I want to create (approximately) the double sigmoid in the shown figure as
a function in terms of the parameters X,Y,Z, a,b,c and d.
Any idea? Thanks.
This question seems to have gone ignored, so try something like this:
k = 1 # adjust this for "sharpness"
s(x) = (tanh(k * x) + 1) / 2
f(x) = X + (Y-X) * s(x-b) + (Z-Y) * s(x-c)
Here's an example plot.
Related
I want to calculate an equation within a controller(Arduino)
y = -0.0000000104529251928664x^3 + 0.0000928316793270531x^2 - 0.282333029643959x + 297.661280719026
Now the decimal values of the coefficients are important because "x" varies in thousands so cube term cannot be ignored. I have tried manipulating the equation in excel to reduce the coefficients but R^2 is lost in the process and I would like to avoid that.
Max variable size available in Arduino is 4byte. And on google search, I was not able to find an appropriate solution.
Thank you for your time.
Since
-0.0000000104529251928664 ^ (1/3) = - 0.0021864822
0.0000928316793270531 ^ (1/2) = 0.00963491978
The formula
y = -0.0000000104529251928664x^3 + 0.0000928316793270531x^2 - 0.282333029643959x + 297.661280719026
Can be rewritten:
y = -(0.0021864822 * x)^3 + (0.00963491978 * x)^2 - 0.282333029643959 * x + 297.661280719026
Rounding all coefficients to 10 decimal places, we get:
y = -(0.0021864822 * x)^3 + (0.00963491978 * x)^2 - 0.2823330296 * x + 297.6612807
But I don't know Arduino, I'm not sure what the correct number of decimal places is, nor do I know what the compiler will accept or refuse.
I'm given a string 2*x + 5 - (3*x-2)=x + 5 and I need to solve for x. My thought process is that I'd convert it to an expression tree, something like,
=
/ \
- +
/\ /\
+ - x 5
/\ /\
* 5 * 2
/\ /\
2 x 3 x
But how do I actually reduce the tree from here? Any other ideas?
You have to reduce it using axioms from algebra
a * (b + c) -> (a * b) + (a * c)
This is done by checking the types of each node in the pass tree. Once the thing is fully expanded into terms, you can then check they are actually linear, etc.
The values in the tree will be either variables or numbers. It isn't very neat to represent these as classes inheriting from some AbstractTreeNode class however, because cplusplus doesn't have multiple dispatch. So it is better to do it the 'c' way.
enum NodeType {
Number,
Variable,
Addition //to represent the + and *
}
struct Node {
NodeType type;
//union {char*, int, Node*[2]} //psuedo code, but you need
//something kind of like this for the
//variable name ("x") and numerical value
//and the children
}
Now you can query they types of a node and its children using switch case.
As I said earlier - c++ idiomatic code would use virtual functions but lack the necessary multiple dispatch to solve this cleanly. (You would need to store the type anyway)
Then you group terms, etc and solve the equation.
You can have rules to normalise the tree, for example
constant + variable -> variable + constant
Would put x always on the left of a term. Then x * 2 + x * 4 could be simplified more easily
var * constant + var * constant -> (sum of constants) * var
In your example...
First, simplify the '=' by moving the terms (as per the rule above)
The right hand side will be -1 * (x + 5), becoming -1 * x + -1 * 5. The left hand side will be harder - consider replacing a - b with a + -1 * b.
Eventually,
2x + 5 + -3x + 2 + -x + -5 = 0
Then you can group terms ever which way you want. (By scanning along, etc)
(2 + -3 + -1) x + 5 + 2 + -5 = 0
Sum them up and when you have mx + c, solve it.
Assuming you have a first order equation, check all the leaves on each side. On each side, have two bins: one to add up all the leaves containing a multiple of X and one for all the leaves containing a multiples of a constant. Either add to a bin or multiply each bin as you step up the tree along each branch from the leaves. You will end up with something that is conceptually like
a*x + b = c*x + d
At that point, you can just solve
x = (d - b) / (a - c)
Assuming the equation can reduce to f(x) = 0, and f(x) = a * x + b.
You can transform all the leaves in expression tree to f(x), for example : 2 -> 0 * x + 2, 3 * x -> 3 * x + 0, then you can do arithmetic operations of f(x) in expression tree. finally solve the equation f(x) = 0.
If the function is much more complicated than polynomial, you can do a binary search on x, and using the expression tree to calculate the left and right side of equation.
How can I maximize the following equation in respect to $\tau$ in Mathematica 9:
$$max_\tau \sqrt{(1 - \tau)y^i} + \sqrt{\tau y}$$
I want to find something like
$$\tau^i = \frac{y}{y^i + y}$$
Let x = Τ then f(x, y) = sqrt((1-x)*y^c) + sqrt(xy)
I'll assume that c is a constant, so there are only two independent variables here.
So take the first partial derivative w.r.t. x and set that equal to zero.
Wolfram Alpha can help you with that.
I'm creating a neural network using the backpropagation technique for learning.
I understand we need to find the derivative of the activation function used. I'm using the standard sigmoid function
f(x) = 1 / (1 + e^(-x))
and I've seen that its derivative is
dy/dx = f(x)' = f(x) * (1 - f(x))
This may be a daft question, but does this mean that we have to pass x through the sigmoid function twice during the equation, so it would expand to
dy/dx = f(x)' = 1 / (1 + e^(-x)) * (1 - (1 / (1 + e^(-x))))
or is it simply a matter of taking the already calculated output of f(x), which is the output of the neuron, and replace that value for f(x)?
Dougal is correct. Just do
f = 1/(1+exp(-x))
df = f * (1 - f)
The two ways of doing it are equivalent (since mathematical functions don't have side-effects and always return the same input for a given output), so you might as well do it the (faster) second way.
A little algebra can simplify this so that you don't have to have df call f.
df = exp(-x)/(1+exp(-x))^2
derivation:
df = 1/(1+e^-x) * (1 - (1/(1+e^-x)))
df = 1/(1+e^-x) * (1+e^-x - 1)/(1+e^-x)
df = 1/(1+e^-x) * (e^-x)/(1+e^-x)
df = (e^-x)/(1+e^-x)^2
You can use the output of your sigmoid function and pass it to your SigmoidDerivative function to be used as the f(x) in the following:
dy/dx = f(x)' = f(x) * (1 - f(x))
I already googled for the problem but only found either 2D solutions or formulas that didn't work for me (found this formula that looks nice: http://www.ogre3d.org/forums/viewtopic.php?f=10&t=55796 but seems not to be correct).
I have given:
Vec3 cannonPos;
Vec3 targetPos;
Vec3 targetVelocityVec;
float bulletSpeed;
what i'm looking for is time t such that
targetPos+t*targetVelocityVec
is the intersectionpoint where to aim the cannon to and shoot.
I'm looking for a simple, inexpensive formula for t (by simple i just mean not making many unnecessary vectorspace transformations and the like)
thanks!
The real problem is finding out where in space that the bullet can intersect the targets path. The bullet speed is constant, so in a certain amount of time it will travel the same distance regardless of the direction in which we fire it. This means that it's position after time t will always lie on a sphere. Here's an ugly illustration in 2d:
This sphere can be expressed mathematically as:
(x-x_b0)^2 + (y-y_b0)^2 + (z-z_b0)^2 = (bulletSpeed * t)^2 (eq 1)
x_b0, y_b0 and z_b0 denote the position of the cannon. You can find the time t by solving this equation for t using the equation provided in your question:
targetPos+t*targetVelocityVec (eq 2)
(eq 2) is a vector equation and can be decomposed into three separate equations:
x = x_t0 + t * v_x
y = y_t0 + t * v_y
z = z_t0 + t * v_z
These three equations can be inserted into (eq 1):
(x_t0 + t * v_x - x_b0)^2 + (y_t0 + t * v_y - y_b0)^2 + (z_t0 + t * v_z - z_b0)^2 = (bulletSpeed * t)^2
This equation contains only known variables and can be solved for t. By assigning the constant part of the quadratic subexpressions to constants we can simplify the calculation:
c_1 = x_t0 - x_b0
c_2 = y_t0 - y_b0
c_3 = z_t0 - z_b0
(v_b = bulletSpeed)
(t * v_x + c_1)^2 + (t * v_y + c_2)^2 + (t * v_z + c_3)^2 = (v_b * t)^2
Rearrange it as a standard quadratic equation:
(v_x^2+v_y^2+v_z^2-v_b^2)t^2 + 2*(v_x*c_1+v_y*c_2+v_z*c_3)t + (c_1^2+c_2^2+c_3^2) = 0
This is easily solvable using the standard formula. It can result in zero, one or two solutions. Zero solutions (not counting complex solutions) means that there's no possible way for the bullet to reach the target. One solution will probably happen very rarely, when the target trajectory intersects with the very edge of the sphere. Two solutions will be the most common scenario. A negative solution means that you can't hit the target, since you would need to fire the bullet into the past. These are all conditions you'll have to check for.
When you've solved the equation you can find the position of t by putting it back into (eq 2). In pseudo code:
# setup all needed variables
c_1 = x_t0 - x_b0
c_2 = y_t0 - y_b0
c_3 = z_t0 - z_b0
v_b = bulletSpeed
# ... and so on
a = v_x^2+v_y^2+v_z^2-v_b^2
b = 2*(v_x*c_1+v_y*c_2+v_z*c_3)
c = c_1^2+c_2^2+c_3^2
if b^2 < 4*a*c:
# no real solutions
raise error
p = -b/(2*a)
q = sqrt(b^2 - 4*a*c)/(2*a)
t1 = p-q
t2 = p+q
if t1 < 0 and t2 < 0:
# no positive solutions, all possible trajectories are in the past
raise error
# we want to hit it at the earliest possible time
if t1 > t2: t = t2
else: t = t1
# calculate point of collision
x = x_t0 + t * v_x
y = y_t0 + t * v_y
z = z_t0 + t * v_z