SE 3S03 - Tutorial 5. Alicia Marinache. Week of Feb 09, 2015

Transcription

1 SE 3S03 - Tutorial 5 Department of Computer Science McMaster University Week of Feb 09, 2015 Acknowledgments: The material of these slides is based on [1] (chapter 2) 1/16

2 Outline 2/16

3 I I I I 3/16

4 Definition Let (P, ) be an empirical relational system, where P is a non-empty countable set and is an equivalence relation on P. Let (R, =) be numerical mathematical structure with R its carrier set and = is its identity relation. Let µ : P! R be a real value function. The system ((P, ), (R, =),µ)isanominal scale i 8(p, q p, q 2 P : p q, µ(p) =µ(q)) I Nominal: Latin nomine, name I Labels, grouping I No quantitative value, either in or out 4/16

5 - examples 5/16

6 - examples I Error fault type: code, design, specification 5/16

7 - examples I Error fault type: code, design, specification 8 >< 1 if x is specification fault I µ 1 (x) = 2 if x is design fault >: 3 if xiscodefault 5/16

8 - examples I Error fault type: code, design, specification 8 >< 1 if x is specification fault I µ 1 (x) = 2 if x is design fault >: 3 if xiscodefault I Example 2:... 5/16

9 - examples I Error fault type: code, design, specification 8 >< 1 if x is specification fault I µ 1 (x) = 2 if x is design fault >: 3 if xiscodefault I Example 2:... I Admissible : 5/16

10 - examples I Error fault type: code, design, specification 8 >< 1 if x is specification fault I µ 1 (x) = 2 if x is design fault >: 3 if xiscodefault I Example 2:... I Admissible : any two mappings µ 1,µ 2 can be related by... 5/16

11 Definition Let (P, ) be an empirical relational system, where P is a non-empty countable set and is an empirical relation describing ranking properties on P. Let (R, ) benumerical mathematical structure with R its carrier set and its partial order. Let µ : P! R be a real value function. The system ((P, ), (R, ), µ)isanordinal scale i 1. 8(p, q p, q 2 P : p q, µ(p) µ(q)) 2. 8(p, q, r p, q, r 2 P : p q ^ q r, p r) (Transitivity) 3. 8(p, q p, q 2 P : p q _ q p) (Completeness) 6/16

12 I Ordinal: Latin ordin - related to order in a sequence I Ranking, ordering I Mappings must preserve the ordering I Arithmetic operations allowed? How about mean, median, mode? I Example: 7/16

13 I Ordinal: Latin ordin - related to order in a sequence I Ranking, ordering I Mappings must preserve the ordering I Arithmetic operations allowed? How about mean, median, mode? I Example: Fault priority: low, average, high, urgent; the relation is more important I µ 1 (x) = 7/16

14 I Ordinal: Latin ordin - related to order in a sequence I Ranking, ordering I Mappings must preserve the ordering I Arithmetic operations allowed? How about mean, median, mode? I Example: Fault priority: low, average, high, urgent; the relation is more important 8 1 if xislow >< 3 if x is average I µ 1 (x) = 5 if x is high >: 10 if x is urgent I Admissible : 7/16

15 I Ordinal: Latin ordin - related to order in a sequence I Ranking, ordering I Mappings must preserve the ordering I Arithmetic operations allowed? How about mean, median, mode? I Example: Fault priority: low, average, high, urgent; the relation is more important 8 1 if xislow >< 3 if x is average I µ 1 (x) = 5 if x is high >: 10 if x is urgent I Admissible : any two mappings µ 1,µ 2 can be related by... 7/16

16 Definition Let (PxP, ) be an algebraic di erence structure. Let µ : P! R be a real value function. The system ((PXP, ), (RX R, ),µ)isaninterval scale i 1. 8(a, b, c, d a, b, c, d 2 P :(a, b) (c, d), µ(a) µ(b) µ(c) µ(d)) 2. If another function, g, satisfies property (1), 9, 2 R,, >0s.t. g(x) = µ(x) + holds 8/16

17 I Does it preserve order?... 9/16

18 I Does it preserve order?... I What else it preserves?... 9/16

19 I Does it preserve order?... I What else it preserves?... I Can I add / subtract? Why? 9/16

20 I Does it preserve order?... I What else it preserves?... I Can I add / subtract? Why? I Can I multiply / divide? Why? 9/16

21 I Does it preserve order?... I What else it preserves?... I Can I add / subtract? Why? I Can I multiply / divide? Why? I Example: we can add the notion of di erence in... for an ordinal scale. 9/16

22 I Does it preserve order?... I What else it preserves?... I Can I add / subtract? Why? I Can I multiply / divide? Why? I Example: we can add the notion of di erence in... for an ordinal scale. 9/16

23 Definition Let (P,, ) be an empirical relational system, where P is a non-empty countable set, is an empirical relation describing ranking properties on P, and is a binary operation on P. Let(R,,+)benumericalmathematical structure with R its carrier set, its partial order, and + the addition on R. Letµ : P! R be a real value function. The system ((P,, ), (R,, +),µ)isaratio scale i 1. 8(p, q p, q 2 P : p q, µ(p) µ(q)) 2. 8(p, q p, q 2 P : µ(p q) =µ(p)+µ(q)) 10/16

24 I The mapping (µ) inaratioscalepreserves: 11/16

25 I The mapping (µ) inaratioscalepreserves:ordering, di erence, and ratio I Preservation of ratio is made possible by the existence of a zero element, the lack of the measured attribute I Measurements start at zero and increase in equal intervals, called units I Example: length of software code. What is zero? Units for length: 11/16

26 I The mapping (µ) inaratioscalepreserves:ordering, di erence, and ratio I Preservation of ratio is made possible by the existence of a zero element, the lack of the measured attribute I Measurements start at zero and increase in equal intervals, called units I Example: length of software code. What is zero? Units for length: # of lines, KLOC, # of characters, # of executable statements 11/16

27 I The type of scale determine what kind of analysis we can perform on the s I Consider these statements: 1. The # of errors discovered during integration testing was The cost of fixing each error is at least A semantic error takes twice as long to fix than an syntactic error. 4. A semantic error is twice as complex as a syntactic error. I A statement involving is meaningful if its truth value is invariant of of allowable scales. 12/16

28 1. Fred is twice as tall as Jane. What type of scale is it used? Is the statement meaningful? Prove it. 2. The temperature in Tokyo (40 C) today is twice that in London (20 C). Is it meaningful? Can you think of a way to transform the statement to make it meaningful? 3. Failure x is twice as critical as failure y. Is it a meaningful statement? Why? 4. One manager measures the months spent on project X since work started (Jan 1, 2014) and one manager measures the years spent on same project X since the contract has been signed (May 1, 2013). What types of scales are? Is there an admissible transformation between them? 13/16

29 1. We define speed of software w.r.t. an empirical binary relation faster than and we observe that A is faster than B, which is faster than C. 2. Define the simplest scale... 14/16

30 1. We define speed of software w.r.t. an empirical binary relation faster than and we observe that A is faster than B, which is faster than C. 2. Define the simplest scale Let µ and µ 0 be 2 admissible measures. Describe the relation between the 2 measures. 4. Is the statement Program A is faster than both B and C meaningful? Why? 5. Is the statement Program B is faster than both A and C meaningful? Why? 6. We define µ as: µ(a) = 3; µ(b) = 2; µ(c) = 1. Is the statement Program A is more than twice as fast as Program C a meaningful statement? Why? 14/16

31 I Understanding the scale means understanding what analytics can be performed on the s. I Meaningfulness does not mean useful, practical, or easy to collect. A statement is meaningful if its truth value is invariant of of the scale. 15/16

32 References I Appendix References N. Fenton; S.L. Pfleeger Software Metrics A Rigorous and Practical Approach. PWS, /16