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Modeling Social Data, Lecture 2: Introduction to Counting

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Modeling Social Data, Lecture 2: Introduction to Counting

  1. 1. Introduction to Counting APAM E4990 Modeling Social Data Jake Hofman Columbia University January 30, 2013 Jake Hofman (Columbia University) Intro to Counting January 30, 2013 1 / 28
  2. 2. Why counting? Jake Hofman (Columbia University) Intro to Counting January 30, 2013 2 / 28
  3. 3. Why counting? sta·tis·tic noun 1. A function of a random sample of data Jake Hofman (Columbia University) Intro to Counting January 30, 2013 3 / 28
  4. 4. Why counting? p( y support | x age ) Jake Hofman (Columbia University) Intro to Counting January 30, 2013 4 / 28
  5. 5. Why counting? ?p( y support | x1, x2, x3, . . . age, sex, race, party ) Jake Hofman (Columbia University) Intro to Counting January 30, 2013 4 / 28
  6. 6. Why counting? Problem: Traditionally difficult to obtain reliable estimates due to small sample sizes or sparsity (e.g., ∼ 100 age × 2 sex × 5 race × 3 party = 3000 groups, but typical surveys collect ∼ 1,000s of responses) Jake Hofman (Columbia University) Intro to Counting January 30, 2013 5 / 28
  7. 7. Why counting? Potential solution: Sacrifice granularity for precision, by binning observations into larger, but fewer, groups (e.g., bin age into a few groups: 18-29, 30-49, 50-64, 65+) Jake Hofman (Columbia University) Intro to Counting January 30, 2013 5 / 28
  8. 8. Why counting? Potential solution: Develop more sophisticated methods that generalize well from small samples (e.g., fit a model: support ∼ β0 + β1age + β2age2 + . . .) Jake Hofman (Columbia University) Intro to Counting January 30, 2013 5 / 28
  9. 9. Why counting? (Partial) solution: Obtain larger samples through other means, so we can just count and divide to make estimates via relative frequencies (e.g., with ∼ 1M responses, we have 100s per group and can estimate support within a few percentage points) Jake Hofman (Columbia University) Intro to Counting January 30, 2013 6 / 28
  10. 10. Why counting? The good: Shift away from sophisticated statistical methods on small samples to simple methods on large samples Jake Hofman (Columbia University) Intro to Counting January 30, 2013 7 / 28
  11. 11. Why counting? The bad: Even simple methods (e.g., counting) are computationally challenging at large scales (1M is easy, 1B a bit less so, 1T gets interesting) Jake Hofman (Columbia University) Intro to Counting January 30, 2013 7 / 28
  12. 12. Why counting? Claim: Solving the counting problem at scale enables you to investigate many interesting questions in the social sciences Jake Hofman (Columbia University) Intro to Counting January 30, 2013 7 / 28
  13. 13. Learning to count This week: Counting at small/medium scales on a single machine Jake Hofman (Columbia University) Intro to Counting January 30, 2013 8 / 28
  14. 14. Learning to count This week: Counting at small/medium scales on a single machine Following weeks: Counting at large scales in parallel Jake Hofman (Columbia University) Intro to Counting January 30, 2013 8 / 28
  15. 15. Counting, the easy way Split / Apply / Combine1 http://bit.ly/sactutorial • Load dataset into memory • Split: Arrange observations into groups of interest • Apply: Compute distributions and statistics within each group • Combine: Collect results across groups 1 http://bit.ly/splitapplycombine Jake Hofman (Columbia University) Intro to Counting January 30, 2013 9 / 28
  16. 16. The generic group-by operation Split / Apply / Combine for each observation as (group, value): place value in bucket for corresponding group for each group: apply a function over values in bucket output group and result Jake Hofman (Columbia University) Intro to Counting January 30, 2013 10 / 28
  17. 17. The generic group-by operation Split / Apply / Combine for each observation as (group, value): place value in bucket for corresponding group for each group: apply a function over values in bucket output group and result Useful for computing arbitrary within-group statistics when we have required memory (e.g., conditional distribution, median, etc.) Jake Hofman (Columbia University) Intro to Counting January 30, 2013 10 / 28
  18. 18. Why counting? Jake Hofman (Columbia University) Intro to Counting January 30, 2013 11 / 28
  19. 19. Example: Anatomy of the long tail Dataset Users Items Rating levels Observations Movielens 100K 10K 10 10M Netflix 500K 20K 5 100M Jake Hofman (Columbia University) Intro to Counting January 30, 2013 12 / 28
  20. 20. Example: Anatomy of the long tail Dataset Users Items Rating levels Observations Movielens 100K 10K 10 10M Netflix 500K 20K 5 100M Jake Hofman (Columbia University) Intro to Counting January 30, 2013 12 / 28
  21. 21. Example: Movielens 0 1,000,000 2,000,000 3,000,000 1 2 3 4 5 Rating Numberofratings Jake Hofman (Columbia University) Intro to Counting January 30, 2013 13 / 28
  22. 22. Example: Movielens 0 1,000,000 2,000,000 3,000,000 1 2 3 4 5 Rating Numberofratings group by rating value for each group: count # ratings Jake Hofman (Columbia University) Intro to Counting January 30, 2013 14 / 28
  23. 23. Example: Movielens 1 2 3 4 5 Mean Rating by Movie Density Jake Hofman (Columbia University) Intro to Counting January 30, 2013 15 / 28
  24. 24. Example: Movielens group by movie id for each group: compute average rating 1 2 3 4 5 Mean Rating by Movie Density Jake Hofman (Columbia University) Intro to Counting January 30, 2013 16 / 28
  25. 25. Example: Movielens 0% 25% 50% 75% 100% 0 3,000 6,000 9,000 Movie Rank CDF Jake Hofman (Columbia University) Intro to Counting January 30, 2013 17 / 28
  26. 26. Example: Movielens 0% 25% 50% 75% 100% 0 3,000 6,000 9,000 Movie Rank CDF group by movie id for each group: count # ratings sort by group size cumulatively sum group sizes Jake Hofman (Columbia University) Intro to Counting January 30, 2013 18 / 28
  27. 27. Example: Movielens 0 2,000 4,000 6,000 8,000 100 10,000 User eccentricity Numberofusers Jake Hofman (Columbia University) Intro to Counting January 30, 2013 19 / 28
  28. 28. Example: Movielens join movie ranks to ratings group by user id for each group: compute median movie rank 0 2,000 4,000 6,000 8,000 100 10,000 User eccentricity Numberofusers Jake Hofman (Columbia University) Intro to Counting January 30, 2013 20 / 28
  29. 29. Example: Anatomy of the long tail Dataset Users Items Rating levels Observations Movielens 100K 10K 10 10M Netflix 500K 20K 5 100M What do we do when the full dataset exceeds available memory? Jake Hofman (Columbia University) Intro to Counting January 30, 2013 21 / 28
  30. 30. Example: Anatomy of the long tail Dataset Users Items Rating levels Observations Movielens 100K 10K 10 10M Netflix 500K 20K 5 100M What do we do when the full dataset exceeds available memory? Sampling? Unreliable estimates for rare groups Jake Hofman (Columbia University) Intro to Counting January 30, 2013 21 / 28
  31. 31. Example: Anatomy of the long tail Dataset Users Items Rating levels Observations Movielens 100K 10K 10 10M Netflix 500K 20K 5 100M What do we do when the full dataset exceeds available memory? Random access from disk? 1000x more storage, but 1000x slower2 2 Numbers every programmer should know Jake Hofman (Columbia University) Intro to Counting January 30, 2013 21 / 28
  32. 32. Example: Anatomy of the long tail Dataset Users Items Rating levels Observations Movielens 100K 10K 10 10M Netflix 500K 20K 5 100M What do we do when the full dataset exceeds available memory? Streaming Read data one observation at a time, storing only needed state Jake Hofman (Columbia University) Intro to Counting January 30, 2013 21 / 28
  33. 33. The combinable group-by operation Streaming for each observation as (group, value): if new group: initialize result update result for corresponding group as function of existing result and current value for each group: output group and result Jake Hofman (Columbia University) Intro to Counting January 30, 2013 22 / 28
  34. 34. The combinable group-by operation Streaming for each observation as (group, value): if new group: initialize result update result for corresponding group as function of existing result and current value for each group: output group and result Useful for computing a subset of within-group statistics with a limited memory footprint (e.g., min, mean, max, variance, etc.) Jake Hofman (Columbia University) Intro to Counting January 30, 2013 22 / 28
  35. 35. Example: Movielens 0 1,000,000 2,000,000 3,000,000 1 2 3 4 5 Rating Numberofratings for each rating: counts[movie id]++ Jake Hofman (Columbia University) Intro to Counting January 30, 2013 23 / 28
  36. 36. Example: Movielens for each rating: totals[movie id] += rating counts[movie id]++ for each group: totals[movie id] / counts[movie id] 1 2 3 4 5 Mean Rating by Movie Density Jake Hofman (Columbia University) Intro to Counting January 30, 2013 24 / 28
  37. 37. Yet another group-by operation Per-group histograms for each observation as (group, value): histogram[group][value]++ for each group: compute result as a function of histogram output group and result Jake Hofman (Columbia University) Intro to Counting January 30, 2013 25 / 28
  38. 38. Yet another group-by operation Per-group histograms for each observation as (group, value): histogram[group][value]++ for each group: compute result as a function of histogram output group and result We can recover arbitrary statistics if we can afford to store counts of all distinct values within in each group Jake Hofman (Columbia University) Intro to Counting January 30, 2013 25 / 28
  39. 39. The group-by operation For arbitrary input data: Memory Scenario Distributions Statistics N Small dataset Yes General V*G Small distributions Yes General G Small # groups No Combinable V Small # outcomes No No 1 Large # both No No N = total number of observations G = number of distinct groups V = largest number of distinct values within group Jake Hofman (Columbia University) Intro to Counting January 30, 2013 26 / 28
  40. 40. Examples (w/ 8GB RAM) Median rating by movie for Netflix N ∼ 100M ratings G ∼ 20K movies V ∼ 10 half-star values V *G ∼ 200K, store per-group histograms for arbitrary statistics (scales to arbitrary N, if you’re patient) Jake Hofman (Columbia University) Intro to Counting January 30, 2013 27 / 28
  41. 41. Examples (w/ 8GB RAM) Median rating by video for YouTube N ∼ 10B ratings G ∼ 1B videos V ∼ 10 half-star values V *G ∼ 10B, fails because per-group histograms are too large to store in memory G ∼ 1B, but no (exact) calculation for streaming median Jake Hofman (Columbia University) Intro to Counting January 30, 2013 27 / 28
  42. 42. Examples (w/ 8GB RAM) Mean rating by video for YouTube N ∼ 10B ratings G ∼ 1B videos V ∼ 10 half-star values G ∼ 1B, use streaming to compute combinable statistics Jake Hofman (Columbia University) Intro to Counting January 30, 2013 27 / 28
  43. 43. The group-by operation For pre-grouped input data: Memory Scenario Distributions Statistics N Small dataset Yes General V*G Small distributions Yes General G Small # groups No Combinable V Small # outcomes Yes General 1 Large # both No Combinable N = total number of observations G = number of distinct groups V = largest number of distinct values within group Jake Hofman (Columbia University) Intro to Counting January 30, 2013 28 / 28

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