Multidimensional Scaling

Multidimensional Scaling (MDS) is a linear embedding method used when we only know the pairwise distances between data points. For linear systems, MDS works well with as little as 10 points and the system is 2 dimensional. For this program, you choose the window size or embedding for the system and MDS identifies what the true embedding should be. The following program uses Logistic map data when r = 3.2 but you can change r to see how well MDS works as the map bifurcates and eventually becomes chaotic and nonlinear at 4. MDS breaks down at this point and the embedding defaults to just 1 over the embedding you choose.

# Requires Numpy and Scipy libraries

from numpy import *
import scipy.linalg
import random

r=3.2 # parameter on Logistic map
n = 10 # the number of timeseries points to use
d = 40 # your embedding choice

# Create a time series
def timeseries(number,throw, d):
  xList = []
  dataX0 = []
  t = 0
  while t < d:
    xList.append(.1)
    t += 1
  while t <= number+throw+d:
    x = r * xList[t-1] * (1 - xList[t-1]) # Logistic Map
    xList.append(x)
    if t > throw:
      y = xList[t-1]
      dataX0.append(x)
    t = t + 1
  return dataX0

# Construct an n x n centering matrix
# The form is P = I - (1/n) U where U is a matrix of all ones
def centering_matrix(n):
  P = eye(n) - 1/float(n) * ones((n,n))
  return P

def create_data(number, d):
  dataX0 = timeseries(number, 500, d)
  # Create a sliding window of size d
  data = []
  i = d
  while i < len(dataX0):
    data.append([])
    j = i - d
    while j < i:
      data[len(data)-1].append(dataX0[j])
      j = j + 1
    i = i + 1
  return data

def main():
  print "n =",n, ": number of timeseries points used"
  print "d =",d, ": embedding"

  P = centering_matrix(n)
  # Create the data with the number of dimensions
  data = create_data(n, d)
  X = pairwise_distances(data)
  A = -1/2.0 * P * X * P
  # Calculate the eigenvalues/vectors
  [vals, vectors] = scipy.linalg.eig(A)
  # Sort the values
  vals = sort(vals)
  embedding = 0
  for x in vals:
    if x > 10**(-10):
      embedding += 1
  print "mds embedding =",embedding

# Compute the pairwise distance between vector x and y
def metric(x1, y1):
  d = 2
  summ = []
  i = 0
  while i < len(x1):
    # in this case use euclidean distance
    summ.append((x1[i] - y1[i])**d)
    i = i + 1
  return sum(summ) ** (1 / float(d))

# Return a matrix of pairwise distances
def pairwise_distances(data):
  distances = []
  i = 0
  while i < len(data):
    distances.append([])
    x1 = data[i]
    j = 0
    while j < len(data):
      y1 = data[j]
      distances[i].append(metric(x1, y1)**2)
      j = j + 1
    i = i + 1
  return distances

main()

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