribute values from different sources so rces are different  possible reasons: different representations, different scales, e.g., metric vs. British units Data Mining - R. Akerkar 16
  17. 17. Handling Redundancy in Data Integration Redundant data occur often when integration of multiple databases  The same attribute may have different names in different databases  One attribute may be a “derived” attribute in another table e g derived table, e.g., annual revenue Redundant data may be able to be detected by correlation analysis Careful integration of the data from multiple sources may help reduce/avoid redundancies and inconsistencies and improve mining speed and quality Data Mining - R. Akerkar 17
  18. 18. Correlation Analysis Redundancies can be detected by this method. Given two attributes, such analysis can measure how strongly one attribute implies the other, based on available data. The correlation between attributes A and B can be measured by Where n is number of tuples, and are respective mean values of A and B, and and are the respective standard deviations of A and B Data Mining - R. Akerkar 18
  19. 19. Correlation Analysis If the resulting value of the equation is greater than 0, then A and B are positively correlated.  i.e. the l i th values of A i f increase as th values of B i the l f increase.  The higher the value, the more each attribute implies the other.  Hence, high value may indicate that A (or B) may be removed as redundancy. redundancy If the resulting value is equal to zero, then A and B are independent  There is no correlation between them. If the resulting value is less than zero, then A and B are negatively correlated.  i.e. the values of one attribute increase as the values of other attribute decrease.  Each attribute discourages the other. Data Mining - R. Akerkar 19
  20. 20. Correlation Analysis Above are three possible relationships between data. The graphs of high positive and negative correlation are approaching a value of 1 and -1 respectively. The graph showing no correlation has a value of 0. Data Mining - R. Akerkar 20
  21. 21. Categorical Data To find the correlation between two categorical attributes we make use of contingency tables. g y Let us consider the following: Let there be 4 car manufacturers given by the set {A,B,C,D} and let there be three segments of cars manufactured by these companies given by the set {S,M,L}, {S M L} where S stands for small cars M stands for cars, medium sized cars and L stands for Large cars. Observer collects data about the cars passing by, manufactured by these companies and categorize them according to their sizes. Data Mining - R. Akerkar 21
  22. 22.  For finding the correlation between car manufacturers and the size of cars that they manufacture we formulate a hypothesis, that the size of car manufactured and the companies that manufacture the cars are independent of each other other. In other terms, we are saying that there is absolutely no correlation between the car manufacturing company and the size of the cars that they manufacture. Such a hypothesis in statistical terms is called the null hypothesis and is denoted by Ho Ho. Null hypothesis: The car size and car manufacturers are attributes independent of each other. other Data Mining - R. Akerkar 22
  23. 23. Data Transformation Smoothing: remove noise from data (binning and regression) Aggregation: summarization data cube construction summarization, E.g. Daily sales data aggregated to compute monthly and annual total amount. Generalization: concept hierarchy climbing Normalization is useful for classification algorithms involving neural nets, clustering etc.. Normalization: attribute data are scaled to fall within a small, specified range such as – 1.0 to 1.0  min-max normalization  z-score normalization  normalization by decimal scaling Attribute/feature construction  New attributes constructed f from the given ones Data Mining - R. Akerkar 23
  24. 24. Data Transformation: Normalization min-max normalization (This type of normalization transforms the data into a desired range, usually [0,1]. ) v  minA v  (new _ maxA  new _ minA)  new _ minA maxA  minA where, [minA, maxA] is the initial range and [new_minA, new_maxA] is the ,[ , ] g [ , ] new range. e.g.: If v = 73600 in [12000, 98000] Then v’ = 0.716 in the range [0, 1]. Here value for “income” is transformed to 0.716 It preserves the relationship among th original d t values. th l ti hi the i i l data l Data Mining - R. Akerkar 24
  25. 25. z-score normalizationBy using this type of normalization, the mean of the transformed set of data points is reduced to zero. For this, the mean and standard deviation of the initial set of data values are required. The t Th transformation formula is f ti f l i v  mean A v stand _ dev AWhere, meanA and std_devA are the mean and standard deviation of the initial data values.e.g.: If meanIncome = 54000, and std_devIncome = 16000, then v = 76000 transformed to v’ =1.225.This is useful when the actual min and max of attribute are unknown. Data Mining - R. Akerkar 25
  26. 26. Normalisation by Decimal Scaling This type of scaling transforms the data into a range between [- 1,1]. The transformation formula is v v j 10Where j is the smallest integer such that Max(| v |)<1 e.g.: Suppose recorded value of A is in initial range [-991, 99], k is 3, and v = -991 becomes v = -0.991. The Th mean absolute value of A is 991. b l t l f i 991 To normalise, we divide each value by 1000 (i.e. j = 3) so -991 normalises -0.991 Data Mining - R. Akerkar 26
  27. 27. Exercise 2 Using the data for Age in previous Question, answer the following:a) Use min-max normalization to transform the value 35 for age into the range [0.0; 1.0].b) Use z-score normalization to transform the value 35 for age, where the standard deviation of age is 12.94.c) Use normalization by decimal scaling to transform the value 35 for age.d) Comment on which method you would prefer to use for the given data, giving reasons as to why. Data Mining - R. Akerkar 27
  28. 28. What Is Prediction? Prediction is similar to classification  First, construct a model  Second, use model to predict unknown value  Major method for prediction is regression  Linear and multiple regression  Non-linear regression Prediction is different from classification P di ti i diff tf l ifi ti  Classification refers to predict categorical class label  Prediction models continuous-valued functions E.g. A model to predict the salary of university graduate with 15 years of work experience. Data Mining - R. Akerkar 28
  29. 29. Regression Regression shows a relationship between the average values of two variables. Thus regression is very useful in estimating and predicting the average value of one variable for a given value of other variable. The estimate or prediction may be made with the help of a regression line. There are two types of variables in regression analysis- independent variable and dependent variable. The variable whose value is to be predicted is called dependent variable and th variable whose value i used f prediction i i bl d the i bl h l is d for di ti is called independent variable. Data Mining - R. Akerkar 29
  30. 30.  Linear regression: If the regression curve is a straight g g g line, then there is a linear regression between two variables. Linear regression models a random variable, Y (called variable response variable) as a linear function of another random variable, X ( called a predictor variable) Y=+X  Two parameters ,  and  specify the line and are to be estimated by using the data at hand. (regression coefficients)  The variance of Y is assumed to be constant.  Th coefficients can be solved f The ffi i t b l d for b th method of by the th d f least squares (minimizes the error between the actual data and the estimate of the line. ) Data Mining - R. Akerkar 30
  31. 31. Linear Regression Given s samples or data points of the form (x1, y1), (x2, y2) …(xs, ys) The regression coefficients can be estimated as, Where is the average of x1, x2 … and is the average of y1, y2,… Data Mining - R. Akerkar 31
  32. 32. Multiple Regression Multiple regression: Y =  + 1 X1 + 2 X2. p g  Many nonlinear functions can be transformed into the above.  The regression analysis for studying more than two variables at a time time.  It involves more than one predictor variable.  Method of least square can be applied to solve for , 1, and 2. Data Mining - R. Akerkar 32
  33. 33. Non-Linear Regression If the curve of regression is not a straight line, i.e., a first degree equation in the variables x and y, then it is called a non-linear regression or curvilinear regression Consider a cubic polynomial relationship,  Y =  +  1X +  2X2 +  3X3. To convert above equation in linear form, we define new variable:  X1 = X, X2 = X2, X3 = X3 Thus we get, g  Y =  +  1X1 +  2 X2+  3 X3. This is solvable by the method of least squares squares. Data Mining - R. Akerkar 33
  34. 34. Exercise 3 Following table shows a set of X Y paired data where X is the number Years Experience Salary (in $ 1000s) of years of work experience of a 3 30 college graduates and Y is the 8 57 corresponding salary of the 9 64 graduate. 13 72 Draw a graph of the data. Do X 3 36 and Y seem to have a linear 6 43 relationship? 11 59 Also, predict the salary of a 21 90 college graduate with 10 years of 1 20 experience. experience 16 83 Data Mining - R. Akerkar 34
  35. 35. Assignment X Y Midterm exam Final exam The following table shows the midterm and 72 84 final exam grades obtained for students in a 50 63 data mining course. 81 77 74 781. Plot the data. Do X and Y seem to have a 94 90 linear relationship? 86 752. Use the method of least squares to find an 59 49 equation for the prediction of a student’s student s 83 79 final grade based on the student’s midterm 65 77 grade in the course. 33 523. Predict the final grade of a student who g 88 74 received an 86 on the midterm exam. 81 90 Data Mining - R. Akerkar 35