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fastmath.stats

Statistics functions.

  • Descriptive statistics.
  • Correlation / covariance
  • Outliers
  • Confidence intervals
  • Extents
  • Effect size
  • Student's t-test
  • Histogram
  • ACF/PACF
  • Bootstrap
  • Binary measures

All 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

Statistics functions.

* Descriptive statistics.
* Correlation / covariance
* Outliers
* Confidence intervals
* Extents
* Effect size
* Student's t-test
* Histogram
* ACF/PACF
* Bootstrap
* Binary measures

All 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)
raw docstring

acfclj

(acf data)
(acf data lags)

Calculate acf (autocorrelation function) for given number of lags or a list of lags.

If lags is omitted function returns maximum possible number of lags.

See also acf-ci, pacf, pacf-ci

Calculate acf (autocorrelation function) for given number of lags or a list of lags.

If lags is omitted function returns maximum possible number of lags.

See also [[acf-ci]], [[pacf]], [[pacf-ci]]
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acf-ciclj

(acf-ci data lags)
(acf-ci data lags alpha)

acf with added confidence interval data.

:cis contains list of calculated ci for every lag.

[[acf]] with added confidence interval data.

`:cis` contains list of calculated ci for every lag.
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adjacent-valuesclj

(adjacent-values vs)
(adjacent-values vs estimation-strategy)
(adjacent-values vs q1 q3)

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]].

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]].
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ameasureclj

(ameasure group1 group2)

Vargha-Delaney A measure for two populations a and b

Vargha-Delaney A measure for two populations a and b
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binary-measuresclj

(binary-measures truth prediction)
(binary-measures truth prediction true-value)

Subset of binary measures. See binary-measures-all.

Following keys are returned: [:tp :tn :fp :fn :accuracy :fdr :f-measure :fall-out :precision :recall :sensitivity :specificity :prevalance]

Subset of binary measures. See [[binary-measures-all]].

Following keys are returned: `[:tp :tn :fp :fn :accuracy :fdr :f-measure :fall-out :precision :recall :sensitivity :specificity :prevalance]`
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binary-measures-allclj

(binary-measures-all truth prediction)
(binary-measures-all truth prediction true-value)

Collection of binary measures.

  • truth - list of ground truth values
  • prediction - list of predicted values
  • true-value - optional, what is true in truth and prediction

true-value can be one of:

  • nil - values are treating as booleans
  • any sequence - values from sequence will be treated as true
  • map - conversion will be done according to provided map (if there is no correspondin key, value is treated as false)

https://en.wikipedia.org/wiki/Precision_and_recall

Collection of binary measures.

* `truth` - list of ground truth values
* `prediction` - list of predicted values
* `true-value` - optional, what is true in `truth` and `prediction`

`true-value` can be one of:

* `nil` - values are treating as booleans
* any sequence - values from sequence will be treated as `true`
* map - conversion will be done according to provided map (if there is no correspondin key, value is treated as `false`)

https://en.wikipedia.org/wiki/Precision_and_recall
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bootstrapclj

(bootstrap vs)
(bootstrap vs samples)
(bootstrap vs samples size)

Generate set of samples of given size from provided data.

Default samples is 50, number of size defaults to 1000

Generate set of samples of given size from provided data.

Default `samples` is 50, number of `size` defaults to 1000
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bootstrap-ciclj

(bootstrap-ci vs)
(bootstrap-ci vs alpha)
(bootstrap-ci vs alpha samples)
(bootstrap-ci vs alpha samples stat-fn)

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.

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.
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ciclj

(ci vs)
(ci vs alpha)

T-student based confidence interval for given data. Alpha value defaults to 0.98.

Last value is mean.

T-student based confidence interval for given data. Alpha value defaults to 0.98.

Last value is mean.
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cliffs-deltaclj

(cliffs-delta group1 group2)

Cliff's delta effect size

Cliff's delta effect size
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cohens-dclj

(cohens-d group1 group2)

Cohen's d effect size for two groups

Cohen's d effect size for two groups
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cohens-d-origclj

(cohens-d-orig group1 group2)

Original version of Cohen's d effect size for two groups

Original version of Cohen's d effect size for two groups
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correlationclj

(correlation vs1 vs2)

Correlation of two sequences.

Correlation of two sequences.
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covarianceclj

(covariance vs1 vs2)

Covariance of two sequences.

Covariance of two sequences.
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covariance-matrixclj

(covariance-matrix vss)

Generate covariance matrix from seq of seqs. Row order.

Generate covariance matrix from seq of seqs. Row order.
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demeanclj

(demean vs)

Subtract mean from sequence

Subtract mean from sequence
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estimate-binsclj

(estimate-bins vs)
(estimate-bins vs bins-or-estimate-method)

Estimate number of bins for histogram.

Possible methods are: :sqrt :sturges :rice :doane :scott :freedman-diaconis (default).

Estimate number of bins for histogram.

Possible methods are: `:sqrt` `:sturges` `:rice` `:doane` `:scott` `:freedman-diaconis` (default).
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estimation-strategies-listclj

source

extentclj

(extent vs)

Return extent (min, max, mean) values from sequence

Return extent (min, max, mean) values from sequence
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glass-deltaclj

(glass-delta group1 group2)

Glass's delta effect size for two groups

Glass's delta effect size for two groups
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hedges-gclj

(hedges-g group1 group2)

Hedges's g effect size for two groups

Hedges's g effect size for two groups
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hedges-g*clj

(hedges-g* group1 group2)

Less biased Hedges's g effect size for two groups

Less biased Hedges's g effect size for two groups
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histogramclj

(histogram vs)
(histogram vs bins-or-estimate-method)
(histogram vs bins [mn mx])

Calculate histogram.

Returns map with keys:

  • :size - number of bins
  • :step - distance between bins
  • :bins - list of pairs of range lower value and number of hits
  • :min - min value
  • :max - max value
  • :samples - number of used samples

For estimation methods check estimate-bins.

Calculate histogram.

Returns map with keys:

* `:size` - number of bins
* `:step` - distance between bins
* `:bins` - list of pairs of range lower value and number of hits
* `:min` - min value
* `:max` - max value
* `:samples` - number of used samples

For estimation methods check [[estimate-bins]].
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iqrclj

(iqr vs)
(iqr vs estimation-strategy)

Interquartile range.

Interquartile range.
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jensen-shannon-divergenceclj

(jensen-shannon-divergence vs1 vs2)

Jensen-Shannon divergence of two sequences.

Jensen-Shannon divergence of two sequences.
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kendall-correlationclj

(kendall-correlation vs1 vs2)

Kendall's correlation of two sequences.

Kendall's correlation of two sequences.
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kullback-leibler-divergenceclj

(kullback-leibler-divergence vs1 vs2)

Kullback-Leibler divergence of two sequences.

Kullback-Leibler divergence of two sequences.
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kurtosisclj

(kurtosis vs)

Calculate kurtosis from sequence.

Calculate kurtosis from sequence.
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mad-extentclj

(mad-extent vs)

-/+ median-absolute-deviation and median

 -/+ median-absolute-deviation and median
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maximumclj

(maximum vs)

Maximum value from sequence.

Maximum value from sequence.
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meanclj

(mean vs)

Calculate mean of vs

Calculate mean of `vs`
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medianclj

(median vs)

Calculate median of vs. See median-3.

Calculate median of `vs`. See [[median-3]].
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median-3clj

(median-3 a b c)

Median of three values. See median.

Median of three values. See [[median]].
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median-absolute-deviationclj

(median-absolute-deviation vs)

Calculate MAD

Calculate MAD
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minimumclj

(minimum vs)

Minimum value from sequence.

Minimum value from sequence.
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modeclj

(mode vs)

Find the value that appears most often in a dataset vs.

See also modes.

Find the value that appears most often in a dataset `vs`.

See also [[modes]].
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modesclj

(modes vs)

Find the values that appears most often in a dataset vs.

Returns sequence with all most appearing values in increasing order.

See also mode.

Find the values that appears most often in a dataset `vs`.

Returns sequence with all most appearing values in increasing order.

See also [[mode]].
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momentclj

(moment vs)
(moment vs order)
(moment vs
        order
        {:keys [absolute? center mean?]
         :or {absolute? false center nil mean? true}})

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 central (default: nil = mean)
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 central (default: `nil` = mean)
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outliersclj

(outliers vs)
(outliers vs estimation-strategy)
(outliers vs q1 q3)

Find outliers defined as values outside outer fences.

Let Q1 is 25-percentile and Q3 is 75-percentile. IQR is (- Q3 Q1).

  • LIF (Lower Outer Fence) equals (- Q1 (* 1.5 IQR)).
  • UIF (Upper Outer Fence) equals (+ Q3 (* 1.5 IQR)).

Returns sequence.

Optional estimation-strategy argument can be set to change quantile calculations estimation type. See [[estimation-strategies]].

Find outliers defined as values outside outer fences.

Let Q1 is 25-percentile and Q3 is 75-percentile. IQR is `(- Q3 Q1)`.

* LIF (Lower Outer Fence) equals `(- Q1 (* 1.5 IQR))`.
* UIF (Upper Outer Fence) equals `(+ Q3 (* 1.5 IQR))`.

Returns sequence.

Optional `estimation-strategy` argument can be set to change quantile calculations estimation type. See [[estimation-strategies]].
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pacfclj

(pacf data)
(pacf data lags)

Caluclate pacf (partial autocorrelation function) for given number of lags.

If lags is omitted function returns maximum possible number of lags.

pacf returns also lag 0 (which is 0.0).

See also acf, acf-ci, pacf-ci

Caluclate pacf (partial autocorrelation function) for given number of lags.

If lags is omitted function returns maximum possible number of lags.

`pacf` returns also lag `0` (which is `0.0`).

See also [[acf]], [[acf-ci]], [[pacf-ci]]
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pacf-ciclj

(pacf-ci data lags)
(pacf-ci data lags alpha)

pacf with added confidence interval data.

[[pacf]] with added confidence interval data.
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pearson-correlationclj

(pearson-correlation vs1 vs2)

Pearson's correlation of two sequences.

Pearson's correlation of two sequences.
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percentileclj

(percentile vs p)
(percentile vs p estimation-strategy)

Calculate percentile of a vs.

Percentile p is from range 0-100.

See docs.

Optionally you can provide estimation-strategy to change interpolation methods for selecting values. Default is :legacy. See more here

See also quantile.

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]].
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percentile-bc-extentclj

(percentile-bc-extent vs)
(percentile-bc-extent vs p)
(percentile-bc-extent vs p1 p2)
(percentile-bc-extent vs p1 p2 estimation-strategy)

Return bias corrected percentile range and mean for bootstrap samples. See https://projecteuclid.org/euclid.ss/1032280214

p - calculates extent of bias corrected p and 100-p (default: p=2.5)

Set estimation-strategy to :r7 to get the same result as in R coxed::bca.

Return bias corrected percentile range and mean for bootstrap samples.
See https://projecteuclid.org/euclid.ss/1032280214

`p` - calculates extent of bias corrected `p` and `100-p` (default: `p=2.5`)

Set `estimation-strategy` to `:r7` to get the same result as in R `coxed::bca`.
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percentile-bca-extentclj

(percentile-bca-extent vs)
(percentile-bca-extent vs p)
(percentile-bca-extent vs p1 p2)
(percentile-bca-extent vs p1 p2 estimation-strategy)
(percentile-bca-extent vs p1 p2 accel estimation-strategy)

Return bias corrected percentile range and mean for bootstrap samples. Also accounts for variance variations throught the accelaration parameter. See https://projecteuclid.org/euclid.ss/1032280214

p - calculates extent of bias corrected p and 100-p (default: p=2.5)

Set estimation-strategy to :r7 to get the same result as in R coxed::bca.

Return bias corrected percentile range and mean for bootstrap samples. Also accounts for variance
 variations throught the accelaration parameter.
See https://projecteuclid.org/euclid.ss/1032280214

`p` - calculates extent of bias corrected `p` and `100-p` (default: `p=2.5`)

Set `estimation-strategy` to `:r7` to get the same result as in R `coxed::bca`.
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percentile-extentclj

(percentile-extent vs)
(percentile-extent vs p)
(percentile-extent vs p1 p2)
(percentile-extent vs p1 p2 estimation-strategy)

Return percentile range and median.

p - calculates extent of p and 100-p (default: p=25)

Return percentile range and median.

`p` - calculates extent of `p` and `100-p` (default: `p=25`)
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percentilesclj

(percentiles vs ps)
(percentiles vs ps estimation-strategy)

Calculate percentiles of a vs.

Percentiles are sequence of values from range 0-100.

See docs.

Optionally you can provide estimation-strategy to change interpolation methods for selecting values. Default is :legacy. See more here

See also quantile.

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]].
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population-stddevclj

(population-stddev vs)
(population-stddev vs u)

Calculate population standard deviation of vs.

See stddev.

Calculate population standard deviation of `vs`.

See [[stddev]].
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population-varianceclj

(population-variance vs)
(population-variance vs u)

Calculate population variance of vs.

See variance.

Calculate population variance of `vs`.

See [[variance]].
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quantileclj

(quantile vs q)
(quantile vs q estimation-strategy)

Calculate quantile of a vs.

Quantile q is from range 0.0-1.0.

See docs for interpolation strategy.

Optionally you can provide estimation-strategy to change interpolation methods for selecting values. Default is :legacy. See more here

See also percentile.

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]].
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quantilesclj

(quantiles vs qs)
(quantiles vs qs estimation-strategy)

Calculate quantiles of a vs.

Quantilizes is sequence with values from range 0.0-1.0.

See docs for interpolation strategy.

Optionally you can provide estimation-strategy to change interpolation methods for selecting values. Default is :legacy. See more here

See also percentiles.

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]].
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second-momentcljdeprecated

source

semclj

(sem vs)

Standard error of mean

Standard error of mean
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sem-extentclj

(sem-extent vs)

-/+ sem and mean

 -/+ sem and mean
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skewnessclj

(skewness vs)

Calculate kurtosis from sequence.

Calculate kurtosis from sequence.
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spearman-correlationclj

(spearman-correlation vs1 vs2)

Spearman's correlation of two sequences.

Spearman's correlation of two sequences.
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standardizeclj

(standardize vs)

Normalize samples to have mean = 0 and stddev = 1.

Normalize samples to have mean = 0 and stddev = 1.
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stats-mapclj

(stats-map vs)
(stats-map vs estimation-strategy)

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]].

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]].
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stddevclj

(stddev vs)
(stddev vs u)

Calculate standard deviation of vs.

See population-stddev.

Calculate standard deviation of `vs`.

See [[population-stddev]].
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stddev-extentclj

(stddev-extent vs)

-/+ stddev and mean

 -/+ stddev and mean
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sumclj

(sum vs)

Sum of all vs values.

Sum of all `vs` values.
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ttest-one-sampleclj

(ttest-one-sample xs)
(ttest-one-sample xs
                  {:keys [alpha sides mu]
                   :or {alpha 0.05 sides :two-sided mu 0.0}})

One-sample Student's t-test

  • alpha - significance level (default: 0.05)
  • sides - one of: :two-sided, :one-sided-less (short: :one-sided) or :one-sided-greater
  • mu - mean (default: 0.0)
One-sample Student's t-test

* `alpha` - significance level (default: `0.05`)
* `sides` - one of: `:two-sided`, `:one-sided-less` (short: `:one-sided`) or `:one-sided-greater`
* `mu` - mean (default: `0.0`)
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ttest-two-samplesclj

(ttest-two-samples xs ys)
(ttest-two-samples
  xs
  ys
  {:keys [alpha sides mu paired? equal-variances?]
   :or {alpha 0.05 sides :two-sided mu 0.0 paired? false equal-variances? false}
   :as params})

Two-sample Student's t-test

  • alpha - significance level (default: 0.05)
  • sides - one of: :two-sided, :one-sided-less (short: :one-sided) or :one-sided-greater
  • mu - mean (default: 0.0)
  • paired? - unpaired or paired test, boolean (default: false)
  • equal-variances? - unequal or equal variances, boolean (default: false)
Two-sample Student's t-test

* `alpha` - significance level (default: `0.05`)
* `sides` - one of: `:two-sided`, `:one-sided-less` (short: `:one-sided`) or `:one-sided-greater`
* `mu` - mean (default: `0.0`)
* `paired?` - unpaired or paired test, boolean (default: `false`)
* `equal-variances?` - unequal or equal variances, boolean (default: `false`)
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varianceclj

(variance vs)
(variance vs u)

Calculate variance of vs.

See population-variance.

Calculate variance of `vs`.

See [[population-variance]].
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