Variable Transformation
Utility functions for variable transformation.
morph(b, r, p, center) morph.identity()
b |
Positive real number. May be missing. |
r |
Non-negative real number. May be missing. If |
p |
Real number strictly greater than 2. May be missing. If
|
center |
Real scalar or vector. May be missing. If
|
The morph function facilitates using variable transformations
by providing functions to (using X for the original random
variable with the pdf f.X, and Y for the transformed
random variable with the pdf f.Y):
Calculate the log unnormalized probability density for Y induced by the transformation.
Transform an arbitrary function of X to a function of Y.
Transform values of X to values of Y.
Transform values of Y to values of X (the inverse transformation).
for a select few transformations.
morph.identity implements the identity transformation,
Y=X.
The parameters r, p, b and center specify the
transformation function. In all cases, center gives the center
of the transformation, which is the value c in the equation
Y = f(X - c).
If no parameters are specified, the identity transformation, Y=X, is used.
The parameters r, p and b specify a function
g, which is a monotonically increasing bijection from the
non-negative reals to the non-negative reals. Then
f(X) = g(|X|) * X / |X|
where |X| represents the Euclidean norm of the vector X. The inverse function is given by
f^{-1}(Y) = g^{-1}(|Y|) * Y / |Y|.
The parameters r and p are used to define the function
g1(x) = x + (x-r)^p * I(x > r)
where I(•) is the indicator
function. We require that r is non-negative and p is
strictly greater than 2. The parameter b is used to define the
function
g2(x) = (exp(b * x) - exp(1) / 3) * I(x > 1 / b) + (x^3 * b^3 exp(1) / 6 + x * b * exp(1) / 2) * I(x <= 1 / b).
We require that b is positive.
The parameters r, p and b specify f^{-1} in
the following manner:
If one or both of r and p is specified, and b
is not specified, then
f^{-1}(X) = g1(|X|) * X / |X|.
If only
r is specified, p = 3 is used. If only p is specified,
r = 0 is used.
If only b is specified, then
f^{-1}(X) = g2(|X|) * X / |X|.
If one or both of r and p is specified, and b is
also specified, then
f^{-1}(X) = g2(g1(|X|)) * X / |X|.
a list containing the functions
outfun(f), a function that operates on functions.
outfun(f) returns the function function(state, ...)
f(inverse(state), ...).
inverse, the inverse transformation function.
transform, the transformation function.
lud, a function that operates on functions. As input,
lud takes a function that calculates a log unnormalized
probability density, and returns a function that calculates the
log unnormalized density by transforming a random variable using the
transform function. lud(f) = function(state, ...)
f(inverse(state), ...) + log.jacobian(state), where
log.jacobian represents the function that calculate the log
Jacobian of the transformation. log.jacobian is not returned.
The equations for the returned transform function (see below)
do not have a general analytical solution when p is not equal
to 3. This implementation uses numerical approximation to calculate
transform when p is not equal to 3. If computation
speed is a factor, it is advisable to use p=3. This is not a
factor when using morph.metrop, as transform is
only called once during setup, and not at all while running the Markov chain.
# use an exponential transformation, centered at 100. b1 <- morph(b=1, center=100) # original log unnormalized density is from a t distribution with 3 # degrees of freedom, centered at 100. lud.transformed <- b1$lud(function(x) dt(x - 100, df=3, log=TRUE)) d.transformed <- Vectorize(function(x) exp(lud.transformed(x))) ## Not run: curve(d.transformed, from=-3, to=3, ylab="Induced Density") ## End(Not run)
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