Intervaled Real Fn


Intervaled Real Fn is a direct subtype of Timed Real To Real Function
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with functions Intervaled Real Fn Functions, keys Intervaled Real Fn keys and example object IntervRealFn

TYPE INCLUSION RELATIONSHIPS

Timed Real To Real Function

Intervaled Real Fn

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AVAILABLE FUNCTIONS

Create

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AVAILABLE CREATE FUNCTION KEYS

Functions

Included

Nodes

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TYPICAL OBJECTS OF TYPE Intervaled Real Fn

IntervRealFn

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This type represents a function ƒ that maps a sequence d , x₁ , x₂ , ... , xᵤ consisting of a date d followed by u real variables (i.e. single numbers) x₁ , x₂ , ... , xᵤ into v real variables y₁ , y₂ , ... , yᵥ and constructed (see details below) with the help of r functions ƒ₁ , ƒ₂ , ... , ƒᵣ, each of which maps the same u real variables x₁ , x₂ , ... , xᵤ into v real variables y₁ , y₂ , ... , yᵥ.
Schematically:
ƒ: (d , x₁, x₂, ... , xᵤ) → (y₁ , y₂ , ... , yᵥ)
where d is date and all other variables are numbers.
The functions ƒ₁ , ƒ₂ , ... , ƒᵣ are expected to be objects of which the type is a subtype of Real To Real Function

The formula that defines f requires the prior partition of the time axis in contiguous non-overlapping time intervals:
(-∞ , d₁) , [d₁ , d₂) , ... , [dᵣ₋₁ , ∞)
where d₁ , d₂ , ... , dᵣ₋₁ are an increasing sequence of dates.

It is also possible to specify the exact boundary behavior of some or all of these intervals, so that they are not necessarily closed on their left and open on their right.

Note the times on the time axis are represented as dates rather than numbers and therefore only the discrete points that represent dates are considered.

Given this time axis partition, we associate each function ƒᵢ with the iᵗʰ interval [dᵢ₋₁ , dᵢ)
where it is assumed that d₀ = -∞ and dᵣ = ∞
so that depending on which interval the date d belongs, the function output ƒ(d ; x₁, x₂, ... , xᵤ) equals the output of corresponding function ƒᵢ(x₁ , x₂ , ... , xᵤ)

So the first function ƒ₁ is defined on the date interval (-∞ , d₁), the second function ƒ₂ on the date interval [d₁ , d₂) and the last function ƒᵣ on the date interval [dᵣ₋₁ , ∞)

Notationally, the function formula is given by (in the case of the default boundary behavior):
ƒ( d ; x₁ , x₂ , ..., xᵤ ) = ƒᵢ( x₁ , x₂ , ..., xᵤ ) if dᵢ₋₁ <= d < dᵢ for i = 1, ..., r