計算趨勢線並預測未來結果

Question

我正在爲我正在編寫的網站編寫一些分析模塊，我需要估計當前小時後的總視圖。我有每分鐘直至當前分鐘的數據，因此，如果時間是12:28，我將有一個數組，它看起來是這樣的：

0: "21410" 
1: "21886" 
2: "21837" 
3: "21895" 
4: "21564" 
5: "21714" 
6: "21571" 
7: "21324" 
8: "21310" 
9: "21390" 
10: "21764" 
11: "21598" 
12: "21493" 
13: "21352" 
14: "21478" 
15: "21058" 
16: "20942" 
17: "20825" 
18: "21321" 
19: "20950" 
20: "21039" 
21: "21117" 
22: "20733" 
23: "20773" 
24: "20929" 
25: "20900" 
26: "20687" 
27: "20999" 

目前我投射小時的值是這樣的：

(60/minsSoFar)*totalSoFar 

這工作相當好，但我寧願做更多的數學。我想計算一下最適合我迄今爲止的數據，並預測到第60分鐘的時間。這將考慮加速和減速。

用我目前使用的方法，我實際上假設趨勢是一條直線。我將如何計算多項式或功率趨勢的公式？

我在NodeJS上寫這個，所以JavaScript會很理想，但我也會採用僞代碼！

下面是情況下，更容易的格式陣列你想讓它：

[21410, 21886, 21837, 21895, 21564, 21714, 21571, 21324, 21310, 21390, 21764, 21598, 21493, 21352, 21478, 21058, 20942, 20825, 21321, 20950, 21039, 21117, 20733, 20773, 20929, 20900, 20687, 20999] 

感謝您的幫助！

Answer 1

你可以做一條線least-squares fit。

function LineFitter() 
{ 
    this.count = 0; 
    this.sumX = 0; 
    this.sumX2 = 0; 
    this.sumXY = 0; 
    this.sumY = 0; 
} 

LineFitter.prototype = { 
    'add': function(x, y) 
    { 
     this.count++; 
     this.sumX += x; 
     this.sumX2 += x*x; 
     this.sumXY += x*y; 
     this.sumY += y; 
    }, 
    'project': function(x) 
    { 
     var det = this.count * this.sumX2 - this.sumX * this.sumX; 
     var offset = (this.sumX2 * this.sumY - this.sumX * this.sumXY)/det; 
     var scale = (this.count * this.sumXY - this.sumX * this.sumY)/det; 
     return offset + x * scale; 
    } 
}; 

function linearProject(data, x) 
{ 
    var fitter = new LineFitter(); 
    for (var i = 0; i < data.length; i++) 
    { 
     fitter.add(i, data[i]); 
    } 
    return fitter.project(x); 
} 

例子：

>>> linearProject([ 
     21410, 21886, 21837, 21895, 21564, 21714, 21571, 21324, 21310, 21390, 
     21764, 21598, 21493, 21352, 21478, 21058, 20942, 20825, 21321, 20950, 
     21039, 21117, 20733, 20773, 20929, 20900, 20687, 20999 
    ], 60); 
19489.614121510676 

---

做了二次多項式類似的東西是有點複雜：

function SquareFitter() 
{ 
    this.count = 0; 
    this.sumX = 0; 
    this.sumX2 = 0; 
    this.sumX3 = 0; 
    this.sumX4 = 0; 
    this.sumY = 0; 
    this.sumXY = 0; 
    this.sumX2Y = 0; 
} 

SquareFitter.prototype = { 
    'add': function(x, y) 
    { 
     this.count++; 
     this.sumX += x; 
     this.sumX2 += x*x; 
     this.sumX3 += x*x*x; 
     this.sumX4 += x*x*x*x; 
     this.sumY += y; 
     this.sumXY += x*y; 
     this.sumX2Y += x*x*y; 
    }, 
    'project': function(x) 
    { 
     var det = this.count*this.sumX2*this.sumX4 - this.count*this.sumX3*this.sumX3 - this.sumX*this.sumX*this.sumX4 + 2*this.sumX*this.sumX2*this.sumX3 - this.sumX2*this.sumX2*this.sumX2; 
     var offset = this.sumX*this.sumX2Y*this.sumX3 - this.sumX*this.sumX4*this.sumXY - this.sumX2*this.sumX2*this.sumX2Y + this.sumX2*this.sumX3*this.sumXY + this.sumX2*this.sumX4*this.sumY - this.sumX3*this.sumX3*this.sumY; 
     var scale = -this.count*this.sumX2Y*this.sumX3 + this.count*this.sumX4*this.sumXY + this.sumX*this.sumX2*this.sumX2Y - this.sumX*this.sumX4*this.sumY - this.sumX2*this.sumX2*this.sumXY + this.sumX2*this.sumX3*this.sumY; 
     var accel = this.sumY*this.sumX*this.sumX3 - this.sumY*this.sumX2*this.sumX2 - this.sumXY*this.count*this.sumX3 + this.sumXY*this.sumX2*this.sumX - this.sumX2Y*this.sumX*this.sumX + this.sumX2Y*this.count*this.sumX2; 
     return (offset + x*scale + x*x*accel)/det; 
    } 
}; 

function squareProject(data) 
{ 
    var fitter = new SquareFitter(); 
    for (var i = 0; i < data.length; i++) 
    { 
     fitter.add(i, data[i]); 
    } 
    return fitter.project(60); 
} 

例2：

>>> squareProject([ 
     21410, 21886, 21837, 21895, 21564, 21714, 21571, 21324, 21310, 21390, 
     21764, 21598, 21493, 21352, 21478, 21058, 20942, 20825, 21321, 20950, 
     21039, 21117, 20733, 20773, 20929, 20900, 20687, 20999 
    ], 60); 
19282.85862700518 

我可以做更高階的多項式，但表達式會變得更長。對於任意的程度，你將不得不看矩陣。