Simple math to get signal out of your data noise - Anomaly Detection - Toufic Boubez - Metafor Software - Velocity Santa Clara 2014-06-25

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Some Simple Math to Get Signalout of your data noise

#VelocityConf25.06.2014

Toufic Boubez, Ph.D.Co-Founder, CTOMetafor Software

[email protected]

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Preamble

• I lied: There is no “simple” math for Anomaly Detection!

• I usually beat up on parametric, Gaussian, supervised techniques– This talk is to show some alternatives– Only enough time to cover a couple (four, really) of relatively simple but

very useful techniques– Oh, and I will actually still start up by beating up on the usual suspects,

but don’t despair, there’s good stuff towards the end

• Note: all real data• Note: no y-axis labels on charts – on purpose!!• Note to self: remember to SLOW DOWN!• Note to self: mention the cats!! Everybody loves cats!!

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• Co-Founder/CTO Metafor Software• Co-Founder/CTO Layer 7 Technologies

– Acquired by Computer Associates in 2013– I escaped

• Co-Founder/CTO Saffron Technology• IBM Chief Architect for SOA/Web Services• Co-Author, Co-Editor: WS-Trust, WS-

SecureConversation, WS-Federation, WS-Policy• Building large scale software systems for >20 years (I’m

older than I look, I know!)

Toufic intro – who I am

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Wall of Charts™

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The WoC side-effects: alert fatigue

“Alert fatigue is the single biggest problem we have right now … We need to be more intelligent about our alerts or we’ll all go insane.”

- John Vincent (@lusis) (#monitoringsucks)

We have forensic tools for analytics after the fact BUT we need to KNOW that something has happened!

We need alerts!

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Watching screens cannot scale

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Time to turn things over to the machines!

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Attempt #1: static thresholds …

• Roots in manufacturing process QC

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… are based on Gaussian distributions

• Make assumptions about probability distributions and process behaviour– Data is normally distributed with a useful and

usable mean and standard deviation– Data is probabilistically “stationary”

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Three-Sigma Rule

• Three-sigma rule– ~68% of the values lie within 1 std deviation of the mean– ~95% of the values lie within 2 std deviations– 99.73% of the values lie within 3 std deviations: anything

else is an outlier

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Aaahhhh

• The mysterious red lines explained

mean

3s

3s

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Stationary Gaussian distributions are powerful

• Because far far in the future, in a galaxy far far away:– I can make the same predictions because the

statistical properties of the data haven’t changed– I can easily compare different metrics since they

have similar statistical properties• Let’s do this!!• BUT…• Cue in DRAMATIC MUSIC

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Then THIS happens

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3-sigma rule alerts

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Or worse, THIS happens!

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3-sigma rule alerts

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WTF!? So what gives!?

• Remember this?

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Histogram – probability distribution

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Histogram – probability distribution

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Attempts #2, #3, etc: mo’ better thresholds

• Static thresholds ineffective on dynamic data– Thresholds use the (static) mean as predictor and

alert if data falls more than 3 sigma away• Need “moving” or “adaptive” thresholds:

– Value of mean changes with time to accommodate new data values, trends, periodicity

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Moving Averages “big idea”

• At any point in time in a well-behaved time series, your next value should not significantly deviate from the general trend of your data

• Mean as a predictor is too static, relies on too much past data (ALL of the data!)

• Instead of overall mean use a finite window of past values, predict most likely next value

• Alert if actual value “significantly” (3 sigmas?) deviates from predicted value

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Moving Averages typical method

• Generate a “smoothed” version of the time series– Average over a sliding (moving) window

• Compute the squared error between raw series and its smoothed version

• Compute a new effective standard deviation (sigma’) by smoothing the squared error

• Generate a moving threshold:– Outliers are 3-sigma’ outside the new, smoothed data!

• Ta-da!

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Simple and Weighted Moving Averages

• Simple Moving Average– Average of last N values in your time series

• S[t] <- sum(X[t-(N-1):t])/N– Each value in the window contributes equally to

prediction– …INCLUDING spikes and outliers

• Weigthed Moving Average– Similar to SMA but assigns linearly (arithmetically)

decreasing weights to every value in the window– Older values contribute less to the prediction

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Exponential Smoothing techniques

• Exponential Smoothing– Similar to weighted average, but with weights decay exponentially over

the whole set of historic samples• S[t]=αX[t-1] + (1-α)S[t-1]

– Does not deal with trends in data• DES

– In addition to data smoothing factor (α), introduces a trend smoothing factor (β)

– Better at dealing with trending– Does not deal with seasonality in data

• TES, Holt-Winters– Introduces additional seasonality factor– … and so on

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Example 1

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Holt-Winters predictions

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Example 2

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Exponential smoothing predictions

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Hmmmm, so are we doomed?

• No!• ALL smoothing predictive methods work best

with normally distributed data!• But there are lots of other non-Gaussian

based techniques– We can only scratch the surface in this talk

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Trick #1: Histogram!

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THIS is normal

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This isn’t

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Neither is this

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Trick #2: Kolmogorov-Smirnov test

• Non-parametric test– Compare two probability

distributions– Makes no assumptions (e.g.

Gaussian) about the distributions of the samples

– Measures maximum distance between cumulative distributions

– Can be used to compare periodic/seasonal metric periods (e.g. day-to-day or week-to-week)

http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test

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KS with windowing

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KS Test on difficult data

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Trick #3: Box Plots / Tukey

• Again, need non-parametric method:– Does not rely on mean and standard deviation

• When you can’t count on good old Gaussian:– Median is always a great alternative to the mean– Quartiles are an alternative to standard deviation

• Q1 = 25% Quartile (25% of the data)• Q2 = 50% Quartile == Median (50% of the data)• Q3 = 75% Quartile (75% of the data)• Interquartile Range (IQR) = Q3 – Q1

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Example: box plots and fences for a Gaussian

http://en.wikipedia.org/wiki/Interquartile_range

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IQR method for streaming time series

• IQR method works well for some non-normal distributions– Generates continuously adaptive fences at(Q1 - 1.5xIRQ) and (Q3 + 1.5xIQR)– Adjusted box plot uses fences at(Q1 - 1.5xIRQ) and (Q3 + 1.5x IQR)

• Method:– As time series is streaming, for every window:

• Re-compute quartiles• Re-compute IQR, fences• Determine if any outliers• Repeat

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Example 1

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Example 1 – Good

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Example 2

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Example 2 – Bad

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Trick #4: Diffing/Derivatives

• Often, even when the data itself is not stationary, its derivatives tends to be!

• Most frequently, first difference is sufficient:dS(t) <- S(t+1) – S(t)

• Can then perform some analytics on first difference

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CPU time series

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Its first difference – possible random walk?

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Trick #5: Neural Networks

• Really?• No time – To Be Continued!

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We’re not doomed, but: Know your data!!

• You need to understand the statistical properties of your data, and where it comes from, in order to determine what kind of analytics to use.– Your data is very important!– You spend time collecting it so spend time analyzing it!

• A large amount of data center data is non-Gaussian– Guassian statistics won’t work– Use appropriate techniques

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More?

• Only scratched the surface• I want to talk more about algorithms, analytics,

current issues, etc, in more depth, but time’s up!!– Come talk to me or email me if interested.

• Office Hour: Tomorrow at 11:30 Booth #801• Thank you!

[email protected]@tboubez

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Oh yeah, and we’re hiring!

In Vancouver, Canada