Formulas 5. Mathematical functions



abs (x)

absolute value

round (x)

nearest integer; round (1.5) = 2

floor (x)

round down: highest integer value not greater than x

ceiling (x)

round up: lowest integer value not less than x

sqrt (x)

square root: √x, x ≥ 0

min (x, ...)

the minimum of a series of numbers, e.g. min (7.2, 5, 3) = 5

max (x, ...)

the maximum of a series of numbers, e.g. max (7.2, 5, 3) = 7.2

imin (x, ...)

the location of the minimum, e.g. imin (7.2, 5, 3) = 2

imax (x, ...)

the location of the maximum, e.g. imax (7.2, 5, 3) = 1

sin (x)

sine

cos (x)

cosine

tan (x)

tangent

arcsin (x)

arcsine, 1 ≤ x ≤ 1

arccos (x)

arccosine, 1 ≤ x ≤ 1

arctan (x)

arctangent

arctan2 (y, x)

argument angle

sinc (x)

sinus cardinalis: sin (x) / x

sincpi (x)

sinc_{π}: sin (πx) / (πx)

exp (x)

exponentiation: e^{x}; same as e^x

ln (x)

natural logarithm, base e

log10 (x)

logarithm, base 10

log2 (x)

logarithm, base 2

sinh (x)

hyperbolic sine: (e^{x}  e^{x}) / 2

cosh (x)

hyperbolic cosine: (e^{x} + e^{x}) / 2

tanh (x)

hyperbolic tangent: sinh (x) / cosh (x)

arcsinh (x)

inverse hyperbolic sine: ln (x + √(1+x^{2}))

arccosh (x)

inverse hyperbolic cosine: ln (x + √(x^{2}−1))

arctanh (x)

inverse hyperbolic tangent

sigmoid (x)

R → (0,1): 1 / (1 + e^{−x}) or 1 − 1 / (1 + e^{x})

invSigmoid (x)

(0,1) → R: ln (x / (1 − x))

erf (x)

the error function: 2/√π _{0}∫^{x} exp(t^{2}) dt

erfc (x)

the complement of the error function: 1  erf (x)

randomUniform (min, max)

a uniform random real number between min (inclusive) and max (exclusive)

randomInteger (min, max)

a uniform random integer number between min and max (inclusive)

randomGauss (μ, σ)

a Gaussian random real number with mean μ and standard deviation σ

randomPoisson (mean)

a Poisson random real number

randomGamma (shape, rate)

a random number drawn from a Gamma distribution with shape parameter α and rate parameter β, which is defined as f(x; α, β) = (1 / Γ (α)) β^{α} x^{α−1} e^{−β x}, for x > 0, α > 0 and β > 0, following the method by Marsaglia & Tsang (2000)

random_initializeWithSeedUnsafelyButPredictably (seed)

can be used in a script to create a reproducible sequence of random numbers (warning: this exceptional situation will continue to exist throughout Praat until you call the following function)

random_initializeSafelyAndUnpredictably ()

undoes the exceptional situation caused by the previous function

lnGamma (x)

logarithm of the Γ function

gaussP (z)

the area under the Gaussian distribution between −∞ and z

gaussQ (z)

the area under the Gaussian distribution between z and +∞: the onetailed "statistical significance p" of a value that is z standard deviations away from the mean of a Gaussian distribution

invGaussQ (q)

the value of z for which
gaussQ
(z) = q

chiSquareP (
chiSquare
, df
)

the area under the χ^{2} distribution between 0 and chiSquare, for
df
degrees of freedom

chiSquareQ (
chiSquare
, df
)

the area under the χ^{2} distribution between
chiSquare
and +∞, for df degrees of freedom: the "statistical significance p" of the χ^{2} difference between two distributions in df+1 dimensions

invChiSquareQ (q, df)

the value of χ^{2} for which
chiSquareQ
(χ^{2}, df) = q

studentP (t, df)

the area under the student Tdistribution from ∞ to t

studentQ (t, df)

the area under the student Tdistribution from t to +∞

invStudentQ (q, df)

the value of t for which
studentQ
(t, df) = q

fisherP (f, df1, df2)

the area under Fisher's Fdistribution from 0 to f

fisherQ (f, df1, df2)

the area under Fisher's Fdistribution from f to +∞

invFisherQ (q, df1, df2)

the value of f for which
fisherQ
(f, df1, df2) = q

binomialP (p, k, n)

the probability that in n experiments, an event with probability p will occur at most k times

binomialQ
(p, k, n)

the probability that in n experiments, an event with probability p will occur at least k times; equals 1 
binomialP
(p, k  1, n)

invBinomialP (P, k, n)

the value of p for which
binomialP
(p, k, n) = P

invBinomialQ
(Q, k, n)

the value of p for which
binomialQ
(p, k, n) = Q

hertzToBark (x)

from acoustic frequency to Barkrate (perceptual spectral frequency; place on basilar membrane): 7 ln (x/650 + √(1 + (x/650)^{2}))

barkToHertz (x)

650 sinh (x / 7)

hertzToMel (x)

from acoustic frequency to perceptual pitch: 550 ln (1 + x / 550)

melToHertz (x)

550 (exp (x / 550)  1)

hertzToSemitones (x)

from acoustic frequency to a logarithmic musical scale, relative to 100 Hz: 12 ln (x / 100) / ln 2

semitonesToHertz (x)

100 exp (x ln 2 / 12)

erb (f)

the perceptual equivalent rectangular bandwidth (ERB) in hertz, for a specified acoustic frequency (also in hertz): 6.23·10^{6} f^{2} + 0.09339 f + 28.52

hertzToErb (x)

from acoustic frequency to ERBrate: 11.17 ln ((x + 312) / (x + 14680)) + 43

erbToHertz (x)

(14680 d  312) / (1  d) where d = exp ((x  43) / 11.17)

phonToDifferenceLimens
(x)

from perceptual loudness (intensity sensation) level in phon, to the number of intensity difference limens above threshold: 30 · ((61/60)^{ x} − 1).

differenceLimensToPhon
(x)

the inverse of the previous: ln (1 + x / 30) / ln (61 / 60).

beta (x, y)

besselI (n, x)

besselK (n, x)
For functions with arrays, see Scripting 5.7. Vectors and matrices.
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