Here is a fast and simple way of analyzing statistics using Scala. For more detail, see the GitHub repository (https://github.com/xxxnell/flip).

1. !
Fast and Simple
Statistics with Scala
@xxxnell
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11. Problems of KDE
4 Slow: or more
4 Large memory consumption:
4 Require a prior knowledge of dataset
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12. animation link 12

13. 'Adaptive histogram (Sketch)' solves
the problems:
4 Fast:
4 Lightweight:
4 Does NOT require a prior knowledge of dataset
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15. animation link 15

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19. Category of probability distributions
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20. Functions of Dist (simply D)
def probability[A](dist: D[A], start: A, end: A): Double
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21. Functions of Histogram (simply H)
def probability[A](hist: H[A], start: A, end: A): Double
def update[A](hist: H[A], as: List[A]): H[A]
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22. Functions of Sketch (simply S)
def probability[A](sketch: S[A], start: A, end: A): Double
def update[A](sketch: S[A], as: List[A]): H[A]
def narrowUpdate[A](sketch: S[A], as: List[A]): S[A]
def deepUpdate[A](sketch: S[A], as: List[A]): S[A]
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23. Functions of Sketch (simply S)
def probability[A](sketch: S[A], start: A, end: A): Double
def narrowUpdate[A](sketch: S[A], as: List[A]): S[A]
def deepUpdate[A](sketch: S[A], as: List[A]): S[A]
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24. import flip.implicits._
// get 100 random variables from standard normal distribution
val underlying0 = NumericDist.normal(0.0, 1.0)
val (underlying1, samples) = underlying0.samples(100)
// update samples to sketch
val sketch0 = Sketch.empty[Double]
val sketch1 = samples.foldLeft(sketch0) {
case (sketch, sample) ⇒ sketch.update(sample)
}
// get probability for interval [0.0, 1.0]
println("result: " + sketch1.probability(0.0, 1.0))
println("expected: " + underlying1.probability(0.0, 1.0))
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25. import flip.implicits._
// get 100 random variables from standard normal distribution
val underlying0 = NumericDist.normal(0.0, 1.0)
val (underlying1, samples) = underlying0.samples(100)
// update samples to sketch
val sketch0 = Sketch.empty[Double]
val sketch1 = samples.foldLeft(sketch0) {
case (sketch, sample) ⇒ sketch.update(sample)
}
// get probability for interval [0.0, 1.0]
println("result: " + sketch1.probability(0.0, 1.0))
println("expected: " + underlying1.probability(0.0, 1.0))
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26. import flip.implicits._
// get 100 random variables from standard normal distribution
val underlying0 = NumericDist.normal(0.0, 1.0)
val (underlying1, samples) = underlying0.samples(100)
// update samples to sketch
val sketch0 = Sketch.empty[Double]
val sketch1 = samples.foldLeft(sketch0) {
case (sketch, sample) ⇒ sketch.update(sample)
}
// get probability for interval [0.0, 1.0]
println("result: " + sketch1.probability(0.0, 1.0))
println("expected: " + underlying1.probability(0.0, 1.0))
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27. import flip.implicits._
// get 100 random variables from standard normal distribution
val underlying0 = NumericDist.normal(0.0, 1.0)
val (underlying1, samples) = underlying0.samples(100)
// update samples to sketch
val sketch0 = Sketch.empty[Double]
val sketch1 = samples.foldLeft(sketch0) {
case (sketch, sample) ⇒ sketch.update(sample)
}
// get probability for interval [0.0, 1.0]
println("result: " + sketch1.probability(0.0, 1.0))
println("expected: " + underlying1.probability(0.0, 1.0))
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28. // probability for interval [0.0, 1.0]
sketch.probability(0.0, 1.0)
// probability density at 0.0
sketch.pdf(0.0)
// median
sketch.median
// 100 random samples
sketch.samples(100)
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29. Rolling one die
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30. Rolling two dice
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31. Rolling two dice
for {
n1 ← diceDist
n2 ← diceDist
} yield n1 + n2
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32. Probability distribution is monad
// premises
def pure[A](a: A): Dist[A]
def flatMap[A, B](f: Dist[A], g: A ⇒ Dist[B]): Dist[B]
// proposition
def map[A](f: Dist[A], g: A ⇒ B): Dist[B]
= flatMap(f, (a: A) ⇒ pure(g(a)))
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33. pure
def pure[A](a: A): Dist[A]
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34. flatMap
def flatMap[A, B](f: Dist[A], g: A ⇒ Dist[B]): Dist[B]
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35. Experiment result of flatMap
sketch.flatMap(x ⇒ Normal(x, 1.5))
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36. map is domain transformation
// translation transformation
sketch.map(x ⇒ x + 1)
// scaling transformation
sketch.map(x ⇒ x * 2)
// reflection transformation
sketch.map(x ⇒ x * -1)
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37. Experiment result of map
sketch.map(x ⇒ math.exp(x))
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38. Average speed for three different speed
models
for {
speedA ← speedSketchA
speedB ← speedSketchB
speedC ← speedSketchC
} yield (speedA + speedB + speedC) / 3
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39. Average height for male and female
for {
gender ← genderDist
height ← gender match {
case Male ⇒ maleHeightDist
case Female ⇒ femaleHeightDist
}
} yield height
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40. !
Flip: Fast, Lightweight library for
Information and Probability
4 Most fast and lightweight
4 Pure-functional
4 Unique and high-level open source
4 GitHub: https://github.com/xxxnell/flip
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41. Conclusion
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42. Further Readings
4 Kernel Density Estimation in Spark
4 A frequentist approach to probability
4 Probability Distribution Monad (code)
4 Foundations of the Giry Monad
4 Platform for statistical modeling
4 A library for probabilistic modeling on TF
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