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Efficiently Estimating Statistics of Points of Interest on Maps – Wang, He, Liu (2016)

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- 1. Eﬃciently Estimating Statistics of Points of Interest on Maps – Wang, He, Liu (2016) Alex Klibisz, alex.klibisz.com, UTK STAT645 November 10, 2016
- 2. Motivation Point-of-interest (PoI) data is valuable. Google, Foursquare, Baidu, etc. collect user activity and feedback for PoIs. This data can help businesses understand market, determine locations, consumer preferences, etc. Public-facing APIs have restrictions Number of PoIs returned, query frequency, area size, no guarantee of underlying sampling. Propose methods to sample and approximate aggregate statistics Emphasize query eﬃciency, aggregate statistic error.
- 3. Notation A – Area of interest (e.g. a city). P – Set of PoIs in A (e.g. all hotels). k – Maximum number of PoIs returned (API constraint). Fully-accessible region – when queried, returns < k PoIs. If k = 50, a region is fully-accessible when a query returns 49 PoIs. δ – minimum acceptable lat. and lon. precision of map APIs. Hit-ratio – probability of sampling a non-empty sub-region from A. ρ is the fraction of non-empty sub-regions in A → 1 ρ queries to get a non-empty sub-region.
- 4. Problem Statement Given 1. Area of interest A containing PoIs P. 2. API-speciﬁc query restrictions. Estimate 1. Sum Aggregate: the sum of an attribute across P. 2. Average Aggregate: the average of an attribute across P. 3. PoI Distribution: distribution of an attribute across P 4. n(A), number of PoIs in A.1 1Not directly presented how to estimate this, but used for evaluation.
- 5. Datasets Baidu and Foursquare datasets used from two other publications. No clear indication of time span.
- 6. Algorithms 1. Random Region Zoom-In, RRZI 2. RRZI w/ Count Information, RRZIC 3. Uniform Region Sampling, RRZI(C)_URS 4. Metropolis-Hastings Weighted Region Sampling, RRZIC_MHWRS
- 7. RRZI Algorithm 1. Sample region Q from A at random. 2. Divide Q into two sub-regions Q0, Q1 without overlap. 3. Randomly select a non-empty sub-region as the next region to query. 4. Query the selected region. 5. Repeat until a fully-accessible sub-region is found. Characteristics Typically run RRZI until m fully-accessible sub-regions are found. How to divide Q into Q0, Q1? – Equations (1), (2). How to determine if Q0, Q1 are empty? – Store prior PoIs. Correcting for sampling bias? – Counter τ records probability of sampling each sub-region. Maximum number of queries? – Hmax = log(Lx /δ) + log(Ly /δ) Lx and Ly are x, y dimensions, δ is degree granularity. Binary search over a 2-d array Seems like random binary search until a fully-accessible sub-region is found.
- 8. RRZI Example Important to note RRZI is repeated m = 3 times. Evaluation show that as m increases, estimation improves.
- 9. RRZI Estimators Sum Estimator, Proven Consistent pg. 5 The average of some attribute over all PoIs in fully-accessible regions, standardized by the probability of picking the PoI’s region. Conﬁdence Interval (variance deﬁned equation 4.)
- 10. RRZI Estimators Distribution Estimator
- 11. RRZIC Algorithm Context Some APIs provide the count z of PoIs in a queried region. Use the count to improve RRZI: Choose the next sub-region with probability z0 z and z1 z . → The larger sub-region is more likely to be chosen to query next. → The number of PoIs in the next-explored region is more stable. → Sampling is closer to uniform and error is reduced. Estimators now standardize by the known count in each region, n(ri ) instead of probability of choosing the region. Question: why not always pick the region with greater z? You would end up with the same FA region every time. Seems like semi-sorted binary search now.
- 12. Mix Methods to overcome Size Constraints Context Some APIs constrain the size of the queried region. 3◦ x3◦ (lat, lon) query fails on Foursquare. Introduce mix-methods URS and MHWRS to overcome size constraints with clever sampling. Intuition Subdivide A before running RRZI and RRZIC. Improved sampling makes it more query-eﬃcient and lowers error.
- 13. Uniform Random Sampling (RRZI_URS, RRZIC_URS) Uniform Random Sampling Step 1. Apply L recursive region divisions to get set of sub-regions BL, |BL| = 2L , B∗ L is the set of non-empty sub-regions. L tuned such that the sub-regions meet size constraint. Continue with RRZI or RRZIC using regions from URS 2. Randomly select nonempty b from BL 3. Sample fully-accessible region(s) from b using RRZI(b) and RRZIC(b) (instead of RRZI(A) and RRZIC(A)). Characteristics Estimator functions are similar; standardize w.r.t BL instead of A. (Generally) more query-eﬃcient: URS requires |BL| |B∗ L | queries to ﬁnd a non-empty region; non-URS requires L. |BL| |B∗ L | < L for small values of L (few division steps). Arrives at a non-empty query more quickly → undersamples dense regions → higher error.
- 14. Metropolis-Hastings Based Weighted Region Sampling (RRZIC_MHWRS) Modify the Sampling Step to Improve Error 1. Sample non-empty region b from BL following distribution π = (πb = n(b) n(A) : b ∈ B∗ L ) Draw more samples from dense regions. 2. Sample a fully-accessible region with RRZIC. Move to Another Region (maybe) 3. Sample next region b∗ , and move to it with probability min(n(b∗ ) n(b) , 1) If b∗ is larger than b, it will always be moved to. Characteristics Only works if you know the count of the region. More query-eﬃcient for same reasons as URS. MHWRS falls into Markov-Chain Monte Carlo techniques.
- 15. Algorithms Comparison Parameter L required to determine sub-region size.
- 16. Evaluation Tests 1. Estimate n(A) (number of PoIs in area A). 2. Estimating average and distribution statistics. Baselines 1. Nearest-Neighbor Search 2. Random Region Sampling Hypothesis 1. RRZIC_MHWRS will be most eﬃcient if PoI count is available. 2. RRZI_URS will be most eﬃcient otherwise.
- 17. Estimating n(A) Normalized root-mean-squared error for n(A) estimate using RRZI. m ↑, error ↓ k ↑, error ↓ Not obvious how they actually estimate n(A).
- 18. Estimating n(A) How many RRZI_URS queries to reach a fully-accessible region? L = 0 models RRZI. L ↑, sub-region size ↓ Local minimum around 10-15.
- 19. Estimating n(A) How many RRZI_URS queries to sample a non-empty sub-region? Small L → large sub-region → less likely to be empty.
- 20. Estimating n(A) How does decreasing sub-region size aﬀect error for n(A)? Smaller regions → lower error until L = 20
- 21. Estimating n(A) 2 How many queries to decrease error to 0.1? Baseline methods require ~150K queries. RRZI and RRZI_URC require ~20K and ~50K queries. 2The lines between cities don’t really represent anything.
- 22. Estimating n(A) Test Interpretations 1. RRZI, RRZI_URC methods reach a low error much sooner. 2. Tuning hyper-parameter L is important. 3. Not obvious how the proposed methods compute the n(A) estimate.
- 23. Estimating average and distribution statistics “Correct” data for Foursquare. Leaving out Baidu evaluation for brevity.
- 24. Estimating average and distribution statistics Average and distribution root-mean-squared error for RRZI, RRZIC, RRZI_URS, RRZIC_MHWRS up to 10K queries. RRZIC_MHWRS is best in all cases.
- 25. Estimating average and distribution statistics How many queries needed for error < 0.1 for average number of Foursquare check-ins? RRZIC_MHWRS is best in all cases. RRZI_URS is best if PoI count is unavailable.
- 26. Estimating average and distribution statistics Test Interpretations 1. True PoI count is very nice to have. 2. Modiﬁed sampling methods reach a low error much more eﬃciently. 3. Modiﬁed sampling methods are more query eﬃcient; possible they get more meaningful data more quickly.
- 27. Omitted for Brevity Real Applications Present data collected from Foursquare, Google, Baidu using the proposed methods. Related Work Describe Nearest-Neighbor Search and Random Region Sampling drawbacks for this task.
- 28. Contributions, Questions Contributions 1. A practical guide to overcoming data limitations. 2. Clear improvement on prior “state-of-the-art” methods (NNS, RRS). 3. Clever sampling methods to reach low errors very eﬃciently. Questions 1. What’s the shelf-life of PoI estimates? How often would you have to re-query to maintain accurate estimates? 2. Do the ground-truth data and estimates come from the same time window? If not, is it valid to compare data from diﬀerent points in time? Is it useful to use data from all time?3 3. Is it possible that the decreased error for mix-methods is a product of more eﬃcient querying? 4. Didn’t fully understand role of CDS_UNI and CDS_NOR in section 4.2. 3At a quick glance, most of Foursquare’s venue statistics can but don’t necessarily require a bounded time range.

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