/// <summary>
/// 提供正態分布的數據和圖片
/// </summary>
public class StandardDistribution
{
/// <summary>
/// 樣本數據
/// </summary>
public List<double> Xs { get; private set; }
public StandardDistribution(List<double> Xs)
{
this.Xs = Xs;
Average = Xs.Average();
Variance = GetVariance(Xs);
if (Variance == 0) throw new Exception("方差為0");//此時不需要統計 因為每個樣本數據都相同，可以在界面做相應提示
StandardVariance = Math.Sqrt(Variance);
}
/// <summary>
/// 方差/標准方差的平方
/// </summary>
public double Variance { get; private set; }
/// <summary>
/// 標准方差
/// </summary>
public double StandardVariance { get; private set; }
/// <summary>
/// 算數平均值/數學期望
/// </summary>
public double Average { get; private set; }
/// <summary>
/// 1/ (2π的平方根)的值
/// </summary>
public static double InverseSqrt2PI = 1 / Math.Sqrt(2 * Math.PI);
/// <summary>
/// 獲取指定X值的Y值 計算正太分布的公式
/// </summary>
/// <param name="x"></param>
/// <returns></returns>
public double GetGaussianDistributionY(double x)
{
double PowOfE = -(Math.Pow(Math.Abs(x - Average), 2) / (2 * Variance));
double result = (StandardDistribution.InverseSqrt2PI / StandardVariance) * Math.Pow(Math.E, PowOfE);
return result;
}
/// <summary>
/// 獲取正太分布的坐標<x,y>
/// </summary>
/// <returns></returns>
public List<Tuple<double, double>> GetGaussianDistributionYs()
{
List<Tuple<double, double>> XYs = new List<Tuple<double, double>>();
Tuple<double, double> xy = null;
foreach (double x in Xs)
{
xy = new Tuple<double, double>(x, GetGaussianDistributionY(x));
XYs.Add(xy);
}
return XYs;
}
/// <summary>
/// 獲取整型列表的方差
/// </summary>
/// <param name="src">要計算方差的數據列表</param>
/// <returns></returns>
public static double GetVariance(List<double> src)
{
double average = src.Average();
double SumOfSquares = 0;
src.ForEach(x => { SumOfSquares += Math.Pow(x - average, 2); });
return SumOfSquares / src.Count;//方差
}
/// <summary>
/// 獲取整型列表的方差
/// </summary>
/// <param name="src">要計算方差的數據列表</param>
/// <returns></returns>
public static float GetVariance(List<float> src)
{
float average = src.Average();
double SumOfSquares = 0;
src.ForEach(x => { SumOfSquares += Math.Pow(x - average, 2); });
return (float)SumOfSquares / src.Count;//方差
}
/// <summary>
/// 畫學生成績的正態分布
/// </summary>
/// <param name="Width"></param>
/// <param name="Height"></param>
/// <param name="Scores">分數，Y值</param>
/// <param name="familyName"></param>
/// <returns></returns>
public Bitmap GetGaussianDistributionGraph(int Width, int Height,int TotalScore, string familyName = "宋體")
{
//橫軸 分數；縱軸 正態分布的值
Bitmap bitmap = new Bitmap(Width, Height);
Graphics gdi = Graphics.FromImage(bitmap);
gdi.Clear(Color.White);
gdi.SmoothingMode = SmoothingMode.HighQuality;
gdi.TextRenderingHint = TextRenderingHint.ClearTypeGridFit;
gdi.PixelOffsetMode = PixelOffsetMode.HighQuality;
List<Tuple<double, double>> Scores = GetGaussianDistributionYs().OrderBy(x => x.Item1).ToList();//排序 方便后面點與點之間的連線 保證 分數低的 在左邊
float YHeight = 0.8F * Height;// 相對左下角 YHeight*0.9F 將表示 maxY
float XWidth = 0.9F * Width;//相對左下角 XWidth*0.9F 將表示 maxX
float marginX = (Width - XWidth) / 2F;//x軸相對左右圖片邊緣的像素
float marginY = (Height - YHeight) / 2F;//y軸相對上下圖片邊緣的像素
PointF leftTop = new PointF(marginX, marginY);
PointF leftBottom = new PointF(marginX, marginY + YHeight);//坐標軸的左下角
PointF rightBottom = new PointF(marginX + XWidth, marginY + YHeight);//坐標軸的右下角
gdi.DrawLine(Pens.Gray, leftBottom, rightBottom);//x軸
gdi.DrawLine(Pens.Gray, leftBottom, leftTop);//Y軸
//兩個箭頭 四條線 6個坐標 另需4個坐標
PointF YArrowLeft = new PointF(leftTop.X - 5, leftTop.Y + 5);
PointF YArrowRight = new PointF(leftTop.X + 5, leftTop.Y + 5);
PointF XArrowTop = new PointF(rightBottom.X - 5, rightBottom.Y - 5);
PointF XArrowBottom = new PointF(rightBottom.X - 5, rightBottom.Y + 5);
gdi.DrawLine(Pens.Gray, leftTop, YArrowLeft);
gdi.DrawLine(Pens.Gray, leftTop, YArrowRight);
gdi.DrawLine(Pens.Gray, rightBottom, XArrowTop);
gdi.DrawLine(Pens.Gray, rightBottom, XArrowBottom);
float unitX = 0.0F;//X軸轉換比率
float unitY = 0.0F;//Y軸轉換比率
List<PointF> pointFs = ConvertToPointF(Scores, XWidth * 0.9F, YHeight * 0.9F, leftTop, out unitX, out unitY);//將分數和概率 轉換成 坐標
gdi.DrawCurve(Pens.Black, pointFs.ToArray(), 0.0F);//基數樣條
//平均分 與 Y軸平行
PointF avg_top = new PointF(leftTop.X + (float)Average * unitX, leftTop.Y);
PointF avg_bottom = new PointF(leftTop.X + (float)Average * unitX, leftBottom.Y);
gdi.DrawLine(Pens.Black, avg_top, avg_bottom);
gdi.DrawString(string.Format("{0}", ((float)Average ).ToString("F2")), new Font("宋體", 11), Brushes.Black, avg_bottom.X, avg_bottom.Y-25);
//將期望和方差寫在橫軸下方中間
PointF variance_pf = new PointF(leftBottom.X+(XWidth/2)-120, avg_bottom.Y + 25);
gdi.DrawString(string.Format("期望：{0}；方差：{1}", ((float)Average).ToString("F2"), Variance.ToString("F2")), new Font("宋體", 11), Brushes.Black, variance_pf.X, variance_pf.Y);
//將最小分數 和 最大分數 分成9段 標記在坐標軸橫軸上
double minX = Scores.Min(x => x.Item1);
double maxX = Scores.Max(x => x.Item1);
double perSegment = TotalScore/10;// (maxX - minX) / 9F;//每一段表示的分數
List<double> segs = new List<double>();//每一個分段分界線橫軸的值
segs.Add(leftBottom.X + (float)minX * unitX);
for (int i = 1; i < 11; i++)
{
segs.Add(leftBottom.X + (float)minX * unitX + perSegment * i * unitX);
}
for (int i = 0; i < 11; i++)
{
gdi.DrawPie(Pens.Black, (float)segs[i] - 1, leftBottom.Y - 1, 2, 2, 0, 360);
gdi.DrawString(string.Format("{0}", ((minX + perSegment * (i))).ToString("F0")), new Font("宋體", 11), Brushes.Black, (float)segs[i] - 15, leftBottom.Y + 5);
}
return bitmap;
}
/// <summary>
/// 將數據轉換為坐標
/// </summary>
/// <param name="Scores"></param>
/// <param name="X">最長利用橫軸</param>
/// <param name="Y">最長利用縱軸 </param>
/// <param name="leftTop">左上角原點</param>
/// <returns></returns>
private static List<PointF> ConvertToPointF(List<Tuple<double, double>> Scores, float X, float Y, PointF leftTop, out float unitX, out float unitY)
{
double maxY = Scores.Max(x => x.Item2);
double maxX = Scores.Max(x => x.Item1);
List<PointF> result = new List<PointF>();
float paddingY = Y * 0.01F;
float paddingX = X * 0.01F;
unitY = (float)((Y - paddingY) / maxY);//單位縱軸表示出來需要的高度 計算出來的縱坐標需要 leftTop.Y+(Y-item2*unitY)+paddingY
unitX = (float)((X - paddingX) / maxX);//單位橫軸表示出來需要的寬度 計算出來的橫坐標需要 leftTop.X+item1*unitX
PointF pf = new PointF();
foreach (Tuple<double, double> item in Scores)
{
pf = new PointF(leftTop.X + (float)item.Item1 * unitX, leftTop.Y + (Y - (float)item.Item2 * unitY) + paddingY);
result.Add(pf);
}
return result;
}
}
調用：
StandardDistribution mathX = new StandardDistribution(scores);
Bitmap bitmap = mathX.GetGaussianDistributionGraph(800, 480, totalScore);
bitmap.Save("tt.jpg", System.Drawing.Imaging.ImageFormat.Jpeg);
測試數據生成的正態分布圖：
