RateDistortion (version 1.01)

MaximumCost: Compute the maximum cost for an information source and cost function.

Description

This function simply returns the maximum possible cost for a given cost function, over the space of the input and output alphabets. This function is useful for determining an appropriate scaling constant for the cost function.

Usage

MaximumCost(x, y, rho.fn, ...)

Arguments

x

The input or source alphabet for the channel. Either a vector or a matrix. For a vector, each element corresponds to one symbol in the input alphabet. For a matrix, each row corresponds to one symbol.

y

The output alphabet for the channel. This can either be a vector or a matrix.

rho.fn

The cost function for the channel, defining rho.fn(x, y) over the domain of x and y. This function should accept vectorized arguments.

...

Optional arguments that are passed to the cost function.

Value

The function returns a single numeric value that corresponds to the maximum cost over the space of the channel input and output alphabets.

Details

When calling the routine BlahutAlgorithm, it is possible that cost functions with a maximum cost substantially larger than 1 may lead to numerical under- or over-flow issues. This can be resolved by normalizing the cost function so that its maximum value is one. Note that multiplying a cost funciton by a positive scalar does not alter the resulting optimal channel. This scaling constant can be passed to the routine BlahutAlgorithm as the argument rho.scale.

See Also

BlahutAlgorithm

Examples

# Define a discretized Gaussian information source
x <- seq(from = -10, to = 10, length.out = 100)
Px <- dnorm(x, mean = 0, sd = 3)
Px <- Px / sum(Px) # Ensure that probability sums to 1
y <- x # The destination alphabet is the same as the source

# Define a quadratic cost function
cost.function <- function(x, y) {
    (y - x)^2
}

MaximumCost(x, y, cost.function)