JavaScript中的Cubic回归（最佳拟合线） [英] Cubic Regression (best fit line) in JavaScript

问题描述

我正在努力寻找可以让我做立方回归的JavaScript代码。会自己写，但我对多项式数学的理解是不理想的。

I'm having the worst time trying to find a JavaScript code that could allow me to do cubic regressions. Would write it myself, but my understanding of polynomial math is, well, suboptimal.

所以，这就是我要找的东西。给定一个数组数组的输入，其中内部数组是[x，y]，该函数将给出一个带有四个参数的数组形式的输出 - [a，b，c，d]，其中a ，b，c和d是等式的参数y = ax ^ 3 + bx ^ 2 + cx + d。

So, here's what I'm looking for. Given an input of an array of arrays, where the internal array would be [x,y], the function would give me an output in the form of an array with four parameters - [a,b,c,d], where a, b, c, and d are parameters of the equation y = ax^3 + bx^2 + cx + d.

示例：
输入是一个这样的数组[[2,5]，[5,10]，[07,15]，[12,20]，[20,25]，[32,30]，[50,35]]。

Example: Input is an array like this [[2,5],[5,10],[07,15],[12,20],[20,25],[32,30],[50,35]].

基本上是表格的表示形式：

Which essentially is the representation of a table:

|    x   |   y    |
|-----------------|
|   02   |   05   |
|   05   |   10   |
|   07   |   15   |
|   12   |   20   |
|   20   |   25   |
|   32   |   30   |
|   50   |   35   |

现在，输出将是[0.000575085，-0.058861065,2.183957502,1.127605507]。这些是三次函数的a，b，c和d参数。

Now, the output would be [0.000575085,-0.058861065,2.183957502,1.127605507]. These are the a, b, c, and d parameters of the cubic function.

（仅供参考，我使用Excel的LINEST函数并在上面运行它得到的输出使用数组函数{1,2,3}）的数字集。

(FYI, the output I got by using Excel's LINEST function and running it on the above set of numbers using an array function {1,2,3}).

如何做到这一点？非常感谢任何指导。

How could this be done? Huge thanks in advance for any guidance.

最好，
汤姆

推荐答案

这是使用 numeric.js 库解决该立方体的真实工作代码 uncmin 无约束最小化器作为最小二乘问题（ jsbin here ）：

Here's a real, working bit of code to solve that cubic using the numeric.js library's uncmin unconstrained minimiser as a least squares problem (jsbin here):

var data_x = [2,5,7,12,20,32,50];
var data_y = [5,10,15,20,25,30,35];

var cubic = function(params,x) {
  return params[0] * x*x*x +
    params[1] * x*x +
    params[2] * x +
    params[3];
};

var objective = function(params) {
  var total = 0.0;
  for(var i=0; i < data_x.length; ++i) {
    var resultThisDatum = cubic(params, data_x[i]);
    var delta = resultThisDatum - data_y[i];
    total += (delta*delta);
  }
  return total;
};

var initial = [1,1,1,1];
var minimiser = numeric.uncmin(objective,initial);

console.log("initial:");
for(var j=0; j<initial.length; ++j) {
  console.log(initial[j]);  
}

console.log("minimiser:");
for(var j=0; j<minimiser.solution.length; ++j) {
  console.log(minimiser.solution[j]);
}

我得到的结果：

 0.0005750849851827991
-0.05886106462847641
 2.1839575020602164
 1.1276055079334206

要解释：我们有一个函数'cubic'，它评估一组参数 params 的一般三次函数和一个值 X 。该函数被包装以创建目标函数，该函数接受一组参数并通过目标函数运行我们的数据集中的每个x值并计算平方和。此函数从numeric.js传递给 uncmin ，并带有一组初始值; uncmin 做了很多工作并返回一个解决方案属性包含优化参数集的对象。

To explain: we have a function 'cubic', which evaluates the general cubic function for a set of parameters params and a value x. This function is wrapped to create the objective function, which takes a set of params and runs each x value from our data set through the target function and calculates the sum of the squares. This function is passed to uncmin from numeric.js with a set of initial values; uncmin does the hard work and returns an object whose solution property contains the optimised parameter set.

要做到这一点而没有全局变量（顽皮！），你可以拥有一个目标函数工厂：

To do this without the global variables (naughty!), you can have an objective function factory thus:

var makeObjective = function(targetFunc,xlist,ylist) {
  var objective = function(params) {
    var total = 0.0;
    for(var i=0; i < xlist.length; ++i) {
      var resultThisDatum = targetFunc(params, xlist[i]);
      var delta = resultThisDatum - ylist[i];
      total += (delta*delta);
    }
    return total;
  };
  return objective;
};

您可以用它来制造目标函数：

Which you can use to manufacture objective functions:

var objective = makeObjective(cubic, data_x, data_y); // then carry on as before

知道如何做到这一点实际上会对我们有很大的帮助人们，所以我很高兴这已经出现了。

Knowing how to do this practically would be of great help to a lot of people, so I'm glad this has come up.

编辑：澄清立方

var cubic = function(params,x) {
  return params[0] * x*x*x +
    params[1] * x*x +
    params[2] * x +
    params[3];
};

Cubic被定义为一个函数，它接受一个参数数组 params 和值 x 。给定 params ，我们可以定义一个函数 f（x）。对于一个立方体，即 f（x）= ax ^ 3 + bx ^ 2 + cx + d 所以有4个参数（ [0] 到 [3] ），给定这4个参数值我们只有一个函数 f（x） 1输入 x 。

Cubic is being defined as a function which takes an array of parameters params and a value x. Given params, we can define a function f(x). For a cubic, that is f(x) = a x^3 + b x^2 + c x + d so there are 4 parameters ([0] to [3]), and given those 4 param values we have a single function f(x) with 1 input x.

该代码的结构允许您将 cubic 替换为具有相同结构的另一个函数;它可以是线性，带有2个参数：

The code is structured to allow you to replace cubic with another function of the same structure; it could be linear with 2 parameters:

var linear = function(params, x) {
    return params[0]*x + params[1];
};

其余代码将查看 params 为了知道需要修改多少个参数。

The rest of the code will look at the length of params in order to know how many parameters need modifying.

注意这整段代码试图找到产生曲线的参数值集合最适合所有数据;如果你想找到适合某些数据的最后4个点，你只能传递 data_x 和 data_y 。

Note that this whole piece of code is trying to find the set of parameter values which produce a curve which best fits all the data; if you wanted to find a fit for the last 4 points of some data, you would pass only those values in data_x and data_y.

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