General syntax
odin
is a “Domain
Specific Language”; a mini-language that solves a specific problem;
in this case representing systems of difference or differential
equations. It is syntactically R (i.e., it can be parsed with R’s
parser) but it is not itself R. Only a subset of expressions and syntax
are supported.
Every line in odin
code must be an
assignment or a relationship (there
are some minor exceptions below).
An assignment looks like
a <- expression
while a relationship looks like
b ~ Distribution(...)
where b
will be an entry from data (introduced by
data()
), Distribution
will be a monty
distribution function (see below), and ...
will be
arguments to this function, which might come from data or from model
variables. See vignette("fitting")
for a high-level
introduction to this interface.
odin2
supports many functions that you’d expect to see
for constructing dynamical models. These include most common
mathematical operations and some that are quite obscure. The support for
stochastic models and comparison to data comes from
monty
.
Basic operations
-
+
– Plus: Both infix (a + b
) and prefix (+a
) versions supported (e.g.,1 + 2
→3
) -
-
– Minus: Both infix (a - b
) and prefix (-a
) versions supported (e.g.,10 - 1
→9
) -
*
– Multiply: Multiply two numbers together (e.g.,2 * 6
→12
) -
/
– Divide: Divide two numbers (e.g.,12 / 6
→2
) -
^
– Power: Raise the first number to the power of the second. Either number may be a floating point number (e.g.,2.3 ^ 1.2
→2.716898
) -
(
– Parenthesis: Group expressions together (e.g.,(1 + 5) * 2
→12
)
Conditionals
You can only use conditionals with if
as an inline
expression, for example
a <- if (9 > 10) 1 else 2
would result in a
being assigned as 2
(this
works in R normally too!).
Because general flow control is not supported, you cannot write:
if (9 > 10) {
a <- 1
} else {
a <- 2
}
Operators
A number of logical-returning operators exist, primarily to support
the if
statement; all the usual comparison operators exist
(though not vectorised |
or &
).
-
>
– Greater than (e.g.,1 > 2
→FALSE
) -
<
– Less than (e.g.,1 < 2
→TRUE
) -
>=
– Greater than or equal to (e.g.,1 >= 2
→FALSE
) -
<=
– Less than or equal to (e.g.,1 <= 2
→TRUE
) -
==
– Is exactly equal to (e.g.,1 == 1
→TRUE
) -
!=
– Is not exactly equal to (e.g.,1 != 2
→TRUE
) -
&&
– Boolean AND (e.g.,(1 == 1) && (2 > 1)
→TRUE
) -
||
– Boolean OR (e.g.,(1 == 1) && (2 > 1)
→TRUE
)
Be wary of strict equality with ==
or !=
as
numbers may be floating point numbers, which have some surprising
properties for the uninitiated, for example
sqrt(3)^2 == 3
## [1] FALSE
Mathematical functions and constants
-
%%
– Modulo: Finds the remainder after division of one number by another (e.g.,123 %% 100
→23
) -
%/%
– Integer divide: Different to floating point division, effectively the full number of times one number divides into another (e.g.,20 %/% 7
→2
) -
abs
– Absolute value (e.g.,abs(-1)
→1
) -
sign
– Sign function: Returns the sign of its argument as either -1, 0 or 1, which may be useful for multiplying by another argument (e.g.,sign(-100)
→-1
) -
round
– Round a number (e.g.,round(1.23)
→1
;round(1.23, 1)
→1.2
) -
floor
– Floor of a number: Largest integer not greater than the provided number (e.g.,floor(6.5)
→6
) -
ceiling
– Ceiling of a number: Smallest integer not less than the provided number (e.g.,ceiling(6.5)
→7
) -
trunc
– Truncate a number: Round a number towards zero -
max
– Maximum: Returns maximum of two numbers (e.g.,max(2, 6)
→6
) -
min
– Minimum (e.g.,min(2, 6)
→2
) -
exp
– Exponential function (e.g.,exp(1)
→2.718282
) -
expm1
– Computes exp(x) - 1 accurately for small |x| (e.g.,exp(1)
→1.718282
) -
log
– Logarithmic function in base e (e.g.,log(1)
→0
) -
log2
– Logarithmic function in base 2 (e.g.,log2(1024)
→10
) -
log10
– Logarithmic function in base 10 (e.g.,log10(1000)
→3
) -
log1p
– Computes log(x + 1) accurately for small |x| (e.g.,log1p(1)
→0.6931472
) -
sqrt
– Square root function (e.g.,sqrt(4)
→2
) -
beta
– Beta function (e.g.,beta(3, 5)
→0.00952381
) -
lbeta
– Log beta function (e.g.,lbeta(3, 5)
→-4.65396
) -
choose
– Binomial coefficients (e.g.,choose(60, 3)
→34220
) -
lchoose
– Log binomial coefficients (e.g.,choose(60, 3)
→10.44057
) -
gamma
– Gamma function (e.g.,gamma(10)
→362880
) -
lgamma
– Log gamma function (e.g.,lgamma(10)
→12.80183
)
The exact behaviour for %%
and %/%
for
floating point numbers and negative numbers is complicated - please see
?Arithmetic
. The rules for operators in odin
try to follow those in R as closely as possible.
The constant pi
can be used, along with all the usual
trig functions:
-
cos
– Cosine function -
sin
– Sine function -
tan
– Tangent function -
acos
– Arc-cosine function -
asin
– Arc-sin function -
atan
– Arc-tangent function -
atan2
– Two-argument arc-tangent function -
cosh
– Hyperbolic cosine function -
sinh
– Hyperbolic sine function -
tanh
– Hyperbolic tangent function -
acosh
– Hyperbolic arc-cosine function -
asinh
– Hyperbolic arc-sine function -
atanh
– Hyperbolic arc-tangent function
Arrays
Use of arrays implies a “for
-loop” in the generated
code. For example, you might write a vectorised version of the logistic map
as:
update(y[]) <- r[i] * y[i] * (1 - y[i])
which will expand to code equivalent to
for (i in 1:length(y)) {
y_next[i] <- r[i] * y[i] * (1 - y[i])
}
The loop extent here (over the entire range of y
) is
determined by the left hand side expression (y[]
). This
enables use of i
on the right hand side to index as loop
progresses.
The indices on the right hand side can be i
,
j
, k
, l
, i5
,
i6
, i7
or i8
(odin supports
arrays up to 8 dimensions: do let us know if you need more for some
reason).
Arrays can have more than one dimension, for example the expression:
ay[, ] <- a[i, j] * y[j]
involves loops over two dimensions because we loop over the whole
extent of ay
which is a matrix. This is roughly equivalent
to:
Note here that y
is accessed using j
, even
though it is only a vector. This is because the loop extents are
generated by the left hand side.
Array size
Every array variable requires a dim()
assignment. For example, in the above examples we might
have:
where y
is defined to be a 1-dimensional array of length
10 and ay
is a matrix (2-dimensional array) with
nr
rows and nc
columns. The extents of arrays
must be determined at the first system initialisation,
and this is checked during parse.
If you have different arrays with the same dimensions, you can also
use dim()
on the right-hand side, to copy from an array you
have set the dimensions of elsewhere. For example:-
You can also combine arrays on the left-hand side to group arrays with the same dimensions together. These 5 arrays will all have the same dimensions:-
Special functions for arrays
We provide several functions for retrieving dimensions from an array
-
length
– Length: get the full length of an array. For a single dimensional array this is obvious, for a multidimensional array it is the product over all dimensions. -
nrow
– Number of rows: number of rows in a matrix or number of elements in the first dimension of a multidimensional array -
ncol
– Number of columns: number of columns in a matrix or number of elements in the second dimension of a multidimensional array
We do not currently offer a function for accessing the size of higher
dimensions, please let us know if this is an issue (see
vignette("migrating")
)
Frequently, you will want to take a sum over an array, or part of an
array, using sum
. To sum over all elements of an array, use
sum()
with the name of the array you would like to sum
over:
If m
is a matrix you can compute the sums over the
second column by writing:
m1_tot <- sum(m[, 2])
This partial sum approach is frequently used within implicit loops:
m_col_totals[] <- sum(m[, i])
You can use this approach to compute a matrix-vector product :
ax_tmp[, ] <- a[i, j] * x[j]
ax[] <- sum(a[, i])
Distribution functions
We support distribution functions in two places:
First, for discrete-time models we support sampling from a distribution at each time step. For example:
a <- Normal(0, 1)
will assign a
to a draw from the standard normal
distribution. You cannot use these functions from continuous time
models. You cannot sample from stochastic functions in a continuous time
(ODE) model.
Second, for comparison to data, for example:
a ~ Normal(0, 1)
will add a log likelihood term looking up the log density of the data
element a
against a standard normal distribution. This form
can be used in both discrete-time and continuous-time models. For more
information, see vignette("fitting")
.
Some distributions have several versions; these are distinguished by the arguments to the functions. For example:
draw from a Gamma distribution with a shape of 2 and a rate of 0.1, while
draw from a Gamma distribution with a shape of 2 and a scale of 10.
The currently supported distributions are (alphabetically):
-
Beta
– the beta distribution with parametersa
andb
(vsrbeta
’sshape1
andshape2
) -
BetaBinomial
– the beta-binomial distribution with two forms:-
size
,prob
(the mean probability of success),rho
(dispersion parameter) (default) -
size
,a
,b
-
-
Binomial
– the binomial distribution with parameterssize
andprob
-
Cauchy
– the Cauchy distribution with parameterslocation
andscale
. Note that as the Cauchy distribution does not have a defined mean, you can not run a model with Cauchy draws in deterministic mode. -
Exponential
– the exponential distribution with two forms:-
rate
(default); this is the same parameterisation asrexp
-
mean
which is the inverse of rate. NOTE: we may change this toscale
soon
-
-
Gamma
– the gamma distribution with two forms:-
shape
,rate
(default) -
shape
,scale
-
-
Hypergeometric
– the hypergeometric distribution with parametersm
(number of white balls),n
(number of black balls), andk
(number of samples), and we return the number of white balls. We may support alternative parametrisations of this distribution in future (this version is the same parametrisation asrhyper
) -
LogNormal
– the log-normal distribution with parametersmeanlog
andsdlog
, the mean and standard deviation of the distribution on the log scale -
NegativeBinomial
– the negative binomial distribution with two forms:-
size
,prob
(default) -
size
,mu
(the mean)
-
-
Normal
– the normal distribution with parametersmean
,sd
-
Poisson
– the Poisson distribution with parameterlambda
(the mean) -
TruncatedNormal
– the truncated normal distribution with parametersmean
,sd
,min
andmax
. For a one-sided truncated normal distribution, you can setmin = -Inf
ormax = Inf
. Note thatmean
andsd
are not the mean and standard deviation of the truncated normal distribution, but are the mean and standard deviation of the normal distribution that has been truncated. -
Uniform
– the uniform distribution with parametersmin
andmax
-
Weibull
– the Weibull distribution with parametersshape
andscale
In the future, we plan support for additional distributions, please
let us know if we are missing any that you need. The support for these
functions comes from monty
and we will link here to the
docs in that package once they exist for additional details.
Semantics of random number draws
Stochastic functions are called for each element in an array they are assigned to, at each time. So here:
x[] <- Normal(0, 1)
x
will be filled with each element having a different
draw from a standard normal. In contrast, in:
a <- Normal(0, 1)
x[] <- a
x
will be a vector where every element is the same, the
result of a single draw from a standard normal.
Special functions
There are some special odin functions that may appear on the right hand side and which must be the only function used in the expression.
Parameters
The function parameter()
introduces a “parameter”;
something that you will initialise your system with, or update after
initialisation. This is the primary mechanism for controlling how
systems behave. The parameter
function accepts
arguments:
-
default
: The first argument, typically unnamed, holds the default value if none is provided at initialisation -
constant
: Logical, indicating if the parameter cannot be changed after being initially set. This must beTRUE
for things leading into array extents -
differentiate
: Logical, indicating if the likelihood (from comparison to data) should be differentiated with respect to this parameter. -
type
: The data type for the variable, as a string. Must be one ofreal
(the default),integer
orlogical
. -
rank
: The number of dimensions of the parameter. This is only used when assigning todim()
(see below)
For example:
a <- parameter()
Or:
n <- parameter(12, constant = TRUE, type = "integer")
There are some interactions among the differentiate
argument combined with constant
or type
:
- If a parameter is differentiable (
differentiate = TRUE
) it may not be constant! - If any parameter is differentiable, the default
value for
constant
isTRUE
, and all non-constant parameters must be differentiable. Otherwise the default value forconstant
isFALSE
- Only parameters with
type = "real"
can be used withdifferentiate = TRUE
If your parameter has its dimensions determined by the size data you give it, you need to write it slightly specially:
a <- parameter()
dim(a) <- parameter(rank = 2)
The rank
argument here is required because otherwise we
have no information on the number of dimensions that a
has;
here by saying rank = 2
we specify that a
is a
matrix. We might change this interface in future, the implementation
here fairly closely matches that in odin1.
Data
If your model compares to data (i.e., it uses ~
) then it
needs data. These are specified similarly to parameter()
,
though at present no arguments are supported.
d <- data()
Unlike parameter()
, you will have a series of data
elements, each corresponding to an observation at a different point in
time in a time series. See vignettes("fitting")
for more
details.
Interpolation
You can create variables that interpolate against time. This is useful in a few contexts, for example:
- A piecewise constant function that represents the level of some external factor
- A smooth function that represents an environmental input
Currently all interpolation functions are scalar valued meaning that at each time a single output is produced.
The usage is:
interpolate(time, value, mode)
-
time
is a vector representing time values -
value
is a vector representing the series you would like to interpolate, the same length astime
-
mode
is a string, one ofconstant
,linear
orspline
Once complete we will show usage of interpolating functions in their own vignette.
Restricted names
You cannot assign to a name that is reserved in:
-
C++ -
includes useful words such as
new
andswitch
-
C - largely a
subset of C++’s words, but also excludes
restrict
-
JavaScript
- includes useful words such as
default
andexport
- A few words restricted by odin itself:
time
,dt
,parameter
,data
,interpolate
,delay
,initial
,deriv
,update
,output
,dim
,config
,state
,state_next
,state_deriv
,shared
,internal
,pi
. We may reduce this list in future.
In addition, odin restricts a few prefixes; a name cannot start with
odin_
, interpolate_
, delay_
or
adjoint_
.