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The math behind big systems analysis.


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This presentations discusses the challenges in processing real-time, high-frequency time-series data for anomaly detection.

Published in: Data & Analytics

The math behind big systems analysis.

  1. 1. A tour through mathematical methods on systems telemetry Math in Big Systems If it was a simple math problem, we’d have solved all this by now.
  2. 2. The many faces of Theo Schlossnagle @postwait CEO Circonus
  3. 3. Choosing an approach is premature… Problems How do we determine the type of signal? How do we manage off-line modeling? How do we manage online fault detection? How do we reconcile so users don’t hate us? How we we solve this in a big-data context?
  4. 4. Picking an Approach Statistical? Machine learning? Supervised? Ad-hoc? ontological? (why it is what it is)
  5. 5. tl;dr Apply PhDs Apply PhDs Rinse Wash Repeat
  6. 6. Garbage in, category out. Classification Understanding a signal We found to be quite ad-hoc At least the feature extraction
  7. 7. A year of service… I should be able to learn something. API requests/second 1 year
  8. 8. A year of service… I should be able to learn something. API requests 1 year
  9. 9. A year of service… I should be able to learn something. API requests 1 year Δv Δt , ! Δv ≥ 0
  10. 10. Some data goes both ways… Complicating Things Imagine disk space used… it makes sense as a gauge (how full) it makes sense as rate (fill rate)
  11. 11. Error + error + guessing = success How we categorize Human identify a variety of categories. Devise a set of ad-hoc features. Bayesian model of features to categories. Human tests.
  12. 12. Many signals have significant noise around their averages Signal Noise A single “obviously wrong” measurement… is often a reasonable outlier.
  13. 13. A year of service… I should be able to learn something. API requests/second 1 year
  14. 14. At a resolution where we witness: “uh oh” API requests/second 4 weeks
  15. 15. But, are there two? three? API requests/second 4 weeks Is that super interesting?
  16. 16. Bring the noise! API requests/second 2 days
  17. 17. Think about what this means… statistically API requests/second 1 year envelope of ±1 std dev
  18. 18. Lies, damned lies, and statistics Simple Truths Statistics are only really useful with p-values are low. p ≤ 0.01 : very strong presumption against null hyp. 0.01 < p ≤ 0.05 : strong presumption against null hyp. 0.05 < p ≤ 0.1 : low presumption against null hyp. p > 0.1 : no presumption against the null hyp. from xkcd #882 by Randall Munroe
  19. 19. What does a p-value have to do with applying stats? The p-value problem It turns out a lot of measurement data (passive) is very infrequent. 60% of the time… it works every time.
  20. 20. Our low frequencies lead us to questions of doubt… Given a certain statistical model: How many few points need to be seen before we are sufficiently confident that it does not fit the model (presumption against the null hypothesis)? With few, we simply have outliers or insignificant aberrations.
  21. 21. Solving the Frequency Problem More data, more often… (obviously) 1. sample faster (faster from the source) 2. analyze wider (more sources) OR
  22. 22. Increasing frequency is the only option at times. Signals of Importance Without large-scale systems We must increase frequency
  23. 23. What do mean that my mean is mean? Why can’t math be nice to people? Most algorithms require measuring residuals from a mean Mean means Calculating means is “easy” There are some pitfalls
  24. 24. Newer data should influence our model. Signals change The model needs to adapt. Exponentially decaying averages are quite common in online control systems and used as a basis for creating control charts. Sliding windows are a bit more expensive.
  25. 25. EWM vs. SWM ❖ EWM : ST ❖ S1 = V1 ❖ ST = 훼VT + (1-훼)ST-1 ❖ Low computation overhead ❖ Low memory usage ❖ Hard to repeat offline ❖ SMW tracking the last N values : ST ❖ ST = ST-1 - VT-N/n + VT/n ❖ Low computational overhead ❖ High memory usage ❖ Easy to repeat offline
  26. 26. Repeatable outcomes are needed In our system… We need our online algorithms to match our offline algorithms. This is because human beings get pissed off when they can’t repeat outcomes that woke them up in the middle of the night. EWM: not repeatable SWM: expensive in online application
  27. 27. Repeatable, low-cost sliding windows Our solution: lurching windows fixed rolling windows of fixed windows SWM + EWM
  28. 28. Putting it all together… How to test if we don’t match our model?
  29. 29. Hypothesis Testing
  30. 30. The CUSUM Method
  31. 31. Applying CUSUM API requests/second 4 weeks CUSUM Control Chart
  32. 32. Can we do better? Investigations The CUSUM method has some issues. It’s challenging when signals are noise or of variable rate. We’re looking into the Tukey test: • compares all possible pairs of means • test is conservative in light of uneven sample sizes
  33. 33. All that work and you tell me *now* that I don’t have a normal distribution? Most statistical methods assume a normal distribution. Your telemetry data has never been evenly distributed Statistics suck. All this “deviation from the mean” assumes some symmetric distribution.
  34. 34. Think about what this means… statistically Rather obviously not a normal distribution The average minus the standard deviation is less than zero on a measurement that can only be ≥ 0.
  35. 35. With all the data, we have a complete picture of population distribution. What if we had all the data? … full distribution Now we can do statistics that isn’t hand-wavy.
  36. 36. High volume data requires a different strategy What happens when we get what we asked for? 10,000 measurements per second? more? on each stream… with millions of streams.
  37. 37. Let’s understand the scope of the problem. First some realities This is 10 billion to 1 trillion measurements per second. At least a million independent models. We need to cheat.
  38. 38. When we have too much, simplify… Information compression We need to look to transform the data. Add error in the value space. Add error in the time space.
  39. 39. Summarization & Extraction ❖ Take our high-velocity stream ❖ Summarize as a histogram over 1 minute (error) ❖ Extract useful less-dimensional characteristics ❖ Apply CUSUM and Tukey tests on characteristics
  40. 40. Lorem Ipsum Dolor Modes & moments. Strong indicators of shifts in workload
  41. 41. Lorem Ipsum Dolor Quantiles… Useful if you understand the problem domain and the expected distribution.
  42. 42. Lorem Ipsum Dolor Inverse Quantiles… Q: “What quantile is 5ms of latency?” Useful if you understand the problem domain and the expected distribution.
  43. 43. Thank ou! and thank you to the real brains behind Circonus’ analysis systems: Heinrich and George.