C 特征:样条衍生物(）给出奇怪的衍生物
c++ Eigen: spline derivatives() gives strange derivatives
我正在尝试掌握如何在特征中使用的花键，特别是我想在某个时候找到样条线插值及其第一个和第二个衍生物的值。找到插值值很容易，但是当我尝试计算衍生物时，我会得到奇怪的值。
我尝试按照手册中的derivatives命令的说明(http://eigen.tuxfamily.org/dox/dox/unsupported/classeigen_1spline.html#af3586ab192995959e01bfe701bfe7daa4015555555c6(，以及以下<classeigen_1spline.html#af3586ab1929959e>它输出
derivative at x = 0: 0 0 32
derivative at x = 1: 1 8 32
derivative at x = 2: 4 16 32
derivative at x = 3: 9 24 32
derivative at x = 4: 16 32 32
也就是说，插值是正确的(c.f. x = 3(，但是衍生物不是，它们以系统的方式脱离，所以我认为我在做错事。由于这些跟随x^2，因此衍生物应为 0,2,4,6,8，第二阶导数应为 2。
关于如何解决这个问题的想法？
编辑1
将x^2更改为x^2 + 1产生相同的衍生物，因此至少检查了。但是将x^2更改为x^3是错误的，但是以稍微不同的方式错误，输出将是：
derivative at x = 2: 8 48 192
derivative at x = 3: 27 108 288
derivative at x = 4: 64 192 384
这是错误的，应该是6, 9, 12。
还运行x^2情况，但是将HE输入向量更改为0,1,...9产生的导数与使用原始输入向量相同，但是第二阶导数变为稳定的200，这也是错误的。我看不出为什么第二阶导数应取决于输入点的数量。
解决了它。你很近。您所要做的就是扩展衍生品用
-
1 / (x_max - x_min)(第一个导数( -
1 / (x_max - x_min)^2(第二个衍生物(。
tldr ：在安装样条时，x值在0到1之间，但是您没有扩展y值。
而不是拟合x^2的样条曲线，您实际上安装了：
x_norm = (x - x_min) / (x_max - x_min)
y = x_norm**2
因此，使用链条规则，y = x_norm**2的第一个衍生物是2x / (x_max - x_min)，第二个导数为2 / (x_max - x_min)**2。
完整的示例代码：
#include <iostream>
#include <Eigen/Core>
#include <unsupported/Eigen/Splines>
using namespace Eigen;
using namespace std;
VectorXd normalize(const VectorXd &x) {
VectorXd x_norm;
x_norm.resize(x.size());
const double min = x.minCoeff();
const double max = x.maxCoeff();
for (int k = 0; k < x.size(); k++) {
x_norm(k) = (x(k) - min)/(max - min);
}
return x_norm;
}
int main() {
typedef Spline<double, 1, 3> Spline1D;
typedef SplineFitting<Spline1D> Spline1DFitting;
const Vector4d x{0, 1, 2, 4};
const Vector4d y = (x.array().square()); // x^2
const auto knots = normalize(x); // Normalize x to be between 0 and 1
const double scale = 1 / (x.maxCoeff() - x.minCoeff());
const double scale_sq = scale * scale;
Spline1D spline = Spline1DFitting::Interpolate(y.transpose(), 3, knots);
cout << "1st deriv at x = 0: " << spline.derivatives(0.00, 1)(1) * scale << endl;
cout << "1st deriv at x = 1: " << spline.derivatives(0.25, 1)(1) * scale << endl;
cout << "1st deriv at x = 2: " << spline.derivatives(0.50, 1)(1) * scale << endl;
cout << "1st deriv at x = 3: " << spline.derivatives(0.75, 1)(1) * scale << endl;
cout << "1st deriv at x = 4: " << spline.derivatives(1.00, 1)(1) * scale << endl;
cout << endl;
/**
* IMPORTANT NOTE: Eigen's spline module is not documented well. Once you fit a spline
* to find the derivative of the fitted spline at any point u [0, 1] you call:
*
* spline.derivatives(u, 1)(1)
* ^ ^ ^
* | | |
* | | +------- Access the result
* | +---------- Derivative order
* +------------- Parameter u [0, 1]
*
* The last bit `(1)` is if the spline is 1D. And value of `1` for the first
* order. `2` for the second order. Do not forget to scale the result.
*
* For higher dimensions, treat the return as a matrix and grab the 1st or
* 2nd column for the first and second derivative.
*/
cout << "2nd deriv at x = 0: " << spline.derivatives(0.00, 2)(2) * scale_sq << endl;
cout << "2nd deriv at x = 1: " << spline.derivatives(0.25, 2)(2) * scale_sq << endl;
cout << "2nd deriv at x = 2: " << spline.derivatives(0.50, 2)(2) * scale_sq << endl;
cout << "2nd deriv at x = 3: " << spline.derivatives(0.75, 2)(2) * scale_sq << endl;
cout << "2nd deriv at x = 4: " << spline.derivatives(1.00, 2)(2) * scale_sq << endl;
return 0;
}
示例输出：
1st deriv at x = 0: 4.52754e-16
1st deriv at x = 1: 2
1st deriv at x = 2: 4
1st deriv at x = 3: 6
1st deriv at x = 4: 8
2nd deriv at x = 0: 2
2nd deriv at x = 1: 2
2nd deriv at x = 2: 2
2nd deriv at x = 3: 2
2nd deriv at x = 4: 2
编辑：有关计算底部任何顺序的b-splines的工作.h文件。
免责声明：这不是我问题的答案，因为它实际上是标题中所述的，而是一个有一些评论的工作。
在与用户@paul H.进行审议之后(请参阅评论(，我意识到我对花样的理解有限可能会引起我的困惑。经过对本特征文档的审查，derivative()命令确实确实按预期起作用，因此使我的问题误解了。derivative()计算 spline 的衍生物，而不是拟合函数的衍生物，如我的意图。我没有找到一种方法来使特征从拟合中输出功能衍生物，而且我认为它并不是为此而设计的。但是，一旦使用一些用于计算衍生物的标准算法获得拟合点后，可以很容易地计算衍生物。
我写了以下.h文件，用于在我认为值得分享的过程中计算花键及其衍生物。为了方便起见，评论得很好。
请注意，该程序使用1个索引而不是花样的0个点，因此用于例如二次B-Spline order应将其设置为4。这是一个小怪癖，可以通过更改计算以匹配Wikipedia来容易固定。
bsplines.h
//
// Header-file for calculating splines using standard libraries
//
// usage:
//
// x bsplines class constructs a set of splines up to some given order,
// on some given knot sequence, the splines are stored in a vector,
// such that splines[a][b] accesses the spline of order a and index b
// x get<some_member>() is an accessor that returns a pointer to some
// data member of the spline
// x calcsplines() calculates spline values as well as first and second
// order derivatives on some predefined grid
// x calcspline() returns the spline value as well as first and second
// derivatives in some point. This alborithm is slower than the grid
// one, due to unnecessary recalculations of intermediate results
// x writesplines() writes the splines and their derivatives to a file
// x for more details se the class declaration below
// TODO:
// x change to 0-indexation
// x introduce the possibility of calculating higher order derivatives
// recursively
//
// change log:
//
// 1.0 - initial release
// 1.1 - reworked grid such that the class now expects separate
// grid and knot files.
// - added the ability to calculate spline value in a point
// rather than calculate values on a grid
// - added a feature to change knots and grid
// 1.1.1 - reworked how returning single values works
// 1.1.2 - enabled swapping grid
//
// Note:
//
// This file uses 1-indexation rathar than 0-indexation, hence a qubic spline
// would be k = 4. Someone should eventually fix this as this is non-standard.
//
// Also, while only standard libraries are used here, you might want to check out
// some linear algebra package (e.g. Armadillo or Eigen) if you're going to use the
// splines in a context where you need linear algebraic operations.
//
// Originally developed by David Andersson
//
#include <iomanip>
#include <sstream>
#include <iostream>
#include <vector>
#include <algorithm>
#include <fstream>
#include <functional>
using namespace std;
typedef unsigned int uint;
class bsplines // class for bsplines
{
// private section
uint order; // order of spline
uint gridpts; // number of grid points
uint knotpts; // number of knot points
double tolerance; // tolerance for float comparisons
vector<double> knots; // knot sequence
vector<double> grid; // grid points
class spline // a member spline in the set of splines
{
int index; // the spline index, or number
vector<double> vals; // spline values
vector<double> d1; // spline first derivatives
vector<double> d2; // spline second derivatives
double tval; // same, but in one point
double td1;
double td2;
friend bsplines; // for ease of access
public:
};
vector<vector <spline>> splines; // the set of splines
// puclic section
public:
void readknots(string); // read knots from file
void readknotsnorm(string); // read knots from file and normalize
void readgrid(string); // read grid from file
void swapgrid(string); // reads and swaps new grid from file
void writesplines(); // write spline vals and derivs to file
void buildsplines(); // build the set of splines
void calcsplines(); // calculate spline vals and derivs
void printknots(); // print knot sequence
void printgrid(); // print grid
void printgridsize(); // print gridsize
void printvals(uint,uint); // print values of a spline
vector <double> calcspline(uint,uint,double); // calculate spline in point
// accessors // returns pointer to member
vector <double>* getknots(){return &knots;}
vector <double>* getgrid(){return &grid;}
uint* getknotpts(){return &knotpts;}
uint* getgridpts(){return &gridpts;}
uint getnosplines(uint m){return splines[m].size();}
vector <spline>* getsplines(uint m){return &splines[m];}
vector <double>* getvals(uint m, uint n){return &splines[m][n].vals;}
vector <double>* getd1(uint m, uint n){return &splines[m][n].d1;}
vector <double>* getd2(uint m, uint n){return &splines[m][n].d2;}
// constructor // sets up the spline class
bsplines (string iknots, string igrid, uint iorder, double itol)
:order(iorder), tolerance(itol)
{
readknots(iknots);
readgrid(igrid);
buildsplines();
}
};
void bsplines::buildsplines()
{
{
for (uint l = 1; l <= order; l++)
{
vector <spline> splinevec;
for (uint k = 0; k < knotpts - l; k++)
{
spline tmp;
tmp.index = k;
tmp.vals.reserve(gridpts);
tmp.d1.reserve(gridpts);
tmp.d2.reserve(gridpts);
splinevec.push_back(tmp);
}
splines.push_back(splinevec);
}
}
}
vector <double> bsplines::calcspline(uint m, uint n, double x)
{
// first order splines // exceptions handles infinities
for (auto& sp : splines[0])
{
uint i = sp.index;
if (x > knots[i+1])
sp.tval = 0;
else if ((x >= knots[i] && x < knots[i+1]) || x == knots.back())
sp.tval = 1;
else
sp.tval = 0;
}
// higher order splines
for (uint o = 1; o < order; o++)
{
uint oo = o+1; // compensating for 1-indexation
for (auto& sp : splines[o])
{
uint i = sp.index;
double t1 = knots[i+oo-1] - knots[i];
double t2 = knots[i+oo] - knots[i+1];
double c = 0;
if (abs(t1) > tolerance)
c += (x - knots[i]) / t1 * splines[o-1][i].tval;
if (abs(t2) > tolerance)
c += (knots[i+oo] - x) / t2 * splines[o-1][i+1].tval;
sp.tval = c;
}
}
uint o = order - 1;
// first order derivatives
for (auto& sp : splines[o])
{
uint i = sp.index;
double t1 = knots[i+order-1] - knots[i];
double t2 = knots[i+order] - knots[i+1];
double c = 0;
if (abs(t1) > tolerance)
c += 1.0 / t1 * splines[o-1][i].tval;
if (abs(t2) > tolerance)
c -= 1.0 / t2 * splines[o-1][i+1].tval;
c *= (order-1);
sp.td1 = c;
}
// second order derivatives
for (auto& sp : splines[o])
{
uint i = sp.index;
double t1 = (knots[i+order-1] - knots[i+0]) * (knots[i+order-2] - knots[i+0]);
double t2 = (knots[i+order-1] - knots[i+0]) * (knots[i+order-1] - knots[i+1]);
double t3 = (knots[i+order-0] - knots[i+1]) * (knots[i+order-1] - knots[i+1]);
double t4 = (knots[i+order-0] - knots[i+1]) * (knots[i+order-0] - knots[i+2]);
double c = 0;
if (abs(t1) > tolerance)
c += 1.0 / t1 * splines[o-2][sp.index].tval;
if (abs(t2) > tolerance)
c -= 1.0 / t2 * splines[o-2][sp.index+1].tval;
if (abs(t3) > tolerance)
c -= 1.0 / t3 * splines[o-2][sp.index+1].tval;
if (abs(t4) > tolerance)
c += 1.0 / t4 * splines[o-2][sp.index+2].tval;
c *= (order-1)*(order-2);
sp.td2 = c;
}
vector <double> retvals = {splines[m][n].tval, splines[m][n].td1, splines[m][n].td2};
return retvals;
}
void bsplines::calcsplines()
{
// first order splines
for (auto& sp : splines[0])
{
uint i = sp.index;
for (auto& x : grid)
{
if (x > knots[i+1])
sp.vals.push_back(0);
else if ((x >= knots[i] && x < knots[i+1]) || x == knots.back())
sp.vals.push_back(1);
else
sp.vals.push_back(0);
}
}
// higher order splines
for (uint o = 1; o < order; o++)
{
uint oo = o+1; // compensating for 1-indexation
for (auto& sp : splines[o])
{
uint i = sp.index;
double t1 = knots[i+oo-1] - knots[i];
double t2 = knots[i+oo] - knots[i+1];
for (auto& x : grid)
{
uint k = &x - &grid[0];
double c = 0;
if (abs(t1) > tolerance)
c += (x - knots[i]) / t1 * splines[o-1][i].vals[k];
if (abs(t2) > tolerance)
c += (knots[i+oo] - x) / t2 * splines[o-1][i+1].vals[k];
sp.vals.push_back(c);
}
}
}
uint o = order - 1; // use this one when accessing splines;
// first order derivatives
for (auto& sp : splines[o])
{
uint i = sp.index;
double t1 = knots[i+order-1] - knots[i];
double t2 = knots[i+order] - knots[i+1];
for (auto& x : grid)
{
uint k = &x - &grid[0];
double c = 0;
if (abs(t1) > tolerance)
c += 1.0 / t1 * splines[o-1][i].vals[k];
if (abs(t2) > tolerance)
c -= 1.0 / t2 * splines[o-1][i+1].vals[k];
c *= (order-1);
sp.d1.push_back(c);
}
}
// second order derivatives
for (auto& sp : splines[o])
{
uint i = sp.index;
double t1 = (knots[i+order-1] - knots[i+0]) * (knots[i+order-2] - knots[i+0]);
double t2 = (knots[i+order-1] - knots[i+0]) * (knots[i+order-1] - knots[i+1]);
double t3 = (knots[i+order-0] - knots[i+1]) * (knots[i+order-1] - knots[i+1]);
double t4 = (knots[i+order-0] - knots[i+1]) * (knots[i+order-0] - knots[i+2]);
for (auto& x : grid)
{
uint k = &x - &grid[0];
double c = 0;
if (abs(t1) > tolerance)
c += 1.0 / t1 * splines[o-2][sp.index].vals[k];
if (abs(t2) > tolerance)
c -= 1.0 / t2 * splines[o-2][sp.index+1].vals[k];
if (abs(t3) > tolerance)
c -= 1.0 / t3 * splines[o-2][sp.index+1].vals[k];
if (abs(t4) > tolerance)
c += 1.0 / t4 * splines[o-2][sp.index+2].vals[k];
c *= (order-1)*(order-2);
sp.d2.push_back(c);
}
}
}
void bsplines::readknots(string knotfile)
{
double x;
ifstream readknots(knotfile);
while (readknots >> x)
knots.push_back(x);
for (uint k = 0; k < order - 1; k++)
{
knots.insert(knots.begin(),knots.front());
knots.insert(knots.end(),knots.back());
}
knotpts = knots.size();
}
void bsplines::readknotsnorm(string knotfile)
{
double x;
knots.reserve(knotpts + 2*(order - 1));
ifstream readknots(knotfile);
while (readknots >> x)
knots.push_back(x);
auto minmax = minmax_element(begin(knots), end(knots));
double min = *(minmax.first);
double max = *(minmax.second);
for (auto& el : knots)
el = (el - min) / (max-min);
}
void bsplines::readgrid(string gridfile)
{
double x;
ifstream readgrid(gridfile);
while (readgrid >> x)
grid.push_back(x);
gridpts = grid.size();
}
void bsplines::swapgrid(string gridfile)
{
grid = {};
double x;
ifstream readgrid(gridfile);
while (readgrid >> x)
grid.push_back(x);
gridpts = grid.size();
}
void bsplines::printknots()
{
cout << "content in knot vector: " << endl;
for (auto& el : knots)
cout << el << " ";
cout << endl;
}
void bsplines::printgrid()
{
cout << "content in grid vector: " << endl;
for (auto& el : grid)
cout << el << " ";
cout << endl;
}
void bsplines::printgridsize()
{
cout << "number of grid points: " << endl << grid.size() << endl;
}
void bsplines::printvals(uint m, uint n)
{
cout << "content in spline (B" << m << "," << n << ") vals vector: " << endl;
for (auto& el : splines[n][m].vals)
cout << el << " ";
cout << endl;
}
void bsplines::writesplines()
{
for (uint o = 0; o < order; o++)
for (auto& sp : splines[o])
{
uint i = sp.index;
ostringstream namestream;
namestream << "B(" << fixed << setprecision(1) << i << ","
<< fixed << setprecision(1) << o << ").csv";
string filename = namestream.str();
ofstream fs;
fs.open(filename);
if (o < order - 1)
{
for (uint k = 0; k < sp.vals.size(); k++)
fs << sp.vals[k] << "," << 0 << "," << 0 << endl;
fs.close();
}
else
{
for (uint k = 0; k < sp.vals.size(); k++)
fs << sp.vals[k] << "," << sp.d1[k] << "," << sp.d2[k] << endl;
fs.close();
}
cout << "write " << sp.vals.size() << " numbers to " << filename << endl;
}
}
编辑：更新.h-file。
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