incanter.stats — incanter/incanter-core 1.9.3

This is the core statistical library for Incanter. It provides probability functions (cdf, pdf, quantile), random number generation, statistical tests, basic modeling functions, similarity/association measures, and more.

This library is built on Parallel Colt (http://sites.google.com/site/piotrwendykier/software/parallelcolt), an extension of the Colt numerics library (http://acs.lbl.gov/~hoschek/colt/).

auto-correlation clj

(auto-correlation x lag)

(auto-correlation x lag mean variance)

Returns the auto-correlation of x with given lag, mean, and variance. If no mean or variance is provided, the they are calculated from x.

References: http://incanter.org/docs/parallelcolt/api/cern/jet/stat/tdouble/DoubleDescriptive.html

benford-test clj

(benford-test coll)

Performs Benford's Law test using chisq-test.

Argument: coll: -- a sequence of numbers

Returns: :X-sq -- the Pearson X-squared test statistics :p-value -- the p-value for the test statistic :df -- the degress of freedom

Reference: http://data-sorcery.org/2009/06/21/chi-square-goodness-of-fit/ http://en.wikipedia.org/wiki/Benford%27s_Law

bigrams clj

(bigrams s)

bootstrap clj

(bootstrap
  data
  statistic
  &
  {:keys [size replacement smooth? smooth-sd]
   :or {replacement true smooth? false smooth-sd (/ (sqrt (count data)))}})

Returns a bootstrap sample of the given statistic on the given data.

Arguments: data -- vector of data to resample from statistic -- a function that returns a value given a vector of data

Options: :size -- the number of bootstrap samples to return :smooth -- (default false) smoothing option :smooth-sd -- (default (/ (sqrt (count data)))) determines the standard deviation of the noise to use for smoothing :replacement -- (default true) determines if sampling of the data should be done with replacement

References:

1. Clifford E. Lunneborg, Data Analysis by Resampling Concepts and Applications, 2000, pages 105-117
2. http://en.wikipedia.org/wiki/Bootstrapping_(statistics)

Examples:

;; example from Data Analysis by Resampling Concepts and Applications ;; Clifford E. Lunneborg (pages 119-122)

(use '(incanter core stats charts))

;; weights (in grams) of 50 randoincanter. sampled bags of preztels (def weights [464 447 446 454 450 457 450 442 433 452 449 454 450 438 448 449 457 451 456 452 450 463 464 453 452 447 439 449 468 443 433 460 452 447 447 446 450 459 466 433 445 453 454 446 464 450 456 456 447 469])

;; calculate the sample median, 450 (median weights)

;; generate bootstrap sample (def t* (bootstrap weights median :size 2000))

;; view histogram of bootstrap histogram (view (histogram t*))

;; calculate the mean of the bootstrap median ~ 450.644 (mean t*)

;; calculate the standard error ~ 1.083 (def se (sd t*))

;; 90% standard normal CI ~ (448.219 451.781) (plus (median weights) (mult (quantile-normal [0.05 0.95]) se))

;; 90% symmetric percentile CI ~ (449.0 452.5) (quantile t* :probs [0.05 0.95])

;; 90% non-symmetric percentile CI ~ (447.5 451.0) (minus (* 2 (median weights)) (quantile t* :probs [0.95 0.05]))

;; calculate bias (- (mean t*) (median weights)) ;; ~ 0.644

;; example with smoothing ;; Newcomb's speed of light data

(use '(incanter core stats charts))

;; A numeric vector giving the Third Series of measurements of the ;; passage time of light recorded by Newcomb in 1882. The given ;; values divided by 1000 plus 24 give the time in millionths of a ;; second for light to traverse a known distance. The 'true' value is ;; now considered to be 33.02.

(def speed-of-light [28 -44 29 30 24 28 37 32 36 27 26 28 29 26 27 22 23 20 25 25 36 23 31 32 24 27 33 16 24 29 36 21 28 26 27 27 32 25 28 24 40 21 31 32 28 26 30 27 26 24 32 29 34 -2 25 19 36 29 30 22 28 33 39 25 16 23])

;; view histogram of data to see outlier observations (view (histogram speed-of-light :nbins 30))

(def samp (bootstrap speed-of-light median :size 10000)) (view (histogram samp :density true :nbins 30)) (mean samp) (quantile samp :probs [0.025 0.975])

(def smooth-samp (bootstrap speed-of-light median :size 10000 :smooth true)) (view (histogram smooth-samp :density true :nbins 30)) (mean smooth-samp) (quantile smooth-samp :probs [0.025 0.975])

category-col-summarizer clj

(category-col-summarizer col ds)

Returns a summarizer function which takes a category column and returns a list of the top 5 columns by volume, and a count of remaining rows

cdf-beta clj

(cdf-beta x
          &
          {:keys [alpha beta lower-tail?]
           :or {alpha 1 beta 1 lower-tail? false}})

Returns the Beta cdf of the given value of x. It will return a sequence of values, if x is a sequence. This is equivalent to R's pbeta function.

Options: :alpha (default 1) :beta (default 1) :lower-tail (default true)

See also: pdf-beta and sample-beta

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Beta.html http://en.wikipedia.org/wiki/Beta_distribution http://en.wikipedia.org/wiki/Cumulative_distribution_function

Example: (cdf-beta 0.5 :alpha 1 :beta 2) (cdf-beta 0.5 :alpha 1 :beta 2 :lower-tail false)

cdf-binomial clj

(cdf-binomial x
              &
              {:keys [size prob lower-tail?]
               :or {size 1 prob 1/2 lower-tail? true}})

Returns the Binomial cdf of the given value of x. It will return a sequence of values, if x is a sequence. Equivalent to R's pbinom.

Options: :size (default 1) :prob (default 1/2) :lower-tail (default true)

See also: pdf-binomial and sample-binomial

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Binomial.html http://en.wikipedia.org/wiki/Binomial_distribution http://en.wikipedia.org/wiki/Cumulative_distribution_function

Example: (cdf-binomial 10 :prob 1/4 :size 20)

cdf-chisq clj

(cdf-chisq x & {:keys [df lower-tail?] :or {df 1 lower-tail? true}})

Returns the Chi Square cdf of the given value of x. It will return a sequence of values, if x is a sequence. Equivalent to R's pchisq function.

Options: :df (default 1) :lower-tail (default true)

See also: pdf-chisq and sample-chisq

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/ChiSquare.html http://en.wikipedia.org/wiki/Chi_square_distribution http://en.wikipedia.org/wiki/Cumulative_distribution_function

Example: (cdf-chisq 5.0 :df 2) (cdf-chisq 5.0 :df 2 :lower-tail false)

cdf-empirical clj

(cdf-empirical x)

Returns a step-function representing the empirical cdf of the given data. Equivalent to R's ecdf function.

The following description is from the ecdf help in R: The e.c.d.f. (empirical cumulative distribution function) Fn is a step function with jumps i/n at observation values, where i is the number of tied observations at that value. Missing values are ignored.

For observations 'x'= (x1,x2, ... xn), Fn is the fraction of observations less or equal to t, i.e.,

Fn(t) = #{x_i <= t} / n = 1/n sum(i=1,n) Indicator(xi <= t).

Examples: (use '(incanter core stats charts))

(def exam1 [192 160 183 136 162 165 181 188 150 163 192 164 184 189 183 181 188 191 190 184 171 177 125 192 149 188 154 151 159 141 171 153 169 168 168 157 160 190 166 150])

;; the ecdf function returns an empirical cdf function for the given data (def ecdf (cdf-empirical exam1))

;; plot the data's empirical cdf (view (scatter-plot exam1 (map ecdf exam1)))

cdf-exp clj

(cdf-exp x & {:keys [rate seed] :or {rate 1 seed (Date.)}})

Returns the Exponential cdf of the given value of x. It will return a sequence of values, if x is a sequence. Equivalent to R's pexp.

Options: :rate (default 1)

See also: pdf-exp and sample-exp

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Exponential.html http://en.wikipedia.org/wiki/Exponential_distribution http://en.wikipedia.org/wiki/Cumulative_distribution_function

Example: (cdf-exp 2.0 :rate 1/2)

cdf-f clj

(cdf-f x & {:keys [df1 df2 lower-tail?] :or {df1 1 df2 1 lower-tail? true}})

Returns the F-distribution cdf of the given value, x. It will return a sequence of values, if x is a sequence. This is equivalent to R's pf function.

Options: :df1 (default 1) :df2 (default 1) :lower-tail? (default true)

See also: pdf-f and quantile-f

References: http://en.wikipedia.org/wiki/F_distribution http://mathworld.wolfram.com/F-Distribution.html http://en.wikipedia.org/wiki/Cumulative_distribution_function

Example: (cdf-f 1.0 :df1 5 :df2 2)

cdf-gamma clj

(cdf-gamma x
           &
           {:keys [shape scale rate lower-tail?]
            :or {shape 1 lower-tail? true}})

Returns the Gamma cdf for the given value of x. It will return a sequence of values, if x is a sequence. This is equivalent to R's pgamma function.

Options: :shape (k) (default 1) :scale (θ) (default 1 or 1/rate, if :rate is specified) :rate (β) (default 1/scale, if :scale is specified) :lower-tail (default true)

See also: pdf-gamma and sample-gamma

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Gamma.html http://en.wikipedia.org/wiki/Gamma_distribution http://en.wikipedia.org/wiki/Cumulative_distribution_function

Example: (cdf-gamma 10 :shape 1 :scale 2) (cdf-gamma 3 :shape 1 :lower-tail false)

cdf-neg-binomial clj

(cdf-neg-binomial x
                  &
                  {:keys [size prob lower-tail?]
                   :or {size 10 prob 1/2 lower-tail? true}})

Returns the Negative Binomial cdf of the given value of x. It will return a sequence of values, if x is a sequence. Equivalent to R's dnbinom.

Options: :size (default 10) :prob (default 1/2) :lower-tail? (default true)

See also: pdf-neg-binomial and sample-neg-binomial

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/NegativeBinomial.html http://en.wikipedia.org/wiki/Negative_binomial_distribution http://en.wikipedia.org/wiki/Cumulative_distribution_function

Example: (cdf-neg-binomial 10 :prob 1/2 :size 20)

cdf-normal clj

(cdf-normal x & {:keys [mean sd seed] :or {mean 0 sd 1 seed (Date.)}})

Returns the Normal cdf of the given value, x. It will return a sequence of values, if x is a sequence. This is equivalent to R's pnorm function.

Options: :mean (default 0) :sd (default 1)

See also: pdf-normal, quantile-normal, sample-normal

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Normal.html http://en.wikipedia.org/wiki/Normal_distribution http://en.wikipedia.org/wiki/Cumulative_distribution_function

Example: (cdf-normal 1.96 :mean -2 :sd (sqrt 0.5))

cdf-poisson clj

(cdf-poisson x & {:keys [lambda lower-tail?] :or {lambda 1 lower-tail? true}})

Returns the Poisson cdf of the given value of x. It will return a sequence of values, if x is a sequence. Equivalent R's ppois.

Options: :lambda (default 1) :lower-tail (default true)

See also: pdf-poisson and sample-poisson

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Poisson.html http://en.wikipedia.org/wiki/Poisson_distribution http://en.wikipedia.org/wiki/Cumulative_distribution_function

Example: (cdf-poisson 5 :lambda 10)

cdf-t clj

(cdf-t x & {:keys [df lower-tail?] :or {df 1 lower-tail? true}})

Returns the Student's t cdf for the given value of x. It will return a sequence of values, if x is a sequence. Equivalent to R's pt function.

Options: :df (default 1)

See also: pdf-t, quantile-t, and sample-t

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/StudentT.html http://en.wikipedia.org/wiki/Student-t_distribution http://en.wikipedia.org/wiki/Cumulative_distribution_function

Example: (cdf-t 1.2 :df 10)

cdf-uniform clj

(cdf-uniform x & {:keys [min max seed] :or {min 0.0 max 1.0 seed (Date.)}})

Returns the Uniform cdf of the given value of x. It will return a sequence of values, if x is a sequence. This is equivalent to R's punif function.

Options: :min (default 0) :max (default 1)

See also: pdf-uniform and sample-uniform

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/DoubleUniform.html http://en.wikipedia.org/wiki/Uniform_distribution http://en.wikipedia.org/wiki/Cumulative_distribution_function

Example: (cdf-uniform 5) (cdf-uniform 5 :min 1 :max 10)

cdf-weibull clj

(cdf-weibull x & options)

Returns the Weibull cdf for the given value of x. It will return a sequence of values, if x is a sequence.

Options: :shape (default 1) :scale (default 1)

See also: pdf-weibull and sample-weibull

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Distributions.html http://en.wikipedia.org/wiki/Weibull_distribution http://en.wikipedia.org/wiki/Cumulative_distribution_function

Example: (cdf-weibull 10 :shape 1 :scale 0.2)

chebyshev-distance clj

(chebyshev-distance a b)

In the limiting case of Lp reaching infinity we obtain the Chebyshev distance.

chisq-test clj

(chisq-test & {:keys [x y correct table probs freq] :or {correct true}})

Performs chi-squared contingency table tests and goodness-of-fit tests.

If the optional argument :y is not provided then a goodness-of-fit test is performed. In this case, the hypothesis tested is whether the population probabilities equal those in :probs, or are all equal if :probs is not given.

If :y is provided, it must be a sequence of integers that is the same length as x. A contingency table is computed from x and :y. Then, Pearson's chi-squared test of the null hypothesis that the joint distribution of the cell counts in a 2-dimensional contingency table is the product of the row and column marginals is performed. By default the Yates' continuity correction for 2x2 contingency tables is performed, this can be disabled by setting the :correct option to false.

Options: :x -- a sequence of numbers. :y -- a sequence of numbers :table -- a contingency table. If one dimensional, the test is a goodness-of-fit :probs (when (nil? y) -- (repeat n-levels (/ n-levels))) :freq (default nil) -- if given, these are rescaled to probabilities :correct (default true) -- use Yates' correction for continuity for 2x2 contingency tables

Returns: :X-sq -- the Pearson X-squared test statistics :p-value -- the p-value for the test statistic :df -- the degress of freedom

Examples: (use '(incanter core stats)) (chisq-test :x [1 2 3 2 3 2 4 3 5]) ;; X-sq 2.6667 ;; create a one-dimensional table of this data (def table (matrix [1 3 3 1 1])) (chisq-test :table table) ;; X-sq 2.6667 (chisq-test :table (trans table)) ;; throws exception

(chisq-test :x [1 0 0 0 1 1 1 0 0 1 0 0 1 1 1 1]) ;; 0.25

(use '(incanter core stats datasets)) (def math-prog (to-matrix (get-dataset :math-prog))) (def x (sel math-prog :cols 1)) (def y (sel math-prog :cols 2)) (chisq-test :x x :y y) ;; X-sq = 1.24145, df=1, p-value = 0.26519 (chisq-test :x x :y y :correct false) ;; X-sq = 2.01094, df=1, p-value = 0.15617

(def table (matrix [[31 12] [9 8]])) (chisq-test :table table) ;; X-sq = 1.24145, df=1, p-value = 0.26519 (chisq-test :table table :correct false) ;; X-sq = 2.01094, df=1, p-value = 0.15617 ;; use the detabulate function to create data rows corresponding to the table (def detab (detabulate :table table)) (chisq-test :x (sel detab :cols 0) :y (sel detab :cols 1))

;; look at the hair-eye-color data ;; turn the count data for males into a contingency table (def male (matrix (sel (get-dataset :hair-eye-color) :cols 3 :rows (range 16)) 4)) (chisq-test :table male) ;; X-sq = 41.280, df = 9, p-value = 4.44E-6 ;; turn the count data for females into a contingency table (def female (matrix (sel (get-dataset :hair-eye-color) :cols 3 :rows (range 16 32)) 4)) (chisq-test :table female) ;; X-sq = 106.664, df = 9, p-value = 7.014E-19,

;; supply probabilities to goodness-of-fit test (def table [89 37 30 28 2]) (def probs [0.40 0.20 0.20 0.19 0.01]) (chisq-test :table table :probs probs) ;; X-sq = 5.7947, df = 4, p-value = 0.215

;; use frequencies instead of probabilities (def freq [40 20 20 15 5]) (chisq-test :table table :freq freq) ;; X-sq = 9.9901, df = 4, p-value = 0.04059

References: http://www.itl.nist.gov/div898/handbook/eda/section3/eda35f.htm http://en.wikipedia.org/wiki/Pearson's_chi-square_test http://en.wikipedia.org/wiki/Yates'_chi-square_test

choose-singletype-col-summarizer clj

(choose-singletype-col-summarizer col-type)

Takes in a type, and returns a suitable column summarizer

concordant-pairs clj

(concordant-pairs a b)

http://en.wikipedia.org/wiki/Concordant_pairs

concordant? clj

(concordant? [[a1 b1] [a2 b2] & more])

Given two pairs of numbers, checks if they are concordant.

correlation clj

(correlation mat)

(correlation x y)

Returns the sample correlation of x and y, or the correlation matrix of the given matrix.

Examples:

References: http://incanter.org/docs/parallelcolt/api/cern/jet/stat/tdouble/DoubleDescriptive.html http://en.wikipedia.org/wiki/Correlation

correlation-linearity-test clj

(correlation-linearity-test a b)

http://en.wikipedia.org/wiki/Correlation_ratio It is worth noting that if the relationship between values of and values of overline y_x is linear (which is certainly true when there are only two possibilities for x) this will give the same result as the square of the correlation coefficient, otherwise the correlation ratio will be larger in magnitude. It can therefore be used for judging non-linear relationships.

correlation-ratio clj

(correlation-ratio & xs)

http://en.wikipedia.org/wiki/Correlation_ratio

In statistics, the correlation ratio is a measure of the relationship between the statistical dispersion within individual categories and the dispersion across the whole population or sample. i.e. the weighted variance of the category means divided by the variance of all samples.

Example

Suppose there is a distribution of test scores in three topics (categories):

• Algebra: 45, 70, 29, 15 and 21 (5 scores)
• Geometry: 40, 20, 30 and 42 (4 scores)
• Statistics: 65, 95, 80, 70, 85 and 73 (6 scores).

Then the subject averages are 36, 33 and 78, with an overall average of 52.

The sums of squares of the differences from the subject averages are 1952 for Algebra, 308 for Geometry and 600 for Statistics, adding to 2860, while the overall sum of squares of the differences from the overall average is 9640. The difference between these of 6780 is also the weighted sum of the square of the differences between the subject averages and the overall average:

5(36 − 52)2 + 4(33 − 52)2 + 6(78 − 52)2 = 6780

This gives

eta^2 =6780/9640=0.7033

suggesting that most of the overall dispersion is a result of differences between topics, rather than within topics. Taking the square root

eta = sqrt 6780/9640=0.8386

Observe that for η = 1 the overall sample dispersion is purely due to dispersion among the categories and not at all due to dispersion within the individual categories. For a quick comprehension simply imagine all Algebra, Geometry, and Statistics scores being the same respectively, e.g. 5 times 36, 4 times 33, 6 times 78.

cosine-similarity clj

(cosine-similarity a b)

http://en.wikipedia.org/wiki/Cosine_similarity http://www.appliedsoftwaredesign.com/cosineSimilarityCalculator.php

The Cosine Similarity of two vectors a and b is the ratio: a dot b / ||a|| ||b||

Let d1 = {2 4 3 1 6} Let d2 = {3 5 1 2 5}

Cosine Similarity (d1, d2) = dot(d1, d2) / ||d1|| ||d2||

dot(d1, d2) = (2)(3) + (4)(5) + (3)(1) + (1)(2) + (6)*(5) = 61

||d1|| = sqrt((2)^2 + (4)^2 + (3)^2 + (1)^2 + (6)^2) = 8.12403840464

||d2|| = sqrt((3)^2 + (5)^2 + (1)^2 + (2)^2 + (5)^2) = 8

Cosine Similarity (d1, d2) = 61 / (8.12403840464) * (8) = 61 / 64.9923072371 = 0.938572618717

covariance clj

(covariance mat)

(covariance x y)

Returns the sample covariance of x and y.

Examples: ;; create some data that covaries (def x (sample-normal 100)) (def err (sample-normal 100)) (def y (plus (mult 5 x) err)) ;; estimate the covariance of x and y (covariance x y)

References: http://incanter.org/docs/parallelcolt/api/cern/jet/stat/tdouble/DoubleDescriptive.html http://en.wikipedia.org/wiki/Covariance

cumulative-mean clj

(cumulative-mean coll)

Returns a sequence of cumulative means for the given collection. For instance The first value equals the first value of the argument, the second value is the mean of the first two arguments, the third is the mean of the first three arguments, etc.

Examples: (cumulative-mean (sample-normal 100))

damerau-levenshtein-distance clj

(damerau-levenshtein-distance a b)

detabulate clj

(detabulate & {:keys [table row-labels col-labels]})

Take a contingency table of counts and returns a matrix of observations.

Examples:

(use '(incanter core stats datasets))

(def by-gender (group-on (get-dataset :hair-eye-color) 2)) (def table (matrix (sel (first by-gender) :cols 3) 4))

(detabulate :table table) (tabulate (detabulate :table table))

;; example 2 (def data (matrix [[1 0] [1 1] [1 1] [1 0] [0 0] [1 1] [1 1] [1 0] [1 1]])) (tabulate data)

(tabulate (detabulate :table (:table (tabulate data))))

dice-coefficient clj

(dice-coefficient a b)

http://en.wikipedia.org/wiki/Dice%27s_coefficient Dice's coefficient (also known as the Dice coefficient) is a similarity measure related to the Jaccard index.

dice-coefficient-str clj

(dice-coefficient-str a b)

http://en.wikipedia.org/wiki/Dice%27s_coefficient

When taken as a string similarity measure, the coefficient may be calculated for two strings, x and y using bigrams. Here nt is the number of character bigrams found in both strings, nx is the number of bigrams in string x and ny is the number of bigrams in string y. For example, to calculate the similarity between:

night nacht

We would find the set of bigrams in each word:

{ni,ig,gh,ht} {na,ac,ch,ht}

Each set has four elements, and the intersection of these two sets has only one element: ht.

Plugging this into the formula, we calculate, s = (2 · 1) / (4 + 4) = 0.25.

discordant-pairs clj

(discordant-pairs a b)

http://en.wikipedia.org/wiki/Discordant_pairs

discordant? clj

euclidean-distance clj

(euclidean-distance a b)

http://en.wikipedia.org/wiki/Euclidean_distance

the Euclidean distance or Euclidean metric is the ordinary distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space (or even any inner product space) becomes a metric space. The associated norm is called the Euclidean norm. Older literature refers to the metric as Pythagorean metric.

f-test clj

(f-test x y)

Test for different variances between 2 samples

Argument: x : 1st sample to test y : 2nd sample to test

Options:

References: http://en.wikipedia.org/wiki/F-test http://people.richland.edu/james/lecture/m170/ch13-f.html

first-digit clj

(first-digit x)

gamma-coefficient clj

(gamma-coefficient a b)

http://www.statsdirect.com/help/nonparametric_methods/kend.htm The gamma coefficient is given as a measure of association that is highly resistant to tied data (Goodman and Kruskal, 1963)

get-counts clj

(get-counts digits)

hamming-distance clj

(hamming-distance a b)

http://en.wikipedia.org/wiki/Hamming_distance

In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. Put another way, it measures the minimum number of substitutions required to change one string into the other, or the number of errors that transformed one string into the other.

indicator clj

(indicator pred coll)

Returns a sequence of ones and zeros, where ones are returned when the given predicate is true for corresponding element in the given collection, and zero otherwise.

Examples: (use 'incanter.stats)

(indicator #(neg? %) (sample-normal 10))

;; return the sum of the positive values in a normal sample (def x (sample-normal 100)) (sum (mult x (indicator #(pos? %) x)))

jaccard-distance clj

(jaccard-distance a b)

http://en.wikipedia.org/wiki/Jaccard_index The Jaccard distance, which measures dissimilarity between sample sets, is complementary to the Jaccard coefficient and is obtained by subtracting the Jaccard coefficient from 1, or, equivalently, by dividing the difference of the sizes of the union and the intersection of two sets by the size of the union.

jaccard-index clj

(jaccard-index a b)

http://en.wikipedia.org/wiki/Jaccard_index

The Jaccard index, also known as the Jaccard similarity coefficient (originally coined coefficient de communauté by Paul Jaccard), is a statistic used for comparing the similarity and diversity of sample sets.

The Jaccard coefficient measures similarity between sample sets, and is defined as the size of the intersection divided by the size of the union of the sample sets.

kendalls-tau clj

(kendalls-tau a b)

http://en.wikipedia.org/wiki/Kendall_tau_rank_correlation_coefficient http://www.statsdirect.com/help/nonparametric_methods/kend.htm http://mail.scipy.org/pipermail/scipy-dev/2009-March/011589.html best explanation and example is in "cluster analysis for researchers" page 165. http://www.amazon.com/Cluster-Analysis-Researchers-Charles-Romesburg/dp/1411606175

kendalls-tau-distance clj

kendalls-w clj

(kendalls-w)

http://en.wikipedia.org/wiki/Kendall%27s_W http://faculty.chass.ncsu.edu/garson/PA765/friedman.htm

Suppose that object i is given the rank ri,j by judge number j, where there are in total n objects and m judges. Then the total rank given to object i is

Ri = sum Rij

and the mean value of these total ranks is

Rbar = 1/2 m (n + 1)

The sum of squared deviations, S, is defined as

S=sum1-n (Ri - Rbar)

and then Kendall's W is defined as[1]

W= 12S / m^2(n^3-n)

If the test statistic W is 1, then all the survey respondents have been unanimous, and each respondent has assigned the same order to the list of concerns. If W is 0, then there is no overall trend of agreement among the respondents, and their responses may be regarded as essentially random. Intermediate values of W indicate a greater or lesser degree of unanimity among the various responses.

Legendre[2] discusses a variant of the W statistic which accommodates ties in the rankings and also describes methods of making significance tests based on W.

[{:observation [1 2 3]} {} ... {}] -> W

kurtosis clj

(kurtosis x)

Returns the kurtosis of the data, x. "Kurtosis is a measure of the "peakedness" of the probability distribution of a real-valued random variable. Higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent modestly-sized deviations." (Wikipedia)

Examples:

(kurtosis (sample-normal 100000)) ;; approximately 0 (kurtosis (sample-gamma 100000)) ;; approximately 6

References: http://incanter.org/docs/parallelcolt/api/cern/jet/stat/tdouble/DoubleDescriptive.html http://en.wikipedia.org/wiki/Kurtosis

lee-distance clj

(lee-distance a b q)

http://en.wikipedia.org/wiki/Lee_distance

In coding theory, the Lee distance is a distance between two strings x1x2...xn and y1y2...yn of equal length n over the q-ary alphabet {0,1,…,q-1} of size q >= 2. It is metric.

If q = 2 or q = 3 the Lee distance coincides with the Hamming distance.

The metric space induced by the Lee distance is a discrete analog of the elliptic space.

levenshtein-distance clj

(levenshtein-distance a b)

http://en.wikipedia.org/wiki/Levenshtein_distance

internal representation is a table d with m+1 rows and n+1 columns

where m is the length of a and m is the length of b.

In information theory and computer science, the Levenshtein distance is a metric for measuring the amount of difference between two sequences (i.e., the so called edit distance). The Levenshtein distance between two strings is given by the minimum number of operations needed to transform one string into the other, where an operation is an insertion, deletion, or substitution of a single character.

For example, the Levenshtein distance between "kitten" and "sitting" is 3, since the following three edits change one into the other, and there is no way to do it with fewer than three edits:

1. kitten → sitten (substitution of 's' for 'k')
2. sitten → sittin (substitution of 'i' for 'e')
3. sittin → sitting (insert 'g' at the end).

The Levenshtein distance has several simple upper and lower bounds that are useful in applications which compute many of them and compare them. These include:

• It is always at least the difference of the sizes of the two strings.
• It is at most the length of the longer string.
• It is zero if and only if the strings are identical.
• If the strings are the same size, the Hamming distance is an upper bound on the Levenshtein distance.

linear-model clj

(linear-model y x & {:keys [intercept] :or {intercept true}})

Returns the results of performing a OLS linear regression of y on x.

Arguments: y is a vector (or sequence) of values for the dependent variable x is a vector or matrix of values for the independent variables

Options: :intercept (default true) indicates weather an intercept term should be included

Returns: a map, of type ::linear-model, containing: :design-matrix -- a matrix containing the independent variables, and an intercept columns :coefs -- the regression coefficients :t-tests -- t-test values of coefficients :t-probs -- p-values for t-test values of coefficients :coefs-ci -- 95% percentile confidence interval :fitted -- the predicted values of y :residuals -- the residuals of each observation :std-errors -- the standard errors of the coeffients :sse -- the sum of squared errors, also called the residual sum of squares :ssr -- the regression sum of squares, also called the explained sum of squares :sst -- the total sum of squares (proportional to the sample variance) :r-square -- coefficient of determination

Examples: (use '(incanter core stats datasets charts)) (def iris (to-matrix (get-dataset :iris) :dummies true)) (def y (sel iris :cols 0)) (def x (sel iris :cols (range 1 6))) (def iris-lm (linear-model y x)) ; with intercept term

(keys iris-lm) ; see what fields are included (:coefs iris-lm) (:sse iris-lm) (quantile (:residuals iris-lm)) (:r-square iris-lm) (:adj-r-square iris-lm) (:f-stat iris-lm) (:f-prob iris-lm) (:df iris-lm)

(def x1 (range 0.0 3 0.1)) (view (xy-plot x1 (cdf-f x1 :df1 4 :df2 144)))

References: http://en.wikipedia.org/wiki/OLS_Regression http://en.wikipedia.org/wiki/Coefficient_of_determination

mahalanobis-distance clj

(mahalanobis-distance x & {:keys [y W centroid]})

Returns the Mahalanobis distance between x, which is either a vector or matrix of row vectors, and the centroid of the observations in the matrix :y.

Arguments: x -- either a vector or a matrix of row vectors

Options: :y -- Defaults to x, must be a matrix of row vectors which will be used to calculate a centroid :W -- Defaults to (solve (covariance y)), if an identity matrix is provided, the mahalanobis-distance function will be equal to the Euclidean distance. :centroid -- Defaults to (map mean (trans y))

References: http://en.wikipedia.org/wiki/Mahalanobis_distance

Examples:

(use '(incanter core stats charts))

;; generate some multivariate normal data with a single outlier. (def data (bind-rows (bind-columns (sample-mvn 100 :sigma (matrix [[1 0.9] [0.9 1]]))) [-1.75 1.75]))

;; view a scatter plot of the data (let [[x y] (trans data)] (doto (scatter-plot x y) (add-points [(mean x)] [(mean y)]) (add-pointer -1.75 1.75 :text "Outlier") (add-pointer (mean x) (mean y) :text "Centroid") view))

;; calculate the distances of each point from the centroid. (def dists (map first (mahalanobis-distance data))) ;; view a bar-chart of the distances (view (bar-chart (range 102) dists))

;; Now contrast with the Euclidean distance. (def dists (map first (mahalanobis-distance data :W (matrix [[1 0] [0 1]])))) ;; view a bar-chart of the distances (view (bar-chart (range 102) dists))

;; another example (mahalanobis-distance [-1.75 1.75] :y data) (mahalanobis-distance [-1.75 1.75] :y data :W (matrix [[1 0] [0 1]]))

manhattan-distance clj

(manhattan-distance a b)

http://en.wikipedia.org/wiki/Manhattan_distance

usual metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the (absolute) differences of their coordinates. The taxicab metric is also known as rectilinear distance, L1 distance or l1 norm (see Lp space), city block distance, Manhattan distance, or Manhattan length

mean clj

(mean x)

Returns the mean of the data, x.

Examples: (mean (sample-normal 100))

References: http://incanter.org/docs/parallelcolt/api/cern/jet/stat/tdouble/DoubleDescriptive.html http://en.wikipedia.org/wiki/Mean

median clj

(median x)

Returns the median of the data, x.

Examples: (median (sample-normal 100))

References: http://incanter.org/docs/parallelcolt/api/cern/jet/stat/tdouble/DoubleDescriptive.html http://en.wikipedia.org/wiki/Median

minkowski-distance clj

(minkowski-distance a b p)

http://en.wikipedia.org/wiki/Minkowski_distance http://en.wikipedia.org/wiki/Lp_space

The Minkowski distance is a metric on Euclidean space which can be considered as a generalization of both the Euclidean distance and the Manhattan distance.

Minkowski distance is typically used with p being 1 or 2. The latter is the Euclidean distance, while the former is sometimes known as the Manhattan distance.

In the limiting case of p reaching infinity we obtain the Chebyshev distance.

n-grams clj

(n-grams n s)

Returns a set of the unique n-grams in a string. this is using actual sets here, discards duplicate n-grams?

normalized-kendall-tau-distance clj

(normalized-kendall-tau-distance a b)

http://en.wikipedia.org/wiki/Kendall_tau_distance Kendall tau distance is the total number of discordant pairs.

numeric-col-summarizer clj

(numeric-col-summarizer col ds)

Returns a summarizer function which takes a purely numeric column with no non-numeric values

odds-ratio clj

(odds-ratio p1 p2)

http://en.wikipedia.org/wiki/Odds_ratio

Definition in terms of group-wise odds

The odds ratio is the ratio of the odds of an event occurring in one group to the odds of it occurring in another group, or to a sample-based estimate of that ratio.

Suppose that in a sample of 100 men, 90 have drunk wine in the previous week, while in a sample of 100 women only 20 have drunk wine in the same period. The odds of a man drinking wine are 90 to 10, or 9:1, while the odds of a woman drinking wine are only 20 to 80, or 1:4 = 0.25:1. The odds ratio is thus 9/0.25, or 36, showing that men are much more likely to drink wine than women.

Relation to statistical independence

If X and Y are independent, their joint probabilities can be expressed in terms of their marginal probabilities. In this case, the odds ratio equals one, and conversely the odds ratio can only equal one if the joint probabilities can be factored in this way. Thus the odds ratio equals one if and only if X and Y are independent.

pairings clj

(pairings a b)

Creates pairs by matching a1 with b1, a2 with b2, etc. and returns all pairs of those pairs without matching a pair with itself.

pairs clj

(pairs a b)

Returns unique pairs of a and b where members of a and b can not be paired with the corresponding slot in the other list.

pdf-beta clj

(pdf-beta x & {:keys [alpha beta seed] :or {alpha 1 beta 1 seed (Date.)}})

Returns the Beta pdf of the given value of x. It will return a sequence of values, if x is a sequence. This is equivalent to R's dbeta function.

Options: :alpha (default 1) :beta (default 1)

See also: cdf-beta and sample-beta

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Beta.html http://en.wikipedia.org/wiki/Beta_distribution http://en.wikipedia.org/wiki/Probability_density_function

Example: (pdf-beta 0.5 :alpha 1 :beta 2)

pdf-binomial clj

(pdf-binomial x & {:keys [size prob seed] :or {size 1 prob 1/2 seed (Date.)}})

Returns the Binomial pdf of the given value of x. It will return a sequence of values, if x is a sequence. Equivalent to R's dbinom.

Options: :size (default 1) :prob (default 1/2)

See also: cdf-binomial and sample-binomial

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Binomial.html http://en.wikipedia.org/wiki/Binomial_distribution http://en.wikipedia.org/wiki/Probability_density_function

Example: (pdf-binomial 10 :prob 1/4 :size 20)

pdf-chisq clj

(pdf-chisq x & {:keys [df seed] :or {df 1 seed (Date.)}})

Returns the Chi Square pdf of the given value of x. It will return a sequence of values, if x is a sequence. Equivalent to R's dchisq function.

Options: :df (default 1)

See also: cdf-chisq and sample-chisq

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/ChiSquare.html http://en.wikipedia.org/wiki/Chi_square_distribution http://en.wikipedia.org/wiki/Probability_density_function

Example: (pdf-chisq 5.0 :df 2)

pdf-exp clj

(pdf-exp x & {:keys [rate seed] :or {rate 1 seed (Date.)}})

Returns the Exponential pdf of the given value of x. It will return a sequence of values, if x is a sequence. Equivalent to R's dexp.

Options: :rate (default 1)

See also: cdf-exp and sample-exp

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Exponential.html http://en.wikipedia.org/wiki/Exponential_distribution http://en.wikipedia.org/wiki/Probability_density_function

Example: (pdf-exp 2.0 :rate 1/2)

pdf-f clj

(pdf-f x & {:keys [df1 df2] :or {df1 1 df2 1}})

Returns the F pdf of the given value, x. It will return a sequence of values, if x is a sequence. This is equivalent to R's df function.

Options: :df1 (default 1) :df2 (default 1)

See also: cdf-f and quantile-f

References: http://en.wikipedia.org/wiki/F_distribution http://mathworld.wolfram.com/F-Distribution.html http://en.wikipedia.org/wiki/Probability_density_function

Example: (pdf-f 1.0 :df1 5 :df2 2)

pdf-gamma clj

(pdf-gamma x & {:keys [shape scale rate seed] :or {shape 1 seed (Date.)}})

Returns the Gamma pdf for the given value of x. It will return a sequence of values, if x is a sequence. This is equivalent to R's dgamma function.

Options: :shape (k) (default 1) :scale (θ) (default 1 or 1/rate, if :rate is specified) :rate (β) (default 1/scale, if :scale is specified)

See also: cdf-gamma and sample-gamma

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Gamma.html http://en.wikipedia.org/wiki/Gamma_distribution http://en.wikipedia.org/wiki/Probability_density_function

Example: (pdf-gamma 10 :shape 1 :scale 2)

pdf-neg-binomial clj

(pdf-neg-binomial x
                  &
                  {:keys [size prob seed] :or {size 10 prob 1/2 seed (Date.)}})

Returns the Negative Binomial pdf of the given value of x. It will return a sequence of values, if x is a sequence. Equivalent to R's dnbinom.

Options: :size (default 10) :prob (default 1/2)

See also: cdf-neg-binomial and sample-neg-binomial

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/NegativeBinomial.html http://en.wikipedia.org/wiki/Negative_binomial_distribution http://en.wikipedia.org/wiki/Probability_density_function

Example: (pdf-neg-binomial 10 :prob 1/2 :size 20)

pdf-normal clj

(pdf-normal x & {:keys [mean sd seed] :or {mean 0 sd 1 seed (Date.)}})

Returns the Normal pdf of the given value, x. It will return a sequence of values, if x is a sequence. This is equivalent to R's dnorm function.

Options: :mean (default 0) :sd (default 1)

See also: cdf-normal, quantile-normal, sample-normal

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Normal.html http://en.wikipedia.org/wiki/Normal_distribution http://en.wikipedia.org/wiki/Probability_density_function

Example: (pdf-normal 1.96 :mean -2 :sd (sqrt 0.5))

pdf-poisson clj

(pdf-poisson x & {:keys [lambda seed] :or {lambda 1 seed (Date.)}})

Returns the Poisson pdf of the given value of x. It will return a sequence of values, if x is a sequence. Equivalent to R's dpois.

Options: :lambda (default 1)

See also: cdf-poisson and sample-poisson

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Poisson.html http://en.wikipedia.org/wiki/Poisson_distribution http://en.wikipedia.org/wiki/Probability_density_function

Example: (pdf-poisson 5 :lambda 10)

pdf-t clj

(pdf-t x & {:keys [df seed] :or {df 1 seed (Date.)}})

Returns the Student's t pdf for the given value of x. It will return a sequence of values, if x is a sequence. Equivalent to R's dt function.

Options: :df (default 1)

See also: cdf-t, quantile-t, and sample-t

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/StudentT.html http://en.wikipedia.org/wiki/Student-t_distribution http://en.wikipedia.org/wiki/Probability_density_function

Example: (pdf-t 1.2 :df 10)

pdf-uniform clj

(pdf-uniform x & {:keys [min max seed] :or {min 0.0 max 1.0 seed (Date.)}})

Returns the Uniform pdf of the given value of x. It will return a sequence of values, if x is a sequence. This is equivalent to R's dunif function.

Options: :min (default 0) :max (default 1)

See also: cdf-uniform and sample-uniform

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/DoubleUniform.html http://en.wikipedia.org/wiki/Uniform_distribution http://en.wikipedia.org/wiki/Probability_density_function

Example: (pdf-uniform 5) (pdf-uniform 5 :min 1 :max 10)

pdf-weibull clj

(pdf-weibull x & options)

Returns the Weibull pdf for the given value of x. It will return a sequence of values, if x is a sequence.

Options: :scale (default 1) :shape (default 1)

See also: cdf-weibull and sample-weibull

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Distributions.html http://en.wikipedia.org/wiki/Weibull_distribution http://en.wikipedia.org/wiki/Probability_density_function

Example: (pdf-weibull 2 :alpha 1 :beta 0.5)

permute clj

(permute x)

(permute x y)

If provided a single argument, returns a permuted version of the given collection. (permute x) is the same as (sample x).

If provided two arguments, returns two lists that are permutations across the given collections. In other words, each of the new collections will contain elements from both of the given collections. Useful for permutation tests or randomization tests.

Examples:

(permute (range 10)) (permute (range 10) (range 10 20))

predict clj

(predict model x)

Takes a linear-model and an x value (either a scalar or vector) and returns the predicted value based on the linear-model.

principal-components clj

(principal-components x & options)

Performs a principal components analysis on the given data matrix. Equivalent to R's prcomp function.

Returns: A map with the following fields: :std-dev -- the standard deviations of the principal components (i.e. the square roots of the eigenvalues of the correlation matrix, though the calculation is actually done with the singular values of the data matrix. :rotation -- the matrix of variable loadings (i.e. a matrix whose columns contain the eigenvectors).

Examples:

(use '(incanter core stats charts datasets)) ;; load the iris dataset (def iris (to-matrix (get-dataset :iris))) ;; run the pca (def pca (principal-components (sel iris :cols (range 4)))) ;; extract the first two principal components (def pc1 (sel (:rotation pca) :cols 0)) (def pc2 (sel (:rotation pca) :cols 1))

;; project the first four dimension of the iris data onto the first ;; two principal components (def x1 (mmult (sel iris :cols (range 4)) pc1)) (def x2 (mmult (sel iris :cols (range 4)) pc2))

;; now plot the transformed data, coloring each species a different color (doto (scatter-plot (sel x1 :rows (range 50)) (sel x2 :rows (range 50)) :x-label "PC1" :y-label "PC2" :title "Iris PCA") (add-points (sel x1 :rows (range 50 100)) (sel x2 :rows (range 50 100))) (add-points (sel x1 :rows (range 100 150)) (sel x2 :rows (range 100 150))) view)

;; alternatively, the :group-by option can be used in scatter-plot (view (scatter-plot x1 x2 :group-by (sel iris :cols 4) :x-label "PC1" :y-label "PC2" :title "Iris PCA"))

References: http://en.wikipedia.org/wiki/Principal_component_analysis

product-marginal-test clj

(product-marginal-test j)

the joint PMF of independent variables is equal to the product of their marginal PMFs.

quantile clj

(quantile x
          &
          {:keys [probs]
           :or {probs (DoubleArrayList. (double-array [0.0 0.25 0.5 0.75
                                                       1.0]))}})

Returns the quantiles of the data, x. By default it returns the min, 25th-percentile, 50th-percentile, 75th-percentile, and max value.

Options: :probs (default [0.0 0.25 0.5 0.75 1.0])

Examples: (quantile (sample-normal 100)) (quantile (sample-normal 100) :probs [0.025 0.975]) (quantile (sample-normal 100) :probs 0.975)

References: http://incanter.org/docs/parallelcolt/api/cern/jet/stat/tdouble/DoubleDescriptive.html http://en.wikipedia.org/wiki/Quantile

quantile-normal clj

(quantile-normal probability & {:keys [mean sd] :or {mean 0 sd 1}})

Returns the inverse of the Normal CDF for the given probability. It will return a sequence of values, if given a sequence of probabilities. This is equivalent to R's qnorm function.

Options: :mean (default 0) :sd (default 1)

Returns: a value x, where (cdf-normal x) = probability

See also: pdf-normal, cdf-normal, and sample-normal

References: http://incanter.org/docs/parallelcolt/api/cern/jet/stat/tdouble/Probability.html http://en.wikipedia.org/wiki/Normal_distribution http://en.wikipedia.org/wiki/Quantile

Example: (quantile-normal 0.975) (quantile-normal [0.025 0.975] :mean -2 :sd (sqrt 0.5))

quantile-t clj

(quantile-t probability & {:keys [df] :or {df 1}})

Returns the inverse of the Student's t CDF for the given probability (i.e. the quantile). It will return a sequence of values, if x is a sequence of probabilities. This is equivalent to R's qt function.

Options: :df (default 1)

Returns: a value x, where (cdf-t x) = probability

See also: pdf-t, cdf-t, and sample-t

References: http://incanter.org/docs/parallelcolt/api/cern/jet/stat/tdouble/Probability.html http://en.wikipedia.org/wiki/Student-t_distribution http://en.wikipedia.org/wiki/Quantile

Example: (quantile-t 0.975) (quantile-t [0.025 0.975] :df 25) (def df [1 2 3 4 5 6 7 8 9 10 20 50 100 1000]) (map #(quantile-t 0.025 :df %) df)

rank-index clj

(rank-index x)

Given a seq, returns a map where the keys are the values of the seq and the values are the positional rank of each member o the seq.

sample clj multimethod

Returns a sample of the given size from the given collection. If replacement is set to false it returns a set, otherwise it returns a list.

Arguments: coll -- collection or dataset to be sampled from

Options: :size -- (default (count x) sample size :replacement (default true) -- sample with replacement

Examples: (sample (range 10)) ; permutation of numbers zero through ten (sample [:red :green :blue] :size 10) ; choose 10 items that are either :red, :green, or :blue. (sample (seq "abcdefghijklmnopqrstuvwxyz") :size 4 :replacement false) ; choose 4 random letters.

sample-beta clj

(sample-beta size & {:keys [alpha beta] :or {alpha 1 beta 1}})

Returns a sample of the given size from a Beta distribution. This is equivalent to R's rbeta function.

Options: :alpha (default 1) :beta (default 1) These default values produce a Uniform distribution.

See also: pdf-beta and cdf-beta

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Beta.html http://en.wikipedia.org/wiki/Beta_distribution

Example: (sample-beta 1000 :alpha 1 :beta 2)

sample-binomial clj

(sample-binomial samplesize & {:keys [size prob] :or {size 1 prob 1/2}})

Returns a sample of the given size from a Binomial distribution. Equivalent to R's rbinom.

Options: :size (default 1) :prob (default 1/2)

See also: pdf-binomial and cdf-binomial

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Binomial.html http://en.wikipedia.org/wiki/Binomial_distribution

Example: (sample-binomial 1000 :prob 1/4 :size 20)

sample-chisq clj

(sample-chisq size & {:keys [df] :or {df 1}})

Returns a sample of the given size from a Chi Square distribution Equivalent to R's rchisq function.

Options: :df (default 1)

See also: pdf-chisq and cdf-chisq

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/ChiSquare.html http://en.wikipedia.org/wiki/Chi_square_distribution

Example: (sample-chisq 1000 :df 2)

sample-dirichlet clj

(sample-dirichlet size alpha)

Examples: (use '(incanter core stats charts))

;; a total of 1447 adults were polled to indicate their preferences for ;; candidate 1 (y1=727), candidate 2 (y2=583), or some other candidate or no ;; preference (y3=137).

;; the counts y1, y2, and y3 are assumed to have a multinomial distribution ;; If a uniform prior distribution is assigned to the multinomial vector ;; theta = (th1, th2, th3), then the posterior distribution of theta is ;; proportional to g(theta) = th1^y1 * th2^y2 * th3^y3, which is a ;; dirichlet distribution with parameters (y1+1, y2+1, y3+1) (def theta (sample-dirichlet 1000 [(inc 727) (inc 583) (inc 137)])) ;; view means, 95% CI, and histograms of the proportion parameters (mean (sel theta :cols 0)) (quantile (sel theta :cols 0) :probs [0.0275 0.975]) (view (histogram (sel theta :cols 0))) (mean (sel theta :cols 1)) (quantile (sel theta :cols 1) :probs [0.0275 0.975]) (view (histogram (sel theta :cols 1))) (mean (sel theta :cols 2)) (quantile (sel theta :cols 2) :probs [0.0275 0.975]) (view (histogram (sel theta :cols 2)))

;; view a histogram of the difference in proportions between the first ;; two candidates (view (histogram (minus (sel theta :cols 0) (sel theta :cols 1))))

sample-exp clj

(sample-exp size & {:keys [rate] :or {rate 1}})

Returns a sample of the given size from a Exponential distribution. Equivalent to R's rexp.

Options: :rate (default 1)

See also: pdf-exp and cdf-exp

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Exponential.html http://en.wikipedia.org/wiki/Exponential_distribution

Example: (sample-exp 1000 :rate 1/2)

sample-gamma clj

(sample-gamma size & {:keys [shape scale rate] :or {shape 1}})

Returns a sample of the given size from a Gamma distribution. This is equivalent to R's rgamma function.

Options: :shape (k) (default 1) :scale (θ) (default 1 or 1/rate, if :rate is specified) :rate (β) (default 1/scale, if :scale is specified)

See also: pdf-gamma, cdf-gamma, and quantile-gamma

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Gamma.html http://en.wikipedia.org/wiki/Gamma_distribution

Example: (sample-gamma 1000 :shape 1 :scale 2)

sample-inv-wishart clj

(sample-inv-wishart & {:keys [scale p df] :or {p 2}})

Returns a p-by-p symmetric distribution drawn from an inverse-Wishart distribution

Options: :p (default 2) -- number of dimensions of resulting matrix :df (default p) -- degree of freedoms (aka n), df <= p :scale (default (identity-matrix p)) -- positive definite matrix (aka V)

Examples: (use 'incanter.stats) (sample-inv-wishart :df 10 :p 4)

;; calculate the mean of 1000 wishart matrices, should equal (mult df scale) (div (reduce plus (for [_ (range 1000)] (sample-wishart :p 4))) 1000)

References: http://en.wikipedia.org/wiki/Inverse-Wishart_distribution

sample-multinomial clj

(sample-multinomial size & {:keys [probs categories] :or {probs [0.5 0.5]}})

Returns a sequence representing a sample from a multinomial distribution.

Arguments: size -- number of values to return

Options: :categories (default [0 1]) -- the values returned :probs (default [0.5 0.5]) -- the probabilities associated with each category

References: http://en.wikipedia.org/wiki/Multinomial_distribution#Sampling_from_a_multinomial_distribution

Examples: (use '(incanter core stats charts))

(sample-multinomial 10) (sample-multinomial 10 :probs [0.25 0.5 0.25])

;; estimate sample proportions (def sample-size 1000.0) (def categories [:red :yellow :blue :green]) (def data (to-dataset (sample-multinomial sample-size :categories categories :probs [0.5 0.25 0.2 0.05])))

;; check the sample proportions (view (pie-chart categories (map #(div (count ($ :col-0 ($where {:col-0 %} data))) sample-size) categories)))

sample-mvn clj

(sample-mvn size & {:keys [mean sigma]})

Returns a sample of the given size from a Multivariate Normal distribution. This is equivalent to R's mvtnorm::rmvnorm function.

Arguments: size -- the size of the sample to return

Options: :mean (default (repeat (ncol sigma) 0)) :sigma (default (identity-matrix (count mean)))

Examples:

(use '(incanter core stats charts)) (def mvn-samp (sample-mvn 1000 :mean [7 5] :sigma (matrix [[2 1.5] [1.5 3]]))) (covariance mvn-samp) (def means (map mean (trans mvn-samp)))

;; plot scatter-plot of points (def mvn-plot (scatter-plot (sel mvn-samp :cols 0) (sel mvn-samp :cols 1))) (view mvn-plot) ;; add centroid to plot (add-points mvn-plot [(first means)] [(second means)])

;; add regression line to scatter plot (def x (sel mvn-samp :cols 0)) (def y (sel mvn-samp :cols 1)) (def lm (linear-model y x)) (add-lines mvn-plot x (:fitted lm))

References: http://en.wikipedia.org/wiki/Multivariate_normal

sample-neg-binomial clj

(sample-neg-binomial samplesize & {:keys [size prob] :or {size 10 prob 1/2}})

Returns a sample of the given size from a Negative Binomial distribution. Equivalent to R's rnbinom.

Options: :size (default 10) :prob (default 1/2)

See also: pdf-neg-binomial and cdf-neg-binomial

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/NegativeBinomial.html http://en.wikipedia.org/wiki/Negative_binomial_distribution

Example: (sample-neg-binomial 1000 :prob 1/2 :size 20)

sample-normal clj

(sample-normal size & {:keys [mean sd] :or {mean 0 sd 1}})

Returns a sample of the given size from a Normal distribution This is equivalent to R's rnorm function.

Options: :mean (default 0) :sd (default 1)

See also: pdf-normal, cdf-normal, quantile-normal

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Normal.html http://en.wikipedia.org/wiki/Normal_distribution

Example: (sample-normal 1000 :mean -2 :sd (sqrt 0.5))

sample-permutations clj

(sample-permutations n x)

(sample-permutations n x y)

If provided a two arguments (n x), it returns a list of n permutations of x. If provided three (n x y) arguments, returns a list with two with n permutations of each arguments, where each permutation is drawn from the pooled arguments.

Arguments: n -- number of randomized versions of the original two groups to return x -- group 1 y -- (default nil) group 2

Examples:

(use '(incanter core stats)) (sample-permutations 10 (range 10)) (sample-permutations 10 (range 10) (range 10 20))

;; extended example with plant-growth data (use '(incanter core stats datasets charts))

;; load the plant-growth dataset (def data (to-matrix (get-dataset :plant-growth)))

;; break the first column of the data into groups based on treatment (second column). (def groups (group-on data 1 :cols 0))

;; define a function for the statistic of interest (defn means-diff [x y] (minus (mean x) (mean y)))

;; calculate the difference in sample means between the two groups (def samp-mean-diff (means-diff (first groups) (second groups))) ;; 0.371

;; create 500 permuted versions of the original two groups (def permuted-groups (sample-permutations 1000 (first groups) (second groups)))

;; calculate the difference of means of the 500 samples (def permuted-means-diffs1 (map means-diff (first permuted-groups) (second permuted-groups)))

;; use an indicator function that returns 1 when the randomized means diff is greater ;; than the original sample mean, and zero otherwise. Then take the mean of this sequence ;; of ones and zeros. That is the proportion of times you would see a value more extreme ;; than the sample mean (i.e. the p-value). (mean (indicator #(> % samp-mean-diff) permuted-means-diffs1)) ;; 0.088

;; calculate the 95% confidence interval of the null hypothesis. If the ;; sample difference in means is outside of this range, that is evidence ;; that the two means are statistically significantly different. (quantile permuted-means-diffs1 :probs [0.025 0.975]) ;; (-0.606 0.595)

;; Plot a histogram of the permuted-means-diffs using the density option, ;; instead of the default frequency, and then add a normal pdf curve with ;; the mean and sd of permuted-means-diffs data for a visual comparison. (doto (histogram permuted-means-diffs1 :density true) (add-lines (range -1 1 0.01) (pdf-normal (range -1 1 0.01) :mean (mean permuted-means-diffs1) :sd (sd permuted-means-diffs1))) view)

;; compare the means of treatment 2 and control (def permuted-groups (sample-permutations 1000 (first groups) (last groups))) (def permuted-means-diffs2 (map means-diff (first permuted-groups) (second permuted-groups))) (def samp-mean-diff (means-diff (first groups) (last groups))) ;; -0.4939 (mean (indicator #(< % samp-mean-diff) permuted-means-diffs2)) ;; 0.022 (quantile permuted-means-diffs2 :probs [0.025 0.975]) ;; (-0.478 0.466)

;; compare the means of treatment 1 and treatment 2 (def permuted-groups (sample-permutations 1000 (second groups) (last groups))) (def permuted-means-diffs3 (map means-diff (first permuted-groups) (second permuted-groups))) (def samp-mean-diff (means-diff (second groups) (last groups))) ;; -0.865 (mean (indicator #(< % samp-mean-diff) permuted-means-diffs3)) ;; 0.002 (quantile permuted-means-diffs3 :probs [0.025 0.975]) ;; (-0.676 0.646)

(doto (box-plot permuted-means-diffs1) (add-box-plot permuted-means-diffs2) (add-box-plot permuted-means-diffs3) view)

Further Reading: http://en.wikipedia.org/wiki/Resampling_(statistics)

sample-poisson clj

(sample-poisson size & {:keys [lambda] :or {lambda 1}})

Returns a sample of the given size from a Poisson distribution. Equivalent to R's rpois.

Options: :lambda (default 1)

See also: pdf-poisson and cdf-poisson

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Poisson.html http://en.wikipedia.org/wiki/Poisson_distribution

Example: (sample-poisson 1000 :lambda 10)

sample-t clj

(sample-t size & {:keys [df] :or {df 1}})

Returns a sample of the given size from a Student's t distribution. Equivalent to R's rt function.

Options: :df (default 1)

See also: pdf-t, cdf-t, and quantile-t

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/StudentT.html http://en.wikipedia.org/wiki/Student-t_distribution

Example: (cdf-t 1000 :df 10)

sample-uniform clj

(sample-uniform size
                &
                {:keys [min max integers seed]
                 :or {min 0.0 max 1.0 integers false seed (Date.)}})

Returns a sample of the given size from a Uniform distribution. This is equivalent to R's runif function.

Options: :min (default 0) :max (default 1) :integers (default false)

See also: pdf-uniform and cdf-uniform

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/DoubleUniform.html http://en.wikipedia.org/wiki/Uniform_distribution

Example: (sample-uniform 1000) (sample-uniform 1000 :min 1 :max 10)

sample-weibull clj

(sample-weibull size & options)

Returns a sample of the given size from a Weibull distribution. Caller may supply a custom seed or a pre-existing random number generator, otherwise a global default generator is used.

Options: :shape (default 1) :scale (default 1) :seed (default nil), seed-value for new random number gen :rng (default nil), existing random number gen

See also: pdf-weibull, cdf-weibull

References: http://incanter.org/docs/parallelcolt/api/cern/jet/random/tdouble/Distributions.html http://en.wikipedia.org/wiki/Weibull_distribution

Example: (sample-weibull 1000 :shape 1 :scale 0.2)

sample-wishart clj

(sample-wishart & {:keys [scale p df] :or {p 2}})

Returns a p-by-p symmetric distribution drawn from a Wishart distribution

Options: :p (default 2) -- number of dimensions of resulting matrix :df (default p) -- degree of freedoms (aka n), df <= p :scale (default (identity-matrix p)) -- positive definite matrix (aka V)

Examples: (use 'incanter.stats) (sample-wishart :df 10 :p 4)

;; calculate the mean of 1000 wishart matrices, should equal (mult df scale) (div (reduce plus (for [_ (range 1000)] (sample-wishart :p 4))) 1000)

References: http://en.wikipedia.org/wiki/Wishart_distribution

scalar-abs clj

(scalar-abs x)

Fast absolute value function

sd clj

(sd x)

Returns the sample standard deviation of the data, x. Equivalent to R's sd function.

Examples: (sd (sample-normal 100))

References: http://incanter.org/docs/parallelcolt/api/cern/jet/stat/tdouble/DoubleDescriptive.html http://en.wikipedia.org/wiki/Standard_deviation

simple-ci clj

(simple-ci coll)

Get the confidence interval for the data.

simple-p-value clj

(simple-p-value coll mu)

Returns the p-value for the data contained in coll.

simple-regression clj

(simple-regression y x & {:keys [intercept] :or {intercept true}})

A stripped version of linear-model that returns a map containing only the coefficients.

simple-t-test clj

(simple-t-test coll mu)

Perform a simple t-test on the data contained in coll.

skewness clj

(skewness x)

Returns the skewness of the data, x. "Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable." (Wikipedia)

Examples:

(skewness (sample-normal 100000)) ;; approximately 0 (skewness (sample-gamma 100000)) ;; approximately 2

References: http://incanter.org/docs/parallelcolt/api/cern/jet/stat/tdouble/DoubleDescriptive.html http://en.wikipedia.org/wiki/Skewness

sorensen-index clj

(sorensen-index a b)

http://en.wikipedia.org/wiki/S%C3%B8rensen_similarity_index#cite_note-4

The Sørensen index, also known as Sørensen’s similarity coefficient, is a statistic used for comparing the similarity of two samples. where A and B are the species numbers in samples A and B, respectively, and C is the number of species shared by the two samples.

The Sørensen index is identical to Dice's coefficient which is always in [0, 1] range. Sørensen index used as a distance measure, 1 − QS, is identical to Hellinger distance and Bray–Curtis dissimilarity.

The Sørensen coefficient is mainly useful for ecological community data (e.g. Looman & Campbell, 1960[3]). Justification for its use is primarily empirical rather than theoretical (although it can be justified theoretically as the intersection of two fuzzy sets[4]). As compared to Euclidean distance, Sørensen distance retains sensitivity in more heterogeneous data sets and gives less weight to outliers

This function assumes you pass in a and b as sets.

The sorensen index extended to abundance instead of incidence of species is called the Czekanowski index.

spearmans-rho clj

(spearmans-rho a b)

http://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient

In statistics, Spearman's rank correlation coefficient or Spearman's rho, is a non-parametric measure of correlation – that is, it assesses how well an arbitrary monotonic function could describe the relationship between two variables, without making any other assumptions about the particular nature of the relationship between the variables. Certain other measures of correlation are parametric in the sense of being based on possible relationships of a parameterised form, such as a linear relationship.

square-devs-from-mean clj

(square-devs-from-mean x)

(square-devs-from-mean x m)

takes either a sample or a sample and a precalculated mean. returns the squares of the difference between each observation and the sample mean.

sum-of-square-devs-from-mean clj

(sum-of-square-devs-from-mean x)

(sum-of-square-devs-from-mean x m)

takes either a sample or a sample and a precalculated mean.

returns the sum of the squares of the difference between each observation and the sample mean.

sum-variance-test clj

(sum-variance-test vs)

The variance of the sum of n independent variables is equal to the sum of their variances.

(variance-independence-test [[1 2 3 4] [1 2 3 4]]) -> 5/2

summarizable? clj

(summarizable? col ds)

Takes in a column name (or number) and a dataset. Returns true if the column can be summarized, and false otherwise

summarizer-fn clj

(summarizer-fn col ds)

Takes in a column (number or name) and a dataset. Returns a function to summarize the column if summarizable, and a string describing why the column can't be summarized in the event that it can't

summary clj

(summary ds)

Takes in a dataset. Returns a summary of that dataset (as a map of maps), having automatically figured out the relevant datatypes of columns. Will be slightly forgiving of mangled data in columns.

sweep clj

(sweep x & {:keys [stat fun] :or {stat mean fun minus}})

Return an array obtained from an input array by sweeping out a summary statistic. Based to R's sweep function.

Arguments: x is an sequence

Options: :stat (default mean) the statistic to sweep out :fun (defaul minus) the function used to sweep the stat out

Example:

(use '(incanter core stats))

(def x (sample-normal 30 :mean 10 :sd 5)) (sweep x) ;; center the data around mean (sweep x :stat sd :fun div) ;; divide data by its sd

t-test clj

(t-test
  x
  &
  {:keys [y mu paired conf-level alternative var-equal]
   :or {paired false alternative :two-sided conf-level 0.95 var-equal false}})

Argument: x : sample to test

Options: :y (default nil) :mu (default (mean y) or 0) population mean :alternative (default :two-sided) other choices :less :greater :var-equal TODO (default false) variance equal :paired TODO (default false) paired test :conf-level (default 0.95) for returned confidence interval

Examples:

(t-test (range 1 11) :mu 0) (t-test (range 1 11) :mu 0 :alternative :less) (t-test (range 1 11) :mu 0 :alternative :greater)

(t-test (range 1 11) :y (range 7 21)) (t-test (range 1 11) :y (range 7 21) :alternative :less) (t-test (range 1 11) :y (range 7 21) :alternative :greater) (t-test (range 1 11) :y (conj (range 7 21) 200))

References: http://en.wikipedia.org/wiki/T_test http://www.socialresearchmethods.net/kb/stat_t.php

tabulate clj

(tabulate x & options)

Cross-tabulates the values of the given numeric matrix.

Returns a hash-map with the following fields: :table -- the table of counts for each combination of values, this table is only returned if x has two-columns :levels -- a sequence of sequences, where each sequence list the levels (possible values) of the corresponding column of x. :margins -- a sequence of sequences, where each sequence represents the marginal total for each level of the corresponding column of x. :counts -- a hash-map, where vectors of unique combinations of the cross-tabulated levels are the keys and the values are the total count of each combination. :N -- the grand-total for the contingency table

Examples:

(use '(incanter core stats)) (tabulate [1 2 3 2 3 2 4 3 5]) (tabulate (sample-poisson 100 :lambda 5))

(use '(incanter core stats datasets)) (def math-prog (to-matrix (get-dataset :math-prog))) (tabulate (sel math-prog :cols [1 2]))

(def data (matrix [[1 0 1] [1 1 1] [1 1 1] [1 0 1] [0 0 0] [1 1 1] [1 1 1] [1 0 1] [1 1 0]])) (tabulate data)

(def data (matrix [[1 0] [1 1] [1 1] [1 0] [0 0] [1 1] [1 1] [1 0] [1 1]])) (tabulate data)

tanimoto-coefficient clj

(tanimoto-coefficient a b)

http://en.wikipedia.org/wiki/Jaccard_index

The cosine similarity metric may be extended such that it yields the Jaccard coefficient in the case of binary attributes. This is the Tanimoto coefficient.

variance clj

(variance x)

Returns the sample variance of the data, x. Equivalent to R's var function.

Examples: (variance (sample-normal 100))

References: http://incanter.org/docs/parallelcolt/api/cern/jet/stat/tdouble/DoubleDescriptive.html http://en.wikipedia.org/wiki/Sample_variance#Population_variance_and_sample_variance

within clj

(within z x y)

y is within z of x in metric space.