1. MATLAB Logical Indexing Greg Reese, Ph.D Research Computing Support Group Academic Technology Services Miami University

2. MATLAB Logical Indexing © 2010-2013 Greg Reese. All rights reserved 2

3. 3 Logical indexing One method of selecting elements in a matrix or vector is to choose those that meet some criterion, e.g., – the elements that are greater than zero –the elements that are not more than three standard deviations from the mean MATLAB does this by logical indexing and relational operators

4. 4 Logical indexing MATLAB has the usual relational operators To get more information, type help ops SymbolOperation ==Equal ~=Not equal <Less than >Greater than <=Less than or equal >=Greater than or equal

5. 5 Logical indexing The result of comparing two scalars with a relational operator is a logical value. A logical value can only be true or false MATLAB represents true by the number 1 and false by 0 Can also use the keywords true and false

6. 6 Logical indexing Example >> x=5; >> x>5 ans = 0 >> x<=5 ans = 1 >> x==5 ans = 1 >> x~=5 ans = 0

7. 7 Logical indexing When compare vector to vector, or vector to scalar, MATLAB Does elementwise comparison – Compares scalar to every element of vector – Compares corresponding elements of vectors Vectors must be same dimension Result is vector of logical values Result has same dimension as vector in comparison

8. 8 Logical indexing Example >> x=8:12 x = 8 9 10 11 12 >> x>10 ans = 0 0 0 1 1 >> x==11 ans = 0 0 0 1 0 >> x>=7 ans = 1 1 1 1 1

9. 9 Logical indexing Tip It helps to picture in your mind that the result of a logical comparison 1.Is a vector 2.Has a 0 or 1 corresponding to each original element >> x=8:12 x = 8 9 10 11 12 >> x>10 ans = 0 0 0 1 1

10. 10 Logical indexing Try It >> times = [ 9.7 9.9 10.1 12.8 ]; Mark times less than nine >> times < 9 ans = 0 0 0 0 Mark times greater than 12 >> times > 12 ans = 0 0 0 1 Mark times equal to 9.9 >> times == 9.9 ans = 0 1 0 0 ==, not =

11. 11 Logical indexing MATLAB provides logical operations such as AND, OR, NOT, etc. Can compare two scalars If compare scalar to vector or vector to vector, do elementwise comparison –If two vectors, must both be same dimension To get more information, type help ops

12. 12 Logical indexing SymbolOperation &AND – true if both elements are nonzero, false otherwise |OR – true if one or both elements are nonzero, false otherwise xor()EXCLUSIVE OR – true if exactly one element is nonzero, false otherwise ~NOT – makes nonzero element zero and zero element one any()True if any element of a vector is nonzero, false otherwise all()True if all elements of a vector are nonzero, false otherwise

13. 13 Logical indexing Example Child – 12 or less years Teenager – more than 12 and less than 20 years Adult – 20 or more years >> age=[45 47 15 13 11] age = 45 47 15 13 11

14. 14 Logical indexing Example age = 45 47 15 13 11 Who is a teenager? >> age>=13 ans = 1 1 1 1 0 >> age<=19 ans = 0 0 1 1 1 >> age>=13 & age<=19 ans = 0 0 1 1 0

15. 15 Logical indexing Example >> age=[45 47 15 13 11] age = 45 47 15 13 11 Who is not a teenager? >> ~(age>=13 & age<=19) ans = 1 1 0 0 1 Who is an adult or a child? >> age>19 | age<13 ans = 1 1 0 0 1

16. 16 Logical indexing Example >> age=[45 47 15 13 11] age = 45 47 15 13 11 Are there any teenagers? >> any( age >= 13 & age <= 19 ) ans = 1 Are all the people teenagers? >> all( age >= 13 & age <= 19 ) ans = 0

17. 17 Logical indexing For even more power you can use logical values as subscripts in a vector or matrix. This is called logical indexing or logical subscripting. To perform logical subscripting on a vector x, pass it (in parentheses) a logical vector of the same dimension. The result is a vector of all the elements of x for which the logical vector is true.

18. 18 Logical indexing Example >> numbers = 1:8 numbers = 1 2 3 4 5 6 7 8 % find even numbers greater than 4 >> matches =... rem( numbers, 2 ) == 0 & numbers > 4 matches = 0 0 0 0 0 1 0 1 >> numbers(matches) ans = 6 8

19. 19 Logical indexing Example numbers = 1 2 3 4 5 6 7 8 matches = 0 0 0 0 0 1 0 1 Think of numbers(matches) as pulling out of numbers all elements that have a 1 in the corresponding element of matches numbers = 1 2 3 4 5 6 7 8 matches = 0 0 0 0 0 1 0 1 >> numbers(matches) ans = 6 8

20. 20 Logical indexing Try It >> age=[45 47 15 13 11] age = 45 47 15 13 11 Mark teenagers with 1, non-teens with 0 >> age>=13 & age<=19 ans = 0 0 1 1 0 How old are the teenagers? >> age( age >= 13 & age <= 19) ans = 15 13

21. 21 Logical indexing Tip If you’re going to use the results of a calculation a lot, compute it once and save it in a variable.

22. 22 Logical indexing Example >> age=[45 47 15 13 11]; >> weight=[202 151 113 125 94]; >> teenager = age>=13 & age<=19; >> age(teenager) ans = 15 13 >> weight(teenager) ans = 113 125

23. 23 Logical indexing Tip Since relational and logical operations return 1 if they meet a criterion and 0 if they don’t, you can count the number of elements in a vector that meet a criterion by finding the sum of the logical vector result. The function sum returns the sum of a vector’s elements

24. 24 Logical indexing Example >> age=[45 47 15 13 11]; >> weight=[202 151 113 125 94]; How many teenagers are there? >> age >= 13 & age <= 19 ans = 0 0 1 1 0 >> sum( age >= 13 & age <= 19 ) ans = 2

25. 25 Logical indexing Example >> age=[45 47 15 13 11]; >> weight=[202 151 113 125 94]; How many people weigh more than 200 lbs? >> weight > 200 ans = 1 0 0 0 0 >> sum( weight > 200 ) ans = 1

26. 26 Logical indexing Try It >> age=[45 47 15 13 11]; >> weight=[202 151 113 125 94]; How many adults are there? How many children are there? >> sum( age >= 20 ) ans = 2 >> sum( age < 13 ) ans = 1

27. 27 Logical indexing Try It >> age=[45 47 15 13 11]; >> weight=[202 151 113 125 94]; How many adults weigh more than 200 lbs? How many children weigh less than 100 lbs? >> sum( age >= 20 & weight > 200 ) ans = 1 >> sum( age < 13 & weight < 100 ) ans = 1

28. 28 Logical indexing Tip One good use of logical indexing is the detection and removal of outliers – Outliers are whacky data values, i.e., numbers way off the mean – Conventional definition is any number more than 3 standard deviations from mean, i.e., x is an outlier if

29. 29 Logical indexing Tip >> data = randn( 1, 100 ); >> [ theMax maxIx ] = max( data ) theMax = 3.5784 maxIx = 9 >> [ theMin minIx ] = min( data ) theMin = -2.9443 minIx = 35 % put in new min/max but keep old ones >> data(maxIx+1) = 988.64; >> data(minIx+1) = -2000;

30. 30 Logical indexing Tip - show outliers >> data( abs(data-mean(data)) > 3*std(data) ) ans = 1.0e+003 * 0.9886 -2.0000

31. 31 Logical indexing Remember, can delete elements by specifying indexes to delete and setting to [], e.g., >> v=2:2:10 v = 2 4 6 8 10 >> v([2 5]) = [] v = 2 6 8

32. 32 Logical indexing Can delete elements that meet certain criteria in an analogous way: v(logical_vector) = [] Example Delete all elements that are a multiple of 4 >> v=2:2:10 v = 2 4 6 8 10 >> v(rem(v,4)==0) = [] v = 2 6 10

33. 33 Logical indexing Tip - can remove outliers % show outliers >> data( abs(data-mean(data))>3*std(data) ) ans = 1.0e+003 * 0.9886 -2.0000 % remove outliers >> data( abs(data-mean(data))>3*std(data) )=[]; >> length( data ) ans = 98 >> max( data ) ans = 3.5784 >> min( data ) ans = -2.9443

34. 34 Logical indexing The find function is related to logical indexing. It returns the indexes of the elements of a vector that are nonzero. You can use those indexes to select the elements of a vector that meet a criterion find is most helpful when used on logical vectors

35. 35 Logical indexing Example >> age=[45 47 15 13 11]; >> find( age >= 13 & age <= 19 ) ans = 3 4 How old are the teenagers? >> age( find( age>=13 & age <=19 )) ans = 15 13

36. 36 Logical indexing Example – use find >> age=[45 47 15 13 11]; >> weight=[202 151 113 125 94]; How old is the second teenager? >> indexes = find(age >= 13 & age <= 19) indexes = 3 4 >> age( indexes(2) ) ans = 13

37. 37 Logical indexing Try It – use find >> age=[45 47 15 13 11]; >> weight=[202 151 113 125 94]; How much does the second teenager weigh? >> indexes = find( age >= 13 & age <= 19 ) indexes = 3 4 >> weight( indexes(2) ) ans = 125

38. 38 Logical Indexing NaN (Not a Number) Built-in constant (can also write nan) Represents result of mathematically undefined operations, such as 0 / 0 or ∞ - ∞ (infinity minus infinity) Any logical or relational comparison of two NaN’s returns false, except ~= isnan(x) returns true if x is NaN and false otherwise

39. 39 Logical Indexing NaN used to mark missing data points Example Two students didn’t show up for a quiz: >> grades=[3 10 8 NaN 0 8 7 5 NaN 6]; >> isnan( grades ) ans = 0 0 0 1 0 0 0 0 1 0 >> find( isnan( grades ) ) ans = 4 9

40. 40 Logical Indexing NaN messes up numerical computations. Remove NaN’s from data first Example >> grades=[3 10 8 NaN 0 8 7 5 NaN 6]; >> mean(grades) ans = NaN >> validGrades = grades( ~isnan(grades) ) validGrades = 3 10 8 0 8 7 5 6 >> mean( validGrades ) ans = 5.8750

41. 41 Logical indexing Tip If you're going to work a lot with data that has missing values, the statistics toolbox can help you. It has functions that compute common statistics and automatically ignore NaN's. They are: nancov, nanmax, nanmean, nanmedian, nanmin, nanstd, nansum, nanvar Example >> v = [ 10 NaN 6 2 NaN ]; >> mean( v ) ans = NaN >> nanmean( v ) ans = 6

42. 42 Logical Indexing In the fall of 2004 non-freshmen students at Miami University were surveyed. One of the questions was to write the number of hours of anti-alcohol education the students had received at Miami. The responses are numbers from 0 to 4 with 0 to 3 being the actual number of hours and 4 representing more than 3 hours. Some students did not respond at all. Load the answers to the question with the command >> allAnswers = load( 'survey.txt' )

43. 43 Logical Indexing Try It In the survey, students who didn’t answer the question are represented by NaN. Answer the following questions (without nanxxx functions): How many students were in the survey? How many students did not answer the survey question?

44. 44 Logical indexing >> allAnswers=load('survey.txt'); % Number of students in survey? >> length(allAnswers) ans = 531 % Number of students not responding? >> sum(isnan(allAnswers)) ans = 59

45. 45 Logical Indexing Try It Make a vector for only the students who answered. Use it to answer the following questions about the amount of college alcohol-prevention education the students had. Are all the values in your new vector legal, i.e., between 0 and 4 inclusive?

46. 46 Logical indexing >> good=allAnswers( ~isnan(allAnswers) ); % Are all responses legal (>=0 and <=4)? >> all(good>=0 & good<=4 ) ans = 1

47. 47 Logical Indexing Try It What percentage didn’t have any prevention education? For every student who had the maximum amount of prevention education, how many had none? What was the average number of hours of prevention education for students who had 1, 2, or 3 such hours?

48. 48 Logical indexing % Percentage uneducated? >> 100*sum(good==0)/length(good) ans = 39.6186 % uneducated / maximally educated? >> sum(good==0)/sum(good==4) ans = 1.8515 % average for those with some but not all >> mean( good( good>=1 & good<=3 ) ) ans = 2.0435

49. 49 Logical indexing FINAL POINT Note that we have answered all the questions 1.Without using if-statements 2.Without using loops Logical indexing makes cleaner, faster code!

50. 50 Logical indexing Questions?

51. 51 The End