Here is my test code to find 1st clipping area on the screen.
Two subroutines and dummy loops in the code to compare the performance of them.
point_in_neon (NEON version) and point_in (Regular version) does the same thing:
find out the first clipping area (contains given point) in given list and return -1 if there is no matching area.
I expected NEON version is faster than regular version.
Unfortunately, it is slower than regular version. Is there another way to speed it up?
The compiler command is:
${CC} -O2 -ftree-vectorize -o vcomp vcomp.c
Thanks,
#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <string.h>
#include <assert.h>
#include <math.h>
#include <sys/time.h>
#include <arm_neon.h>
#define WIDTH (4096)
#define HEIGHT (4096)
#define CLIPS (32)
static inline uint64_t now(void) {
struct timeval tv;
gettimeofday(&tv,NULL);
return tv.tv_sec*1000000+tv.tv_usec;
}
typedef struct _rect_t {
int32_t x;
int32_t y;
uint32_t width;
uint32_t height;
} rect_t;
typedef struct _point_t {
int32_t x;
int32_t y;
} point_t;
int32_t inline point_in_neon(const point_t *pt, const rect_t rs[4]) {
const int32_t right[4]={
rs[0].x+rs[0].width-1,
rs[1].x+rs[1].width-1,
rs[2].x+rs[2].width-1,
rs[3].x+rs[3].width-1
}, bottom[4]={
rs[0].y+rs[0].height-1,
rs[1].y+rs[1].height-1,
rs[2].y+rs[2].height-1,
rs[3].y+rs[3].height-1
};
int32x4_t p, r;
uint32x4_t t;
uint32_t res[4];
//p = <Xp, Xp, Xp, Xp>
p=vld1q_dup_s32(&pt->x);
//r = <Left0, Left1, Left2, Left3>
r=vld1q_lane_s32(&rs[0].x, r, 0);
r=vld1q_lane_s32(&rs[1].x, r, 1);
r=vld1q_lane_s32(&rs[2].x, r, 2);
r=vld1q_lane_s32(&rs[3].x, r, 3);
//t = (p >= r)
t=vcgeq_s32(p, r);
//r = <Right0, Right1, Right2, Right3>
r=vld1q_s32(&right);
//t = t & (r >= p)
t=vandq_u32(t, vcgeq_s32(r, p));
//p = <Yp, Yp, Yp, Yp>
p=vld1q_dup_s32(&pt->y);
//r = <Top0, Top1, Top2, Top3>
r=vld1q_lane_s32(&rs[0].y, r, 0);
r=vld1q_lane_s32(&rs[1].y, r, 1);
r=vld1q_lane_s32(&rs[2].y, r, 2);
r=vld1q_lane_s32(&rs[3].y, r, 3);
//t = t & (p >= r)
t=vandq_u32(t, vcgeq_s32(p, r));
//r = <Bottom0, Bottom1, Bottom2, Bottom3>
r=vld1q_s32(&bottom);
//t = t & (r >= p)
t=vandq_u32(t, vcgeq_s32(r, p));
vst1q_u32(res, t);
if(res[0])
return 0;
else if(res[1])
return 1;
else if(res[2])
return 2;
else if(res[3])
return 3;
return -1;
}
int32_t inline point_in(const point_t *pt, const rect_t *rs, uint32_t len) {
int32_t i;
for(i=0;i<len;i++) {
int32_t right=rs[i].x+rs[i].width-1,
bottom=rs[i].y+rs[i].height-1;
if(pt->x>=rs[i].x && pt->x<=right &&
pt->y>=rs[i].y && pt->y<=bottom)
return i;
}
return -1;
}
int32_t main(int32_t argc, char *argv[]) {
rect_t rs[CLIPS];
int32_t i, j;
uint64_t ts0, ts1;
int32_t res[2][CLIPS];
srand((unsigned int)time(NULL));
for(i=0;i<CLIPS;i++) {
rs[i].x=rand()%WIDTH;
rs[i].y=rand()%HEIGHT;
rs[i].width=rand()%WIDTH;
rs[i].height=rand()%HEIGHT;
}
memset(res, 0, sizeof(res));
ts0=now();
for(i=0;i<HEIGHT;i++) {
for(j=0;j<WIDTH;j++) {
point_t p={i, j};
int32_t idx=point_in(&p, rs, CLIPS);
if(idx>=0)
res[0][idx]=1;
}
}
ts0=now()-ts0;
ts1=now();
for(i=0;i<HEIGHT;i++) {
for(j=0;j<WIDTH;j++) {
int32_t k, idx;
point_t p={i, j};
for(k=0, idx=-1;k<CLIPS/4;k++) {
idx=point_in_neon(&p, &rs[k*4]);
if(idx>=0)
break;
}
if(idx>=0)
res[1][k*4+idx]=1;
}
}
ts1=now()-ts1;
/*
for(i=0;i<CLIPS;i++) {
if(res[0][i]!=res[1][i]) {
printf("error.\n");
return 1;
}
}
*/
printf("regular = %lu\n", ts0);
printf("neon = %lu\n", ts1);
return 0;
}
According to Peter Cordes's suggestion, I replaced data loding parts of point_in_neon subroutine with vld4q_s32 intrinsic and subsequent right and bottom calculation can be vectorized. Now the code is shorter and faster than regular version.
int32_t inline point_in_neon(const point_t *pt, const rect_t rs[4]) {
int32x4x4_t r;
int32x4_t right, bottom, p;
uint32x4_t t;
uint32_t res[4];
/*
r.val[0] = <X0, X1, X2, X3>
r.val[1] = <Y0, Y1, Y2, Y3>
r.val[2] = <Width0, Width1, Width2, Width3>
r.val[3] = <Height0, Height1, Height2, Height3>
*/
r=vld4q_s32(rs);
//right = <Right0, Right1, Right2, Right3>
right=vsubq_s32(vaddq_s32(r.val[0], r.val[2]), vdupq_n_s32(1));
//bottom = <Bottom0, Bottom1, Bottom2, Bottom3>
bottom=vsubq_s32(vaddq_s32(r.val[1], r.val[3]), vdupq_n_s32(1));
//p = <Xp, Xp, Xp, Xp>
p=vld1q_dup_s32(&pt->x);
//t = (p >= left)
t=vcgeq_s32(p, r.val[0]);
//t = t & (right >= p)
t=vandq_u32(t, vcgeq_s32(right, p));
//p = <Yp, Yp, Yp, Yp>
p=vld1q_dup_s32(&pt->y);
//t = t & (p >= top)
t=vandq_u32(t, vcgeq_s32(p, r.val[1]));
//t = t & (r >= bottom)
t=vandq_u32(t, vcgeq_s32(bottom, p));
vst1q_u32(res, t);
if(res[0])
return 0;
else if(res[1])
return 1;
else if(res[2])
return 2;
else if(res[3])
return 3;
return -1;
}
Starting with your original point_in method, we can clean up a little bit here by removing the -1's, and changing <= to <.
int32_t inline point_in(const point_t *pt, const rect_t *rs, uint32_t len) {
int32_t i;
for(i=0; i < len; i++)
{
// this is pointless - change your data structures so that
// the rect stores minx/maxx, miny/maxy instead!
int32_t right = rs[i].x + rs[i].width;
int32_t bottom= rs[i].y + rs[i].height;
bool cmp0 = pt->x >= rs[i].x;
bool cmp1 = pt->y >= rs[i].y;
bool cmp2 = pt->x < right;
bool cmp3 = pt->y < bottom;
if(cmp0 & cmp1 & cmp2 & cmp3)
return i;
}
return -1;
}
Next obvious thing to point out:
// your screen size...
#define WIDTH (4096)
#define HEIGHT (4096)
// yet your structures use uint32 as storage???
typedef struct _rect_t {
int32_t x;
int32_t y;
uint32_t width;
uint32_t height;
} rect_t;
typedef struct _point_t {
int32_t x;
int32_t y;
} point_t;
If you can get away with using 16bit integers, this will go at twice the speed (because you can fit 8x 16bit numbers in a SIMD register, v.s. 4x 32bit). Whilst we're at it, we might as well change the data layout to structure of array at the same time.
I'm also going to hoist the pointless p.x + width out, and store it as xmax/ymax instead (removes duplicated computation in your loops).
typedef struct rect_x8_t {
int16x8_t x;
int16x8_t y;
int16x8_t xmax; //< x + width
int16x8_t ymax; //< y + height
} rect_x8_t;
typedef struct point_x8_t {
int16x8_t x;
int16x8_t y;
} point_x8_t;
On the assumption you don't have a number of clips that's divisible by 8, we'll need to pad the number slightly (not a big deal)
// assuming this has already been initialised
rect_t rs[CLIPS];
// how many batches of 8 do we need?
uint32_t CLIPS8 = (CLIPS / 8) + (CLIPS & 7 ? 1 : 0);
// allocate in batches of 8
rect_x8_t rs8[CLIPS8] = {};
// I'm going to do this rubbishly as an pre-process step.
// I don't care too much about efficiency here...
for(uint32_t i = 0; i < CLIPS; ++i) {
rs8[i / 8].x[i & 7] = rs[i].x;
rs8[i / 8].y[i & 7] = rs[I].y;
rs8[i / 8].xmax[i & 7] = rs[i].x + rs[i].width;
rs8[i / 8].ymax[i & 7] = rs[i].y + rs[i].height;
}
I have a couple of concerns here:
for(i=0;i<HEIGHT;i++) {
for(j=0;j<WIDTH;j++) {
// This seems wrong? Shouldn't it be p = {j, i} ?
point_t p={i, j};
int32_t idx=point_in(&p, rs, CLIPS);
// I'm not quite sure what the result says about your
// image data and clip regions???
//
// This seems like a really silly way of asking
// a simple question about the clip regions. The pixels
// don't have any effect here.
if(idx >= 0)
res[0][idx] = 1;
}
}
Anyhow, now refactoring the point_in method to use int16x8_t, we get:
inline int32_t point_in_x8(const point_x8_t pt,
const rect_x8_t* rs,
uint32_t len) {
for(int32_t i = 0; i < len; i++) {
// perform comparisons on 8 rects at a time
uint16x8_t cmp0 = vcgeq_s16(pt.x, rs[i].x);
uint16x8_t cmp1 = vcgeq_s16(pt.y, rs[i].y);
uint16x8_t cmp2 = vcltq_s16(pt.x, rs[i].xmax);
uint16x8_t cmp3 = vcltq_s16(pt.y, rs[I].ymax);
// combine to single comparison value
uint16x8_t cmp01 = vandq_u16(cmp0, cmp1);
uint16x8_t cmp23 = vandq_u16(cmp2, cmp3);
uint16x8_t cmp0123 = vandq_u16(cmp01, cmp23);
// use a horizontal max to see if any lanes are true
if(vmaxvq_u16(cmp0123)) {
for(int32_t j = 0; j < 8; ++j) {
if(cmp0123[j])
return 8*i + j;
}
}
}
return -1;
}
Any additional padded elements in the rect_x8_t structs should end up being ignored (since they should be 0/0, 0/0, which will always end up being false).
Then finally...
for(i = 0; i < HEIGHT; i++) {
point_x8_t p;
// splat the y value
p.y = vld1q_dup_s16(i);
for(j = 0; j < WIDTH; j++) {
// splat the x value
p.x = vld1q_dup_s16(j);
int32_t idx = point_in_x8(p, rs8, CLIPS8);
if(idx >= 0)
res[1][idx] = 1;
}
}
The vld4 instruction actually has a fairly high latency. Given that WIDTH * HEIGHT is actually a very big number, pre-swizzling here (as a pre-processing step) makes a lot more sense imho.
HOWEVER
This whole algorithm could be massively improved by simply ignoring the pixels, and working on CLIP regions directly.
A clip region will be false if it is entirely contained by the preceding clip regions
for(i = 0; i < CLIPS; i++) {
// if region is empty, ignore.
if(rs[i].width == 0 || rs[i].height == 0) {
res[0][i] = 0;
continue;
}
// first region will always be true (unless it's of zero size)
if(i == 0) {
res[0][1] = 1;
continue;
}
uint32_t how_many_intersect = 0;
bool entirely_contained = false;
uint32_t intersection_indices[CLIPS] = {};
// do a lazy test first.
for(j = i - 1; j >= 0; --j) {
// if the last region is entirely contained by preceding
// ones, it will be false. exit loop.
if(region_is_entirely_contained(rs[i], rs[j])) {
res[0][i] = 0;
entirely_contained = true;
j = -1; ///< break out of loop
}
else
// do the regions intersect?
if(region_intersects(rs[i], rs[j])) {
intersection_indices[how_many_intersect] = j;
++how_many_intersect;
}
}
// if one region entirely contains this clip region, skip it.
if(entirely_contained) {
continue;
}
// if you only intersect one or no regions, the result is true.
if(how_many_intersect <= 1) {
res[0][i] = 1;
continue;
}
// If you get here, the result is *probably* true, however
// you will need to split this clip region against the previous
// ones to be fully sure. If all regions are fully contained,
// the answer is false.
// I won't implement it, but something like this:
* split rs[i] against each rs[intersection_indices[]].
* Throw away the rectangles that are entirely contained.
* Each bit that remains should be tested against each rs[intersection_indices[]]
* If you find any split rectangle that isn't contained,
set to true and move on.
}
So I am currently working on a tutorial to implement a very basic raytracer (currently just drawing solid spheres). Said tutorial is located here: http://thingsiamdoing.com/intro-to-ray-tracing/
The tutorial is completely language agnostic and deals only in pseudocode. I attempted to convert this pseudocode into C but have encountered difficulty. My program compiles fine, yet the outputted .ppm image file experiences an early EOF error. The lack of information about the problem has left me stuck.
Here is my C code, which is meant to be a direct translation of the pseudocode:
#include <stdio.h>
#include <math.h>
#define WIDTH 512
#define HEIGHT 512
typedef struct {
float x, y, z;
} vector;
float vectorDot(vector *v1, vector *v2) {
return v1->x * v2->x + v1->y * v2->y + v1->z * v2->z;
}
void writeppm(char *filename, unsigned char *img, int width, int height){
FILE *f;
f = fopen(filename, "w");
fprintf(f, "P6 %d %d %d\n", width, height, 255);
fwrite(img, 3, width*height, f);
fclose(f);
}
float check_ray(px, py, pz, dx, dy, dz, r) {
vector v1 = {px, py, pz};
vector v2 = {dx, dy, dz};
float det, b;
b = -vectorDot(&v1, &v2);
det = b*b - vectorDot(&v1, &v1) + r*r;
if (det<0)
return -1;
det = sqrtf(det);
float t1 = b - det;
float t2 = b + det;
return t1;
}
int main(void) {
int img[WIDTH*HEIGHT*3], distToPlane;
float cameraY, cameraZ, cameraX, pixelWorldX, pixelWorldY, pixelWorldZ, amp, rayX, rayY, rayZ;
for (int px = 0; px<WIDTH; px++) {
for (int py = 0; py<HEIGHT; py++) {
distToPlane = 100;
pixelWorldX = distToPlane;
pixelWorldY = (px - WIDTH / 2) / WIDTH;
pixelWorldZ = (py - HEIGHT / 2) / WIDTH;
rayX = pixelWorldX - cameraX;
rayY = pixelWorldY - cameraY;
rayZ = pixelWorldZ - cameraZ;
amp = 1/sqrtf(rayX*rayX + rayY*rayY + rayZ*rayZ);
rayX *= amp;
rayY *= amp;
rayZ *= amp;
if (check_ray(50, 50, 50, rayX, rayY, rayZ, 50)) {
img[(py + px*WIDTH)*3 + 0] = 0;
img[(py + px*WIDTH)*3 + 1] = 0;
img[(py + px*WIDTH)*3 + 2] = 128;
}
else {
img[(py + px*WIDTH)*3 + 0] = 255;
img[(py + px*WIDTH)*3 + 1] = 255;
img[(py + px*WIDTH)*3 + 2] = 255;
}
}
}
writeppm("image.ppm", "img", WIDTH, HEIGHT);
}
I am fairly confident the error does not lie with my function to write the .ppm file as I have used this for other work and it has been fine.
You may want to remove the quotes from around "img" in the following line of code:
writeppm("image.ppm", "img", WIDTH, HEIGHT);
seeing as how its prototype is void writeppm(char *, unsigned char *, int, int), although I am surprised that your compiler didn't at least give you a warning about a type mismatch.
Also, for the record, I would suggest putting some error checking code (like checking the return value of fwrite, or checking the return value of fopen)--- but that's just me.
Also, if you are not, please compile with all the warning enabled (eg with gcc use -ansi -Wall -pedantic), this will help you catch type mismatches and other little gotcha-ya's
I see two errors in main
int img[WIDTH*HEIGHT*3];
...
writeppm("image.ppm", "img", WIDTH, HEIGHT);
should be
unsigned char img[WIDTH*HEIGHT*3];
...
writeppm("image.ppm", img, WIDTH, HEIGHT);
I am trying to generate an array of n points that are equidistant from each other and lie on a circle in C. Basically, I need to be able to pass a function the number of points that I would like to generate and get back an array of points.
It's been a really long time since I've done C/C++, so I've had a stab at this more to see how I got on with it, but here's some code that will calculate the points for you. (It's a VS2010 console application)
// CirclePoints.cpp : Defines the entry point for the console application.
//
#include "stdafx.h"
#include "stdio.h"
#include "math.h"
int _tmain()
{
int points = 8;
double radius = 100;
double step = ((3.14159265 * 2) / points);
double x, y, current = 0;
for (int i = 0; i < points; i++)
{
x = sin(current) * radius;
y = cos(current) * radius;
printf("point: %d x:%lf y:%lf\n", i, x, y);
current += step;
}
return 0;
}
Try something like this:
void make_circle(float *output, size_t num, float radius)
{
size_t i;
for(i = 0; i < num; i++)
{
const float angle = 2 * M_PI * i / num;
*output++ = radius * cos(angle);
*output++ = radius * sin(angle);
}
}
This is untested, there might be an off-by-one hiding in the angle step calculation but it should be close.
This assumes I understood the question correctly, of course.
UPDATE: Redid the angle computation to not be incrementing, to reduce float precision loss due to repeated addition.
Here's a solution, somewhat optimized, untested. Error can accumulate, but using double rather than float probably more than makes up for it except with extremely large values of n.
void make_circle(double *dest, size_t n, double r)
{
double x0 = cos(2*M_PI/n), y0 = sin(2*M_PI/n), x=x0, y=y0, tmp;
for (;;) {
*dest++ = r*x;
*dest++ = r*y;
if (!--n) break;
tmp = x*x0 - y*y0;
y = x*y0 + y*x0;
x = tmp;
}
}
You have to solve this in c language:
In an x-y Cartesian coordinate system, the circle with centre coordinates (a, b) and radius r is the set of all points (x, y) such that
(x - a)^2 + (y - b)^2 = r^2
Here's a javascript implementation that also takes an optional center point.
function circlePoints (radius, numPoints, centerX, centerY) {
centerX = centerX || 0;
centerY = centerY || 0;
var
step = (Math.PI * 2) / numPoints,
current = 0,
i = 0,
results = [],
x, y;
for (; i < numPoints; i += 1) {
x = centerX + Math.sin(current) * radius;
y = centerY + Math.cos(current) * radius;
results.push([x,y]);
console.log('point %d # x:%d, y:%d', i, x, y);
current += step;
}
return results;
}
I am trying to implement a linear least squares fit onto 2 arrays of data: time vs amplitude. The only technique I know so far is to test all of the possible m and b points in (y = m*x+b) and then find out which combination fits my data best so that it has the least error. However, I think iterating so many combinations is sometimes useless because it tests out everything. Are there any techniques to speed up the process that I don't know about? Thanks.
Try this code. It fits y = mx + b to your (x,y) data.
The arguments to linreg are
linreg(int n, REAL x[], REAL y[], REAL* b, REAL* m, REAL* r)
n = number of data points
x,y = arrays of data
*b = output intercept
*m = output slope
*r = output correlation coefficient (can be NULL if you don't want it)
The return value is 0 on success, !=0 on failure.
Here's the code
#include "linreg.h"
#include <stdlib.h>
#include <math.h> /* math functions */
//#define REAL float
#define REAL double
inline static REAL sqr(REAL x) {
return x*x;
}
int linreg(int n, const REAL x[], const REAL y[], REAL* m, REAL* b, REAL* r){
REAL sumx = 0.0; /* sum of x */
REAL sumx2 = 0.0; /* sum of x**2 */
REAL sumxy = 0.0; /* sum of x * y */
REAL sumy = 0.0; /* sum of y */
REAL sumy2 = 0.0; /* sum of y**2 */
for (int i=0;i<n;i++){
sumx += x[i];
sumx2 += sqr(x[i]);
sumxy += x[i] * y[i];
sumy += y[i];
sumy2 += sqr(y[i]);
}
REAL denom = (n * sumx2 - sqr(sumx));
if (denom == 0) {
// singular matrix. can't solve the problem.
*m = 0;
*b = 0;
if (r) *r = 0;
return 1;
}
*m = (n * sumxy - sumx * sumy) / denom;
*b = (sumy * sumx2 - sumx * sumxy) / denom;
if (r!=NULL) {
*r = (sumxy - sumx * sumy / n) / /* compute correlation coeff */
sqrt((sumx2 - sqr(sumx)/n) *
(sumy2 - sqr(sumy)/n));
}
return 0;
}
Example
You can run this example online.
int main()
{
int n = 6;
REAL x[6]= {1, 2, 4, 5, 10, 20};
REAL y[6]= {4, 6, 12, 15, 34, 68};
REAL m,b,r;
linreg(n,x,y,&m,&b,&r);
printf("m=%g b=%g r=%g\n",m,b,r);
return 0;
}
Here is the output
m=3.43651 b=-0.888889 r=0.999192
Here is the Excel plot and linear fit (for verification).
All values agree exactly with the C code above (note C code returns r while Excel returns R**2).
There are efficient algorithms for least-squares fitting; see Wikipedia for details. There are also libraries that implement the algorithms for you, likely more efficiently than a naive implementation would do; the GNU Scientific Library is one example, but there are others under more lenient licenses as well.
From Numerical Recipes: The Art of Scientific Computing in (15.2) Fitting Data to a Straight Line:
Linear Regression:
Consider the problem of fitting a set of N data points (xi, yi) to a straight-line model:
Assume that the uncertainty: sigmai associated with each yi and that the xi’s (values of the dependent variable) are known exactly. To measure how well the model agrees with the data, we use the chi-square function, which in this case is:
The above equation is minimized to determine a and b. This is done by finding the derivative of the above equation with respect to a and b, equate them to zero and solve for a and b. Then we estimate the probable uncertainties in the estimates of a and b, since obviously the measurement errors in the data must introduce some uncertainty in the determination of those parameters. Additionally, we must estimate the goodness-of-fit of the data to the
model. Absent this estimate, we have not the slightest indication that the parameters a and b in the model have any meaning at all.
The below struct performs the mentioned calculations:
struct Fitab {
// Object for fitting a straight line y = a + b*x to a set of
// points (xi, yi), with or without available
// errors sigma i . Call one of the two constructors to calculate the fit.
// The answers are then available as the variables:
// a, b, siga, sigb, chi2, and either q or sigdat.
int ndata;
double a, b, siga, sigb, chi2, q, sigdat; // Answers.
vector<double> &x, &y, &sig;
// Constructor.
Fitab(vector<double> &xx, vector<double> &yy, vector<double> &ssig)
: ndata(xx.size()), x(xx), y(yy), sig(ssig), chi2(0.), q(1.), sigdat(0.)
{
// Given a set of data points x[0..ndata-1], y[0..ndata-1]
// with individual standard deviations sig[0..ndata-1],
// sets a,b and their respective probable uncertainties
// siga and sigb, the chi-square: chi2, and the goodness-of-fit
// probability: q
Gamma gam;
int i;
double ss=0., sx=0., sy=0., st2=0., t, wt, sxoss; b=0.0;
for (i=0;i < ndata; i++) { // Accumulate sums ...
wt = 1.0 / SQR(sig[i]); //...with weights
ss += wt;
sx += x[i]*wt;
sy += y[i]*wt;
}
sxoss = sx/ss;
for (i=0; i < ndata; i++) {
t = (x[i]-sxoss) / sig[i];
st2 += t*t;
b += t*y[i]/sig[i];
}
b /= st2; // Solve for a, b, sigma-a, and simga-b.
a = (sy-sx*b) / ss;
siga = sqrt((1.0+sx*sx/(ss*st2))/ss);
sigb = sqrt(1.0/st2); // Calculate chi2.
for (i=0;i<ndata;i++) chi2 += SQR((y[i]-a-b*x[i])/sig[i]);
if (ndata>2) q=gam.gammq(0.5*(ndata-2),0.5*chi2); // goodness of fit
}
// Constructor.
Fitab(vector<double> &xx, vector<double> &yy)
: ndata(xx.size()), x(xx), y(yy), sig(xx), chi2(0.), q(1.), sigdat(0.)
{
// As above, but without known errors (sig is not used).
// The uncertainties siga and sigb are estimated by assuming
// equal errors for all points, and that a straight line is
// a good fit. q is returned as 1.0, the normalization of chi2
// is to unit standard deviation on all points, and sigdat
// is set to the estimated error of each point.
int i;
double ss,sx=0.,sy=0.,st2=0.,t,sxoss;
b=0.0; // Accumulate sums ...
for (i=0; i < ndata; i++) {
sx += x[i]; // ...without weights.
sy += y[i];
}
ss = ndata;
sxoss = sx/ss;
for (i=0;i < ndata; i++) {
t = x[i]-sxoss;
st2 += t*t;
b += t*y[i];
}
b /= st2; // Solve for a, b, sigma-a, and sigma-b.
a = (sy-sx*b)/ss;
siga=sqrt((1.0+sx*sx/(ss*st2))/ss);
sigb=sqrt(1.0/st2); // Calculate chi2.
for (i=0;i<ndata;i++) chi2 += SQR(y[i]-a-b*x[i]);
if (ndata > 2) sigdat=sqrt(chi2/(ndata-2));
// For unweighted data evaluate typical
// sig using chi2, and adjust
// the standard deviations.
siga *= sigdat;
sigb *= sigdat;
}
};
where struct Gamma:
struct Gamma : Gauleg18 {
// Object for incomplete gamma function.
// Gauleg18 provides coefficients for Gauss-Legendre quadrature.
static const Int ASWITCH=100; When to switch to quadrature method.
static const double EPS; // See end of struct for initializations.
static const double FPMIN;
double gln;
double gammp(const double a, const double x) {
// Returns the incomplete gamma function P(a,x)
if (x < 0.0 || a <= 0.0) throw("bad args in gammp");
if (x == 0.0) return 0.0;
else if ((Int)a >= ASWITCH) return gammpapprox(a,x,1); // Quadrature.
else if (x < a+1.0) return gser(a,x); // Use the series representation.
else return 1.0-gcf(a,x); // Use the continued fraction representation.
}
double gammq(const double a, const double x) {
// Returns the incomplete gamma function Q(a,x) = 1 - P(a,x)
if (x < 0.0 || a <= 0.0) throw("bad args in gammq");
if (x == 0.0) return 1.0;
else if ((Int)a >= ASWITCH) return gammpapprox(a,x,0); // Quadrature.
else if (x < a+1.0) return 1.0-gser(a,x); // Use the series representation.
else return gcf(a,x); // Use the continued fraction representation.
}
double gser(const Doub a, const Doub x) {
// Returns the incomplete gamma function P(a,x) evaluated by its series representation.
// Also sets ln (gamma) as gln. User should not call directly.
double sum,del,ap;
gln=gammln(a);
ap=a;
del=sum=1.0/a;
for (;;) {
++ap;
del *= x/ap;
sum += del;
if (fabs(del) < fabs(sum)*EPS) {
return sum*exp(-x+a*log(x)-gln);
}
}
}
double gcf(const Doub a, const Doub x) {
// Returns the incomplete gamma function Q(a, x) evaluated
// by its continued fraction representation.
// Also sets ln (gamma) as gln. User should not call directly.
int i;
double an,b,c,d,del,h;
gln=gammln(a);
b=x+1.0-a; // Set up for evaluating continued fraction
// by modified Lentz’s method with with b0 = 0.
c=1.0/FPMIN;
d=1.0/b;
h=d;
for (i=1;;i++) {
// Iterate to convergence.
an = -i*(i-a);
b += 2.0;
d=an*d+b;
if (fabs(d) < FPMIN) d=FPMIN;
c=b+an/c;
if (fabs(c) < FPMIN) c=FPMIN;
d=1.0/d;
del=d*c;
h *= del;
if (fabs(del-1.0) <= EPS) break;
}
return exp(-x+a*log(x)-gln)*h; Put factors in front.
}
double gammpapprox(double a, double x, int psig) {
// Incomplete gamma by quadrature. Returns P(a,x) or Q(a, x),
// when psig is 1 or 0, respectively. User should not call directly.
int j;
double xu,t,sum,ans;
double a1 = a-1.0, lna1 = log(a1), sqrta1 = sqrt(a1);
gln = gammln(a);
// Set how far to integrate into the tail:
if (x > a1) xu = MAX(a1 + 11.5*sqrta1, x + 6.0*sqrta1);
else xu = MAX(0.,MIN(a1 - 7.5*sqrta1, x - 5.0*sqrta1));
sum = 0;
for (j=0;j<ngau;j++) { // Gauss-Legendre.
t = x + (xu-x)*y[j];
sum += w[j]*exp(-(t-a1)+a1*(log(t)-lna1));
}
ans = sum*(xu-x)*exp(a1*(lna1-1.)-gln);
return (psig?(ans>0.0? 1.0-ans:-ans):(ans>=0.0? ans:1.0+ans));
}
double invgammp(Doub p, Doub a);
// Inverse function on x of P(a,x) .
};
const Doub Gamma::EPS = numeric_limits<Doub>::epsilon();
const Doub Gamma::FPMIN = numeric_limits<Doub>::min()/EPS
and stuct Gauleg18:
struct Gauleg18 {
// Abscissas and weights for Gauss-Legendre quadrature.
static const Int ngau = 18;
static const Doub y[18];
static const Doub w[18];
};
const Doub Gauleg18::y[18] = {0.0021695375159141994,
0.011413521097787704,0.027972308950302116,0.051727015600492421,
0.082502225484340941, 0.12007019910960293,0.16415283300752470,
0.21442376986779355, 0.27051082840644336, 0.33199876341447887,
0.39843234186401943, 0.46931971407375483, 0.54413605556657973,
0.62232745288031077, 0.70331500465597174, 0.78649910768313447,
0.87126389619061517, 0.95698180152629142};
const Doub Gauleg18::w[18] = {0.0055657196642445571,
0.012915947284065419,0.020181515297735382,0.027298621498568734,
0.034213810770299537,0.040875750923643261,0.047235083490265582,
0.053244713977759692,0.058860144245324798,0.064039797355015485
0.068745323835736408,0.072941885005653087,0.076598410645870640,
0.079687828912071670,0.082187266704339706,0.084078218979661945,
0.085346685739338721,0.085983275670394821};
and, finally fuinction Gamma::invgamp():
double Gamma::invgammp(double p, double a) {
// Returns x such that P(a,x) = p for an argument p between 0 and 1.
int j;
double x,err,t,u,pp,lna1,afac,a1=a-1;
const double EPS=1.e-8; // Accuracy is the square of EPS.
gln=gammln(a);
if (a <= 0.) throw("a must be pos in invgammap");
if (p >= 1.) return MAX(100.,a + 100.*sqrt(a));
if (p <= 0.) return 0.0;
if (a > 1.) {
lna1=log(a1);
afac = exp(a1*(lna1-1.)-gln);
pp = (p < 0.5)? p : 1. - p;
t = sqrt(-2.*log(pp));
x = (2.30753+t*0.27061)/(1.+t*(0.99229+t*0.04481)) - t;
if (p < 0.5) x = -x;
x = MAX(1.e-3,a*pow(1.-1./(9.*a)-x/(3.*sqrt(a)),3));
} else {
t = 1.0 - a*(0.253+a*0.12); and (6.2.9).
if (p < t) x = pow(p/t,1./a);
else x = 1.-log(1.-(p-t)/(1.-t));
}
for (j=0;j<12;j++) {
if (x <= 0.0) return 0.0; // x too small to compute accurately.
err = gammp(a,x) - p;
if (a > 1.) t = afac*exp(-(x-a1)+a1*(log(x)-lna1));
else t = exp(-x+a1*log(x)-gln);
u = err/t;
// Halley’s method.
x -= (t = u/(1.-0.5*MIN(1.,u*((a-1.)/x - 1))));
// Halve old value if x tries to go negative.
if (x <= 0.) x = 0.5*(x + t);
if (fabs(t) < EPS*x ) break;
}
return x;
}
Here is my version of a C/C++ function that does simple linear regression. The calculations follow the wikipedia article on simple linear regression. This is published as a single-header public-domain (MIT) library on github: simple_linear_regression. The library (.h file) is tested to work on Linux and Windows, and from C and C++ using -Wall -Werror and all -std versions supported by clang/gcc.
#define SIMPLE_LINEAR_REGRESSION_ERROR_INPUT_VALUE -2
#define SIMPLE_LINEAR_REGRESSION_ERROR_NUMERIC -3
int simple_linear_regression(const double * x, const double * y, const int n, double * slope_out, double * intercept_out, double * r2_out) {
double sum_x = 0.0;
double sum_xx = 0.0;
double sum_xy = 0.0;
double sum_y = 0.0;
double sum_yy = 0.0;
double n_real = (double)(n);
int i = 0;
double slope = 0.0;
double denominator = 0.0;
if (x == NULL || y == NULL || n < 2) {
return SIMPLE_LINEAR_REGRESSION_ERROR_INPUT_VALUE;
}
for (i = 0; i < n; ++i) {
sum_x += x[i];
sum_xx += x[i] * x[i];
sum_xy += x[i] * y[i];
sum_y += y[i];
sum_yy += y[i] * y[i];
}
denominator = n_real * sum_xx - sum_x * sum_x;
if (denominator == 0.0) {
return SIMPLE_LINEAR_REGRESSION_ERROR_NUMERIC;
}
slope = (n_real * sum_xy - sum_x * sum_y) / denominator;
if (slope_out != NULL) {
*slope_out = slope;
}
if (intercept_out != NULL) {
*intercept_out = (sum_y - slope * sum_x) / n_real;
}
if (r2_out != NULL) {
denominator = ((n_real * sum_xx) - (sum_x * sum_x)) * ((n_real * sum_yy) - (sum_y * sum_y));
if (denominator == 0.0) {
return SIMPLE_LINEAR_REGRESSION_ERROR_NUMERIC;
}
*r2_out = ((n_real * sum_xy) - (sum_x * sum_y)) * ((n_real * sum_xy) - (sum_x * sum_y)) / denominator;
}
return 0;
}
Usage example:
#define SIMPLE_LINEAR_REGRESSION_IMPLEMENTATION
#include "simple_linear_regression.h"
#include <stdio.h>
/* Some data that we want to find the slope, intercept and r2 for */
static const double x[] = { 1.47, 1.50, 1.52, 1.55, 1.57, 1.60, 1.63, 1.65, 1.68, 1.70, 1.73, 1.75, 1.78, 1.80, 1.83 };
static const double y[] = { 52.21, 53.12, 54.48, 55.84, 57.20, 58.57, 59.93, 61.29, 63.11, 64.47, 66.28, 68.10, 69.92, 72.19, 74.46 };
int main() {
double slope = 0.0;
double intercept = 0.0;
double r2 = 0.0;
int res = 0;
res = simple_linear_regression(x, y, sizeof(x) / sizeof(x[0]), &slope, &intercept, &r2);
if (res < 0) {
printf("Error: %s\n", simple_linear_regression_error_string(res));
return res;
}
printf("slope: %f\n", slope);
printf("intercept: %f\n", intercept);
printf("r2: %f\n", r2);
return 0;
}
The original example above worked well for me with slope and offset but I had a hard time with the corr coef. Maybe I don't have my parenthesis working the same as the assumed precedence? Anyway, with some help of other web pages I finally got values that match the linear trend-line in Excel. Thought I would share my code using Mark Lakata's variable names. Hope this helps.
double slope = ((n * sumxy) - (sumx * sumy )) / denom;
double intercept = ((sumy * sumx2) - (sumx * sumxy)) / denom;
double term1 = ((n * sumxy) - (sumx * sumy));
double term2 = ((n * sumx2) - (sumx * sumx));
double term3 = ((n * sumy2) - (sumy * sumy));
double term23 = (term2 * term3);
double r2 = 1.0;
if (fabs(term23) > MIN_DOUBLE) // Define MIN_DOUBLE somewhere as 1e-9 or similar
r2 = (term1 * term1) / term23;
as an assignment I had to code in C a simple linear regression using RMSE loss function. The program is dynamic and you can enter your own values and choose your own loss function which is for now limited to Root Mean Square Error. But first here are the algorithms I used:
now the code... you need gnuplot to display the chart, sudo apt install gnuplot
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
#include <sys/types.h>
#define BUFFSIZE 64
#define MAXSIZE 100
static double vector_x[MAXSIZE] = {0};
static double vector_y[MAXSIZE] = {0};
static double vector_predict[MAXSIZE] = {0};
static double max_x;
static double max_y;
static double mean_x;
static double mean_y;
static double teta_0_intercept;
static double teta_1_grad;
static double RMSE;
static double r_square;
static double prediction;
static char intercept[BUFFSIZE];
static char grad[BUFFSIZE];
static char xrange[BUFFSIZE];
static char yrange[BUFFSIZE];
static char lossname_RMSE[BUFFSIZE] = "Simple Linear Regression using RMSE'";
static char cmd_gnu_0[BUFFSIZE] = "set title '";
static char cmd_gnu_1[BUFFSIZE] = "intercept = ";
static char cmd_gnu_2[BUFFSIZE] = "grad = ";
static char cmd_gnu_3[BUFFSIZE] = "set xrange [0:";
static char cmd_gnu_4[BUFFSIZE] = "set yrange [0:";
static char cmd_gnu_5[BUFFSIZE] = "f(x) = (grad * x) + intercept";
static char cmd_gnu_6[BUFFSIZE] = "plot f(x), 'data.temp' with points pointtype 7";
static char const *commands_gnuplot[] = {
cmd_gnu_0,
cmd_gnu_1,
cmd_gnu_2,
cmd_gnu_3,
cmd_gnu_4,
cmd_gnu_5,
cmd_gnu_6,
};
static size_t size;
static void user_input()
{
printf("Enter x,y vector size, MAX = 100\n");
scanf("%lu", &size);
if (size > MAXSIZE) {
printf("Wrong input size is too big\n");
user_input();
}
printf("vector's size is %lu\n", size);
size_t i;
for (i = 0; i < size; i++) {
printf("Enter vector_x[%ld] values\n", i);
scanf("%lf", &vector_x[i]);
}
for (i = 0; i < size; i++) {
printf("Enter vector_y[%ld] values\n", i);
scanf("%lf", &vector_y[i]);
}
}
static void display_vector()
{
size_t i;
for (i = 0; i < size; i++){
printf("vector_x[%lu] = %lf\t", i, vector_x[i]);
printf("vector_y[%lu] = %lf\n", i, vector_y[i]);
}
}
static void concatenate(char p[], char q[]) {
int c;
int d;
c = 0;
while (p[c] != '\0') {
c++;
}
d = 0;
while (q[d] != '\0') {
p[c] = q[d];
d++;
c++;
}
p[c] = '\0';
}
static void compute_mean_x_y()
{
size_t i;
double tmp_x = 0.0;
double tmp_y = 0.0;
for (i = 0; i < size; i++) {
tmp_x += vector_x[i];
tmp_y += vector_y[i];
}
mean_x = tmp_x / size;
mean_y = tmp_y / size;
printf("mean_x = %lf\n", mean_x);
printf("mean_y = %lf\n", mean_y);
}
static void compute_teta_1_grad()
{
double numerator = 0.0;
double denominator = 0.0;
double tmp1 = 0.0;
double tmp2 = 0.0;
size_t i;
for (i = 0; i < size; i++) {
numerator += (vector_x[i] - mean_x) * (vector_y[i] - mean_y);
}
for (i = 0; i < size; i++) {
tmp1 = vector_x[i] - mean_x;
tmp2 = tmp1 * tmp1;
denominator += tmp2;
}
teta_1_grad = numerator / denominator;
printf("teta_1_grad = %lf\n", teta_1_grad);
}
static void compute_teta_0_intercept()
{
teta_0_intercept = mean_y - (teta_1_grad * mean_x);
printf("teta_0_intercept = %lf\n", teta_0_intercept);
}
static void compute_prediction()
{
size_t i;
for (i = 0; i < size; i++) {
vector_predict[i] = teta_0_intercept + (teta_1_grad * vector_x[i]);
printf("y^[%ld] = %lf\n", i, vector_predict[i]);
}
printf("\n");
}
static void compute_RMSE()
{
compute_prediction();
double error = 0;
size_t i;
for (i = 0; i < size; i++) {
error = (vector_predict[i] - vector_y[i]) * (vector_predict[i] - vector_y[i]);
printf("error y^[%ld] = %lf\n", i, error);
RMSE += error;
}
/* mean */
RMSE = RMSE / size;
/* square root mean */
RMSE = sqrt(RMSE);
printf("\nRMSE = %lf\n", RMSE);
}
static void compute_loss_function()
{
int input = 0;
printf("Which loss function do you want to use?\n");
printf(" 1 - RMSE\n");
scanf("%d", &input);
switch(input) {
case 1:
concatenate(cmd_gnu_0, lossname_RMSE);
compute_RMSE();
printf("\n");
break;
default:
printf("Wrong input try again\n");
compute_loss_function(size);
}
}
static void compute_r_square(size_t size)
{
double num_err = 0.0;
double den_err = 0.0;
size_t i;
for (i = 0; i < size; i++) {
num_err += (vector_y[i] - vector_predict[i]) * (vector_y[i] - vector_predict[i]);
den_err += (vector_y[i] - mean_y) * (vector_y[i] - mean_y);
}
r_square = 1 - (num_err/den_err);
printf("R_square = %lf\n", r_square);
}
static void compute_predict_for_x()
{
double x = 0.0;
printf("Please enter x value\n");
scanf("%lf", &x);
prediction = teta_0_intercept + (teta_1_grad * x);
printf("y^ if x = %lf -> %lf\n",x, prediction);
}
static void compute_max_x_y()
{
size_t i;
double tmp1= 0.0;
double tmp2= 0.0;
for (i = 0; i < size; i++) {
if (vector_x[i] > tmp1) {
tmp1 = vector_x[i];
max_x = vector_x[i];
}
if (vector_y[i] > tmp2) {
tmp2 = vector_y[i];
max_y = vector_y[i];
}
}
printf("vector_x max value %lf\n", max_x);
printf("vector_y max value %lf\n", max_y);
}
static void display_model_line()
{
sprintf(intercept, "%0.7lf", teta_0_intercept);
sprintf(grad, "%0.7lf", teta_1_grad);
sprintf(xrange, "%0.7lf", max_x + 1);
sprintf(yrange, "%0.7lf", max_y + 1);
concatenate(cmd_gnu_1, intercept);
concatenate(cmd_gnu_2, grad);
concatenate(cmd_gnu_3, xrange);
concatenate(cmd_gnu_3, "]");
concatenate(cmd_gnu_4, yrange);
concatenate(cmd_gnu_4, "]");
printf("grad = %s\n", grad);
printf("intercept = %s\n", intercept);
printf("xrange = %s\n", xrange);
printf("yrange = %s\n", yrange);
printf("cmd_gnu_0: %s\n", cmd_gnu_0);
printf("cmd_gnu_1: %s\n", cmd_gnu_1);
printf("cmd_gnu_2: %s\n", cmd_gnu_2);
printf("cmd_gnu_3: %s\n", cmd_gnu_3);
printf("cmd_gnu_4: %s\n", cmd_gnu_4);
printf("cmd_gnu_5: %s\n", cmd_gnu_5);
printf("cmd_gnu_6: %s\n", cmd_gnu_6);
/* print plot */
FILE *gnuplot_pipe = (FILE*)popen("gnuplot -persistent", "w");
FILE *temp = (FILE*)fopen("data.temp", "w");
/* create data.temp */
size_t i;
for (i = 0; i < size; i++)
{
fprintf(temp, "%f %f \n", vector_x[i], vector_y[i]);
}
/* display gnuplot */
for (i = 0; i < 7; i++)
{
fprintf(gnuplot_pipe, "%s \n", commands_gnuplot[i]);
}
}
int main(void)
{
printf("===========================================\n");
printf("INPUT DATA\n");
printf("===========================================\n");
user_input();
display_vector();
printf("\n");
printf("===========================================\n");
printf("COMPUTE MEAN X:Y, TETA_1 TETA_0\n");
printf("===========================================\n");
compute_mean_x_y();
compute_max_x_y();
compute_teta_1_grad();
compute_teta_0_intercept();
printf("\n");
printf("===========================================\n");
printf("COMPUTE LOSS FUNCTION\n");
printf("===========================================\n");
compute_loss_function();
printf("===========================================\n");
printf("COMPUTE R_square\n");
printf("===========================================\n");
compute_r_square(size);
printf("\n");
printf("===========================================\n");
printf("COMPUTE y^ according to x\n");
printf("===========================================\n");
compute_predict_for_x();
printf("\n");
printf("===========================================\n");
printf("DISPLAY LINEAR REGRESSION\n");
printf("===========================================\n");
display_model_line();
printf("\n");
return 0;
}
Look at Section 1 of this paper. This section expresses a 2D linear regression as a matrix multiplication exercise. As long as your data is well-behaved, this technique should permit you to develop a quick least squares fit.
Depending on the size of your data, it might be worthwhile to algebraically reduce the matrix multiplication to simple set of equations, thereby avoiding the need to write a matmult() function. (Be forewarned, this is completely impractical for more than 4 or 5 data points!)
The fastest, most efficient way to solve least squares, as far as I am aware, is to subtract (the gradient)/(the 2nd order gradient) from your parameter vector. (2nd order gradient = i.e. the diagonal of the Hessian.)
Here is the intuition:
Let's say you want to optimize least squares over a single parameter. This is equivalent to finding the vertex of a parabola. Then, for any random initial parameter, x0, the vertex of the loss function is located at x0 - f(1) / f(2). That's because adding - f(1) / f(2) to x will always zero out the derivative, f(1).
Side note: Implementing this in Tensorflow, the solution appeared at w0 - f(1) / f(2) / (number of weights), but I'm not sure if that's due to Tensorflow or if it's due to something else..