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Temporal GIS Algebra for Raster and Vector Data
Overview of the temporal algebra elements and concepts.
Temporal topology relations
equals A ------
B ------
during A ----
B ------
contains A ------
B ----
starts A ----
B ------
started A ------
B ----
finishs A ----
B ------
finished A ------
B ----
precedes A ----
B ----
follows A ----
B ----
overlapped A ------
B ------
overlaps A ------
B ------
over booth overlaps and overlapped
The relations must be read A is related to B, like
- A equals B
- A is during B
- A contains B
Topological relations must be specified in {} parenthesis
A {equals} B
Topological relations are always combined with other temporal operators in the {} parenthesis
Temporal operators
Temporal operators that work on time intervals and instances of maps in space time datasets:
Intersection &
Examples with time intervals a and b:
c <- a & b a ------ b ---- c ---- a ---- b ---- c -- a --- b --- c
Union |
Examples with time intervals a and b:
c <- a | b a ------ b ---- c ------ a ---- b ---- c ------ a --- b --- c ------ a --- b --- c
Disjoint Union +
Examples with time intervals a and b:
c <- a + b a ------ b ---- c ------ a ---- b ---- c ------ a --- b --- c ------ a --- b --- c -------
Left Reference =
Examples with time intervals a and b:
c <- a = b a ------ b ---- c ------ a ---- b ---- c ---- a --- b --- c --- a --- b --- c ---
The = operator ignores indexing operators for space time datasets: A[i]
Temporal selection operators to select parts of space time datasets:
Selection :
Inverse selection !:
Temporal operators have no precedence.
Examples:
- Select all maps from space time dataset A that have equal time intervals with space time dataset B and store them in space time datasets C.
C = A : B
C = A {:} B
C = A {=:} B
C = A {equals,:} B
C = A {equals,=:} B
- Select all maps from space time dataset A that have unequal time intervals with space time dataset B and store them in space time datasets C.
C = A !: B
C = A {!:} B
C = A {=!:} B
C = A {equals,!:} B
C = A {equals,=!:} B
- Select all maps from space time dataset A that are during time intervals of space time dataset B and store them in space time datasets C. Use the time interval/instances of A in C.
C = A {during,:} B
C = A {during,=:} B
- Select all maps from space time dataset A that are during time intervals of space time dataset B and store them in space time datasets C. The maps in C will have different time intervals as A or B using the union operator. Hence they are unions between time intervals of A and B.
C = A {during,|:} B
- Select all maps from space time dataset A that are during B and during C and during D. Store the resulting list of maps using the timer intervals/instances of A in E.
E = A {during,=:} B {during,=:} C {during,=:} D
Operator number of maps in interval
Number of maps #
Compute the number of maps from space time dataset B that are during the time intervals of maps from space time dataset A.
A{contains,#}B
This will return a list of integers (scalars) corresponding to the maps of A that contain maps from B.
Combination of temporal operators
-------------- | | : | !: | |---|- --|-----| | & | &: | &!: | | | | |: | |!: | | + | +: | +!: | | = | =: | =!: | --------------
Logical operators
== equal != not equal > greater than >= greater than or equal < less than <= less than or equal && and || or
Temporal functions
Note a and b can either be space time datasets or expressions.
if decision option
if(x, a) a if x not 0
if(x, a, b) a if x not 0, b otherwise
buff_t(a, size) Buffer stds a with granule ("1 month" or 5)
tshift(a, size) Shift stds a with granule ("1 month" or 5)
tsnap(a) Snap time instances and intervals of stds a
start_time() Start time as HH::MM:SS
start_date() Start date as yyyy-mm-DD
start_datetime() Start datetime as yyyy-mm-DD HH:MM:SS
end_time() End time as HH:MM:SS
end_date() End date as yyyy-mm-DD
end_datetime() End datetime as yyyy-mm-DD HH:MM
start_doy() Day of year (doy) from the start time [1 - 366]
start_dow() Day of week (dow) from the start time [1 - 7], the start of the week is Monday == 1
start_year() The year of the start time [0 - 9999]
start_month() The month of the start time [1 - 12]
start_week() Week of year of the start time [1 - 54]
start_day() Day of month from the start time [1 - 31]
start_hour() The hour of the start time [0 - 23]
start_minute() The minute of the start time [0 - 59]
start_second() The second of the start time [0 - 59]
end_doy() Day of year (doy) from the end time [1 - 366]
end_dow() Day of week (dow) from the end time [1 - 7], the start of the week is Monday == 1
end_year() The year of the end time [0 - 9999]
end_month() The month of the end time [1 - 12]
end_week() Week of year of the end time [1 - 54]
end_day() Day of month from the start time [1 - 31]
end_hour() The hour of the end time [0 - 23]
end_minute() The minute of the end time [0 - 59]
end_second() The second of the end time [0 - 59]
Examples:
Select all maps from space time dataset A that have equal time stamps with space time dataset B and C and are ealier that Jan. 1. 2005 and store them in space time dataset D.
D = if(start_date() < "2005-01-01"), A{equals,=:}B{equals,=:}C)
Temporal Vector Algebra
Boolean vector operations:
AND & Intersection (v.overlay operator=and) OR | Union (v.overlay operator=or) DISJOINT OR + Disjoint union (v.patch) XOR ^ Symmetric difference (v.overlay operator=xor) NOT ~ Complement (v.overlay operator=not)
Vector functions:
buff_p(a, size) Buffer the points of all maps from space time vector dataset a with size buff_l(a, size) Buffer the lines of all maps from space time vector dataset a with size buff_a(a, size) Buffer the areas of all maps from space time vector dataset a with size
Examples:
- Create an intersection between all maps of space time dataset A and B that have equal time stamps.
C = A & B
C = A {&} B
C = A {=&} B
C = A {equals,=&} B
- Question: Meeting of two animals that occur when the distance between two sea elephants is less than 10 meters? Space time vector dataset A and B contains the measured positions and time instances/intervals of two sea elephants. The granularity of the space time vector datasets is 1h. We create a new space time vector dataset C that contains the spatio-temporal intersections of the positions of the sea elephants.
C = buff_p(buff_t(A, "2 hours"), 5) {over||equals,&&} buff_p(buff_t(B, "2 hours"), 5)
Combinations of temporal and vector operators
--------------------------------------- | | : | !: | & | | | ^ | ~ | + | |---|----|-----|----|----|----|----|----| | & | &: | &!: | && | &| | &^ | &~ | &+ | | | | |: | |!: | |& | || | |^ | |~ | |+ | | + | +: | +!: | +& | +| | +^ | +~ | ++ | | = | =: | =!: | =& | =| | =^ | =~ | =+ | ---------------------------------------
Temporal Raster Algebra
Arithmetic Operators:
Symbol description precedence % modulus 1 / division 1 * multiplication 1 + addition 2 - subtraction 2
Logical operators:
Symbol description precedence == equal 3 != not equal 3 > greater than 3 >= greater than or equal 3 < less than 3 <= less than or equal 3 && and 4 || or 4
Raster functions:
abs(x) return absolute value of x foat(x) convert x to foating point if decision options: if(x) 1 if x not zero, 0 otherwise if(x,a) a if x not zero, 0 otherwise if(x,a,b) a if x not zero, b otherwise int(x) convert x to integer [ truncates ] log(x) natural log of x log(x,b) log of x base b max(x,y[,z...]) largest value of those listed median(x,y[,z...]) median value of those listed min(x,y[,z...]) smallest value of those listed round(x) round x to nearest integer sqrt(x) square root of x tan(x) tangent of x (x is in degrees) not(x) 1 if x is zero, 0 otherwise pow(x,y) x to the power y rand(a,b) random value x : a <= x < b round(x) round x to nearest integer sin(x) sine of x (x is in degrees) sqrt(x) square root of x tan(x) tangent of x (x is in degrees) xor(x,y) exclusive-or (XOR) of x and y isnull(x) check if x = NULL
Internal variables:
row() current row of moving window col() current col of moving window x() current x-coordinate of moving window y() current y-coordinate of moving window ewres() current east-west resolution nsres() current north-south resolution null() NULL value
Combination of temporal and arithmetical raster operators:
--------------------------------------- | | : | !: | % | / | * | - | + | |---|----|-----|----|----|----|----|----| | & | &: | &!: | &% | &/ | &* | &- | &+ | | | | |: | |!: | |% | |/ | |* | |- | |+ | | + | +: | +!: | +% | +/ | +* | +- | ++ | | = | =: | =!: | =% | =/ | =* | =- | =+ | ---------------------------------------
Combination of temporal and logical raster operators:
------------------------------------------------ | | : | !: | == | != | <= | >= | < | > | |---|----|-----|-----|-----|-----|-----|----|----| | & | &: | &!: | &== | &!= | &<= | &>= | &< | &> | | | | |: | |!: | |== | |!= | |<= | |>= | |< | |> | | + | +: | +!: | +== | +!= | +<= | +>= | +< | +> | | = | =: | =!: | === | =!= | =<= | =>= | =< | => | ------------------------------------------- ----
Note the combination of temporal and logical operators is only possible in reference time mode.
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