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Temporal GIS Algebra for Raster and Vector Data
Temporal Algebra
The idea is to implement a temporal algebra to process space time datasets, the new spatio-temporal datatypes in GRASS GIS 7. Space time datasets represent time series of raster, 3D raster and vector maps. The GRASS GIS temporal framework implements a snapshot approach, so that time and space are separated from each other. Hence, space time datasets are not truly 4 dimensional datatypes. Time and space can only be processes separately from each other. The spatial capabilities of GRASS GIS are used to perform the spatial processing of the time stamped maps that are registered in the space time datasets. Hence the modules r.mapcalc for raster processing and v.buffer, v.over , v.patch for vector map processing are used. The temporal processing is provided by the GRASS GIS temporal framework that supports time intervals and time instances. Mixing time instance and time intervals as well as gaps, overlapping or inclusion of intervals and instances is possible. Hence this framework allows an arbitrary layout of the time dimension,
The only way to process several space time datasets with arbitrary temporal layout is to use their temporal relationships. Or more precisely, the temporal relationships of the time stamped maps that are registered in the space time datasets. The spatial operations between maps (sum, union, ...) can be performed after the temporal related maps are identified. This concept leads the design of the temporal algebra. The algebra must provide the functionality to define spatio-temporal operators that process time and space in a single expression.
It should be possible to select maps based on their temporal relations. It must be possible to temporally shift the time stamps, to create temporal buffer and to snap time instances to create a valid temporal topology. Hence many operations that are common in spatial processing should have an equivalent in the temporal processing.
Temporal selection
Here an example of the temporal algebra that simply selects parts of a space time dataset without the processing raster or vector data:
We define a selection operator : that by default selects parts of a space time dataset that are temporally equal to parts of a second, hence the following expression
C = A : B
means: Select all parts of space time dataset A that are equal to B and store it in space time dataset C. The parts are in this case time stamped maps. In addition we define the inverse selection operator !:, hence the following expression
C = A !: B
means: select all parts of space time time dataset A that are not equal to B and store it in space time dataset (STDS) C.
We defined the {topological relations, selection operator} operator to perform selection with different temporal topology relations. Examples:
C = A {equals,:} B is exactly the same as C = A : B
C = A {equals,!:} B is exactly the same as C = A !: B
We can now define arbitrary topological relations using logical OR to connect them:
C = A {equals||during||overlaps,:} B
Select all parts of A that are equal B, during B or overlaps B.
In addition we define a temporal functions:
Temporal buffer
Buffer time instances or time intervals:
buff_t(STDS, "time interval")
Example:
C = buff_t(A, "1 day")
Buffer all maps of STDS A with a temporal buffer of one day.
Temporal shift
Temporally shift all maps in a space time dataset by an interval:
tshift(STDS, "time interval")
Example:
tshift(A, "2 months")
Temporally shift all maps of space time dataset A by two months.
Temporally snap
Temporally snap time instances to its temporal nearest neighbor in the future and close gaps between time intervals.
tsnap(STDS)
The hash operator
It is of important interest how many maps are located in the time interval of a single map when processing two space time datasets. For this reason the hash operator # was introduced. In addition to the hash operator the temporal relation can be specified that should be analysed.
Example:
C = if(equal, A{contains,#}B > 2, A{contains,:}B)
This expression selects all maps from A that temporally contains at least 2 maps from B and stores them in space time dataset C. The leading equal statement in the if condition specifies the temporal relation between the if and then part of the expression. This is very important, so we do not need to specify a time reference for temporal processing.
Temporal Operators
The temporal algebra defines temporal operators that can be combined later with the spatial operators to perform spatio-temporal operations. The temporal operators process the time instances and intervals of temporal related maps.
AND & Intersection OR | Union DISJOINT OR + Disjoint union LEFT REFERENCE = Use the time stamp of the left space time vector dataset
For example we can compute the intersection, union or disjoint union from time stamps of maps that temporally overlap, or we can just keep the time stamp of the left STDS.
Left Operator Right
A {equal,:} B
Temporal Vector Algebra
The temporal vector algebra includes all functions from the temporal algebra and adds spatial boolean and buffer operations that can be performed on temporal related vector maps that are registered in space time vector datasets.
Spatio-Temporal Operators
These are the spatial operators that are supported:
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)
We combine the temporal topology relations, the temporal operators and the spatial operators to create spatio-temporal operators:
{"list of temporal relations", "temporal operator" "spatial operator"}
The spatial operator can be used stand-alone. In this case the temporal topology relation "equal" and the temporal operator "left reference =" is assumed and used as default. This allows the convenient use of the spatial operators in case of space time vector datasets with equal time stamps.
Example: A spatio intersection between maps of STVDS A and B that are temporally equal, resulting spatial intersection maps are stored in C with time stamps from A:
C = A & B
A spatio-temporal intersection between maps of STVDS A and B that temporally overlap, resulting spatial intersection maps are stored in C with new time stamps resulting from temporal intersection:
C = A {overlap,&&} B
Perform a spatial union and a temporal disjoint union between all maps of STVDS A and B that are temporally equal or overlap and are later than 1990, resulting spatial union maps are stored in C with new time stamps resulting from temporal disjoint union:
C = if(start_year() > 1990, A {equal,overlap,+|} B)
The temporal vector algebra provides point, line and area buffer functions in addition to the operators.
Overview of all temporal algebra elements
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 {} parentheses
A {equals} B
Topological relations are always combined with other temporal operators in the {} parentheses
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 {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 {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.
C = A {during,:} B
- Select all maps from space time dataset A that are during B and during C and during D.
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.
Logical operators
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
Combination of temporal and logical operators:
------------------------------------------------- | | && | || | == | != | <= | >= | < | > | |---|-----|-----|-----|-----|-----|-----|----|----| | & | &&& | &|| | &== | &!= | &<= | &>= | &< | &> | | | | |&& | ||| | |== | |!= | |<= | |>= | |< | |> | | + | +&& | +|| | +== | +!= | +<= | +>= | +< | +> | | = | =&& | =|| | === | =!= | =<= | =>= | =< | => | -------------------------------------------- ----
Temporal functions
Note a and b can either be space time datasets or expressions.
if decision option
if(if, then, else)
if(x, a) a if x not 0; temporal topological relation between if and then is equal
if(x, a, b) a if x not 0, b otherwise; temporal topological relation between if, then and else is equal
if(topo, x, a) a if x not 0; temporal topological relation between if and then is explicit specified by topo
if(topo, x, a, b) a if x not 0, b otherwise; temporal topological relation between if, then and else is explicit specified by topo
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
td(a) Returns a list of time 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 : B : 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
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:
--------------------------------------- | | : | !: | % | / | * | - | + | |---|----|-----|----|----|----|----|----| | & | &: | &!: | &% | &/ | &* | &- | &+ | | | | |: | |!: | |% | |/ | |* | |- | |+ | | + | +: | +!: | +% | +/ | +* | +- | ++ | | = | =: | =!: | =% | =/ | =* | =- | =+ | ---------------------------------------
Examples: Create the sum all maps from STRDS A and B that have equal time stamps and store the new maps in STRDS C.
C = A + B
Same expression with explicit temporal topology and temporal interval processing operators.
C = A {equal,=+} B
Select all cells from STRDS B with equal relations to STRDS A if the cells of A are in the range of [100.0, 1600] of time intervals that have more then 30 days
C = if(equal, (A > 100 && A < 1600) {equal,&&} td(A) > 30, B)
