多维缩放

算法的输入是所有数据在高维情况下两两之间的距离（记i与j的距离为Dij）。

首先我们把所有数据点随机绘制在一张二维图像上，然后计算它们两两之间的距离dij，然后我们计算出它与高维距离Dij的误差，根据这些误差，我们将每对数据点按比例移近或移远，然后重新计算所有dij，不断重复到我们没法减少误差为止。

python代码实现

def scaledown(data,distance=pearson,rate=0.01):
  n=len(data)

  # The real distances between every pair of items
  realdist=[[distance(data[i],data[j]) for j in range(n)] 
             for i in range(0,n)]

  # Randomly initialize the starting points of the locations in 2D
  loc=[[random.random(),random.random()] for i in range(n)]
  fakedist=[[0.0 for j in range(n)] for i in range(n)]
  
  lasterror=None
  for m in range(0,1000):
    # Find projected distances
    for i in range(n):
      for j in range(n):
        fakedist[i][j]=sqrt(sum([pow(loc[i][x]-loc[j][x],2) 
                                 for x in range(len(loc[i]))]))
  
    # Move points
    grad=[[0.0,0.0] for i in range(n)]
    
    totalerror=0
    for k in range(n):
      for j in range(n):
        if j==k: continue
        # The error is percent difference between the distances
        errorterm=(fakedist[j][k]-realdist[j][k])/realdist[j][k]
        
        # Each point needs to be moved away from or towards the other
        # point in proportion to how much error it has
        grad[k][0]+=((loc[k][0]-loc[j][0])/fakedist[j][k])*errorterm
        grad[k][1]+=((loc[k][1]-loc[j][1])/fakedist[j][k])*errorterm

        # Keep track of the total error
        totalerror+=abs(errorterm)
    print totalerror

    # If the answer got worse by moving the points, we are done
    if lasterror and lasterror<totalerror: break
    lasterror=totalerror
    
    # Move each of the points by the learning rate times the gradient
    for k in range(n):
      loc[k][0]-=rate*grad[k][0]
      loc[k][1]-=rate*grad[k][1]

  return loc