def procrustes(data1, data2):
"""Procrustes analysis, a similarity test for two data sets.
Each input matrix is a set of points or vectors (the rows of the matrix)
The dimension of the space is the number of columns of each matrix.
Given two identially sized matrices, procrustes standardizes both
such that:
- trace(AA') = 1 (A' is the transpose, and the product is
a standard matrix product).
- Both sets of points are centered around the origin
Procrustes then applies the optimal transform to the second matrix
(including scaling/dilation, rotations, and reflections) to minimize
M^2 = sum(square(mtx1 - mtx2)), or the sum of the squares of the pointwise
differences between the two input datasets
If two data sets have different dimensionality (different number of
columns), simply add columns of zeros the the smaller of the two.
This function was not designed to handle datasets with different numbers of
datapoints (rows)
Arguments:
- data1: matrix, n rows represent points in k (columns) space
data1 is the reference data, after it is standardised, the data from
data2 will be transformed to fit the pattern in data1
- data2: n rows of data in k space to be fit to data1. Must be the
same shape (numrows, numcols) as data1
- both must have >1 unique points
Returns:
- mtx1: a standardized version of data1
- mtx2: the orientation of data2 that best fits data1.
centered, but not necessarily trace(mtx2*mtx2') = 1
- disparity: a metric for the dissimilarity of the two datasets,
disparity = M^2 defined above
Notes:
- The disparity should not depend on the order of the input matrices,
but the output matrices will, as only the first output matrix is
guaranteed to be scaled such that trace(AA') = 1.
- duplicate datapoints are generally ok, duplicating a data point will
increase it's effect on the procrustes fit.
- the disparity scales as the number of points per input matrix
"""
SMALL_NUM = 1e-6 # used to check for zero values in added dimension
# make local copies
mtx1 = array(data1.copy(),'d')
mtx2 = array(data2.copy(),'d')
num_rows, num_cols = shape(data1)
if (num_rows, num_cols) != shape(data2):
raise ValueError("input matrices must be of same shape")
if (num_rows == 0 or num_cols == 0):
raise ValueError("input matrices must be >0 rows, >0 cols")
# add a dimension to allow reflections (rotations in n + 1 dimensions)
mtx1 = numpy_append(data1, zeros((num_rows, 1)), 1)
mtx2 = numpy_append(data2, zeros((num_rows, 1)), 1)
# standardize each matrix
mtx1 = center(mtx1)
mtx2 = center(mtx2)
if ((not any(mtx1)) or (not any(mtx2))):
raise ValueError("input matrices must contain >1 unique points")
mtx1 = normalize(mtx1)
mtx2 = normalize(mtx2)
# transform mtx2 to minimize disparity (sum( (mtx1[i,j] - mtx2[i,j])^2) )
mtx2 = match_points(mtx1, mtx2)
# WARNING: I haven't proven that after matching the matrices, no point has
# a nonzero component in the added dimension. I believe it is true,
# though, since the unchanged matrix has no points extending into
# that dimension
if any(abs(mtx2[:,-1]) > SMALL_NUM):
raise StandardError("we have accidentially added a dimension to \
the matrix, and the vectors have nonzero components in that dimension")
# strip extra dimension which was added to allow reflections
mtx1 = mtx1[:,:-1]
mtx2 = mtx2[:,:-1]
disparity = get_disparity(mtx1, mtx2)
return mtx1, mtx2, disparity
def procrustes(data1, data2):
"""Procrustes analysis, a similarity test for two data sets.
Each input matrix is a set of points or vectors (the rows of the matrix)
The dimension of the space is the number of columns of each matrix.
Given two identially sized matrices, procrustes standardizes both
such that:
- trace(AA') = 1 (A' is the transpose, and the product is
a standard matrix product).
- Both sets of points are centered around the origin
Procrustes then applies the optimal transform to the second matrix
(including scaling/dilation, rotations, and reflections) to minimize
M^2 = sum(square(mtx1 - mtx2)), or the sum of the squares of the pointwise
differences between the two input datasets
If two data sets have different dimensionality (different number of
columns), simply add columns of zeros the the smaller of the two.
This function was not designed to handle datasets with different numbers of
datapoints (rows)
Arguments:
- data1: matrix, n rows represent points in k (columns) space
data1 is the reference data, after it is standardised, the data from
data2 will be transformed to fit the pattern in data1
- data2: n rows of data in k space to be fit to data1. Must be the
same shape (numrows, numcols) as data1
- both must have >1 unique points
Returns:
- mtx1: a standardized version of data1
- mtx2: the orientation of data2 that best fits data1.
centered, but not necessarily trace(mtx2*mtx2') = 1
- disparity: a metric for the dissimilarity of the two datasets,
disparity = M^2 defined above
Notes:
- The disparity should not depend on the order of the input matrices,
but the output matrices will, as only the first output matrix is
guaranteed to be scaled such that trace(AA') = 1.
- duplicate datapoints are generally ok, duplicating a data point will
increase it's effect on the procrustes fit.
- the disparity scales as the number of points per input matrix
"""
SMALL_NUM = 1e-6 # used to check for zero values in added dimension
# make local copies
# mtx1 = array(data1.copy(),'d')
# mtx2 = array(data2.copy(),'d')
num_rows, num_cols = shape(data1)
if (num_rows, num_cols) != shape(data2):
raise ValueError("input matrices must be of same shape")
if (num_rows == 0 or num_cols == 0):
raise ValueError("input matrices must be >0 rows, >0 cols")
# add a dimension to allow reflections (rotations in n + 1 dimensions)
mtx1 = numpy_append(data1, zeros((num_rows, 1)), 1)
mtx2 = numpy_append(data2, zeros((num_rows, 1)), 1)
# standardize each matrix
mtx1 = center(mtx1)
mtx2 = center(mtx2)
if ((not any(mtx1)) or (not any(mtx2))):
raise ValueError("input matrices must contain >1 unique points")
mtx1 = normalize(mtx1)
mtx2 = normalize(mtx2)
# transform mtx2 to minimize disparity (sum( (mtx1[i,j] - mtx2[i,j])^2) )
mtx2 = match_points(mtx1, mtx2)
# WARNING: I haven't proven that after matching the matrices, no point has
# a nonzero component in the added dimension. I believe it is true,
# though, since the unchanged matrix has no points extending into
# that dimension
if any(abs(mtx2[:,-1]) > SMALL_NUM):
raise Exception("we have accidentially added a dimension to \
the matrix, and the vectors have nonzero components in that dimension")
# strip extra dimension which was added to allow reflections
mtx1 = mtx1[:,:-1]
mtx2 = mtx2[:,:-1]
disparity = get_disparity(mtx1, mtx2)
return mtx1, mtx2, disparity