clojure for data science
TRANSCRIPT
CLOJURE FOR DATASCIENCE
WHY AM I GIVING THIS TALK?I am in the final stages of writing Clojure for Data Science.
It will be published by later this year.http://packtpub.com
AM I QUALIFIED?I co-founded and was CTO of a data analytics company.
I am a software engineer, not a statistician.
WHY IS DATA SCIENCE IMPORTANT?The robots are coming!
The rise of the computational developer.
These trends influence the kinds of systems we are allexpected to build.
WHY CLOJURE?Clojure lends itself to interactive exploration and learning.
It has fantastic data manipulating abstractions.
The JVM hosts many of the workhorse data storage andprocessing frameworks.
WHAT I WILL COVERDistributionsStatisticsVisualisation with QuilCorrelationSimple linear regressionMultivariable linear regression with IncanterBreakCategorical dataBayes classificationLogistic regression with Apache Commons MathClustering with Parkour and Apache Mahout
FOLLOW ALONGThe book's GitHub is available at
http://github.com/clojuredatascience
ch1-introductionch2-statistical-inferencech3-linear-regressionch5-classificationch6-clustering
PART 1: SPOTTING ELECTION FRAUD
LOADING UK ELECTION DATAUsing incanter's excel namespace
(ns cljds.ch1.data (:require [incanter [core :as i] [excel :as xls]] [clojure.java.io :as io]))
(defn uk-data [] (-> (io/resource "UK2010.xls") (str) (xls/read-xls)))
(i/view (uk-data))
IF YOU'RE FOLLOWING ALONGgit clone [email protected]:clojuredatascience/ch1-introduction.git
cd ch1-introduction
script/download-data.sh
lein run -e 1.1
COLUMN NAMES(defn ex-1-1 [] (i/col-names (uk-data)))
;; => ["Press Association Reference" "Constituency Name" "Region" "Election Year" "Electorate" "Votes" "AC" "AD" "AGS" "APNI" "APP" "AWL" "AWP" "BB" "BCP" "Bean" "Best" "BGPV" "BIB" "BIC" "Blue" "BNP" "BP Elvis" "C28" "Cam Soc" "CG" "Ch M" "Ch P" "CIP" "CITY" "CNPG" "Comm" "Comm L" "Con" "Cor D" "CPA" "CSP" "CTDP" "CURE" "D Lab" "D Nat" "DDP" "DUP" "ED" "EIP" "EPA" "FAWG" "FDP" "FFR" "Grn" "GSOT" "Hum" "ICHC" "IEAC" "IFED" "ILEU" "Impact" "Ind1" "Ind2" "Ind3" "Ind4" "Ind5" "IPT" "ISGB" "ISQM" "IUK" "IVH" "IZB" "JAC" "Joy" "JP" "Lab" "Land" "LD" "Lib" "Libert" "LIND" "LLPB" "LTT" "MACI" "MCP" "MEDI" "MEP" "MIF" "MK" "MPEA" "MRLP" "MRP" "Nat Lib" "NCDV" "ND" "New" "NF" "NFP" "NICF" "Nobody" "NSPS" "PBP" "PC" "Pirate" "PNDP" "Poet" "PPBF" "PPE" "PPNV" "Reform" "Respect" "Rest" "RRG" "RTBP" "SACL" "Sci" "SDLP" "SEP" "SF" "SIG" "SJP" "SKGP" "SMA" "SMRA" "SNP" "Soc" "Soc Alt" "Soc Dem" "Soc Lab" "South" "Speaker" "SSP" "TF" "TOC" "Trust" "TUSC" "TUV" "UCUNF" "UKIP" "UPS" "UV" "VCCA" "Vote" "Wessex Reg" "WRP" "You" "Youth" "YRDPL"]
ELECTORATE(defn uk-electorate [] (->> (uk-data) (i/$ "Electorate") (remove nil?))
VARIANCE( +1
n *ni=1 xi μx)2
…EXPLAINED is `(reduce + …)`.*
is "for all xs"*ni=1
is a function of x and the mean of x( +xi μx)2
(defn variance [xs] (let [m (mean xs) n (count xs) square-error (fn [x] (Math/pow (- x m) 2))] (/ (reduce + (map square-error xs)) n)))
HISTOGRAM(require '[incanter.charts :as c])
(defn ex-1-11 [] (-> (uk-electorate) (c/histogram :nbins 20) (i/view)))
POINCARÉ'S BREADPoincaré weighed his bread every day for a year.
He discovered that the weights of the bread followed anormal distribution, but that the peak was at 950g, whereas
loaves of bread were supposed to be regulated at 1kg. Hereported his baker to the authorities.
The next year Poincaré continued to weigh his bread fromthe same baker, who was now wary of giving him the lighterloaves. After a year the mean loaf weight was 1kg, but thistime the distribution had a positive skew. This is consistent
with the baker giving Poincaré only the heaviest of his loaves.The baker was reported to the authorities again
HONEST BAKER(require '[incanter.distributions :as d])
(defn honest-baker [] (let [distribution (d/normal-distribution 1000 30)] (repeatedly #(d/draw distribution))))
(defn ex-1-16 [] (-> (take 10000 (honest-baker)) (c/histogram :nbins 25) (i/view)))
DISHONEST BAKER(defn dishonest-baker [] (let [distribution (d/normal-distribution 950 30)] (->> (repeatedly #(d/draw distribution)) (partition 13) (map (partial apply max)))))
(defn ex-1-17 [] (-> (take 10000 (dishonest-baker)) (c/histogram :nbins 25) (i/view)))
THE IMPORTANCE OF VISUALISATIONAnscombe's Quartet: all have identical mean and variance.
SELECTION(defn filter-election-year [data] (i/$where {"Election Year" {:$ne nil}} data))
(defn filter-victor-constituencies [data] (i/$where {"Con" {:$fn number?} "LD" {:$fn number?}} data))
PROJECTION(->> (uk-data) (filter-election-year) (filter-victor-constituencies) (i/$ ["Region" "Electorate" "Con" "LD"]) (i/add-derived-column "Victors" ["Con" "LD"] +) (i/add-derived-column "Victors Share" ["Victors" "Electorate"] /) (i/view))
TWO VARIABLES: SCATTER PLOTS!(defn ex-1-33 [] (let [data (->> (uk-data) (clean-uk-data) (derive-uk-data))] (-> (scatter-plot ($ "Turnout" data) ($ "Victors Share" data) :x-label "Turnout" :y-label "Victor's Share") (view))))
UK VOTES SCATTER
RUSSIA IS A BIGGER COUNTRY
WE NEED A BETTER VISUALIZATION
BINNING DATA(defn bin [n-bins xs] (let [min-x (apply min xs) range-x (- (apply max xs) min-x) max-bin (dec n-bins) bin-fn (fn [x] (-> x (- min-x) (/ range-x) (* n-bins) int (min max-bin)))] (map bin-fn xs)))
(defn ex-1-10 [] (->> (uk-electorate) (bin 10) (frequencies)))
;; => {0 1, 1 1, 2 4, 3 22, 4 130, 5 320, 6 156, 7 15, 9 1}
A 2D HISTOGRAM(defn histogram-2d [xs ys n-bins] (-> (map vector (bin n-bins xs) (bin n-bins ys)) (frequencies)))
(defn uk-histogram-2d [] (let [data (->> (uk-data) (clean-uk-data) (derive-uk-data))] (histogram-2d ($ "Turnout" data) ($ "Victors Share" data) 5)))
;; => {[2 1] 59, [3 2] 91, [4 3] 32, [1 0] 8, [2 2] 89, [3 3] 101, [4 4] 60, [0 0] 2, [1 1] 22, [2 3] 19, [3 4] 53, [0 1] 6, [1 2] 15, [2 4] 5, [1 3] 2, [0 3] 1, [3 0] 6, [4 1] 3, [3 1] 17, [4 2] 17, [2 0] 23}
VISUALIZATION WITH QUIL(require '[quil.core :as q])
(defn ratio->grayscale [f] (-> f (* 255) (int) (min 255) (max 0) (q/color)))
(defn draw-histogram [data {:keys [n-bins size]}] (let [[width height] size x-scale (/ width n-bins) y-scale (/ height n-bins) max-value (apply max (vals data)) setup (fn [] (doseq [x (range n-bins) y (range n-bins)] (let [v (get data [x y] 0) x-pos (* x x-scale) y-pos (- height (* y y-scale))] (q/fill (ratio->grayscale (/ v max-value))) (q/rect x-pos y-pos x-scale y-scale))))] (q/sketch :setup setup :size size)))
A 2D HISTOGRAM
A COLOUR HEATMAPInterpolate between the colours of the spectrum.
(defn ratio->heat [f] (let [colors [(q/color 0 0 255) ;; blue (q/color 0 255 255) ;; turquoise (q/color 0 255 0) ;; green (q/color 255 255 0) ;; yellow (q/color 255 0 0)] ;; red f (-> f (max 0.000) (min 0.999) (* (dec (count colors))))] (q/lerp-color (nth colors f) (nth colors (inc f)) (rem f 1))))
A FINISHED HEATMAP$> lein run -e 1.36
CREDITProceedings of the National Academy of Sciences, titled
"Statistical Detection of Election Irregularities," a team led bySanta Fe Institute External Professor Stefan Thurner
INFERENCE
WHAT ARE STATISTICS ANYWAY?They are estimates of values, based on a sample.
WHAT ARE PARAMETERS?They are the true values, based on the entire population.
SAMPLING SIZEThe values converge as the sample size increases.
We can often only infer the population parameters.
Sample Population
n NX̄ μX
SX σX
REAGENT(def population-mean 100)(def population-sd 20)(def sample-size 10)
REAGENT ATOMS(require '[reagent.core :as r])
(defn randn [mean sd] (.. js/jStat -normal (sample mean sd)))
(defn normal-distribution [mean sd] (repeatedly #(randn mean sd)))
(def state (r/atom {:sample []}))
(defn update-sample! [state] (swap! state assoc :sample (->> (normal-distribution population-mean population-sd) (map int) (take sample-size))))
CREATE THE WIDGETS(defn new-sample [state] [:button {:on-click #(update-sample! state)} "New Sample"])
(defn sample-list [state] [:div (let [sample (:sample @state)] [:div [:ul (for [n sample] [:li n])] [:dl [:dt "Sample Mean:"] [:dd (mean sample)]]])])
LAY OUT THE INTERFACE(defn layout-interface [] [:div [:h1 "Normal Sample"] [new-sample state] [sample-list state]])
;; Render the root component(defn run [] (r/render-component [layout-interface] (.getElementById js/document "root")))
COMPILE THE CLJS$> lein cljsbuild once
DEMO
STANDARD ERRORIt's the standard deviation of the sample means.
SE = σXn√
STANDARD ERRORLet's see how the standard error changes with sample size.
DEMO
STANDARD DEVIATION PROPORTIONS
SMALL SAMPLESThe standard error is calculated from the population
standard deviation, but we don't know it!
In practice they're assumed to be the same above around 30samples, but there is another distribution that models the
loss of precision with small samples.
T-DISTRIBUTION
CALCULATING THE T-STATISTICBased entirely on our sample statistics
(defn t-statistic [sample test-mean] (let [sample-mean (mean sample) sample-size (count sample) sample-sd (standard-deviation sample)] (/ (- sample-mean test-mean) (/ sample-sd (Math/sqrt sample-size)))))
DEMO
WHY THIS INTEREST IN MEANS?Because often when we want to know if a difference in
populations is statistically significant, we'll compare themeans.
HYPOTHESIS TESTINGBy convention the data is assumed not to support what the
researcher is looking for.
This conservative assumption is called the null hypothesis anddenoted .h0
The alternate hypothesis, , can then only be supported witha given confidence interval.
h1
SIGNIFICANCEThe greater the significance of a result, the more certainty we
have that the null hypothesis can be rejected.
Let's use our range controller to adjust the significancethreshold.
DEMO
PREDICTION
QUESTIONWhat was Olympic swimmer Mark Spitz' competition weight?
POPULATION OF OLYMPIC SWIMMERSThe Guardian has helpfully provided data on the vital
statistics of Olympians
http://www.theguardian.com/sport/datablog/2012/aug/07/olympics-2012-athletes-age-weight-height#data
WEIGHT HISTOGRAM
LOG-WEIGHT HISTOGRAM
LOG-NORMAL DISTRIBUTION"A variable might be modeled as log-normal if it can be
thought of as the multiplicative product of many independentrandom variables, each of which is positive. This is justified by
considering the central limit theorem in the log-domain."
SCATTER PLOT
WHAT IS CORRELATION ANYWAY?We've talked about variance in the context of the normal
distribution.
COVARIANCEHow much things vary together!
COV(X, Y) = ( + )( + )1n*
ni=1 xi μx yi μy
CORRELATIONA few ways of measuring it, depending on whether your data
is continuous or discrete
http://xkcd.com/552/
PEARSON'S CORRELATIONCovariance divided by the product of standard deviations. It
measures linear correlation.
ρX, Y = COV(X,Y)σXσY
(defn pearsons-correlation [x y] (/ (covariance x y) (* (standard-deviation x) (standard-deviation y))))
PEARSON'S CORRELATIONIf is 0, it doesn’t necessarily mean that the variables are not
correlated. Pearson’s correlation only measures linearrelationships.
r
THIS IS A STATISTICThe unknown population parameter for correlation is theGreek letter . We are only able to calculate the sample
statistic .ρ
r
How far we can trust as an estimate of will depend on twofactors:
r ρ
the size of the coefficientthe size of the sample
rX, Y = COV(X,Y)sX sY
SIMPLE LINEAR REGRESSIONy = α + βx
SIMPLE LINEAR REGRESSION(defn slope [x y] (/ (covariance x y) (variance x)))
(defn intercept [x y] (- (mean y) (* (mean x) (slope x y))))
(defn predict [a b x] (+ a (* b x)))
TRAINING A MODEL(defn swimmer-data [] (->> (athlete-data) ($where {"Height, cm" {:$ne nil} "Weight" {:$ne nil} "Sport" {:$eq "Swimming"}})))
(defn ex-3-12 [] (let [data (swimmer-data) heights ($ "Height, cm" data) weights (log ($ "Weight" data)) a (intercept heights weights) b (slope heights weights)] (println "Intercept: " a) (println "Slope: " b)))
MAKING A PREDICTION(predict 1.691 0.0143 185)
;; => 4.3365
(i/exp (predict 1.691 0.0143 185))
;; => 76.44
Corresponding to a predicted weight of 76.4kg
In 1979, Mark Spitz was 79kg.
http://www.topendsports.com/sport/swimming/profiles/spitz-mark.htm
MORE DATA!(defn features [dataset col-names] (->> (i/$ col-names dataset) (i/to-matrix)))
(defn gender-dummy [gender] (if (= gender "F") 0.0 1.0))
(defn ex-3-26 [] (let [data (->> (swimmer-data) (i/add-derived-column "Gender Dummy" ["Sex"] gender-dummy)) x (features data ["Height, cm" "Age" "Gender Dummy"]) y (i/log ($ "Weight" data)) model (s/linear-model y x)] (:coefs model)))
;; => [2.2307529431422637 0.010714697827121089 0.002372188749408574 0.0975412532492026]
MULTIVARIABLE LINEAR REGRESSIONy = + +. . . +θ0 θ1x1 θnxn
LEAST SQUARES
MAKING PREDICTIONSy = xθT
(defn predict [theta x] (-> (cl/t theta) (cl/* x) (first)))
(defn ex-3-27 [] (let [data (->> (swimmer-data) (i/add-derived-column "Gender Dummy" ["Sex"] gender-dummy)) x (features data ["Height, cm" "Age" "Gender Dummy"]) y (i/log ($ "Weight" data)) model (s/linear-model y x)] (i/exp (predict (i/matrix (:coefs model)) (i/matrix [1 185 22 1])))))
;; => 78.46882772631697
HOW CLOSE?The result is around 78.47kg.
Compared to 79kg, we were pretty close!
SUMMARYDistributionsStatisticsVisualisation with QuilCorrelationLinear regressionMultivariate linear regression with Incanter
INTERLUDE
CLASSIFICATION
QUESTIONDid all passengers on the Titanic have an equal chance of
survival?
ON THE DATAData is based on Thomas Cason's Titanic3 dataset.
INSPECT THE DATAClass Survived Name Sex Age
1 1 Allen, Miss. ElisabethWalton
female 29
1 0 Allison, Mr. HudsonJoshua Creighton
male 30
CATEGORICAL VARIABLES Survived Perished
Male 161 682
Female 339 127
STANDARD ERROR FOR A PROPORTIONSE =
p(1 + p)n
‾ ‾‾‾‾‾‾‾‾√(defn standard-error-proportion [p n] (-> (- 1 p) (* p) (/ n) (Math/sqrt)))
= = 0.61161 + 339682 + 127
500809
SE = 0.013
HOW SIGNIFICANT?z =
+p1 p2
SEP1: the proportion of women who survived is = 0.76339
446
P2: the proportion of men who survived = = 0.19161843
SE: 0.013
z = 20.36
This is essentially impossible.
MORE CATEGORIES Survived Perished
First Class 200 123
Second Class 119 158
Third Class 181 528
OUR APPROACH DOESN'T SCALEWe can use a test.χ 2
(defn ex-5-5 [] (let [observations (i/matrix [[200 119 181] [123 158 528]])] (s/chisq-test :table observations)))
How likely is that this distribution occurred via chance?{:X-sq 127.85915643930326, :col-levels (0 1 2), :row-margins {0 500.0, 1 809.0}, :table [matrix] , :p-value 1.7208259588256175E-28, :df 2, :probs nil, :col-margins {0 323.0, 1 277.0, 2 709.0}, :E (123.37662337662337 199.62337662337663 105.80595874713522 171.1940412528648 270.8174178762414 438.1825821237586), :row-levels (0 1), :two-samp? true, :N 1309.0}
P-VALUE"The estimated probability of rejecting the null hypothesis
of a study question when that hypothesis is true."h0
BAYES RULEP(A|B) =
P(B|A)P(A)P(B)
initial degree of belief in A (the prior)P(A)
BAYES TITANICP(survive|f emale) =
P(f emale|survive)P(survive)P(f emale)
P(survive|f emale) = =339500
5001309
4461309
339446
BAYES CLASSIFICATIONP(survive|third, male) =
P(survive)P(third|survive)P(male|P(third, male)
P(perish|third, male) =P(perish)P(third|perish)P(male|per
P(third, male)Because the evidence is the same for all classes, we can
cancel this out.
PARSE THE DATA(titanic-samples)
;; => ({:survived true, :gender :female, :class :first, :embarked "S", :age "20-30"} {:survived true, :gender :male, :class :first, :embarked "S", :age "30-40"} ...)
IMPLEMENTING A NAIVE BAYES MODEL(defn safe-inc [v] (inc (or v 0)))
(defn inc-class-total [model class] (update-in model [class :total] safe-inc))
(defn inc-predictors-count-fn [row class] (fn [model attr] (let [val (get row attr)] (update-in model [class attr val] safe-inc))))
IMPLEMENTING A NAIVE BAYES MODEL(defn assoc-row-fn [class-attr predictors] (fn [model row] (let [class (get row class-attr)] (reduce (inc-predictors-count-fn row class) (inc-class-total model class) predictors))))
(defn naive-bayes [data class-attr predictors] (reduce (assoc-row-fn class-attr predictors) {} data))
NAIVE BAYES MODEL(defn ex-5-6 [] (let [data (titanic-samples)] (pprint (naive-bayes data :survived [:gender :class]))))
…produces the following output…;; {false;; {:class {:third 528, :second 158, :first 123},;; :gender {:male 682, :female 127},;; :total 809},;; true;; {:class {:third 181, :second 119, :first 198},;; :gender {:male 161, :female 337},;; :total 498}}
MAKING PREDICTIONS(defn n [model] (->> (vals model) (map :total) (apply +)))
(defn conditional-probability [model test class] (let [evidence (get model class) prior (/ (:total evidence) (n model))] (apply * prior (for [kv test] (/ (get-in evidence kv) (:total evidence))))))
(defn bayes-classify [model test] (let [probs (map (fn [class] [class (conditional-probability model test class)]) (keys model))] (-> (sort-by second > probs) (ffirst))))
DOES IT WORK?(defn ex-5-7 [] (let [data (titanic-samples) model (naive-bayes data :survived [:gender :class])] (bayes-classify model {:gender :male :class :third})))
;; => false
(defn ex-5-8 [] (let [data (titanic-samples) model (naive-bayes data :survived [:gender :class])] (bayes-classify model {:gender :female :class :first})))
;; => true
WHY NAIVE?Because it assumes all variables are independent. We know
they are not (e.g. being male and in third class) but naivebayes weights all attributes equally.
In practice it works surprisingly well, particularly where thereare large numbers of features.
LOGISTIC REGRESSION
LOGISTIC REGRESSIONLogistic regression uses similar techniques to linear
regression but guarantees an output only between 0 and 1.
(x) = xhθ θT
(x) = g( x)hθ θT
Where the sigmoid function is
g(z) =1
1 + e+z
THE LOGISTIC FUNCTION
THE LOGISTIC FUNCTION(defn logistic-function [theta] (let [tt (matrix/transpose (vec theta)) z (fn [x] (- (matrix/mmul tt (vec x))))] (fn [x] (/ 1 (+ 1 (Math/exp (z x)))))))
INTERPRETATION(let [f (logistic-function [0])] (f [1]) ;; => 0.5
(f [-1]) ;; => 0.5
(f [42]) ;; => 0.5 )
(let [f (logistic-function [0.2]) g (logistic-function [-0.2])] (f [5]) ;; => 0.73
(g [5]) ;; => 0.27 )
COST FUNCTIONCost varies between 0 and (a big number).
(defn cost-function [y y-hat] (- (if (zero? y) (Math/log (max (- 1 y-hat) Double/MIN_VALUE)) (Math/log (max y-hat (Double/MIN_VALUE))))))
(defn logistic-cost [ys y-hats] (avg (map cost-function ys y-hats)))
CONVERTING TITANIC DATA TO FEATURES(defn titanic-features [] (remove (partial some nil?) (for [row (titanic-data)] [(:survived row) (:pclass row) (:sibsp row) (:parch row) (if (nil? (:age row)) 30 (:age row)) (if (= (:sex row) "female") 1.0 0.0) (if (= (:embarked row) "S") 1.0 0.0) (if (= (:embarked row) "C") 1.0 0.0) (if (= (:embarked row) "Q") 1.0 0.0)])))
CALCULATING THE GRADIENT(defn gradient-fn [h-theta xs ys] (let [g (fn [x y] (matrix/mmul (- (h-theta x) y) x))] (->> (map g xs ys) (matrix/transpose) (map avg))))
We transpose to calculate the average for each featureacross all xs rather than average for each x across all
features.
GRADIENT DESCENTThe cost function will be lowest when the parameters are at
their optimum.
APACHE COMMONS MATHProvides heavy-lifting for running tasks like gradient descent.
(:import [org.apache.commons.math3.analysis MultivariateFunction MultivariateVectorFunction] [org.apache.commons.math3.optim InitialGuess MaxEval SimpleBounds OptimizationData SimpleValueChecker PointValuePair] [org.apache.commons.math3.optim.nonlinear.scalar ObjectiveFunction ObjectiveFunctionGradient GoalType] [org.apache.commons.math3.optim.nonlinear.scalar.gradient NonLinearConjugateGradientOptimizer NonLinearConjugateGradientOptimizer$Formula])
CLOJURE'S JAVA INTEROPAn object wrapper to represent a function: too many levels of
indirection?!(defn objective-function [f] (ObjectiveFunction. (reify MultivariateFunction (value [_ v] (apply f (vec v))))))
(defn objective-function-gradient [f] (ObjectiveFunctionGradient. (reify MultivariateVectorFunction (value [_ v] (double-array (apply f (vec v)))))))
GRADIENT DESCENT(defn make-ncg-optimizer [] (NonLinearConjugateGradientOptimizer. NonLinearConjugateGradientOptimizer$Formula/FLETCHER_REEVES (SimpleValueChecker. (double 1e-6) (double 1e-6))))
(defn initial-guess [guess] (InitialGuess. (double-array guess)))
(defn max-evaluations [n] (MaxEval. n))
(defn gradient-descent [f g estimate n] (let [options (into-array OptimizationData [(objective-function f) (objective-function-gradient g) (initial-guess estimate) (max-evaluations n) GoalType/MINIMIZE])] (-> (make-ncg-optimizer) (.optimize options) (.getPoint) (vec))))
RUNNING GRADIENT DESCENT(defn run-logistic-regression [data initial-guess] (let [points (titanic-features) xs (->> points (map rest) (map #(cons 1 %))) ys (map first points)]
(gradient-descent (fn [& theta] (let [f (logistic-function theta)] (logistic-cost (map f xs) ys))) (fn [& theta] (gradient-fn (logistic-function theta) xs ys)) initial-guess 2000)))
PRODUCING A MODEL(defn ex-5-11 [] (let [data (titanic-features) initial-guess (-> data first count (take (repeatedly rand)))] (run-logistic-regression data initial-guess)))
MAKING PREDICTIONS(def theta [0.690807824623404 -0.9033828001369435 -0.3114375278698766 -0.01894319673287219 -0.03100315579768661 2.5894858366033273 0.7939190708193374 1.3711334887947388 0.6672555257828919])
(defn round [x] (Math/round x))
(def logistic-model (logistic-function theta))
(defn ex-5-13 [] (let [data (titanic-features) test (fn [x] (= (round (logistic-model (cons 1 (rest x)))) (round (first x)))) results (frequencies (map test data))] (/ (get results true) (apply + (vals results)))))
;; => 1030/1309
EVALUATING THE CLASSIFIERCross-validation: we want to separate our test and training
data sets
Bias vs variance: your model may fail to generalise
CLUSTERING
CLUSTERINGFind a grouping of a set of objects such that objects in thesame group are more similar to each other than those in
other groups.
SIMILARITY MEASURESMany to choose from: Jaccard, Euclidean.
For text documents the Cosine measure is often chosen.
Good for high-dimensional spacesPositive spaces the similarity is between 0 and 1.
COSINE SIMILARITYcos(θ) =
A Þ B>A>>B>
(defn cosine [a b] (let [dot-product (->> (map * a b) (apply +)) magnitude (fn [d] (->> (map #(Math/pow % 2) d) (apply +) Math/sqrt))] (/ dot-product (* (magnitude a) (magnitude b)))))
CREATING SPARSE VECTORS(def dictionary (atom {:count 0 :words {}}))
(defn add-word-to-dict [dict word] (if (get-in dict [:words word]) dict (-> dict (update-in [:words] assoc word (get dict :count)) (update-in [:count] inc))))
(defn update-words [dict doc word] (let [word-id (-> (swap! dict add-word-to-dict word) (get-in [:words word]))] (update-in doc [word-id] #(inc (or % 0)))))
(defn document-vector [dict ngrams] (r/reduce (partial update-words dict) {} ngrams))
EXAMPLE(->> (split "the quick brown fox jumps over the lazy dog" #"\W+") (document-vector dictionary))
;; => {7 1, 6 1, 5 1, 4 1, 3 1, 2 1, 1 1, 0 2}
@dictionary
;; => {:words {"dog" 7, "lazy" 6, "over" 5, "jumps" 4, "fox" 3, "brown" 2, "quick" 1, "the" 0}, :count 8}
STEMMING / STOPWORDShttp://clojars.org/stemmers
(stemmer/stems "it's lovely that you're musical")
;; => ("love" "music")
WHY? (cosine-sparse (->> "music is the food of love" stemmer/stems (document-vector dictionary)) (->> "war is the locomotive of history" stemmer/stems (document-vector dictionary)))
;; => 0.0
(cosine-sparse (->> "music is the food of love" stemmer/stems (document-vector dictionary)) (->> "it's lovely that you're musical" stemmer/stems (document-vector dictionary)))
;; => 0.8164965809277259
EXAMPLE(->> "it's lovely that you're musical" stemmer/stems (document-vector dictionary))
;; => {0 1, 2 1}
@dictionary
;; => {:count 6, :words {"histori" 5, "locomot" 4, "war" 3, "love" 2, "food" 1, "music" 0}}
MAHOUThttp://mahout.apache.org/
"The Apache Mahout™ project's goal is to build anenvironment for quickly creating scalable preformant
machine learning applications."
GET THE DATAWe're going to be clustering the Reuters dataset.
Follow the readme instructions:brew install mahout
script/download-reuters.shlein run -e 6.7mahout seqdirectory -i data/reuters-txt -o data/reuters-sequencefile
VECTOR REPRESENTATIONEach document is converted into a vector representation.
All vectors share a dictionary providing a unique index foreach word.
SEQUENCEFILESInput:
org.apache.hadoop.io.Textorg.apache.hadoop.io.Text
Output (Vectors):
org.apache.hadoop.io.Textorg.apache.mahout.math.VectorWritable
Output (Dictionary):
org.apache.hadoop.io.Textorg.apache.mahout.math.IntWritable
PARKOURParkour is a Clojure library for interacting with Hadoop.
It provides a thinner layer of abstraction than PigPen andCascalog.
TF-IDFTerm frequency, inverse document frenquency.
tf idf (t, d, D) = tf (t, d) Þ idf (t, D)
WE NEED A UNIQUE IDAnd we need to compute it in parallel.
PARKOUR MAPPING(require '[clojure.core.reducers :as r] '[parkour.mapreduce :as mr])
(defn document->terms [doc] (clojure.string/split doc #"\W+"))
(defn document-count-m "Emits the unique words from each document" {::mr/source-as :vals} [documents] (->> documents (r/mapcat (comp distinct document->terms)) (r/map #(vector % 1))))
SHAPE METADATA:keyvals ;; Re-shape as vectors of key-vals pairs.:keys ;; Just the keys from each key-value pair.:vals ;; Just the values from each key-value pair.
PLAIN OLD FUNCTIONS (->> (document-count-m ["it's lovely that you're musical" "music is the food of love" "war is the locomotive of history"]) (into []))
;; => [["love" 1] ["music" 1] ["music" 1] ["food" 1] ["love" 1] ["war" 1] ["locomot" 1] ["histori" 1]]
AND REDUCING…(require '[parkour.io.dux :as dux] '[transduce.reducers :as tr])
(defn unique-index-r {::mr/source-as :keyvalgroups, ::mr/sink-as dux/named-keyvals} [coll] (let [global-offset (conf/get-long mr/*context* "mapred.task.partition" -1)] (tr/mapcat-state (fn [local-offset [word counts]] [(inc local-offset) (if (identical? ::finished word) [[:counts [global-offset local-offset]]] [[:data [word [[global-offset local-offset] (apply + counts)]]]])]) 0 (r/mapcat identity [coll [[::finished nil]]]))))
CREATING A JOB(require '[parkour.graph :as pg] '[parkour.avro :as mra] '[abracad.avro :as avro])
(def long-pair (avro/tuple-schema [:long :long]))(def index-value (avro/tuple-schema [long-pair :long]))
(defn df-j [dseq] (-> (pg/input dseq) (pg/map #'document-count-m) (pg/partition (mra/shuffle [:string :long])) (pg/reduce #'unique-index-r) (pg/output :data (mra/dsink [:string index-value]) :counts (mra/dsink [:long :long]))))
WRITING TO DISTRIBUTED CACHE(require '[parkour.io.dval :as dval])
(defn calculate-offsets "Build map of offsets from dseq of counts." [dseq] (->> dseq (into []) (sort-by first) (reductions (fn [[_ t] [i n]] [(inc i) (+ t n)]) [0 0]) (into {})))
(defn df-execute [conf dseq] (let [[df-data df-counts] (pg/execute (df-j dseq) conf ̀df) offsets-dval (dval/edn-dval (calculate-offsets df-counts))] ...))
READING FROM DISTRIBUTED CACHE(defn global-id "Use offsets to calculate unique id from global and local offset" [offsets [global-offset local-offset]] (+ local-offset (get offsets global-offset)))
(defn words-idf-m "Calculate the unique id and inverse document frequency for each word" {::mr/sink-as :keys} [offsets-dval n coll] (let [offsets @offsets-dval] (r/map (fn [[word [word-offset df]]] [word (global-id offsets word-offset) (Math/log (/ n df))]) coll)))
(defn make-dictionary [conf df-data df-counts doc-count] (let [offsets-dval (dval/edn-dval (calculate-offsets df-counts))] (-> (pg/input df-data) (pg/map #'words-idf-m offsets-dval doc-count) (pg/output (mra/dsink [words])) (pg/fexecute conf ̀idf) (->> (r/map parse-idf) (into {})) (dval/edn-dval))))
CREATING TEXT VECTORS(import '[org.apache.mahout.math RandomAccessSparseVector])
(defn create-sparse-vector [dictionary [id doc]] (let [vector (RandomAccessSparseVector. (count dictionary))] (doseq [[term freq] (-> doc document->terms frequencies)] (let [term-info (get dictionary term)] (.setQuick vector (:id term-info) (* freq (:idf term-info))))) [id vector]))
(defn create-vectors-m [dictionary coll] (let [dictionary @dictionary] (r/map #(create-sparse-vector dictionary %) coll)))
THE FINISHED JOB(import '[org.apache.hadoop.io Text] '[org.apache.mahout.math VectorWritable])
(defn tfidf [conf dseq dictionary-path vector-path] (let [doc-count (->> dseq (into []) count) [df-data df-counts] (pg/execute (df-j dseq) conf ̀df) dictionary-dval (make-dictionary conf df-data df-counts doc-count)] (write-dictionary dictionary-path dictionary-dval) (-> (pg/input dseq) (pg/map #'create-vectors-m dictionary-dval) (pg/output (seqf/dsink [Text VectorWritable] vector-path)) (pg/fexecute conf ̀vectorize))))
(defn tool [conf input output] (let [dseq (seqf/dseq input) dictionary-path (doto (str output "/dictionary") fs/path-delete) vector-path (doto (str output "/vectors") fs/path-delete)] (tfidf conf dseq dictionary-path vector-path)))
(defn -main [& args] (System/exit (tool/run tool args)))
RUN THE JOB(defn ex-6-14 [] (let [input "data/reuters-sequencefile" output "data/parkour-vectors"] (tool/run vectorizer/tool [input output])))
K-MEANS
K-MEANS
RUNNING CLUSTERINGscript/run-kmeans.sh
#!/bin/bash
WORK_DIR=dataINPUT_DIR=${WORK_DIR}/parkour-vectors
mahout kmeans \ -i ${INPUT_DIR}/vectors \ -c ${WORK_DIR}/clusters-out \ -o ${WORK_DIR}/kmeans-out \ -dm org.apache.mahout.common.distance.CosineDistanceMeasure \ -x 20 -k 5 -cd 0.01 -ow --clustering
mahout clusterdump \ -i ${WORK_DIR}/kmeans-out/clusters-*-final \ -o ${WORK_DIR}/clusterdump.txt \ -d ${INPUT_DIR}/dictionary/part-r-00000 \ -dt sequencefile \ -dm org.apache.mahout.common.distance.CosineDistanceMeasure \ --pointsDir ${WORK_DIR}/kmeans-out/clusteredPoints \ -b 100 -n 20 -sp 0 -e
HOW MANY CLUSTERS?
WHAT DID I LEAVE OUT?Cluster quality measuresSpectral and LDA clusteringCollaborative filtering with MahoutRandom forestsSpark for movie recommendations with SparklingGraph data with Loom and DatomicMapReduce with Cascalog and PigPenAdapting algorithms for massive scaleTime series and forecastingDimensionality reduction, feature selectionMore visualisation techniquesLots more…
BOOKClojure for Data Science will be available in the second half
of the year from .http://packtpub.com
http://cljds.com
SLIDEShttp://github.com/henrygarner/clojure-data-science
THANK YOU!Henry Garner
@henrygarner