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1 © Amit Mitra & Amar Gupta DOMAINS OF MEANING

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2 © Amit Mitra & Amar Gupta Parameter/ Feature Directional?SubtypesValid in (Space) AssociationYPatterns of AssociationsAll Sequencing PatternsOrdinal Dimensionality NDimensionality of state spaceAll Dimensionality of patternOrdinal & subtypes Cohesion/ Separation YPatterns of distinctionAll Ranking patternsNominal & Subtypes Patterns of separation in terms of quantitative differences (eg: differences in military rank) Ordinal & Subtypes Patterns of separation in terms of ratios of separation (eg: Physical distance) Difference scaled & Subtypes LocationYAbsolute locationSpaces with “Nil” value (Ratio scaled and Ordinal with Nil value) Differences in absolute locationOrdinal with Nil value Ratios of absolute locationRatio scaled space Inclusion vs. Exclusion YPatterns of inclusionNominal & Subtypes Patterns of exclusionNominal & Subtypes ExtentYInfiniteNominal & Subtypes FiniteNominal & Subtypes DelimitationYUnbounded (Undelimited)Nominal & Subtypes Bounded (Delimited)Ordinal & Subtypes OpenDifference Scaled & Subtypes ClosedDifference Scaled & Subtypes UNIVERSAL PROPERTIES OF PATTERNS Order of a pattern (pattern of patterns, pattern of pattern of patterns etc.) Degrees of freedom (information carrying capacity) Partition Subtype of Cardinality (No. of participating Objects) May be infinite

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3 © Amit Mitra & Amar Gupta Subtype of NOMINAL DOMAIN ORDINAL DOMAIN DOMAINS WITH BOUNDS UNKNOWN DOMAIN DOMAINS WITH NIL VALUES RATIO SCALED DOMAIN Subtype of DIFFERENCE SCALED DOMAIN Subtype of ALL/NOTHING DOMAINS WITH LOWER BOUNDS DOMAINS WITH UPPER BOUNDS Subtype of Information content limits the meanings domains can convey Comparison, discrimination and equality start here Sequence, magnitude start here Quantitative difference starts here Multiplication & division start here Neutrality starts here The concept of Range starts here The concept of connection/neighborhood & structure start here The concept of existence starts here DENSE DOMAINS The concept of property (attributes, relationships, effects) starts here Effect creates relationship between a value and a temporal object instance Effect switches relationships between temporal object instances and values Subtype of Eg: Physical Space Eg: Time

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4 © Amit Mitra & Amar Gupta NEW DOMAINS FROM OLD Rule 1: Adding meaning (information) to a domain creates a new domain. Therefore… Relationships between domains create new domains –All domains are created from a few fundamental domains (called Primary or Base domains) –Domains derived from relationships between domains are called secondary domains Eg: Velocity may be derived from Length and Time duration, Color Preference may be derived from Color and Preference, Area may be derived from a recursive (multiplication) relationship with Length, Unit Price from the Money and Enumeration domains UOMs of derived domains may be derived from the UOMs of the domains they were created from PERSON VISUAL CAR COLOR PREFERENC E COLOR PREFERENCE PERSON CAR Visualize Car SEE Domains may also be polymorphisms of parent domains; inherit properties of their parent(s) –Eg: Nominal to Difference scaled domain –If Nil is added to a domain, the resulting subtype will inherit all units of measure of the parent, but with zeros reset in every UOM to map to Nil The new UOM is a subtype of the old UOM Eg: The discovery of an absolute zero temperature created the Kelvin measure from the Celsius measure of temperature

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5 © Amit Mitra & Amar Gupta NEW DOMAINS FROM OLD All domains are created from a few fundamental domains (called Primary or Base domains) Rule 2: Multiplication and division relationships create new domains –Eg: Length/Time = velocity; Length x Length = Area Rule 3: Addition and subtraction map back to a subtype of the same domain –Eg: velocity - velocity is velocity (relative velocity); sum of weights is weight –Addition and subtraction between subtypes is permitted Eg: 3 Fruits + 5 Oranges + 2 Apples = 10 Fruit (see suppl. Box 43) –The domain of sums (and differences) adds information on values that were summed (or subtracted) to the parent domain Rule 4: Adding and subtracting values across heterogeneous domains (not even subtypes) is meaningless –Eg: Price + Quantity is a meaningless number, 2 apples + 3 oranges is a meaningless number unless we consider “Fruit”, their common parent See notes on the blackboard on domains of meaning, Box 16 in your text book. Some salient issues: –Domain of gaps is a subtype of its parent domain Inherits UOMs of its parent –Any arithmetic operation in which the Nil value does not cancel out is meaningless in a difference scaled domain Addition is meaningless –The unknown nil value may be arbitrarily mapped to some non-zero number, the result is therefore unknown Averages are meaningful, Subtraction is meaningful: creates the ratio scaled domain of gaps –Enumeration and proportions in exhaustive/non-exhaustive partitions An instance is a special kind of subtype of its parent class –Adding proportions across partitions or across attributes is meaningless

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6 © Amit Mitra & Amar Gupta ORDINAL DOMAIN WITH NIL VALUES DIFFERENCE SCALED DOMAIN MASSLENGTHINFORMATIONMONEY Subtype of PRIMARY DOMAIN PARTITION (NOT EXHAUSTIVE) PRIMARY RATIO SCALED DOMAIN Subtract 2 values in ENUMERATION PREFERENCEDATE-TIME ELECTRIC CHARGE Subtype of TIME DURATION RELIABILITY (RISK) ACCURACYVALIDITY COMPLETENESS (EXHAUSTIVITY) Subtype of All domains are created from a few fundamental domains (called Primary or Base domains) –Domains derived from relationships between base domains are called secondary domains Eg: Velocity may be derived from Length and Time duration, Color Preference may be derived from Color and Preference, Area may be derived from a recursive (multiplication) relationship with Length, Unit Price from the Money and Enumeration domains Physical primary domains were defined in the Tenth General Conference on Weights and Measures (1954) –Mass, Length, Temperature, Electric Charge/current, time duration Add the Date-Time domain, aka Time domain. –Duration is derived from the time domain, a ratio scaled subtype of the domain of gaps (Future: Strong, weak forces also?) –Temperature may be substituted by information Buckingham’s Pi Theorem: If a primary domain is dropped from the list, a secondary domain derived from that primary domain must replace it in order to preserve information content Add Preference –Economic Value (money) is a polymorphism of Preference Add Nominal domains DOMAINS OF INFORMATION QUALITY Subtype of

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7 © Amit Mitra & Amar Gupta DOMAINS OF INFORMATION QUALITY Exhaustivity/Completeness –Bounded above (total) and below (nil) Eg: Whether the roster of employees lists all employee names or not Validity (accuracy of interpretation and meaningfulness within a context) –Relates at least two objects Eg: Is length a measure of time? Can the proportion of effort expended exceed 100% of capacity? Does the length of a column of mercury in a thermometer measure temperature? –Bounded above (total) and below (nil) Either total or nil in a deterministic model Accuracy –Measurement bias Eg: A weighing scale that consistently shows 2 lbs over the real weight of an object Reliability –Consistency of measurement results or meanings Eg: Reliability of weather forecasts; another eg: A weighing scale that consistently shows a positive bias of 2 pounds is inaccurate but reliable; the reliability of the meaning of “man” Cardinality –Enumeration: Polymorphism of cardinality when extent (size) is finite

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8 © Amit Mitra & Amar Gupta Suppl. Box 42

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9 © Amit Mitra & Amar Gupta BLUE RED GREEN INTENSITY COLOR BLUE RED GREEN Nil Intensity (Black) (White) magenta cyan BLUERED GREEN BLACKWHITE Nil Intensity (Black) EXAMPLE: ABSORBING NEW LEARNING – SHIFTING PERSPECTIVES OF COLOR See End Note on Maxwell’s Color Triangle & supplementary Blackboard material

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10 © Amit Mitra & Amar Gupta Subtype of NOMINAL DOMAIN ORDINAL DOMAIN DOMAINS WITH LOWER BOUNDS ORDINAL DOMAIN WITH NIL VALUES RATIO SCALED DOMAIN Subtype of DIFFERENCE SCALED DOMAIN Subtype of class of 0 or more [member of 1] VALUE DOMAIN Join with 0 or more [conjunction of 0 or more] Junction of 1 or more MASSLENGTHMONEY Subtype of PRIMARY DOMAIN PARTITION (DOMAINS THAT ARE CONJUNCTIONS OF NO, i.e., 0, DOMAINS) - EXHAUSTIVITY OF PARTITION UNKNOWN - PRIMARY RATIO SCALED DOMAIN PRIMARY/SECONDARY DOMAIN PARTITION SECONDARY RATIO SCALED DOMAIN Subtract 2 values in 1 [add 1 value of corresponding] Subtract 2 values in [add 1 value of] (inherited by inclusion, i.e., inclusion polymorphic) Subtype of Difference between 2 value in 1 BINARY DOMAIN (Distinguishes only nil from non-nil values) NIL VALUE Subtype of member of 0 or more [contains 1] Subtract 2 values in 1 [add 1 value of corresponding] DOMAIN OF GAPS OF SEQUENCE DOMAIN OF RATIO SCALED GAPS PREFERENCE DATE-TIME ELECTRIC CHARGE Subtype of (inherited by inclusion, i.e., inclusion polymorphic) Subtype of TIME DOMAIN Subtype of Subtype of DOMAIN Partition of [partitioned by] Partition of [partitioned by] VALUE DIFFERENCE DOMAINS Junction of 0 Subtype of cartesian product of 0 or more Join with 0 or more [conjunction of 0 or more] Partitioning criterion Subtype of Subtract 2 values in 1 [add 1 value of corresponding] INFORMATION ENUMERATIONRELIABILITY (RISK) ACCURACYVALIDITYCOMPLETENESS (EXHAUSTIVITY) Subtype of

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11 © Amit Mitra & Amar Gupta Subtype of NOMINAL DOMAIN ORDINAL DOMAIN DOMAINS WITH LOWER BOUNDS ORDINAL DOMAIN WITH NIL VALUES RATIO SCALED DOMAIN Subtype of DIFFERENCE SCALED DOMAIN Subtype of class of 0 or more [member of 1] VALUE DOMAIN Join with 0 or more [conjunction of 0 or more] Junction of 1 or more MASSLENGTHINFORMATIONMONEY Subtype of PRIMARY DOMAIN PARTITION (DOMAINS THAT ARE CONJUNCTIONS OF NO, i.e., 0, DOMAINS) - EXHAUSTIVITY OF PARTITION UNKNOWN - PRIMARY RATIO SCALED DOMAIN PRIMARY/SECONDARY DOMAIN PARTITION SECONDARY RATIO SCALED DOMAIN Subtract 2 values in 1 [add 1 value of corresponding] Subtract 2 values in [add 1 value of] (inherited by inclusion, i.e., inclusion polymorphic) ENUMERATION Subtype of Difference between 2 value in 1 BINARY DOMAIN (Distinguishes only nil from non-nil values) NIL VALUE Subtype of member of 0 or more [contains 1] Subtract 2 values in 1 [add 1 value of corresponding] DOMAIN OF GAPS OF SEQUENCE DOMAIN OF RATIO SCALED GAPS PREFERENCE DATE-TIME ELECTRIC CHARGE Subtype of (inherited by inclusion, i.e., inclusion polymorphic) Subtype of TIME DOMAIN Subtype of Subtype of DOMAIN Partition of [partitioned by] Partition of [partitioned by] VALUE DIFFERENCE DOMAINS Junction of 0 Subtype of cartesian product of 0 or more Join with 0 or more [conjunction of 0 or more] Partitioning criterion Subtype of Subtract 2 values in 1 [add 1 value of corresponding] RELIABILITY (RISK) ACCURACYVALIDITYCOMPLETENESS (EXHAUSTIVITY) Subtype of

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12 © Amit Mitra & Amar Gupta Rule Expression (All Kinds) (Only) Ordinal or Nominal Rule Expression (Only) Nominal Rule Expression Subset of Exclude rules with Arithmetic operators Exclude rules with ranking operations

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13 © Amit Mitra & Amar Gupta Quantitative (arithmetic) Rule Expression Nominal Rule Expression Ordinal Rule Expression Subtype of Inherit Classification information; Add ranking information Inherit classification and ranking information; Add quantitative information

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14 © Amit Mitra & Amar Gupta Cartesian Product of Domains PREFERENCE COLOR (Sequence does not matter. Colors can be arranged in any order along this axis) Green Red LikeDislikeNeutral Blue

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15 © Amit Mitra & Amar Gupta Money Number of pieces Money per piece Hole

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16 © Amit Mitra & Amar Gupta Subtype of NOMINAL DOMAIN ORDINAL DOMAIN WITH NIL VALUES DOMAINS WITH LOWER BOUNDS ORDINAL DOMAIN WITH NIL VALUES RATIO SCALED DOMAIN Subtype of DIFFERENCE SCALED DOMAIN Subtype of class of 0 or more [member of 1] VALUE DOMAIN Join with 0 or more [conjunction of 0 or more] Junction of 1 or more MASSLENGTHINFORMATIONMONEY Subtype of PRIMARY DOMAIN PARTITION (DOMAINS THAT ARE CONJUNCTIONS OF NO, i.e., 0, DOMAINS) - EXHAUSTIVITY OF PARTITION UNKNOWN - PRIMARY RATIO SCALED DOMAIN PRIMARY/SECONDARY DOMAIN PARTITION SECONDARY RATIO SCALED DOMAIN Subtract 2 values in 1 [add 1 value of corresponding] Subtract 2 values in [add 1 value of] (inherited by inclusion, i.e., inclusion polymorphic) ENUMERATION Subtype of Difference between 2 value in 1 BINARY DOMAIN (Distinguishes only nil from non-nil values) NIL VALUE Subtype of member of 0 or more [contains 1] Subtract 2 values in 1 [add 1 value of corresponding] DOMAIN OF GAPS OF SEQUENCE DOMAIN OF RATIO SCALED GAPS PREFERENCE DATE-TIME ELECTRIC CHARGE Subtype of (inherited by inclusion, i.e., inclusion polymorphic) Subtype of TIME DOMAIN Subtype of Subtype of DOMAIN Partition of [partitioned by] Partition of [partitioned by] VALUE DIFFERENCE DOMAINS Junction of 0 Subtype of cartesian product of 0 or more Join with 0 or more [conjunction of 0 or more] Partitioning criterion Subtype of Subtract 2 values in 1 [add 1 value of corresponding] RELIABILITY (RISK) ACCURACYVALIDITYCOMPLETENESS (EXHAUSTIVITY) Subtype of

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18 © Amit Mitra & Amar Gupta Maxwell’s Color Triangle BLUERED GREENWHITE magenta cyan yellow

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19 © Amit Mitra & Amar Gupta Nominal Color Domain BLUERED GREEN

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20 © Amit Mitra & Amar Gupta Nominal Color Domain BLUE RED GREEN INTENSITY COLOR

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21 © Amit Mitra & Amar Gupta BLUE RED GREEN Nil Intensity (Black) COLOR (White) magenta cyan

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22 © Amit Mitra & Amar Gupta BLUE RED GREEN INTENSITY COLOR BLUE RED GREEN Nil Intensity (Black) (White) magenta cyan BLUERED GREEN BLACKWHITE Nil Intensity (Black)

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23 © Amit Mitra & Amar Gupta BLUE RED GREEN INTENSITY COLOR BLUE RED GREEN Nil Intensity (Black) (White) magenta cyan BLUERED GREEN BLACKWHITE Nil Intensity (Black)

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24 © Amit Mitra & Amar Gupta BLUE RED GREEN INTENSITY COLOR BLUE RED GREEN Nil Intensity (Black) (White) magenta cyan BLUERED GREEN BLACKWHITE Nil Intensity (Black)

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25 © Amit Mitra & Amar Gupta BLUE RED GREEN INTENSITY COLOR BLUE RED GREEN Nil Intensity (Black) (White) magenta cyan BLUERED GREEN BLACKWHITE Nil Intensity (Black)

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26 © Amit Mitra & Amar Gupta Rotation DOMAIN 1 DOMAIN 2 Domain 1 Domain 2

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27 © Amit Mitra & Amar Gupta Types of Formats FORMAT (formatting domain) Visible Format Taste Format Audible Format Tactile Format ScriptSpeech Olfactory Format Sensory PartitionSubtype of Written Alphabetic Language Written Numeric Script Spoken Language Language Language Script at least one must exist Subtype of may contain 0 or more [must be contained in 1 or more] may contain 0 or more [must be contained in 1 or more] at least one must exist may contain 0 or more [must be contained in 1 or more] may contain 0 or more [must be contained in 1 or more] MEANING Expressed in 0 or more [express 1 or more]

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1.3 - 1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by.

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