Algebraic Structure “ Set of values, coupled with one or more finite operations, and a set of laws those operations must obey. “ e.g Sum, Magma, Semigroup, Groups, Monoid, Abelian Group, Semi Lattices, Rings, Monads, etc.
Monoids Monoid Laws : (x <> y) <> z = x <> (y <> z) (associativity) identity <> x = x x <> identity = x trait Monoid[T] extends Semigroup[T]{ def identity : T }
Groups Group Laws: (x <> y) <> z = x <> (y <> z) (associativity) identity <> x = x x <> identity = x (identity) x <> inverse x = identity inverse x <> x = identity (invertibility)
Groups Group Laws (x <> y) <> z = x <> (y <> z) identity <> x = x x <> identity = x x <> inverse x = identity inverse x <> x = identity trait Group[T] extends Monoid[T]{ def inverse (v : T) :T }
Examples - (a min b) min c = a (b min c) with Int. - a max ( b max c) = (a max b) max c ** - a or (b or c) = (a or b) or c - a and (b and c) = (a and b) and c - int addition - set union - harmonic sum - Integer mean - Priority queue
We want to : - Build scalable analytics systems - Leverage distributed computing to perform aggregation on really large data sets. - A lot of operations in analytics are just sorting and counting at the end of the day
class PiorityQueueMonoid[T] (max : Int) (implicit order : Ordering[T] ) extends Monoid[Priorityqueue[T] ] Let’s take a look : PQ1 : 55, 45, 21, 3 PQ2: 100, 80, 40, 3 top-4 (PQ1 U PQ2 ): 100, 80, 55, 45 Priority Queue : Can be empty Two Priority Queues can be “added” in any order Associative + Commutative
class PiorityQueueMonoid[T] (max : Int) (implicit order : Ordering[T] ) extends Monoid[Priorityqueue[T] ] Let’s take a look : PQ1 : 55, 45, 21, 3 PQ2: 100, 80, 40, 3 top-4 (PQ1 U PQ2 ): 100, 80, 55, 45 Priority Queue : Can be empty Two Priority Queues can be “added” in any order Associative + Commutative Makes Scalding go fast, by doing sorting, filtering and extracting in one single “map” step.
Stream Mining Challenges - Update predictions after each observation - Single pass : can’t read old data or replay the stream - Full size of the stream often unknown - Limited time for computation per observation - O(1) memory size
Sketches Probabilistic data structures that store a summary (hashed mostly)of a data set that would be costly to store in its entirety, thus providing most of the time, sublinear algorithmic properties. E.g Bloom Filters, Counter Sketch, KMV counters, Count Min Sketch, HyperLogLog, Min Hashes
Bloom filters Approximate data structure for set membership Behaves like an approximate set BloomFilter.contains(x) => NO | Maybe P(False Positive) > 0 P(False Negative) = 0
Internally : Bit Array of fixed size add(x) : for all element i, b[h(x,i)]=1 contains(x) : TRUE if b[h(x,i)] = = 1 for all i. (Boolean AND => associative) Both are associative => BF can be designed as a Monoid
Bloom filters import com.twitter.algebird._ import com.twitter.algebird.Operators._ // generate 2 lists val A = (1 to 300).toList // Generate a Bloomfilter val NUM_HASHES = 6 val WIDTH = 6000 // bits val SEED = 1 implicit val bfm = new BloomFilterMonoid(NUM_HASHES, WIDTH, SEED) // approximate set with bloomfilter val A_bf = A.map{i => bfm.create(i.toString)}.reduce(_ + _) val approxBool = A_bf.contains(“150”) ---> ApproximateBoolean(true, 0.9995…)
Count Min Sketch Count min sketch Adding an element is a numerical addition Querying uses a MIN function. Both are associative. useful for detecting heavy hitters, topK, LSH We have in Algebird :
HyperLogLog Popular sketch for cardinality estimtion. Gives within a probilistic distribution of an error the number of distinct values in a data set. HLL.size = Approx[Number] Intuition Long runs of trailings 0 in a random bits chain are rare But the more bit chains you look at, the more likely you are to find a long one The longest run of trailing 0-bits seen can be an estimator of the number of unique bit chains observed.
Adding an element uses a Max and Sum function. Both are associative and Monoids. (Max is an ordered semigroup in Algebird really) Querying for an element uses an harmonic mean which is a Monoid. In Algebird :
Many More juicy sketches ... - MinHashes to compute Jaccard similarity - QTree for quantiles estimation. Neat for anomaly detection. - SpaceSaverMonoid, Awesome to find the approximate most frequent and top K elements. - TopKMonoid - SGD, PriorityQueues, Histograms, etc.
-Algebra for analytics by Oscar Boykin (Creator of Algebird) http://speakerdeck.com/johnynek/algebra-for-analytics - Take a look into HLearn https://github.com/mikeizbicki/HLearn - Great intro into Algebird by Michael Noll http://www.michael-noll.com/blog/2013/12/02/twitter-algebird-monoid- monad-for-large-scala-data-analytics/ -Aggregate Knowledge http://research.neustar.biz/2012/10/25/sketch-of- the-day-hyperloglog-cornerstone-of-a-big-data-infrastructure - Probabilistic data structures for web analytics. http://highlyscalable.wordpress.com/2012/05/01/probabilistic- structures-web-analytics-data-mining/ - http://debasishg.blogspot.fr/2014/01/count-min-sketch-data- structure-for.html - http://infolab.stanford.edu/~ullman/mmds/ch3.pdf Links
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