Vector

Functions over vectors are divided into:

• Numerical vector functions operate on numerical vectors.
• Sequence functions operate on vectors whose elements can be of arbitrary type and need not be numerical.
• Vector aggregate functions operate on collections (bags) of vectors.

Notice that the select vector statement provides a powerful mechanism for constructing new vectors through queries in terms of functions.

Numerical vector functions​

The following infix operators over numeric vectors v and w, matrices a and b, and numbers lambda are defined:

operationdescriptionimplemented as function
v + welement-wise additionplus
v .+ w-"-plus
lambda + wadd number to elementsplus
v + lambda-"-plus
v - welement-wise subtractionsminus
v .- w-"-minus
-vnegate elementsuminus
v - lambdasubtract number from elementsminus
lambda - wsubtract elements from numberminus
v * wscalar producttimes
a * bmatrix producttimes
v * bscalar productstimes
a * wscalar productstimes
v .* welement-wise multiplicationselemtimes
lambda * wmultiply elements with numbertimes
v * lambda-"-times
v ./ welement-wise divisionelemdiv
v / lambdadivide elements with numberdiv
lambda / wdivide number with elementsdiv
v .^ lambdaelement-wise exponentelempower

Examples​

[1,2,3]+[4,5,6];[1,2,3].+[4,5,6]
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5+[1,2,3];[1,2,3]+5
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[[1],2,3]+[[4],5,6];[[1],2,3].+[[4],5,6]
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Vector subtraction:

Subtract vector elements:

[1,2,3]-[4,5,6];[1,2,3].-[4,5,6]
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Number and vector subtraction:

5-[1,2,3];[1,2,3]-5
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Vector division:

Number and vector division:

[1,2,3]/2;3/[1,2,3]
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Element division:

[1,2,3]./[4,5,6]
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Nested element division:

[[1],2,3]./[[4],5,6]
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Multiply vector elements:

Scalar product:

[1,2,3]*[4,5,6]
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Number and vector multiplication:

[1,2,3]*5;5*[1,2,3]
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Element multiplication:

[1,2,3].*[4,5,6]
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Nested element multiplication:

[[1],2,3].*[[4],5,[6]]
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Raise each element to a given exponent:

[1,2,3] .^ 2
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Round each element in a vector of numbers to two decimals:

roundto([3.14159,2.71828],2)
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Compute the max value of vector elements:

max([1,2,3]);max(["a","b","c"])
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Compute the min value of vector elements:

min([1,2,3]);min(["a","b","c"])
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Transforming vectors​

The function section(v,l,u),  returns a splice of a vector v, e.g.:

section([1,2,3,4,5],2,4)
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The function skip(v,n) returns the part of vector v starting at position n+1, e.g.:

skip(["a","b","c"],2)
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The function permute(v,indl) permutes the elements of a vector, e.g.:

permute([1,2,3,4,5],[2,4])
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Matrices​

The type Matrix is an alias for type Vector of Vector of Number.

Matrix multiplication is used when the function times (operator *) is applied on matrices. The shapes of a and b must match, e.g.:

[[1,2],[3,4],[5,6]] * [[7,8],[9,10]]
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The scalar product is computed when the function times (operator *) is applied on vectors and on combinations of vectors and matrices, e.g.two scalar multiplications:

[4,5,6] * [[1,2,3],[6,7,8]]
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The function transpose(m) trasposes the matrix m, e.g.:

transpose([[1,2,3],[6,7,8]])
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Vector aggregate functions​

Dimension-wise aggregates over bags of vectors can be computed using the function aggv().

Example:

   aggv((select [i, i + 10]           from Integer i          where i in range(1, 10)), thefunction('mean'))
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note

Notice that the function mean() is overloaded and thefunction() has to be use to get the generic mean() function.

Each dimension in a bag of vector of number can be normalized using one of the normalization functions meansub(), zscore(), or maxmin():

   meansub(Bag of Vector of Number b) -> Bag of Vector of Number   zscore(Bag of Vector of Number b) -> Bag of Vector of Number   maxmin(Bag of Vector of Number b) -> Bag of Vector of Number

meansub() transforms each dimension to a N(0, s) distribution (assuming that the dimension was N(u, s) distributed) by subtracting the mean u of each dimension.

zscore() transforms each dimension to a N(0, 1) distribution by also dividing by the standard deviation of each dimension.

maxmin() transforms each dimension to be on the [0, 1] interval by applying the transformation (w - min) ./ (max - min) to each vector w in bag b where max and min are computed using aggv(b, #' max') and aggv(b, #'min') respectively.

Example:

   meansub((select [i, i/2 + 10]              from Integer i             where i in range(1, 5)))
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Functions​

abs(Vector v)->Vector of Number

The absolute values in vector v

aggv(Bag of Vector bv,Function fn)->Vector of Number

Apply aggregate function fn on each position of the vectors in bv

argmax(Vector v)->Integer

Index of the first largest element in v

argmin(Vector v)->Integer

Index of the first smallest element in v

concat(Vector v,Vector w)->Vector

Concatenate vectors v and w

dim(Vector v)->Integer

Size of vector v

div(Number lambda,Vector w)->Vector of Number

Divide lambda with each element in vector w, lambda / w

div(Vector v,Number lambda)->Vector of Number

Divide elements in vector v with lambda, v / lambda

elemdiv(Vector v,Vector w)->Vector of Number

Divide elements in vector v with elements in w, v ./ w

elempower(Vector v,Number exp)->Vector of Number

Compute power(e,exp) for each element e in vector v, v .^ exp

elemtimes(Vector v,Vector w)->Vector of Number

Multiply vectrors v and w element by element, v .* w

euclid(Vector v,Vector w)->Real

Euclidean distance between vectors v and w

fft(Vector v)->Vector of Complex

Full FFT over a vector

geodist(Vector v,Vector w)->Real

The surface distance in meters between geographic position vectors [latitude, longitude], v and w

ifft(Vector v)->Vector of Real

Inverse full FFT

integer_vector(Vector of Real v)->Vector of Integer

irfft(Vector v)->Vector of Real

manhattan(Vector of Number v,Vector of Number w)->Number

Manhattan distance between vectors v and w

max(Vector v)->Object

The largest element in vector v

maxmin(Bag of Vector of Number b)->Vector of Number

Transform to [0, 1]: Subtract min, divide by (max - min)

maxnorm(Vector of Number v,Vector of Number w)->Real

Maxnorm distance between vectors v and w

mean(Vector v)->Real

Average of vector of numbers v

meansub(Bag of Vector of Number b)->Vector of Number

Transform to N(0, s): Subtract mean(v)

min(Vector v)->Object

The smallest element in vector v

minkowski(Vector of Number v,Vector of Number w,Number r)->Real

Minkowski distance of degree r between vectors v and w

minus(Vector v,Vector w)->Vector of Number r

Subtract elements in vectors v and w, v .- w

minus(Vector v,Number lambda)->Vector of Number

Subtract lambda from each element in vector v, v - lambda

minus(Number lambda,Vector w)->Vector of Number

Subtract lambda from each element in vector w, lambda - w

not_null(Vector v)->Boolean

Are all elements in vector v not null?

numvector(Vector x)->Vector of Number

Cast x to vector of numbers

ones(Number dim)->Vector of Integer

Construct vector of dim 1:s

permute(Vector v,Vector of Integer indl)->Vector

Reorder vector v on index positions in indl

plus(Vector v,Vector w)->Vector of Number r

Add elements in vectors v and w, v .+ w

plus(Number lambda,Vector w)->Vector of Number

Add lambda to each element in vector w, lambda + w

plus(Vector v,Number lambda)->Vector of Number r

Add lambda to each element in vector v, v + lambda

rfft(Vector v)->Vector of Real

RFFT over vectors of numbers

section(Vector v,Number l,Number u)->Vector r

The subvector of vector v starting at position p and ending at u

setf(Vector v,Integer i,Object o)->Boolean

skip(Vector v,Number n)->Vector

Skip first n elements in vector v

stdev(Vector v)->Real

Standard deviation of vector of numbers v

sum(Vector v)->Number

The sum of the numbers in vector v

times(Vector v,Vector w)->Number

Scalar product of vectors v and w, v * W

times(Vector v,Number lambda)->Vector of Number

Multiply elements in vector v with lambda, v * lambda

times(Number lambda,Vector w)->Vector of Number

Multiply lambda with elements in w, lambda * w

times(Vector of Number v,Matrix m)->Vector of Number

Vector-matrix multiplication, v * m

times(Matrix a,Matrix b)->Matrix _v1

Matrix multiplication, a * b

times(Matrix m,Vector of Number w)->Vector of Number

Matrix-vector multiplication, m * v

transpose(Matrix m)->Matrix _v1

Transpose matrix m

uminus(Vector v)->Vector of Number

Negate numbers in vector v, -v

vdiff(Vector v,Vector w)->Vector

Elements in vector v that are not in vector w

vectorize(Bag b)->Vector v

vmean(Bag of Vector of Number bv)->Vector of Number

Mean vector for a given bag of vectors bv

vref(Vector v,Number i)->Object

Value of element i in vector v, same as v[i]

whennull(Vector v,Object dflt)->Vector

Value 'dflt' for nulls in 'v'

whenzero(Vector v,Object dflt)->Vector

Value 'dflt' for zeros in 'v'

zeros(Number dim)->Vector of Integer

Construct vector of dim 0:s

zscore(Bag of Vector of Number b)->Bag of Vector of Number

Transform elements in the vectors in b into normal distributions centered around zero