How to Recognize and Deal with Numeric Representation Error in SAS

1

• How to Recognize and Deal with Numeric
  Representation Error in SAS

Jeff Martin Serono, Inc.
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Outline

• How do computers store numeric data?
• Overview of floating point binary
• How does SAS store numeric data?
• What is numeric representation error?
• How can you deal with it?

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Why Worry About This?

• Brief example
• data temp
• initial_val0.3
• other_val0.13
• differenceinitial_val-other_val
• run
• proc print noobs
• run

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Why Worry About This?

• Output
• initial_ other_
• val val difference
• 0.3 0.3 -5.5511E-17

5
How Do Computers Store Numbers?

• Integer
• Decimal or Packed Decimal
• Floating Point Binary

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How Do Computers Store Numbers?

• Integer
• Pros
• Simple, straightforward usage and computations
• Cons
• Not practical - unable to store fractional values

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How Do Computers Store Numbers?

• Decimal or Packed Decimal
• Pros
• Able to store fractional values
• Cons
• Hardware needs to provide storage for large
  decimals
• Loss of computational efficiency

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How Do Computers Store Numbers?

• Floating Point Binary
• (used by the SAS System)
• Pros
• Able to store fractional values and integer
  values
• Efficient for computing use of binary numbers
• Cons
• Subject to representation error

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How Does Floating Point Binary Work?

• Number values are stored as four types of
  information
• Sign
• Mantissa
• Base
• Exponent

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How Does Floating Point Binary Work?

• Example
• .4321 10 4
• sign mantissa base
  exponent

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How Does SAS Store Numbers?

• Each numeric value is stored in up to 64 bits
  (binary digits) or 8 bytes (groups of 8 bits).
• Different operating systems may distribute the
  bits differently and have different base value
  (usually 2 or 16).
• Windows/DOS/Unix
• 1 bit for the sign,
• 52 bits for the mantissa
• 11 bits for the exponent
• VAX/VMS
• 1 bit for the sign
• 55 bits for the mantissa
• 8 bits for the exponent

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How Does SAS Store Numbers?

• Example
• SEEEEEEE EEEEMMMM MMMMMMMM MMMMMMMM
• byte 1 byte 2 byte 3 byte 4
•  
• MMMMMMMM MMMMMMMM MMMMMMMM MMMMMMMM
• byte 5 byte 6 byte 7 byte 8
• S sign
• E exponent
• M mantissa

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The Problem

• Computers are limited in that they need to use a
  finite set of numbers to represent infinite real
  numbers (note limited bits in previous slide)
• Affects the amount of precision available in
  which to store the numeric value
• Computers use binary arithmetic instead of
  decimal arithmetic

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The Result

• Values that do not have an exact binary
  representation (ie, 0.1, 1/3, etc.) must be
  approximated
• Associative law may not apply (ie, (ab)c may
  not equal a(bc))
• May yield unexpected results
• Use of different procedures or orders of
  operations may yield different numeric results
  (may cause suspicion of programming error)

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What do numeric values in SAS really look like?

• Numeric values in SAS are displayed using
  formatted values rather than the floating point
  representations
• Even using maximum decimal places will not
  display the full value
• Only HEX16. format displays the exact value of a
  numeric variable
• Can help track down representation error

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What do numeric values in SAS really look like?

• Example
• data temp
• initial_val0.3
• other_val0.13
• differenceinitial_val-other_val
• initial_val_hexput(initial_val,hex16.)
• other_val_hexput(other_val,hex16.)
• run
• proc print noobs
• run

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What do numeric values in SAS really look like?

• Output
• initial_ other_
• val val difference
• 0.3 0.3 -5.5511E-17
• initial_val_hex other_val_hex
• 3FD3333333333333 3FD3333333333334

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What can you do?

• Rounding the numbers
• Need to determine the number of significant
  digits to round to
• Can use the SAS ROUND function

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How the ROUND function works

• Scales the value based on the round off unit
• If the value is positive, adds a constant of
  .500000000001 and lowers the result to the
  nearest integer
• If the value is negative, subtracts constant of
  .500000000001 and raises the result to the
  nearest integer
• Scales the value back based on the round off unit

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What else can you do?

• FUZZ the comparisons
• Determine an acceptable difference between two
  values
• If the difference is smaller than the acceptable
  difference, the two values are considered equal

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More on fuzzing

• Options in PROC COMPARE allow you to set a fuzz
  factor when comparing datasets. By using the
  CRITERION and FUZZ options, you can consider
  values within a specified difference to be equal.
• proc compare basebasedata comparecompdata
• criterion.000000000001 fuzz.000000000001
• run
• Can set up a statement in conditionals
• If abs(a-b) le .000000000001 then do

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Conclusions

• Numeric representatation error is a consequence
  of working in a computing environment
• While there is no definitive solution to the
  problem, methods exist to adjust for it and still
  get the most out of your data
• Recognizing that it exists is a step forward in
  dealing with it

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References

• Numeric Precision 101, available from
• http//support.sas.com/techsup/technote/ts654.html
  
•  
• Dealing With Numeric Representation Error,
  available from
• http//support.sas.com/techsup/technote/ts230.html
  
•  
• Numeric Data in SAS Guidelines for Storage and
  Display
• Paul Gorrell
• Social Scientific Systems, Inc.
• Silver Spring, MD