Is there a way to predict a value from a sum of two distributions? I am getting a syntax error on rstan when I try to estimate y here: y ~ binomial(,) + poisson()
library(rstan)
BH_model_block <- "
data{
int y;
int a;
}
parameters{
real <lower = 0, upper = 1> c;
real <lower = 0, upper = 1> b;
}
model{
y ~ binomial(a,b)+ poisson(c);
}
"
BH_model <- stan_model(model_code = BH_model_block)
BH_fit <- sampling(BH_model,
data = list(y = 5,
a = 2),
iter= 1000)
Produces this error:
SYNTAX ERROR, MESSAGE(S) FROM PARSER:
error in 'model2c6022623d56_457bd7ab767c318c1db686d1edf0b8f6' at line 13, column 20
-------------------------------------------------
11:
12: model{
13: y ~ binomial(a,b)+ poisson(c);
^
14: }
-------------------------------------------------
PARSER EXPECTED: ";"
Error in stanc(file = file, model_code = model_code, model_name = model_name, :
failed to parse Stan model '457bd7ab767c318c1db686d1edf0b8f6' due to the above error.
Stan doesn't support integer parameters, so you can't technically do that. For two real variables, it'd look like this:
parameters {
real x;
real y;
}
transformed parameters {
real z = x + y;
}
model {
x ~ normal(0, 1);
y ~ gamma(0.1, 2);
}
Then you get the sum distribution for z. If the variables are discrete, it won't compile.
If you don't need z in the model, then you can do this in the generated quantities block,
generated quantities {
int x = binomial_rng(a, b);
int y = poisson_rng(c);
int z = x + y;
}
The drawback of doing this is that none of the variables are available in the model block. If you need discrete parameters, they need to be marginalized as described in the user's guide chapter on latent discrete parameters (also in the chapter on mixtures and HMMs). This is not so easy with a Poisson, because support isn't bounded. If the expectations of the two discrete distributions is small, then you can do it approximately with a loop over plausible values.
It looked from the example in the original post that z is data. That's a slightly different marginalization over x and y, but you only sum over the x and y such that x + y = z, so the combinatorics are greatly reduced.
An alternative is to substitute the Binomial with a Poisson, and use Poisson additivity:
BH_model_block <- "
data{
int y;
int a;
}
parameters{
real <lower = 0, upper = 1> c;
real <lower = 0, upper = 1> b;
}
model{
y ~ poisson(a * b + c);
}
"
This differs in that if b is not small, the Binomial has a lower variance than the Poisson, but maybe there is overdispersion anyhow?
I am trying to write an iterator with a conditional in Chapel. This works
var x = [1,4,5,2,6,3];
iter dx(x) {
for y in x do yield 2*y;
}
for y in dx(x) {
writeln("y -> ", y);
}
returning
y -> 2
y -> 8
y -> 10
y -> 4
y -> 12
y -> 6
Suppose I want to only return the ones that are greater than 3. None of these will compile. What is the proper syntax?
var x = [1,4,5,2,6,3];
iter dx(x) {
//for y in x do {if x > 3} yield 2*y; // Barf
//for y in x do {if x > 3 yield 2*y }; // Barf
//for y in x do if x > 3 yield 2*y ; // Barf
}
for y in dx(x) {
writeln("y -> ", y);
}
The error is that you are checking against the iterator argument x instead of the current element y in the conditional. Try:
iter dx(x) {
for y in x {
if y > 3 {
yield 2*y;
}
}
}
or in the more concise form:
iter dx(x) {
for y in x do if y > 3 then yield 2*y;
}
Note that when the body of an if statement is a single statement, you may use a then keyword to introduce the body rather than enclosing it in braces { }. Unlike C, the then keyword is required (due to syntactic ambiguities that would occur otherwise).
I have some questions regarding collision angles. I am trying to code physics for a game and I do not want to use any third party library, actually I want to code each and every thing by myself. I know how to detect collisions between two spheres but I can't figure out, how to find the angle of collision/repulsion between the two spherical objects. I've tried reversing the direction of the objects, but no luck. It would be very nice if you link me to an interesting .pdf file teaching physics programming.
There's a lot of ways to deal with collision
Impulsion
To model a impulsion, you can directly act on the speed of each objects, using the law of reflection, you can "reflect" each speed using the "normal of the impact"
so : v1 = v1 - 2 x ( v1 . n2 ) x n2
and v2 = v2 - 2 x ( v2 . n1 ) x n1
v1 and v2 speeds of sphere s1 and s2
n1 and n2 normal at collision point
Penalty
Here, we have 2 object interpenetrating, and we model the fact that they tend to not interpenetrate anymore, so you create a force that is proportional to the penetration using a spring force
I didn't speak about all the ways, but this are the two simplest I know
the angle between two objects in the 2D or 3D coordinate space can be found by
A * B = |A||B|cosɵ
Both A and B are vectors and ɵ is the angle between both vectors.
the below class can be used to solve basic Vector calculations in games
class 3Dvector
{
private:
float x, y, z;
public:
// purpose: Our constructor
// input: ex- our vector's i component
// why- our vector's j component
// zee- our vector's k component
// output: no explicit output
3Dvector(float ex = 0, float why = 0, float zee = 0)
{
x = ex; y = why; z = zee;
}
// purpose: Our destructor
// input: none
// output: none
~3Dvector() { }
// purpose: calculate the magnitude of our invoking vector
// input: no explicit input
// output: the magnitude of our invoking object
float getMagnitude()
{
return sqrtf(x * x + y * y + z * z);
}
// purpose: multiply our vector by a scalar value
// input: num - the scalar value being multiplied
// output: our newly created vector
3Dvector operator*(float num) const
{
return 3Dvector(x * num, y * num, z * num);
}
// purpose: multiply our vector by a scalar value
// input: num - the scalar value being multiplied
// vec - the vector we are multiplying to
// output: our newly created vector
friend 3Dvector operator*(float num, const 3Dvector &vec)
{
return 3Dvector(vec.x * num, vec.y * num, vec.z * num);
}
// purpose: Adding two vectors
// input: vec - the vector being added to our invoking object
// output: our newly created sum of the two vectors
3Dvector operator+(const 3Dvector &vec) const
{
return 3Dvector(x + vec.x, y + vec.y, z + vec.z);
}
// purpose: Subtracting two vectors
// input: vec - the vector being subtracted from our invoking object
// output: our newly created difference of the two vectors
3Dvector operator-(const 3Dvector &vec) const
{
return 3Dvector(x - vec.x, y - vec.y, z - vec.z);
}
// purpose: Normalize our invoking vector *this changes our vector*
// input: no explicit input
// output: none
void normalize3Dvector(void)
{
float mag = sqrtf(x * x + y * y + z * z);
x /= mag; y /= mag; z /= mag
}
// purpose: Dot Product two vectors
// input: vec - the vector being dotted with our invoking object
// output: the dot product of the two vectors
float dot3Dvector(const 3Dvector &vec) const
{
return x * vec.x + y * vec.y + z * vec.z;
}
// purpose: Cross product two vectors
// input: vec- the vector being crossed with our invoking object
// output: our newly created resultant vector
3Dvector cross3Dvector(const 3Dvector &vec) const
{
return 3Dvector( y * vec.z – z * vec.y,
z * vec.x – x * vec.z,
x * vec.y – y * vec.x);
}
};
I shouldn't be answering my own question but I found what I needed, I guess. It may help other people too. I was just fingering the wikipedia's physics section and I got this.
This link solves my question
The angle in a cartesian system can be found this way:
arctan((Ya-Yb)/(Xa-Xb))
Because this is a retangle triangle where you know the catets (diferences of heights and widths). This will calc the tangent. So the arctan will calc the angle thats have this tangent.
I hope I was helpful.
I want to create a struct which is like a CGPoint, but with 3 coordinates instead of 2.
I create it in the following way:
typedef struct {CGFloat x;CGFloat y;CGFloat z;} CG3Vector;
CG_INLINE CG3Vector CG3VectorMake(CGFloat x, CGFloat y, CGFloat z)
{
CG3Vector p; p.x = x; p.y = y; p.z = z; return p;
}
It works fine. But I now want to improve this struct so that it has the constants like for CGPoint: CGPointZero
Also what is the way to introduce the limits for particular components of the struct, like it is for the CGSize, where components are never lower than 0?
Thanks.
You could create constants like this:
const CG3Vector CG3VectorZero = { 0, 0, 0 };
If you want limits, I suppose you can do some checking like this:
CG_INLINE CG3Vector CG3VectorMake(CGFloat x, CGFloat y, CGFloat z)
{
// normalize the values
x = fmod(x, 360);
y = fmod(y, 360);
z = fmod(z, 360);
x = (x < 0) ? 360 + x : x;
y = (y < 0) ? 360 + y : y;
z = (z < 0) ? 360 + z : z;
return (CG3Vector) { x, y, z };
}
A friend and I are going back and forth with brain-teasers and I have no idea how to solve this one. My assumption is that it's possible with some bitwise operators, but not sure.
In C, with bitwise operators:
#include<stdio.h>
int add(int x, int y) {
int a, b;
do {
a = x & y;
b = x ^ y;
x = a << 1;
y = b;
} while (a);
return b;
}
int main( void ){
printf( "2 + 3 = %d", add(2,3));
return 0;
}
XOR (x ^ y) is addition without carry. (x & y) is the carry-out from each bit. (x & y) << 1 is the carry-in to each bit.
The loop keeps adding the carries until the carry is zero for all bits.
int add(int a, int b) {
const char *c=0;
return &(&c[a])[b];
}
No + right?
int add(int a, int b)
{
return -(-a) - (-b);
}
CMS's add() function is beautiful. It should not be sullied by unary negation (a non-bitwise operation, tantamount to using addition: -y==(~y)+1). So here's a subtraction function using the same bitwise-only design:
int sub(int x, int y) {
unsigned a, b;
do {
a = ~x & y;
b = x ^ y;
x = b;
y = a << 1;
} while (a);
return b;
}
Define "best". Here's a python version:
len(range(x)+range(y))
The + performs list concatenation, not addition.
Java solution with bitwise operators:
// Recursive solution
public static int addR(int x, int y) {
if (y == 0) return x;
int sum = x ^ y; //SUM of two integer is X XOR Y
int carry = (x & y) << 1; //CARRY of two integer is X AND Y
return addR(sum, carry);
}
//Iterative solution
public static int addI(int x, int y) {
while (y != 0) {
int carry = (x & y); //CARRY is AND of two bits
x = x ^ y; //SUM of two bits is X XOR Y
y = carry << 1; //shifts carry to 1 bit to calculate sum
}
return x;
}
Cheat. You could negate the number and subtract it from the first :)
Failing that, look up how a binary adder works. :)
EDIT: Ah, saw your comment after I posted.
Details of binary addition are here.
Note, this would be for an adder known as a ripple-carry adder, which works, but does not perform optimally. Most binary adders built into hardware are a form of fast adder such as a carry-look-ahead adder.
My ripple-carry adder works for both unsigned and 2's complement integers if you set carry_in to 0, and 1's complement integers if carry_in is set to 1. I also added flags to show underflow or overflow on the addition.
#define BIT_LEN 32
#define ADD_OK 0
#define ADD_UNDERFLOW 1
#define ADD_OVERFLOW 2
int ripple_add(int a, int b, char carry_in, char* flags) {
int result = 0;
int current_bit_position = 0;
char a_bit = 0, b_bit = 0, result_bit = 0;
while ((a || b) && current_bit_position < BIT_LEN) {
a_bit = a & 1;
b_bit = b & 1;
result_bit = (a_bit ^ b_bit ^ carry_in);
result |= result_bit << current_bit_position++;
carry_in = (a_bit & b_bit) | (a_bit & carry_in) | (b_bit & carry_in);
a >>= 1;
b >>= 1;
}
if (current_bit_position < BIT_LEN) {
*flags = ADD_OK;
}
else if (a_bit & b_bit & ~result_bit) {
*flags = ADD_UNDERFLOW;
}
else if (~a_bit & ~b_bit & result_bit) {
*flags = ADD_OVERFLOW;
}
else {
*flags = ADD_OK;
}
return result;
}
Go based solution
func add(a int, b int) int {
for {
carry := (a & b) << 1
a = a ^ b
b = carry
if b == 0 {
break
}
}
return a
}
same solution can be implemented in Python as follows, but there is some problem about number represent in Python, Python has more than 32 bits for integers. so we will use a mask to obtain the last 32 bits.
Eg: if we don't use mask we won't get the result for numbers (-1,1)
def add(a,b):
mask = 0xffffffff
while b & mask:
carry = a & b
a = a ^ b
b = carry << 1
return (a & mask)
Why not just incremet the first number as often, as the second number?
The reason ADD is implememted in assembler as a single instruction, rather than as some combination of bitwise operations, is that it is hard to do. You have to worry about the carries from a given low order bit to the next higher order bit. This is stuff that the machines do in hardware fast, but that even with C, you can't do in software fast.
Here's a portable one-line ternary and recursive solution.
int add(int x, int y) {
return y == 0 ? x : add(x ^ y, (x & y) << 1);
}
I saw this as problem 18.1 in the coding interview.
My python solution:
def foo(a, b):
"""iterate through a and b, count iteration via a list, check len"""
x = []
for i in range(a):
x.append(a)
for i in range(b):
x.append(b)
print len(x)
This method uses iteration, so the time complexity isn't optimal.
I believe the best way is to work at a lower level with bitwise operations.
In python using bitwise operators:
def sum_no_arithmetic_operators(x,y):
while True:
carry = x & y
x = x ^ y
y = carry << 1
if y == 0:
break
return x
Adding two integers is not that difficult; there are many examples of binary addition online.
A more challenging problem is floating point numbers! There's an example at http://pages.cs.wisc.edu/~smoler/x86text/lect.notes/arith.flpt.html
Was working on this problem myself in C# and couldn't get all test cases to pass. I then ran across this.
Here is an implementation in C# 6:
public int Sum(int a, int b) => b != 0 ? Sum(a ^ b, (a & b) << 1) : a;
Implemented in same way as we might do binary addition on paper.
int add(int x, int y)
{
int t1_set, t2_set;
int carry = 0;
int result = 0;
int mask = 0x1;
while (mask != 0) {
t1_set = x & mask;
t2_set = y & mask;
if (carry) {
if (!t1_set && !t2_set) {
carry = 0;
result |= mask;
} else if (t1_set && t2_set) {
result |= mask;
}
} else {
if ((t1_set && !t2_set) || (!t1_set && t2_set)) {
result |= mask;
} else if (t1_set && t2_set) {
carry = 1;
}
}
mask <<= 1;
}
return (result);
}
Improved for speed would be below::
int add_better (int x, int y)
{
int b1_set, b2_set;
int mask = 0x1;
int result = 0;
int carry = 0;
while (mask != 0) {
b1_set = x & mask ? 1 : 0;
b2_set = y & mask ? 1 : 0;
if ( (b1_set ^ b2_set) ^ carry)
result |= mask;
carry = (b1_set & b2_set) | (b1_set & carry) | (b2_set & carry);
mask <<= 1;
}
return (result);
}
It is my implementation on Python. It works well, when we know the number of bytes(or bits).
def summ(a, b):
#for 4 bytes(or 4*8 bits)
max_num = 0xFFFFFFFF
while a != 0:
a, b = ((a & b) << 1), (a ^ b)
if a > max_num:
b = (b&max_num)
break
return b
You can do it using bit-shifting and the AND operation.
#include <stdio.h>
int main()
{
unsigned int x = 3, y = 1, sum, carry;
sum = x ^ y; // Ex - OR x and y
carry = x & y; // AND x and y
while (carry != 0) {
carry = carry << 1; // left shift the carry
x = sum; // initialize x as sum
y = carry; // initialize y as carry
sum = x ^ y; // sum is calculated
carry = x & y; /* carry is calculated, the loop condition is
evaluated and the process is repeated until
carry is equal to 0.
*/
}
printf("%d\n", sum); // the program will print 4
return 0;
}
The most voted answer will not work if the inputs are of opposite sign. The following however will. I have cheated at one place, but only to keep the code a bit clean. Any suggestions for improvement welcome
def add(x, y):
if (x >= 0 and y >= 0) or (x < 0 and y < 0):
return _add(x, y)
else:
return __add(x, y)
def _add(x, y):
if y == 0:
return x
else:
return _add((x ^ y), ((x & y) << 1))
def __add(x, y):
if x < 0 < y:
x = _add(~x, 1)
if x > y:
diff = -sub(x, y)
else:
diff = sub(y, x)
return diff
elif y < 0 < x:
y = _add(~y, 1)
if y > x:
diff = -sub(y, x)
else:
diff = sub(y, x)
return diff
else:
raise ValueError("Invalid Input")
def sub(x, y):
if y > x:
raise ValueError('y must be less than x')
while y > 0:
b = ~x & y
x ^= y
y = b << 1
return x
Here is the solution in C++, you can find it on my github here: https://github.com/CrispenGari/Add-Without-Integers-without-operators/blob/master/main.cpp
int add(int a, int b){
while(b!=0){
int sum = a^b; // add without carrying
int carry = (a&b)<<1; // carrying without adding
a= sum;
b= carry;
}
return a;
}
// the function can be writen as follows :
int add(int a, int b){
if(b==0){
return a; // any number plus 0 = that number simple!
}
int sum = a ^ b;// adding without carrying;
int carry = (a & b)<<1; // carry, without adding
return add(sum, carry);
}
This can be done using Half Adder.
Half Adder is method to find sum of numbers with single bit.
A B SUM CARRY A & B A ^ B
0 0 0 0 0 0
0 1 1 0 0 1
1 0 1 0 0 1
1 1 0 1 0 0
We can observe here that SUM = A ^ B and CARRY = A & B
We know CARRY is always added at 1 left position from where it was
generated.
so now add ( CARRY << 1 ) in SUM, and repeat this process until we get
Carry 0.
int Addition( int a, int b)
{
if(B==0)
return A;
Addition( A ^ B, (A & B) <<1 )
}
let's add 7 (0111) and 3 (0011) answer will be 10 (1010)
A = 0100 and B = 0110
A = 0010 and B = 1000
A = 1010 and B = 0000
final answer is A.
I implemented this in Swift, I am sure someone will benefit from
var a = 3
var b = 5
var sum = 0
var carry = 0
while (b != 0) {
sum = a ^ b
carry = a & b
a = sum
b = carry << 1
}
print (sum)
You can do it iteratively or recursively. Recursive:-
public int getSum(int a, int b) {
return (b==0) ? a : getSum(a^b, (a&b)<<1);
}
Iterative:-
public int getSum(int a, int b) {
int c=0;
while(b!=0) {
c=a&b;
a=a^b;
b=c<<1;
}
return a;
}
time complexity - O(log b)
space complexity - O(1)
for further clarifications if not clear, refer leetcode or geekForGeeks explanations.
I'll interpret this question as forbidding the +,-,* operators but not ++ or -- since the question specified operator and not character (and also because that's more interesting).
A reasonable solution using the increment operator is as follows:
int add(int a, int b) {
if (b == 0)
return a;
if (b > 0)
return add(++a, --b);
else
return add(--a, ++b);
}
This function recursively nudges b towards 0, while giving a the same amount to keep the sum the same.
As an additional challenge, let's get rid of the second if block to avoid a conditional jump. This time we'll need to use some bitwise operators:
int add(int a, int b) {
if(!b)
return a;
int gt = (b > 0);
int m = -1 << (gt << 4) << (gt << 4);
return (++a & --b & 0)
| add( (~m & a--) | (m & --a),
(~m & b++) | (m & ++b)
);
}
The function trace is identical; a and b are nudged between each add call just like before.
However, some bitwise magic is employed to drop the if statement while continuing to not use +,-,*:
A mask m is set to 0xFFFFFFFF (-1 in signed decimal) if b is positive, or 0x00000000 if b is negative.
The reason for shifting the mask left by 16 twice instead a single shift left by 32 is because shifting by >= the size of the value is undefined behavior.
The final return takes a bit of thought to fully appreciate:
Consider this technique to avoid a branch when deciding between two values. Of the values, one is multiplied by the boolean while the other is multiplied by the inverse, and the results are summed like so:
double naiveFoodPrice(int ownPetBool) {
if(ownPetBool)
return 23.75;
else
return 10.50;
}
double conditionlessFoodPrice(int ownPetBool) {
double result = ownPetBool*23.75 + (!ownPetBool)*10.50;
}
This technique works great in most cases. For us, the addition operator can easily be substituted for the bitwise or | operator without changing the behavior.
The multiplication operator is also not allowed for this problem. This is the reason for our earlier mask value - a bitwise and & with the mask will achieve the same effect as multiplying by the original boolean.
The nature of the unary increment and decrement operators halts our progress.
Normally, we would easily be able to choose between an a which was incremented by 1 and an a which was decremented by 1.
However, because the increment and decrement operators modify their operand, our conditionless code will end up always performing both operations - meaning that the values of a and b will be tainted before we finish using them.
One way around this is to simply create new variables which each contain the original values of a and b, allowing a clean slate for each operation. I consider this boring, so instead we will adjust a and b in a way that does not affect the rest of the code (++a & --b & 0) in order to make full use of the differences between x++ and ++x.
We can now get both possible values for a and b, as the unary operators modifying the operands' values now works in our favor. Our techniques from earlier help us choose the correct versions of each, and we now have a working add function. :)
Python codes:
(1)
add = lambda a,b : -(-a)-(-b)
use lambda function with '-' operator
(2)
add= lambda a,b : len(list(map(lambda x:x,(i for i in range(-a,b)))))