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:
| operation | description | implemented as function |
|---|---|---|
v + w | element-wise addition | plus |
v .+ w | -"- | plus |
lambda + w | add number to elements | plus |
v + lambda | -"- | plus |
v - w | element-wise subtractions | minus |
v .- w | -"- | minus |
-v | negate elements | uminus |
v - lambda | subtract number from elements | minus |
lambda - w | subtract elements from number | minus |
v * w | scalar product | times |
a * b | matrix product | times |
v * b | scalar products | times |
a * w | scalar products | times |
v .* w | element-wise multiplications | elemtimes |
lambda * w | multiply elements with number | times |
v * lambda | -"- | times |
v ./ w | element-wise division | elemdiv |
v / lambda | divide elements with number | div |
lambda / w | divide number with elements | div |
v .^ lambda | element-wise exponent | elempower |
Examples
Vector addition:
Add vector elements:
[1,2,3]+[4,5,6];
[1,2,3].+[4,5,6]
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Number and vector addition:
5+[1,2,3];
[1,2,3]+5
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Add nested vector elements:
[[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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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