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stats.clj
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(ns fastmath.stats
"#### Statistics functions.
* Descriptive statistics.
* Correlation / covariance
* Outliers
* Confidence intervals
* Extents
* Effect size
* Tests
* Histogram
* ACF/PACF
* Bootstrap (see `fastmath.stats.bootstrap`)
* Binary measures
Functions are backed by Apache Commons Math or SMILE libraries. All work with Clojure sequences.
##### Descriptive statistics
All in one function [[stats-map]] contains:
* `:Size` - size of the samples, `(count ...)`
* `:Min` - [[minimum]] value
* `:Max` - [[maximum]] value
* `:Range` - range of values
* `:Mean` - [[mean]]/average
* `:Median` - [[median]], see also: [[median-3]]
* `:Mode` - [[mode]], see also: [[modes]]
* `:Q1` - first quartile, use: [[percentile]], [[quartile]]
* `:Q3` - third quartile, use: [[percentile]], [[quartile]]
* `:Total` - [[sum]] of all samples
* `:SD` - sample standard deviation
* `:Variance` - variance
* `:MAD` - [[median-absolute-deviation]]
* `:SEM` - standard error of mean
* `:LAV` - lower adjacent value, use: [[adjacent-values]]
* `:UAV` - upper adjacent value, use: [[adjacent-values]]
* `:IQR` - interquartile range, `(- q3 q1)`
* `:LOF` - lower outer fence, `(- q1 (* 3.0 iqr))`
* `:UOF` - upper outer fence, `(+ q3 (* 3.0 iqr))`
* `:LIF` - lower inner fence, `(- q1 (* 1.5 iqr))`
* `:UIF` - upper inner fence, `(+ q3 (* 1.5 iqr))`
* `:Outliers` - list of [[outliers]], samples which are outside outer fences
* `:Kurtosis` - [[kurtosis]]
* `:Skewness` - [[skewness]]
Note: [[percentile]] and [[quartile]] can have 10 different interpolation strategies. See [docs](http://commons.apache.org/proper/commons-math/javadocs/api-3.6.1/org/apache/commons/math3/stat/descriptive/rank/Percentile.html)"
(:require [fastmath.core :as m]
[fastmath.random :as r]
[fastmath.distance :as d]
[fastmath.vector :as v]
[fastmath.interpolation :as interp]
[fastmath.optimization :as opt]
[fastmath.kernel :as k])
(:import [org.apache.commons.math3.stat StatUtils]
[org.apache.commons.math3.stat.descriptive.rank Percentile Percentile$EstimationType]
[org.apache.commons.math3.stat.descriptive.moment Kurtosis Skewness]
[org.apache.commons.math3.stat.correlation KendallsCorrelation SpearmansCorrelation PearsonsCorrelation]
[org.apache.commons.math3.stat.regression SimpleRegression]
[org.apache.commons.math3.analysis.integration RombergIntegrator]
[org.apache.commons.math3.analysis UnivariateFunction]))
(set! *unchecked-math* :warn-on-boxed)
(m/use-primitive-operators)
(def ^{:doc "List of estimation strategies for [[percentile]]/[[quantile]] functions."}
estimation-strategies-list {:legacy Percentile$EstimationType/LEGACY
:r1 Percentile$EstimationType/R_1
:r2 Percentile$EstimationType/R_2
:r3 Percentile$EstimationType/R_3
:r4 Percentile$EstimationType/R_4
:r5 Percentile$EstimationType/R_5
:r6 Percentile$EstimationType/R_6
:r7 Percentile$EstimationType/R_7
:r8 Percentile$EstimationType/R_8
:r9 Percentile$EstimationType/R_9})
(defn percentile
"Calculate percentile of a `vs`.
Percentile `p` is from range 0-100.
See [docs](http://commons.apache.org/proper/commons-math/javadocs/api-3.4/org/apache/commons/math3/stat/descriptive/rank/Percentile.html).
Optionally you can provide `estimation-strategy` to change interpolation methods for selecting values. Default is `:legacy`. See more [here](http://commons.apache.org/proper/commons-math/javadocs/api-3.6.1/org/apache/commons/math3/stat/descriptive/rank/Percentile.EstimationType.html)
See also [[quantile]]."
(^double [vs ^double p]
(StatUtils/percentile (m/seq->double-array vs) p))
(^double [vs ^double p estimation-strategy]
(let [^Percentile perc (.withEstimationType (Percentile.) (get estimation-strategies-list estimation-strategy Percentile$EstimationType/LEGACY))]
(.evaluate perc (m/seq->double-array vs) p))))
(defn percentiles
"Calculate percentiles of a `vs`.
Percentiles are sequence of values from range 0-100.
See [docs](http://commons.apache.org/proper/commons-math/javadocs/api-3.4/org/apache/commons/math3/stat/descriptive/rank/Percentile.html).
Optionally you can provide `estimation-strategy` to change interpolation methods for selecting values. Default is `:legacy`. See more [here](http://commons.apache.org/proper/commons-math/javadocs/api-3.6.1/org/apache/commons/math3/stat/descriptive/rank/Percentile.EstimationType.html)
See also [[quantile]]."
([vs] (percentiles vs [25 50 75 100]))
([vs ps] (percentiles vs ps nil))
([vs ps estimation-strategy]
(let [^Percentile perc (.withEstimationType (Percentile.) (or (estimation-strategies-list estimation-strategy) Percentile$EstimationType/LEGACY))]
(.setData perc (m/seq->double-array vs))
(mapv (fn [^double p] (.evaluate perc p)) ps))))
(defn quantile
"Calculate quantile of a `vs`.
Quantile `q` is from range 0.0-1.0.
See [docs](http://commons.apache.org/proper/commons-math/javadocs/api-3.4/org/apache/commons/math3/stat/descriptive/rank/Percentile.html) for interpolation strategy.
Optionally you can provide `estimation-strategy` to change interpolation methods for selecting values. Default is `:legacy`. See more [here](http://commons.apache.org/proper/commons-math/javadocs/api-3.6.1/org/apache/commons/math3/stat/descriptive/rank/Percentile.EstimationType.html)
See also [[percentile]]."
(^double [vs ^double q]
(percentile vs (m/constrain (* q 100.0) 0.0 100.0)))
(^double [vs ^double q estimation-strategy]
(percentile vs (m/constrain (* q 100.0) 0.0 100.0) estimation-strategy)))
(defn quantiles
"Calculate quantiles of a `vs`.
Quantilizes is sequence with values from range 0.0-1.0.
See [docs](http://commons.apache.org/proper/commons-math/javadocs/api-3.4/org/apache/commons/math3/stat/descriptive/rank/Percentile.html) for interpolation strategy.
Optionally you can provide `estimation-strategy` to change interpolation methods for selecting values. Default is `:legacy`. See more [here](http://commons.apache.org/proper/commons-math/javadocs/api-3.6.1/org/apache/commons/math3/stat/descriptive/rank/Percentile.EstimationType.html)
See also [[percentiles]]."
([vs] (quantiles vs [0.25 0.5 0.75 1.0]))
([vs qs]
(percentiles vs (map #(m/constrain (* ^double % 100.0) 0.0 100.0) qs)))
([vs qs estimation-strategy]
(percentiles vs (map #(m/constrain (* ^double % 100.0) 0.0 100.0) qs) estimation-strategy)))
(defn- wquantile-interpolator
[vs ws method]
(let [sorted (sort-by first (map vector vs ws))
probabilities (map second sorted)
data (map first sorted)
data (conj data (first data))
^double wsum (reduce m/fast+ probabilities)
weights (conj (reductions m/fast+ (map (fn [^double p] (/ p wsum)) probabilities)) 0.0)]
(case method
:linear (interp/linear-smile weights data)
:average (let [interp1 (interp/step-before weights data)
interp2 (interp/step-after weights data)]
(fn [^double x] (* 0.5 (+ ^double (interp1 x)
^double (interp2 x)))))
:step (interp/step-before weights data))))
;; based on spatstat.geom::weighted.quantile
(defn wquantile
"Weighted quantile.
Calculation is done using interpolation. There are three methods:
* `:linear` - linear interpolation, default
* `:step` - step interpolation
* `:average` - average of ties
Based on `spatstat.geom::weighted.quantile` from R."
(^double [vs ws ^double q] (wquantile vs ws q :linear))
(^double [vs ws ^double q method]
(let [interp (wquantile-interpolator vs ws method)]
(interp q))))
(defn wquantiles
"Weighted quantiles.
Calculation is done using interpolation. There are three methods:
* `:linear` - linear interpolation, default
* `:step` - step interpolation
* `:average` - average of ties
Based on `spatstat.geom::weighted.quantile` from R."
([vs ws] (wquantiles vs ws [0.25 0.5 0.75 1.0]))
([vs ws qs] (wquantiles vs ws qs :linear))
([vs ws qs method]
(let [interp (wquantile-interpolator vs ws method)]
(mapv interp qs))))
(defn wmedian
"Weighted median.
Calculation is done using interpolation. There are three methods:
* `:linear` - linear interpolation, default
* `:step` - step interpolation
* `:average` - average of ties
Based on `spatstat.geom::weighted.quantile` from R."
(^double [vs ws] (wquantile vs ws 0.5))
(^double [vs ws method] (wquantile vs ws 0.5 method)))
(defn median
"Calculate median of `vs`. See [[median-3]]."
(^double [vs estimation-strategy]
(percentile vs 50.0 estimation-strategy))
(^double [vs]
(percentile vs 50.0)))
(defn median-3
"Median of three values. See [[median]]."
^double [^double a ^double b ^double c]
(m/max (m/min a b) (m/min (m/max a b) c)))
(defn mean
"Calculate mean of `vs`"
(^double [vs] (StatUtils/mean (m/seq->double-array vs))))
(defn geomean
"Geometric mean for positive values only"
^double [vs]
(m/exp (mean (map (fn [^double v] (m/log v)) vs))))
(defn harmean
"Harmonic mean"
^double [vs]
(/ (mean (map (fn [^double v] (/ v)) vs))))
(defn powmean
"Generalized power mean"
^double [vs ^double power]
(cond
(zero? power) (geomean vs)
(m/one? power) (mean vs)
(== power m/THIRD) (m/cb (mean (map #(m/cbrt %) vs)))
(== power 0.5) (m/sq (mean (map #(m/sqrt %) vs)))
(== power 2.0) (m/sqrt (mean (map m/sq vs)))
(== power 3.0) (m/cbrt (mean (map m/cb vs)))
:else (m/pow (mean (map (fn [^double v] (m/pow v power)) vs)) (/ power))))
(defn wmean
"Weighted mean"
(^double [vs] (mean vs))
(^double [vs weights]
(/ ^double (reduce m/fast+ (map m/fast* vs weights))
^double (reduce m/fast+ weights))))
(defn population-variance
"Calculate population variance of `vs`.
See [[variance]]."
(^double [vs]
(StatUtils/populationVariance (m/seq->double-array vs)))
(^double [vs ^double u]
(StatUtils/populationVariance (m/seq->double-array vs) u)))
(defn variance
"Calculate variance of `vs`.
See [[population-variance]]."
(^double [vs]
(StatUtils/variance (m/seq->double-array vs)))
(^double [vs ^double u]
(StatUtils/variance (m/seq->double-array vs) u)))
(defn population-stddev
"Calculate population standard deviation of `vs`.
See [[stddev]]."
(^double [vs]
(m/sqrt (population-variance vs)))
(^double [vs u]
(m/sqrt (population-variance vs u))))
(defn stddev
"Calculate standard deviation of `vs`.
See [[population-stddev]]."
(^double [vs]
(m/sqrt (variance vs)))
(^double [vs u]
(m/sqrt (variance vs u))))
(defn variation
"Coefficient of variation CV = stddev / mean"
^double [vs]
(let [vs (m/seq->double-array vs)]
(/ (stddev vs)
(mean vs))))
(defn median-absolute-deviation
"Calculate MAD"
(^double [vs] (median-absolute-deviation vs nil))
(^double [vs center]
(let [m (double (or center (median vs)))]
(median (map (fn [^double x] (m/abs (- x m))) vs))))
(^double [vs center estimation-strategy]
(let [m (double (or center (median vs estimation-strategy)))]
(median (map (fn [^double x] (m/abs (- x m))) vs) estimation-strategy))))
(def ^{:doc "Alias for [[median-absolute-deviation]]"}
mad median-absolute-deviation)
(defn mean-absolute-deviation
"Calculate mean absolute deviation"
(^double [vs] (mean-absolute-deviation vs nil))
(^double [vs center]
(let [m (double (or center (mean vs)))]
(mean (map (fn [^double x] (m/abs (- x m))) vs)))))
(defn sem
"Standard error of mean"
^double [vs]
(let [s (stddev vs)]
(/ s (m/sqrt (count vs)))))
(defmacro ^:private build-extent
[nm mid ext]
`(defn ~nm
~(str " -/+ " ext " and " mid)
[~'vs]
(let [vs# (m/seq->double-array ~'vs)
m# (~mid vs#)
s# (~ext vs#)]
[(- m# s#) (+ m# s#) m#])))
(build-extent stddev-extent mean stddev)
(build-extent mad-extent median median-absolute-deviation)
(build-extent sem-extent mean sem)
(defn percentile-extent
"Return percentile range and median.
`p` - calculates extent of `p` and `100-p` (default: `p=25`)"
([vs] (percentile-extent vs 25.0))
([vs ^double p] (percentile-extent vs p (- 100.0 p)))
([vs p1 p2] (percentile-extent vs p1 p2 :legacy))
([vs ^double p1 ^double p2 estimation-strategy]
(let [avs (m/seq->double-array vs)]
[(percentile avs p1 estimation-strategy)
(percentile avs p2 estimation-strategy)
(median avs)])))
(defn quantile-extent
"Return quantile range and median.
`q` - calculates extent of `q` and `1.0-q` (default: `q=0.25`)"
([vs] (quantile-extent vs 0.25))
([vs ^double q] (quantile-extent vs q (- 1.0 q)))
([vs q1 q2] (quantile-extent vs q1 q2 :legacy))
([vs ^double q1 ^double q2 estimation-strategy]
(let [avs (m/seq->double-array vs)]
[(quantile avs q1 estimation-strategy)
(quantile avs q2 estimation-strategy)
(median avs)])))
(defn pi
"Returns PI as a map, quantile intervals based on interval size.
Quantiles are `(1-size)/2` and `1-(1-size)/2`"
([vs] (pi vs 0.5))
([vs ^double size] (pi vs size :legacy))
([vs ^double size estimation-strategy]
(let [a (* 0.5 (- 1.0 size))
avs (m/seq->double-array vs)
q1 (quantile avs a estimation-strategy)
q2 (quantile avs (- 1.0 a) estimation-strategy)]
{(m/approx (* a 100.0) 2) q1
(m/approx (* (- 1.0 a) 100.0) 2) q2})))
(defn pi-extent
"Returns PI extent, quantile intervals based on interval size + median.
Quantiles are `(1-size)/2` and `1-(1-size)/2`"
([vs] (pi-extent vs 0.5))
([vs ^double size] (pi-extent vs size :legacy))
([vs ^double size estimation-strategy]
(let [a (* 0.5 (- 1.0 size))]
(quantile-extent vs a (- 1.0 a) estimation-strategy))))
(defn hpdi-extent
"Higher Posterior Density interval + median.
`size` parameter is the target probability content of the interval."
([vs] (hpdi-extent vs 0.95))
([vs ^double size]
(let [avs (m/seq->double-array vs)
nsamp (alength avs)
gap (m/constrain (m/round (* nsamp size)) 1 (dec nsamp))
max-idx (- nsamp gap)]
(java.util.Arrays/sort avs)
(loop [idx (long 0)
min-idx (long 0)
mn Double/MAX_VALUE]
(if (< idx max-idx)
(let [diff (- (aget avs (+ idx gap))
(aget avs idx))]
(if (< diff mn)
(recur (inc idx) idx diff)
(recur (inc idx) min-idx mn)))
[(aget avs min-idx) (aget avs (+ min-idx gap)) (median avs)])))))
(defn iqr
"Interquartile range."
(^double [vs] (iqr vs :legacy))
(^double [vs estimation-strategy]
(let [[^double q1 ^double q3] (percentile-extent vs 25.0 75.0 estimation-strategy)]
(- q3 q1))))
(defn adjacent-values
"Lower and upper adjacent values (LAV and UAV).
Let Q1 is 25-percentile and Q3 is 75-percentile. IQR is `(- Q3 Q1)`.
* LAV is smallest value which is greater or equal to the LIF = `(- Q1 (* 1.5 IQR))`.
* UAV is largest value which is lower or equal to the UIF = `(+ Q3 (* 1.5 IQR))`.
* third value is a median of samples
Optional `estimation-strategy` argument can be set to change quantile calculations estimation type. See [[estimation-strategies]]."
([vs]
(adjacent-values vs :legacy))
([vs estimation-strategy]
(let [avs (m/seq->double-array vs)
[q1 m q3] (percentiles avs [25.0 50.0 75.0] estimation-strategy)]
(adjacent-values avs q1 q3 m)))
([vs ^double q1 ^double q3 ^double m]
(let [avs (m/seq->double-array vs)
iqr (* 1.5 (- q3 q1))
lav-thr (- q1 iqr)
uav-thr (+ q3 iqr)]
(java.util.Arrays/sort avs)
[(first (filter #(>= (double %) lav-thr) avs))
(last (filter #(<= (double %) uav-thr) avs))
m])))
(defn inner-fence-extent
"Returns LIF, UIF and median"
([vs] (inner-fence-extent vs :legacy))
([vs estimation-strategy]
(let [[^double q1 ^double m ^double q3] (percentiles vs [25.0 50.0 75.0] estimation-strategy)
iqr+ (* 1.5 (- q3 q1))]
[(- q1 iqr+) (+ q3 iqr+) m])))
(defn outer-fence-extent
"Returns LOF, UOF and median"
([vs] (outer-fence-extent vs :legacy))
([vs estimation-strategy]
(let [[^double q1 ^double m ^double q3] (percentiles vs [25.0 50.0 75.0] estimation-strategy)
iqr+ (* 3.0 (- q3 q1))]
[(- q1 iqr+) (+ q3 iqr+) m])))
(defn outliers
"Find outliers defined as values outside inner fences.
Let Q1 is 25-percentile and Q3 is 75-percentile. IQR is `(- Q3 Q1)`.
* LIF (Lower Inner Fence) equals `(- Q1 (* 1.5 IQR))`.
* UIF (Upper Inner Fence) equals `(+ Q3 (* 1.5 IQR))`.
Returns sequence.
Optional `estimation-strategy` argument can be set to change quantile calculations estimation type. See [[estimation-strategies]]."
([vs]
(outliers vs :legacy))
([vs estimation-strategy]
(let [avs (m/seq->double-array vs)
q1 (percentile avs 25.0 estimation-strategy)
q3 (percentile avs 75.0 estimation-strategy)]
(outliers avs q1 q3)))
([vs ^double q1 ^double q3]
(let [ ;;avs (m/seq->double-array vs)
iqr (* 1.5 (- q3 q1))
lif-thr (- q1 iqr)
uif-thr (+ q3 iqr)]
;; (java.util.Arrays/sort avs)
(filter #(let [v (double %)]
(or (< v lif-thr)
(> v uif-thr))) vs))))
(declare histogram)
(defn modes
"Find the values that appears most often in a dataset `vs`.
Returns sequence with all most appearing values in increasing order.
See also [[mode]]."
([vs method] (modes vs method {}))
([vs method opts]
(let [avs (m/seq->double-array vs)]
(case method
:histogram (let [{:keys [bins ^double step]} (histogram avs (get opts :bins :rice))
ibins (vec (map-indexed #(conj %2 %1) bins))]
(->> ibins
(map (fn [[^double L ^long fm ^long id]]
(let [^long f1 (second (get ibins (dec id) [0 0]))
^long f2 (second (get ibins (inc id) [0 0]))]
[(+ L (* step (/ (- fm f1)
(- (* 2.0 fm) f1 f2)))) (- fm)])))
(sort-by second)
(map first)))
:pearson (let [mu (mean avs)
m (median avs (:estimation-strategy opts))]
[(- (* 3.0 m) (* 2.0 mu))])
:kde (let [kde (k/kernel-density (get opts :kernel :gaussian) avs (:bandwidth opts))]
(->> (map (fn [^double v] [v (- ^double (kde v))]) vs)
(sort-by second)
(map first)))
(seq ^doubles (StatUtils/mode avs)))))
([vs]
(seq ^doubles (StatUtils/mode (m/seq->double-array vs)))))
(defn mode
"Find the value that appears most often in a dataset `vs`.
For sample from continuous distribution, three algorithms are possible:
* `:histogram` - calculated from [[histogram]]
* `:kde` - calculated from KDE
* `:pearson` - mode = mean-3(median-mean)
* `:default` - discrete mode
Histogram accepts optional `:bins` (see [[histogram]]). KDE method accepts `:kde` for kernel name (default `:gaussian`) and `:bandwidth` (auto). Pearson can accept `:estimation-strategy` for median.
See also [[modes]]."
(^double [vs method] (mode vs method {}))
(^double [vs method opts]
(first (modes vs method opts)))
(^double [vs]
(let [m (StatUtils/mode (m/seq->double-array vs))]
(aget ^doubles m 0))))
(defn minimum
"Minimum value from sequence."
^double [vs]
(if (= (type vs) m/double-array-type)
(smile.math.MathEx/min ^doubles vs)
(reduce clojure.core/min vs)))
(defn maximum
"Maximum value from sequence."
^double [vs]
(if (= (type vs) m/double-array-type)
(smile.math.MathEx/max ^doubles vs)
(reduce clojure.core/max vs)))
(defn span
"Width of the sample, maximum value minus minimum value"
^double [vs]
(let [avs (m/seq->double-array vs)]
(- (maximum avs) (minimum avs))))
(defn extent
"Return extent (min, max, mean) values from sequence"
[vs]
(let [^double fv (first vs)]
(conj (reduce (fn [[^double mn ^double mx] ^double v]
[(min mn v) (max mx v)]) [fv fv] (rest vs)) (mean vs))))
(defn sum
"Sum of all `vs` values."
^double [vs]
(if (= (type vs) m/double-array-type)
(smile.math.MathEx/sum ^doubles vs)
(reduce (fn [^double x ^double y] (+ x y)) 0.0 vs)))
(defn moment
"Calculate moment (central or/and absolute) of given order (default: 2).
Additional parameters as a map:
* `:absolute?` - calculate sum as absolute values (default: `false`)
* `:mean?` - returns mean (proper moment) or just sum of differences (default: `true`)
* `:center` - value of center (default: `nil` = mean)
* `:normalize?` - apply normalization by standard deviation to the order power"
(^double [vs] (moment vs 2.0 nil))
(^double [vs ^double order] (moment vs order nil))
(^double [vs ^double order {:keys [absolute? center mean? normalize?]
:or {mean? true}}]
(let [in (m/seq->double-array vs)
cin (alength in)
out (double-array cin)
nf (if normalize? (m/pow (variance in) (* 0.5 order)) 1.0)
^double center (or center (mean in))
f (cond
(m/one? order) m/fast-identity
(== order 2.0) m/sq
(== order 3.0) m/cb
(== order 4.0) (fn ^double [^double diff] (m/sq (m/sq diff)))
:else (fn ^double [^double diff] (m/pow diff order)))
a (if absolute? m/abs m/fast-identity)]
(loop [idx (int 0)]
(when (< idx cin)
(aset out idx ^double (f (a (- (aget in idx) center))))
(recur (inc idx))))
(/ (if mean? (mean out) (sum out)) nf))))
(def ^{:deprecated "Use [[moment]] function"} second-moment moment)
(defn- yule-skewness
^double [vs ^double u]
(let [[^double q1 ^double q2 ^double q3] (quantiles vs [u 0.5 (- 1.0 u)])]
(/ (+ q3 (* -2.0 q2) q1)
(- q3 q1))))
(defn skewness
"Calculate skewness from sequence.
Possible types: `:G1` (default), `:g1` (`:pearson`), `:b1`, `:B1` (`:yule`), `:B3`, `:skew`, `:mode` or `:median`."
(^double [vs] (skewness vs :G1))
(^double [vs typ]
(let [vs (m/seq->double-array vs)]
(cond
(sequential? typ) (cond
(= :mode (first typ)) (let [[_ method opts] typ]
(/ (- (mean vs) (mode vs method opts)) (stddev vs)))
(#{:B1 :yule} (first typ)) (yule-skewness vs (second typ)))
(= :mode typ) (/ (- (mean vs) (mode vs)) (stddev vs))
(= :median typ) (/ (* 3.0 (- (mean vs) (median vs))) (stddev vs))
(#{:B1 :yule} typ) (yule-skewness vs 0.25)
(= :B3 typ) (let [v (median vs)]
(/ (- (mean vs) v)
(moment vs 1.0 {:absolute? true :center v})))
:else (let [^Skewness k (Skewness.)
n (alength vs)
v (.evaluate k vs)]
(cond
(= :b1 typ) (* v (/ (* (- n 2.0) (dec n)) (* n n)))
(#{:pearson :g1} typ) (* v (/ (- n 2.0) (m/sqrt (* n (dec n)))))
(= :skew typ) (* v (/ (- n 2.0) (* n (m/sqrt (dec n))))) ;; artificial, to match BCa skew definition
:else v))))))
(defn kurtosis
"Calculate kurtosis from sequence.
Possible typs: `:G2` (default), `:g2` (or `:excess`), `:geary` or `:kurt`."
(^double [vs] (kurtosis vs nil))
(^double [vs typ]
(let [vs (m/seq->double-array vs)
n (alength vs) ]
(if (= typ :geary)
(/ (mean-absolute-deviation vs)
(population-stddev vs))
(let [^Kurtosis k (Kurtosis.)
v (.evaluate k vs)]
(cond
(#{:excess :g2} typ) (/ (- (/ (* v (- n 2) (- n 3)) (dec n)) 6.0)
(inc n))
(= :kurt typ) (+ 3.0 (/ (- (/ (* v (- n 2) (- n 3)) (dec n)) 6.0)
(inc n)))
:else v))))))
(defn ci
"T-student based confidence interval for given data. Alpha value defaults to 0.05.
Last value is mean."
([vs] (ci vs 0.05))
([vs ^double alpha]
(let [vsa (m/seq->double-array vs)
cnt (count vs)
dist (r/distribution :t {:degrees-of-freedom (dec cnt)})
^double crit-val (r/icdf dist (- 1.0 (* 0.5 alpha)))
mean-ci (/ (* crit-val (stddev vsa)) (m/sqrt cnt))
mn (mean vsa)]
[(- mn mean-ci) (+ mn mean-ci) mn])))
;; https://ocw.mit.edu/courses/mathematics/18-05-introduction-to-probability-and-statistics-spring-2014/readings/MIT18_05S14_Reading24.pdf
(defn bootstrap-ci
"Bootstrap method to calculate confidence interval.
Alpha defaults to 0.98, samples to 1000.
Last parameter is statistical function used to measure, default: [[mean]].
Returns ci and statistical function value."
{:deprecated "Please use fastmath.stats.boostrap/ci-basic instead"}
([vs] (bootstrap-ci vs 0.98))
([vs alpha] (bootstrap-ci vs alpha 1000))
([vs alpha samples] (bootstrap-ci vs alpha samples mean))
([vs ^double alpha ^long samples stat-fn]
(let [vsa (m/seq->double-array vs)
cnt (count vs)
dist (r/distribution :enumerated-real {:data vsa})
^double m (stat-fn vsa)
deltas (m/seq->double-array (repeatedly samples #(- ^double (mean (r/->seq dist cnt)) m)))
q1 (quantile deltas alpha)
q2 (quantile deltas (- 1.0 alpha))]
[(- m q1) (- m q2) m])))
(defn bootstrap
{:doc "Generate set of samples of given size from provided data.
Default `samples` is 200, number of `size` defaults to sample size."
:deprecated "Please use fastmath.stats.bootstrap/bootstrap instead"}
([vs] (bootstrap vs 200))
([vs samples] (bootstrap vs samples (count vs)))
([vs samples size]
(let [dist (r/distribution :enumerated-real {:data vs})]
(repeatedly samples #(r/->seq dist size)))))
(defn stats-map
"Calculate several statistics of `vs` and return as map.
Optional `estimation-strategy` argument can be set to change quantile calculations estimation type. See [[estimation-strategies]]."
([vs] (stats-map vs :legacy))
([vs estimation-strategy]
(let [avs (m/seq->double-array vs)
sz (alength avs)
mn (smile.math.MathEx/min avs)
mx (smile.math.MathEx/max avs)
sm (smile.math.MathEx/sum avs)
u (/ sm sz)
mdn (median avs)
q1 (percentile avs 25.0 estimation-strategy)
q3 (percentile avs 75.0 estimation-strategy)
iqr (- q3 q1)
sd (stddev avs)
mad (median-absolute-deviation avs)
[lav uav] (adjacent-values avs q1 q3 mdn)]
{:Size sz
:Min mn
:Max mx
:Range (- mx mn)
:Mean u
:Median mdn
:Mode (mode avs)
:Q1 q1
:Q3 q3
:Total sm
:SD sd
:Variance (* sd sd)
:MAD mad
:SEM (/ sd (m/sqrt sz))
:LAV lav
:UAV uav
:IQR iqr
:LOF (- q1 (* 3.0 iqr))
:UOF (+ q3 (* 3.0 iqr))
:LIF (- q1 (* 1.5 iqr))
:UIF (+ q3 (* 1.5 iqr))
:Outliers (outliers avs q1 q3)
:Kurtosis (kurtosis avs)
:Skewness (skewness avs)})))
(defn standardize
"Normalize samples to have mean = 0 and stddev = 1."
[vs]
(seq ^doubles (StatUtils/normalize (m/seq->double-array vs))))
(defn robust-standardize
"Normalize samples to have median = 0 and MAD = 1.
If `q` argument is used, scaling is done by quantile difference (Q_q, Q_(1-q)). Set 0.25 for IQR."
([vs]
(let [avs (m/seq->double-array vs)
mad (median-absolute-deviation avs)
md (median avs)]
(map (fn [^double x] (/ (- x md) mad)) vs)))
([vs ^double q]
(let [[^double q1 ^double md ^double q2] (quantiles vs [q 0.5 (- 1.0 q)])
diff (m/abs (- q2 q1))]
(map (fn [^double x] (/ (- x md) diff)) vs))))
(defn demean
"Subtract mean from sequence"
[vs]
(let [m (mean vs)]
(map (fn [^double v]
(- v m)) vs)))
(defn winsor
"Return winsorized data. Trim is done by using quantiles, by default is set to 0.2."
([vs] (winsor vs 0.2))
([vs quantile] (winsor vs quantile :legacy))
([vs ^double quantile estimation-strategy]
(let [[qlow qmid qhigh] (quantiles (remove m/nan? vs)
[quantile 0.5 (- 1.0 quantile)] estimation-strategy)]
(winsor vs qlow qhigh qmid)))
([vs ^double low ^double high nan]
(let [[^double low ^double high] (if (< low high) [low high] [high low])]
(map (fn [^double v]
(if (m/nan? v)
nan
(m/constrain v low high))) vs))))
(defn trim
"Return trimmed data. Trim is done by using quantiles, by default is set to 0.2."
([vs] (trim vs 0.2))
([vs quantile] (trim vs quantile :legacy))
([vs ^double quantile estimation-strategy]
(let [[qlow qmid qhigh] (quantiles (remove m/nan? vs)
[quantile 0.5 (- 1.0 quantile)] estimation-strategy)]
(trim vs qlow qhigh qmid)))
([vs ^double low ^double high nan]
(let [[^double low ^double high] (if (< low high) [low high] [high low])]
(->> (filter (fn [^double v]
(or (m/nan? v)
(<= low v high))) vs)
(map (fn [^double v] (if (m/nan? v) nan v)))))))
(defn rescale
"Lineary rascale data to desired range, [0,1] by default"
([vs] (rescale vs 0.0 1.0))
([vs ^double low ^double high]
(let [avs (m/seq->double-array vs)
mn (minimum avs)
mx (maximum avs)
n (m/make-norm mn mx low high)]
(map n vs))))
(defn covariance
"Covariance of two sequences."
(^double [[vs1 vs2]] (covariance vs1 vs2))
(^double [vs1 vs2]
(smile.math.MathEx/cov (m/seq->double-array vs1) (m/seq->double-array vs2))))
(defn correlation
"Correlation of two sequences."
(^double [[vs1 vs2]] (correlation vs1 vs2))
(^double [vs1 vs2]
(smile.math.MathEx/cor (m/seq->double-array vs1) (m/seq->double-array vs2))))
(defn spearman-correlation
"Spearman's correlation of two sequences."
(^double [[vs1 vs2]] (spearman-correlation vs1 vs2))
(^double [vs1 vs2]
(.correlation (SpearmansCorrelation.) (m/seq->double-array vs1) (m/seq->double-array vs2))))
(defn pearson-correlation
"Pearson's correlation of two sequences."
(^double [[vs1 vs2]] (pearson-correlation vs1 vs2))
(^double [vs1 vs2]
(.correlation (PearsonsCorrelation.) (m/seq->double-array vs1) (m/seq->double-array vs2))))
(defn kendall-correlation
"Kendall's correlation of two sequences."
(^double [[vs1 vs2]] (kendall-correlation vs1 vs2))
(^double [vs1 vs2]
(.correlation (KendallsCorrelation.) (m/seq->double-array vs1) (m/seq->double-array vs2))))
(defn kullback-leibler-divergence
"Kullback-Leibler divergence of two sequences."
(^double [[vs1 vs2]] (kullback-leibler-divergence vs1 vs2))
(^double [vs1 vs2]
(smile.math.MathEx/KullbackLeiblerDivergence (m/seq->double-array vs1) (m/seq->double-array vs2))))
(defn jensen-shannon-divergence
"Jensen-Shannon divergence of two sequences."
(^double [[vs1 vs2]] (jensen-shannon-divergence vs1 vs2))
(^double [vs1 vs2]
(smile.math.MathEx/JensenShannonDivergence (m/seq->double-array vs1) (m/seq->double-array vs2))))
(defn coefficient-matrix
"Generate coefficient (correlation, covariance, any two arg function) matrix from seq of seqs. Row order.
Default method: pearson-correlation"
([vss] (coefficient-matrix vss pearson-correlation))
([vss measure-fn] (coefficient-matrix vss measure-fn true))
([vss measure-fn symmetric?]
(if symmetric?
(let [avss (map-indexed (fn [id v] [id (m/seq->double-array v)]) vss)
cache (atom {})]
(for [[^long id1 ^doubles a] avss]
(mapv (fn [[^long id2 ^doubles b]]
(let [key (if (< id1 id2) [id1 id2] [id2 id1])]
(if (contains? @cache key)
(@cache key)
(let [cov (measure-fn a b)]
(swap! cache assoc key cov)
cov)))) avss)))
(let [avss (map m/seq->double-array vss)]
(for [^doubles a avss]
(mapv #(measure-fn a ^doubles %) avss))))))
(defn correlation-matrix
"Generate correlation matrix from seq of seqs. Row order.
Possible measures: `:pearson` (default), `:kendall`, `:spearman`, `:kullback-leibler` and `jensen-shannon`."
([vss] (correlation-matrix vss :pearson))
([vss measure]
(let [measure (get {:pearson pearson-correlation
:kendall kendall-correlation
:spearman spearman-correlation
:kullback-leibler kullback-leibler-divergence
:jensen-shannon jensen-shannon-divergence} measure pearson-correlation)]
(coefficient-matrix vss measure true))))
(defn covariance-matrix
"Generate covariance matrix from seq of seqs. Row order."
[vss] (coefficient-matrix vss covariance true))
(defn- maybe-number->seq [v] (if (number? v) (repeat v) v))
(defn me
"Mean error"
(^double [[vs1 vs2-or-val]] (me vs1 vs2-or-val))
(^double [vs1 vs2-or-val]
(mean (map m/fast- vs1 (maybe-number->seq vs2-or-val)))))
(defn mae
"Mean absolute error"
(^double [[vs1 vs2-or-val]] (mae vs1 vs2-or-val))
(^double [vs1 vs2-or-val]
(mean (map (comp m/abs m/fast-) vs1 (maybe-number->seq vs2-or-val)))))
(defn mape
"Mean absolute percentage error"
(^double [[vs1 vs2-or-val]] (mape vs1 vs2-or-val))
(^double [vs1 vs2-or-val]
(mean (map (fn [^double a ^double b]
(m/abs (/ (- a b) a))) vs1 (maybe-number->seq vs2-or-val)))))
(defn rss
"Residual sum of squares"
(^double [[vs1 vs2-or-val]] (rss vs1 vs2-or-val))
(^double [vs1 vs2-or-val]
(sum (map (comp m/sq m/fast-) vs1 (maybe-number->seq vs2-or-val)))))
(defn r2
"R2"
(^double [[vs1 vs2-or-val]] (r2 vs1 vs2-or-val))
(^double [vs1 vs2-or-val]
(- 1.0 (/ (rss vs1 vs2-or-val)
(moment vs1 2 {:mean? false})))))
(defn mse
"Mean squared error"
(^double [[vs1 vs2-or-val]] (mse vs1 vs2-or-val))
(^double [vs1 vs2-or-val]
(mean (map (comp m/sq m/fast-) vs1 (maybe-number->seq vs2-or-val)))))
(defn rmse
"Root mean squared error"
(^double [[vs1 vs2-or-val]] (rmse vs1 vs2-or-val))
(^double [vs1 vs2-or-val]
(m/sqrt (mse vs1 vs2-or-val))))
(defn count=
"Count equal values in both seqs. Same as [[L0]]"
(^long [[vs1 vs2-or-val]] (count= vs1 vs2-or-val))
(^long [vs1 vs2-or-val]
(count (filter (fn [^double v] (zero? v)) (map m/fast- vs1 (maybe-number->seq vs2-or-val))))))
(def ^{:doc "Count equal values in both seqs. Same as [[count==]]"} L0 count=)
(defn L1
"Manhattan distance"
(^double [[vs1 vs2-or-val]] (L1 vs1 vs2-or-val))
(^double [vs1 vs2-or-val]
(d/manhattan vs1 (take (count vs1) (maybe-number->seq vs2-or-val)))))
(defn L2sq
"Squared euclidean distance"
(^double [[vs1 vs2-or-val]] (L2sq vs1 vs2-or-val))
(^double [vs1 vs2-or-val]
(d/euclidean-sq vs1 (take (count vs1) (maybe-number->seq vs2-or-val)))))
(defn L2
"Euclidean distance"
(^double [[vs1 vs2-or-val]] (L2 vs1 vs2-or-val))
(^double [vs1 vs2-or-val]
(d/euclidean vs1 (take (count vs1) (maybe-number->seq vs2-or-val)))))
(defn LInf
"Chebyshev distance"
(^double [[vs1 vs2-or-val]] (LInf vs1 vs2-or-val))
(^double [vs1 vs2-or-val]
(d/chebyshev vs1 (take (count vs1) (maybe-number->seq vs2-or-val)))))
(defn psnr
"Peak signal to noise, `max-value` is maximum possible value (default: max from `vs1` and `vs2`)"
(^double [[vs1 vs2-or-val]] (psnr vs1 vs2-or-val))
(^double [vs1 vs2-or-val]
(let [mx1 (maximum vs1)
^double mx2 (if (number? vs2-or-val) vs2-or-val (maximum vs2-or-val))]
(psnr vs1 vs2-or-val (max mx1 mx2))))
(^double [vs1 vs2-or-val ^double max-value]
(- (* 20.0 (m/log10 max-value))
(* 10.0 (m/log10 (mse vs1 vs2-or-val))))))
;;
(defn- scott-fd-helper
"Calculate number of bins based on width of the bin."
^double [vvs ^double h]
(let [h (if (< h m/EPSILON) (median-absolute-deviation vvs) h)]
(if (pos? h)
(let [[^double mn ^double mx] (extent vvs)]
(m/ceil (/ (- mx mn) h)))
1.0)))
(defn estimate-bins
"Estimate number of bins for histogram.
Possible methods are: `:sqrt` `:sturges` `:rice` `:doane` `:scott` `:freedman-diaconis` (default).
The number returned is not higher than number of samples."
([vs] (estimate-bins vs :freedman-diaconis))
([vs bins-or-estimate-method]
(if-not (keyword? bins-or-estimate-method)
(or bins-or-estimate-method (estimate-bins vs))
(let [n (count vs)]