Example
Let f(x1, x2) be a two-variable function (or signal) which is of the form
- f(x1,x2) = g(x1)
for some one-variable function g which is not constant. This means that f varies, in accordance to g, with the first variable or along the first coordinate. On the other hand, f is constant with respect to the second variable or along the second coordinate. It is only necessary to know the value of one, namely the first, variable in order to determine the value of f. Hence, it is a two-variable function but its intrinsic dimension is one.
A slightly more complicated example is
- f(x1,x2) = g(x1 + x2)
f is still intrinsic one-dimensional, which can be seen by making a variable transformation
- x1 + x2 = y1
- x1 - x2 = y2
which gives
- f(y1,y2) = g(y1)
Since the variation in f can be described by the single variable y1 its intrinsic dimension is one.
For the case that f is constant, its intrinsic dimension is zero since no variable is needed to describe variation. For the general case, when the intrinsic dimension of the two-variable function f is neither zero or one, it is two.
In the literature, functions which are of intrinsic dimension zero, one, or two are sometimes referred to as i0D, i1D or i2D, respectively.
Read more about this topic: Intrinsic Dimension
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