incanter.distributions — incanter/incanter-core 1.9.3

Probability functions (pdf, cdf, draw, etc.) for common distributions, and for collections, sets, and maps.

*test-statistic-iterations* clj

*test-statistic-map* clj

beta-distribution clj

(beta-distribution)

(beta-distribution alpha beta)

Returns a Beta distribution that implements the incanter.distributions.Distribution protocol.

Arguments: alpha (default 1) beta (default 1)

See also: Distribution, pdf, cdf, draw, support

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

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

binomial-distribution clj

(binomial-distribution)

(binomial-distribution n p)

Returns a Binomial distribution that implements the incanter.distributions.Distribution protocol.

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

See also: Distribution, pdf, cdf, draw, support

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

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

chisq-distribution clj

(chisq-distribution)

(chisq-distribution df)

Returns a Chi-square distribution that implements the incanter.distributions.Distribution protocol.

Arguments: df (default 1)

See also: Distribution, pdf, cdf, draw, support

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

Example: (pdf (chisq-distribution 2) 5.0)

colt-extenders clj

combination-distribution clj

(combination-distribution n k)

Create a distribution of all the k-sized combinations of n integers. Can be considered a multivariate distribution over k-dimensions, where each dimension is a discrete random variable on the (0, n] range (though these variables are decidedly non-independent).

A draw from this distribution can also be considered a sample without replacement from any finite set, where the values in the returned vector represent the indices of the items in the set.

Arguments: n The number of possible items from which to select. k The size of a sample (without replacement) to draw.

See also: test-statistic-distribution, integer-distribution, pdf, cdf, draw, support

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

Distribution clj protocol

The distribution protocol defines operations on probability distributions. Distributions may be univariate (defined over scalars) or multivariate (defined over vectors). Distributions may be discrete or continuous.

For a list of types that implement the protocol run (extenders Distribution). Implementations are provided for the various Clojure collection datatypes. See the example below for using the distribution methods on these types.

See also: pdf, cdf, draw, support

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

Examples: (support [1 3 4 2 1 3 4 2]) ; returns the set #{1 2 3 4} (draw [1 3 4 2 1 3 4 2]) ; returns a value from #{1 2 3 4} (pdf [2 1 2] 1) ; returns the value 1/3 (cdf [2 1 2 3] 2) ; returns the value 3/4

cdf clj

(cdf d v)

A function of the incanter.distribution.Distribution protocol.

Returns the value of the cumulative density function for the distribution d at support v.

See also: Distribution, pdf, draw, support

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

Examples: (cdf [2 1 2 3] 2) ; returns the value 3/4

draw clj

(draw d)

A function of the incanter.distribution.Distribution protocol.

Returns a randomly drawn value from the support of distribution d.

See also: Distribution, pdf, cdf, support

Examples: (draw [1 3 4 2 1 3 4 2]) ; returns a value from #{1 2 3 4}

mean clj

(mean d)

mean

pdf clj

(pdf d v)

A function of the incanter.distribution.Distribution protocol.

Returns the value of the probability density/mass function for the distribution d at support v.

See also: Distribution, cdf, draw, support

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

Examples: (pdf [2 1 2] 1) ; returns the value 1/3

support clj

(support d)

**** EXPERIMENTAL **** A function of the incanter.distribution.Distribution protocol.

Returns the support of the probability distribution d. For discrete distributions, the support is a set (i.e. #{1 2 3}). For continuous distributions, the support is a 2 element vector describing the range. For example, the uniform distribution over the unit interval would return the vector [0 1].

This function is marked as experimental to note that the output format might need to adapt to more complex support structures. For example, what would best describe a mixture of continuous distributions?

See also: Distribution, pdf, draw, support

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

Examples: (cdf [2 1 2 3] 2) ; returns the value 3/4

variance clj

(variance d)

variance

exponential-distribution clj

(exponential-distribution)

(exponential-distribution rate)

Returns a Exponential distribution that implements the incanter.distributions.Distribution protocol.

Arguments: rate (default 1)

See also: Distribution, pdf, cdf, draw, support

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

Example: (pdf (exponential-distribution 1/2) 2.0)

f-distribution clj

(f-distribution)

(f-distribution df1 df2)

Returns a F-distribution that implements the incanter.distributions.Distribution protocol.

Arguments: df1 (default 1) df2 (default 1)

See also: Distribution, pdf, cdf, draw, support

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

Example: (pdf (f-distribution 5 2) 1.0)

gamma-distribution clj

(gamma-distribution)

(gamma-distribution shape scale)

Returns a Gamma distribution that implements the incanter.distributions.Distribution protocol.

Arguments: shape (k) (default 1) scale (θ) (default 1)

See also: Distribution, pdf, cdf, draw, support

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

Example: (pdf (gamma-distribution 1 2) 10)

inf+ clj

inf- clj

integer-distribution clj

(integer-distribution)

(integer-distribution end)

(integer-distribution start end)

Create a uniform distribution over a set of integers over the (start, end] interval. An alternative method of creating a distribution would be to just use a sequence of integers (e.g. (draw (range 100000))). For large sequences, like the one in the example, using a sequence will be require realizing the entire sequence before a draw can be taken. This less efficient than computing random draws based on the end points of the distribution.

Arguments: start The lowest end of the interval, such that (>= (draw d) start) is always true. (Default 0) end The value at the upper end of the interval, such that (> end (draw d)) is always true. Note the strict inequality. (Default 1)

See also: pdf, cdf, draw, support

References: http://en.wikipedia.org/wiki/Uniform_distribution_(discrete)

Examples: (pdf (integer-distribution 0 10) 3) ; returns 1/10 for any value (draw (integer-distribution -5 5)) (draw (integer-distribution (bit-shift-left 2 1000))) ; probably a very large value

neg-binomial-distribution clj

(neg-binomial-distribution)

(neg-binomial-distribution size prob)

Returns a Negative binomial distribution that implements the incanter.distributions.Distribution protocol.

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

See also: Distribution, pdf, cdf, draw, support

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

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

next-raw clj

(next-raw rng)

Wrapper fn to pull the next raw value from an cern.jet.random.tdouble.engine.DoubleMersenneTwister prng and other engines.

normal-distribution clj

(normal-distribution)

(normal-distribution mean sd)

Returns a Normal distribution that implements the incanter.distributions.Distribution protocol.

Arguments: mean The mean of the distribution. One of two parameters that summarize the Normal distribution (default 0). sd The standard deviation of the distribution. The second parameter that describes the Normal (default 1).

See also: Distribution, pdf, cdf, draw, support

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

Example: (pdf (normal-distribution -2 (sqrt 0.5)) 1.96)

poisson-distribution clj

(poisson-distribution)

(poisson-distribution lambda)

Returns a Poisson distribution that implements the incanter.distributions.Distribution protocol.

Arguments: lambda (default 1)

See also: Distribution, pdf, cdf, draw, support

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

Example: (pdf (poisson-distribution 10) 5)

roulette-wheel clj

(roulette-wheel freqs)

Perform a roulette wheel selection given a list of frequencies

t-distribution clj

(t-distribution)

(t-distribution df)

Returns a Student-t distribution that implements the incanter.distributions.Distribution protocol.

Arguments: df (default 1)

See also: Distribution, pdf, cdf, draw, support

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

Example: (pdf (t-distribution 10) 1.2)

test-statistic-distribution clj

(test-statistic-distribution test-statistic n k)

Create a distribution of the test-statistic over the possible random samples of treatment units from the possible units.

There are two methods for generating the distribution. The first method is enumerating all possible randomizations and performing the test statistic on each. This gives the exact distribution, but is only feasible for small problems.

The second method uses a combination-distribution to sample for the space of possible treatment assignments and applies the test statistic the sampled randomizations. While the resulting distribution is not exact, it is tractable for larger problems.

The algorithm automatically chooses between the two methods by computing the number of possible randomizations and comparing it to test-statistic-iterations. If the exact distribution requires fewer than test-statistic-iterations the enumeration method is used. Otherwise, it draws test-statistic-iterations total samples for the simulated method.

By default, the algorithm uses parallel computation. This is controlled by the function test-statistic-map, which is bound to pmap by default. Bind it to map to use a single thread for computation.

Arguments: test-statistic A function that takes two vectors and summarizes the difference between them n The number of total units in the pool k The number of treatment units per sample

See also: combination-distribution, pdf, cdf, draw, support

References: http://en.wikipedia.org/wiki/Sampling_distribution http://en.wikipedia.org/wiki/Exact_test http://en.wikipedia.org/wiki/Randomization_test http://en.wikipedia.org/wiki/Lady_tasting_tea

uniform-distribution clj

(uniform-distribution)

(uniform-distribution min max)

Returns a Uniform distribution that implements the incanter.distributions.Distribution protocol.

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

See also: Distribution, pdf, cdf, draw, support

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

Example: (pdf (uniform-distribution 1.0 10.0) 5)

weibull-distribution clj

(weibull-distribution scale shape)

(weibull-distribution scale shape rng)