Mixing
Searching monotonous function with parameter for convexity
$begingroup$
I am looking for a function with the following constraints
$$
f(0, a) = 0 land \
f(1, a) = 1 land \
f'(x, a) > 0 land \
exists a:f''(x, a) > 0 land \
exists a:f''(x, a) < 0 land \
forall x in [0,1]
$$
$a$ should be a parameter that controls the convexity. Ideal would be a function that does not change its either convex or concave for all values of $a$.
I have found $$f(x) = a x^2 + (1-a) x, a in [-1, 1]$$
I am hoping fore more extreme gradients.
Another idea is to have a combination of the circle formula
$$
a (1 - sqrt{1 - x^2}) + (1-a) sqrt{2x - x^2}, a in [0, 1]
$$
Although this is only convex for $a=1$ and concave for $a = 0$ and a bit mixed for other values of $a$ so it's not a real a solution.
Another idea was the distribution function of the gamma distribution but this I was hoping for something easier to grasp.
functions
asked Jan 29 at 16:41
jakob-rjakob-r
1012
$endgroup$
add a comment |
$begingroup$
I am looking for a function with the following constraints
$$
f(0, a) = 0 land \
f(1, a) = 1 land \
f'(x, a) > 0 land \
exists a:f''(x, a) > 0 land \
exists a:f''(x, a) < 0 land \
forall x in [0,1]
$$
$a$ should be a parameter that controls the convexity. Ideal would be a function that does not change its either convex or concave for all values of $a$.
I have found $$f(x) = a x^2 + (1-a) x, a in [-1, 1]$$
I am hoping fore more extreme gradients.
Another idea is to have a combination of the circle formula
$$
a (1 - sqrt{1 - x^2}) + (1-a) sqrt{2x - x^2}, a in [0, 1]
$$
Although this is only convex for $a=1$ and concave for $a = 0$ and a bit mixed for other values of $a$ so it's not a real a solution.
Another idea was the distribution function of the gamma distribution but this I was hoping for something easier to grasp.
functions
asked Jan 29 at 16:41
jakob-rjakob-r
1012
$endgroup$
add a comment |
$begingroup$
I am looking for a function with the following constraints
$$
f(0, a) = 0 land \
f(1, a) = 1 land \
f'(x, a) > 0 land \
exists a:f''(x, a) > 0 land \
exists a:f''(x, a) < 0 land \
forall x in [0,1]
$$
$a$ should be a parameter that controls the convexity. Ideal would be a function that does not change its either convex or concave for all values of $a$.
I have found $$f(x) = a x^2 + (1-a) x, a in [-1, 1]$$
I am hoping fore more extreme gradients.
Another idea is to have a combination of the circle formula
$$
a (1 - sqrt{1 - x^2}) + (1-a) sqrt{2x - x^2}, a in [0, 1]
$$
Although this is only convex for $a=1$ and concave for $a = 0$ and a bit mixed for other values of $a$ so it's not a real a solution.
Another idea was the distribution function of the gamma distribution but this I was hoping for something easier to grasp.
functions
asked Jan 29 at 16:41
jakob-rjakob-r
1012
$endgroup$
I am looking for a function with the following constraints
$$
f(0, a) = 0 land \
f(1, a) = 1 land \
f'(x, a) > 0 land \
exists a:f''(x, a) > 0 land \
exists a:f''(x, a) < 0 land \
forall x in [0,1]
$$
$a$ should be a parameter that controls the convexity. Ideal would be a function that does not change its either convex or concave for all values of $a$.
I have found $$f(x) = a x^2 + (1-a) x, a in [-1, 1]$$
I am hoping fore more extreme gradients.
Another idea is to have a combination of the circle formula
$$
a (1 - sqrt{1 - x^2}) + (1-a) sqrt{2x - x^2}, a in [0, 1]
$$
Although this is only convex for $a=1$ and concave for $a = 0$ and a bit mixed for other values of $a$ so it's not a real a solution.
Another idea was the distribution function of the gamma distribution but this I was hoping for something easier to grasp.
functions
functions
asked Jan 29 at 16:41
jakob-rjakob-r
1012
asked Jan 29 at 16:41
jakob-rjakob-r
1012
asked Jan 29 at 16:41
jakob-rjakob-r
1012
asked Jan 29 at 16:41
jakob-rjakob-r
1012
asked Jan 29 at 16:41
jakob-rjakob-r
1012
1012
add a comment |
add a comment |
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