""" ===================================================== Distance computations (:mod:`scipy.spatial.distance`) ===================================================== .. sectionauthor:: Damian Eads Function Reference ------------------ Distance matrix computation from a collection of raw observation vectors stored in a rectangular array. .. autosummary:: :toctree: generated/ pdist -- pairwise distances between observation vectors. cdist -- distances between between two collections of observation vectors squareform -- convert distance matrix to a condensed one and vice versa Predicates for checking the validity of distance matrices, both condensed and redundant. Also contained in this module are functions for computing the number of observations in a distance matrix. .. autosummary:: :toctree: generated/ is_valid_dm -- checks for a valid distance matrix is_valid_y -- checks for a valid condensed distance matrix num_obs_dm -- # of observations in a distance matrix num_obs_y -- # of observations in a condensed distance matrix Distance functions between two vectors ``u`` and ``v``. Computing distances over a large collection of vectors is inefficient for these functions. Use ``pdist`` for this purpose. .. autosummary:: :toctree: generated/ braycurtis -- the Bray-Curtis distance. canberra -- the Canberra distance. chebyshev -- the Chebyshev distance. cityblock -- the Manhattan distance. correlation -- the Correlation distance. cosine -- the Cosine distance. dice -- the Dice dissimilarity (boolean). euclidean -- the Euclidean distance. hamming -- the Hamming distance (boolean). jaccard -- the Jaccard distance (boolean). kulsinski -- the Kulsinski distance (boolean). mahalanobis -- the Mahalanobis distance. matching -- the matching dissimilarity (boolean). minkowski -- the Minkowski distance. rogerstanimoto -- the Rogers-Tanimoto dissimilarity (boolean). russellrao -- the Russell-Rao dissimilarity (boolean). seuclidean -- the normalized Euclidean distance. sokalmichener -- the Sokal-Michener dissimilarity (boolean). sokalsneath -- the Sokal-Sneath dissimilarity (boolean). sqeuclidean -- the squared Euclidean distance. yule -- the Yule dissimilarity (boolean). References ---------- .. [Sta07] "Statistics toolbox." API Reference Documentation. The MathWorks. http://www.mathworks.com/access/helpdesk/help/toolbox/stats/. Accessed October 1, 2007. .. [Mti07] "Hierarchical clustering." API Reference Documentation. The Wolfram Research, Inc. http://reference.wolfram.com/mathematica/HierarchicalClustering/tutorial/HierarchicalClustering.html. Accessed October 1, 2007. .. [Gow69] Gower, JC and Ross, GJS. "Minimum Spanning Trees and Single Linkage Cluster Analysis." Applied Statistics. 18(1): pp. 54--64. 1969. .. [War63] Ward Jr, JH. "Hierarchical grouping to optimize an objective function." Journal of the American Statistical Association. 58(301): pp. 236--44. 1963. .. [Joh66] Johnson, SC. "Hierarchical clustering schemes." Psychometrika. 32(2): pp. 241--54. 1966. .. [Sne62] Sneath, PH and Sokal, RR. "Numerical taxonomy." Nature. 193: pp. 855--60. 1962. .. [Bat95] Batagelj, V. "Comparing resemblance measures." Journal of Classification. 12: pp. 73--90. 1995. .. [Sok58] Sokal, RR and Michener, CD. "A statistical method for evaluating systematic relationships." Scientific Bulletins. 38(22): pp. 1409--38. 1958. .. [Ede79] Edelbrock, C. "Mixture model tests of hierarchical clustering algorithms: the problem of classifying everybody." Multivariate Behavioral Research. 14: pp. 367--84. 1979. .. [Jai88] Jain, A., and Dubes, R., "Algorithms for Clustering Data." Prentice-Hall. Englewood Cliffs, NJ. 1988. .. [Fis36] Fisher, RA "The use of multiple measurements in taxonomic problems." Annals of Eugenics, 7(2): 179-188. 1936 Copyright Notice ---------------- Copyright (C) Damian Eads, 2007-2008. New BSD License. """ import warnings import numpy as np from numpy.linalg import norm import _distance_wrap def _copy_array_if_base_present(a): """ Copies the array if its base points to a parent array. """ if a.base is not None: return a.copy() elif np.issubsctype(a, np.float32): return np.array(a, dtype=np.double) else: return a def _copy_arrays_if_base_present(T): """ Accepts a tuple of arrays T. Copies the array T[i] if its base array points to an actual array. Otherwise, the reference is just copied. This is useful if the arrays are being passed to a C function that does not do proper striding. """ l = [_copy_array_if_base_present(a) for a in T] return l def _convert_to_bool(X): if X.dtype != np.bool: X = np.bool_(X) if not X.flags.contiguous: X = X.copy() return X def _convert_to_double(X): if X.dtype != np.double: X = np.double(X) if not X.flags.contiguous: X = X.copy() return X def _validate_vector(u, dtype=None): # XXX Is order='c' really necessary? u = np.asarray(u, dtype=dtype, order='c').squeeze() # Ensure values such as u=1 and u=[1] still return 1-D arrays. u = np.atleast_1d(u) if u.ndim > 1: raise ValueError("Input vector should be 1-D.") return u def minkowski(u, v, p): r""" Computes the Minkowski distance between two vectors ``u`` and ``v``, defined as .. math:: {||u-v||}_p = (\sum{|u_i - v_i|^p})^{1/p}. Parameters ---------- u : ndarray An n-dimensional vector. v : ndarray An n-dimensional vector. p : int The order of the norm of the difference :math:`{||u-v||}_p`. Returns ------- d : double The Minkowski distance between vectors ``u`` and ``v``. """ u = _validate_vector(u) v = _validate_vector(v) if p < 1: raise ValueError("p must be at least 1") dist = norm(u - v, ord=p) return dist def wminkowski(u, v, p, w): r""" Computes the weighted Minkowski distance between two vectors ``u`` and ``v``, defined as .. math:: \left(\sum{(w_i |u_i - v_i|^p)}\right)^{1/p}. Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. p : int The order of the norm of the difference :math:`{||u-v||}_p`. w : ndarray The weight vector. Returns ------- d : double The Minkowski distance between vectors ``u`` and ``v``. """ u = _validate_vector(u) v = _validate_vector(v) w = _validate_vector(w) if p < 1: raise ValueError("p must be at least 1") dist = norm(w * (u - v), ord=p) return dist def euclidean(u, v): """ Computes the Euclidean distance between two n-vectors ``u`` and ``v``, which is defined as .. math:: {||u-v||}_2 Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. Returns ------- d : double The Euclidean distance between vectors ``u`` and ``v``. """ u = _validate_vector(u) v = _validate_vector(v) dist = norm(u - v) return dist def sqeuclidean(u, v): """ Computes the squared Euclidean distance between two n-vectors u and v, which is defined as .. math:: {||u-v||}_2^2. Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. Returns ------- d : double The squared Euclidean distance between vectors ``u`` and ``v``. """ u = _validate_vector(u) v = _validate_vector(v) dist = ((u - v) ** 2).sum() return dist def cosine(u, v): r""" Computes the Cosine distance between two n-vectors u and v, which is defined as .. math:: 1 - \frac{uv^T} {||u||_2 ||v||_2}. Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. Returns ------- d : double The Cosine distance between vectors ``u`` and ``v``. """ u = _validate_vector(u) v = _validate_vector(v) dist = 1.0 - np.dot(u, v) / (norm(u) * norm(v)) return dist def correlation(u, v): r""" Computes the correlation distance between two n-vectors ``u`` and ``v``, which is defined as .. math:: 1 - frac{(u - \bar{u}){(v - \bar{v})}^T} {{||(u - \bar{u})||}_2 {||(v - \bar{v})||}_2^T} where :math:`\bar{u}` is the mean of a vectors elements and ``n`` is the common dimensionality of ``u`` and ``v``. Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. Returns ------- d : double The correlation distance between vectors ``u`` and ``v``. """ u = _validate_vector(u) v = _validate_vector(v) umu = u.mean() vmu = v.mean() um = u - umu vm = v - vmu dist = 1.0 - np.dot(um, vm) / (norm(um) * norm(vm)) return dist def hamming(u, v): r""" Computes the Hamming distance between two n-vectors ``u`` and ``v``, which is simply the proportion of disagreeing components in ``u`` and ``v``. If ``u`` and ``v`` are boolean vectors, the Hamming distance is .. math:: \frac{c_{01} + c_{10}}{n} where :math:`c_{ij}` is the number of occurrences of :math:`\mathtt{u[k]} = i` and :math:`\mathtt{v[k]} = j` for :math:`k < n`. Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. Returns ------- d : double The Hamming distance between vectors ``u`` and ``v``. """ u = _validate_vector(u) v = _validate_vector(v) return (u != v).mean() def jaccard(u, v): """ Computes the Jaccard-Needham dissimilarity between two boolean n-vectors u and v, which is .. math:: \frac{c_{TF} + c_{FT}} {c_{TT} + c_{FT} + c_{TF}} where :math:`c_{ij}` is the number of occurrences of :math:`\mathtt{u[k]} = i` and :math:`\mathtt{v[k]} = j` for :math:`k < n`. Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. Returns ------- d : double The Jaccard distance between vectors ``u`` and ``v``. """ u = _validate_vector(u) v = _validate_vector(v) dist = (np.double(np.bitwise_and((u != v), np.bitwise_or(u != 0, v != 0)).sum()) / np.double(np.bitwise_or(u != 0, v != 0).sum())) return dist def kulsinski(u, v): """ Computes the Kulsinski dissimilarity between two boolean n-vectors u and v, which is defined as .. math:: \frac{c_{TF} + c_{FT} - c_{TT} + n} {c_{FT} + c_{TF} + n} where :math:`c_{ij}` is the number of occurrences of :math:`\mathtt{u[k]} = i` and :math:`\mathtt{v[k]} = j` for :math:`k < n`. Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. Returns ------- d : double The Kulsinski distance between vectors ``u`` and ``v``. """ u = _validate_vector(u) v = _validate_vector(v) n = float(len(u)) (nff, nft, ntf, ntt) = _nbool_correspond_all(u, v) return (ntf + nft - ntt + n) / (ntf + nft + n) def seuclidean(u, v, V): """ Returns the standardized Euclidean distance between two n-vectors ``u`` and ``v``. ``V`` is an n-dimensional vector of component variances. It is usually computed among a larger collection vectors. Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. V : ndarray An :math:`n`-dimensional vector. Returns ------- d : double The standardized Euclidean distance between vectors ``u`` and ``v``. """ u = _validate_vector(u) v = _validate_vector(v) V = _validate_vector(V, dtype=np.float64) if V.shape[0] != u.shape[0] or u.shape[0] != v.shape[0]: raise TypeError('V must be a 1-D array of the same dimension ' 'as u and v.') return np.sqrt(((u - v) ** 2 / V).sum()) def cityblock(u, v): """ Computes the Manhattan distance between two n-vectors u and v, which is defined as .. math:: \\sum_i {\\left| u_i - v_i \\right|}. Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. Returns ------- d : double The City Block distance between vectors ``u`` and ``v``. """ u = _validate_vector(u) v = _validate_vector(v) return abs(u - v).sum() def mahalanobis(u, v, VI): r""" Computes the Mahalanobis distance between two n-vectors ``u`` and ``v``, which is defiend as .. math:: (u-v)V^{-1}(u-v)^T where ``VI`` is the inverse covariance matrix :math:`V^{-1}`. Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. Returns ------- d : double The Mahalanobis distance between vectors ``u`` and ``v``. """ u = _validate_vector(u) v = _validate_vector(v) VI = np.atleast_2d(VI) delta = u - v m = np.dot(np.dot(delta, VI), delta) return np.sqrt(m) def chebyshev(u, v): r""" Computes the Chebyshev distance between two n-vectors u and v, which is defined as .. math:: \max_i {|u_i-v_i|}. Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. Returns ------- d : double The Chebyshev distance between vectors ``u`` and ``v``. """ u = _validate_vector(u) v = _validate_vector(v) return max(abs(u - v)) def braycurtis(u, v): r""" Computes the Bray-Curtis distance between two n-vectors ``u`` and ``v``, which is defined as .. math:: \sum{|u_i-v_i|} / \sum{|u_i+v_i|}. The Bray-Curtis distance is in the range [0, 1] if all coordinates are positive, and is undefined if the inputs are of length zero. Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. Returns ------- d : double The Bray-Curtis distance between vectors ``u`` and ``v``. """ u = _validate_vector(u) v = _validate_vector(v, dtype=np.float64) return abs(u - v).sum() / abs(u + v).sum() def canberra(u, v): r""" Computes the Canberra distance between two n-vectors u and v, which is defined as .. math:: \sum_u \frac{|u_i-v_i|} {(|u_i|+|v_i|)}. Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. Returns ------- d : double The Canberra distance between vectors ``u`` and ``v``. Notes ----- Whe u[i] and v[i] are 0 for given i, then the fraction 0/0 = 0 is used in the calculation. """ u = _validate_vector(u) v = _validate_vector(v, dtype=np.float64) olderr = np.seterr(invalid='ignore') try: d = np.nansum(abs(u - v) / (abs(u) + abs(v))) finally: np.seterr(**olderr) return d def _nbool_correspond_all(u, v): if u.dtype != v.dtype: raise TypeError("Arrays being compared must be of the same data type.") if u.dtype == np.int or u.dtype == np.float_ or u.dtype == np.double: not_u = 1.0 - u not_v = 1.0 - v nff = (not_u * not_v).sum() nft = (not_u * v).sum() ntf = (u * not_v).sum() ntt = (u * v).sum() elif u.dtype == np.bool: not_u = ~u not_v = ~v nff = (not_u & not_v).sum() nft = (not_u & v).sum() ntf = (u & not_v).sum() ntt = (u & v).sum() else: raise TypeError("Arrays being compared have unknown type.") return (nff, nft, ntf, ntt) def _nbool_correspond_ft_tf(u, v): if u.dtype == np.int or u.dtype == np.float_ or u.dtype == np.double: not_u = 1.0 - u not_v = 1.0 - v nft = (not_u * v).sum() ntf = (u * not_v).sum() else: not_u = ~u not_v = ~v nft = (not_u & v).sum() ntf = (u & not_v).sum() return (nft, ntf) def yule(u, v): r""" Computes the Yule dissimilarity between two boolean n-vectors u and v, which is defined as .. math:: \frac{R}{c_{TT} + c_{FF} + \frac{R}{2}} where :math:`c_{ij}` is the number of occurrences of :math:`\mathtt{u[k]} = i` and :math:`\mathtt{v[k]} = j` for :math:`k < n` and :math:`R = 2.0 * (c_{TF} + c_{FT})`. Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. Returns ------- d : double The Yule dissimilarity between vectors ``u`` and ``v``. """ u = _validate_vector(u) v = _validate_vector(v) (nff, nft, ntf, ntt) = _nbool_correspond_all(u, v) return float(2.0 * ntf * nft) / float(ntt * nff + ntf * nft) def matching(u, v): r""" Computes the Matching dissimilarity between two boolean n-vectors u and v, which is defined as .. math:: \frac{c_{TF} + c_{FT}}{n} where :math:`c_{ij}` is the number of occurrences of :math:`\mathtt{u[k]} = i` and :math:`\mathtt{v[k]} = j` for :math:`k < n`. Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. Returns ------- d : double The Matching dissimilarity between vectors ``u`` and ``v``. """ u = _validate_vector(u) v = _validate_vector(v) (nft, ntf) = _nbool_correspond_ft_tf(u, v) return float(nft + ntf) / float(len(u)) def dice(u, v): r""" Computes the Dice dissimilarity between two boolean n-vectors ``u`` and ``v``, which is .. math:: \frac{c_{TF} + c_{FT}} {2c_{TT} + c_{FT} + c_{TF}} where :math:`c_{ij}` is the number of occurrences of :math:`\mathtt{u[k]} = i` and :math:`\mathtt{v[k]} = j` for :math:`k < n`. Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. Returns ------- d : double The Dice dissimilarity between vectors ``u`` and ``v``. """ u = _validate_vector(u) v = _validate_vector(v) if u.dtype == np.bool: ntt = (u & v).sum() else: ntt = (u * v).sum() (nft, ntf) = _nbool_correspond_ft_tf(u, v) return float(ntf + nft) / float(2.0 * ntt + ntf + nft) def rogerstanimoto(u, v): r""" Computes the Rogers-Tanimoto dissimilarity between two boolean n-vectors ``u`` and ``v``, which is defined as .. math:: \frac{R} {c_{TT} + c_{FF} + R} where :math:`c_{ij}` is the number of occurrences of :math:`\mathtt{u[k]} = i` and :math:`\mathtt{v[k]} = j` for :math:`k < n` and :math:`R = 2(c_{TF} + c_{FT})`. Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. Returns ------- d : double The Rogers-Tanimoto dissimilarity between vectors `u` and `v`. """ u = _validate_vector(u) v = _validate_vector(v) (nff, nft, ntf, ntt) = _nbool_correspond_all(u, v) return float(2.0 * (ntf + nft)) / float(ntt + nff + (2.0 * (ntf + nft))) def russellrao(u, v): r""" Computes the Russell-Rao dissimilarity between two boolean n-vectors ``u`` and ``v``, which is defined as .. math:: \frac{n - c_{TT}} {n} where :math:`c_{ij}` is the number of occurrences of :math:`\mathtt{u[k]} = i` and :math:`\mathtt{v[k]} = j` for :math:`k < n`. Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. Returns ------- d : double The Russell-Rao dissimilarity between vectors ``u`` and ``v``. """ u = _validate_vector(u) v = _validate_vector(v) if u.dtype == np.bool: ntt = (u & v).sum() else: ntt = (u * v).sum() return float(len(u) - ntt) / float(len(u)) def sokalmichener(u, v): r""" Computes the Sokal-Michener dissimilarity between two boolean vectors ``u`` and ``v``, which is defined as .. math:: \frac{R} {S + R} where :math:`c_{ij}` is the number of occurrences of :math:`\mathtt{u[k]} = i` and :math:`\mathtt{v[k]} = j` for :math:`k < n`, :math:`R = 2 * (c_{TF} + c_{FT})` and :math:`S = c_{FF} + c_{TT}`. Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. Returns ------- d : double The Sokal-Michener dissimilarity between vectors ``u`` and ``v``. """ u = _validate_vector(u) v = _validate_vector(v) if u.dtype == np.bool: ntt = (u & v).sum() nff = (~u & ~v).sum() else: ntt = (u * v).sum() nff = ((1.0 - u) * (1.0 - v)).sum() (nft, ntf) = _nbool_correspond_ft_tf(u, v) return float(2.0 * (ntf + nft)) / float(ntt + nff + 2.0 * (ntf + nft)) def sokalsneath(u, v): r""" Computes the Sokal-Sneath dissimilarity between two boolean vectors ``u`` and ``v``, .. math:: \frac{R} {c_{TT} + R} where :math:`c_{ij}` is the number of occurrences of :math:`\mathtt{u[k]} = i` and :math:`\mathtt{v[k]} = j` for :math:`k < n` and :math:`R = 2(c_{TF} + c_{FT})`. Parameters ---------- u : ndarray An :math:`n`-dimensional vector. v : ndarray An :math:`n`-dimensional vector. Returns ------- d : double The Sokal-Sneath dissimilarity between vectors ``u`` and ``v``. """ u = _validate_vector(u) v = _validate_vector(v) if u.dtype == np.bool: ntt = (u & v).sum() else: ntt = (u * v).sum() (nft, ntf) = _nbool_correspond_ft_tf(u, v) denom = ntt + 2.0 * (ntf + nft) if denom == 0: raise ValueError('Sokal-Sneath dissimilarity is not defined for ' 'vectors that are entirely false.') return float(2.0 * (ntf + nft)) / denom def pdist(X, metric='euclidean', p=2, w=None, V=None, VI=None): r""" Computes the pairwise distances between m original observations in n-dimensional space. Returns a condensed distance matrix Y. For each :math:`i` and :math:`j` (where :math:`i=2 encoding distances as described, X=squareform(v) returns a d by d distance matrix X. The X[i, j] and X[j, i] values are set to v[{n \choose 2}-{n-i \choose 2} + (j-u-1)] and all diagonal elements are zero. """ X = _convert_to_double(np.asarray(X, order='c')) if not np.issubsctype(X, np.double): raise TypeError('A double array must be passed.') s = X.shape if force.lower() == 'tomatrix': if len(s) != 1: raise ValueError("Forcing 'tomatrix' but input X is not a " "distance vector.") elif force.lower() == 'tovector': if len(s) != 2: raise ValueError("Forcing 'tovector' but input X is not a " "distance matrix.") # X = squareform(v) if len(s) == 1: if X.shape[0] == 0: return np.zeros((1, 1), dtype=np.double) # Grab the closest value to the square root of the number # of elements times 2 to see if the number of elements # is indeed a binomial coefficient. d = int(np.ceil(np.sqrt(X.shape[0] * 2))) # Check that v is of valid dimensions. if d * (d - 1) / 2 != int(s[0]): raise ValueError('Incompatible vector size. It must be a binomial ' 'coefficient n choose 2 for some integer n >= 2.') # Allocate memory for the distance matrix. M = np.zeros((d, d), dtype=np.double) # Since the C code does not support striding using strides. # The dimensions are used instead. [X] = _copy_arrays_if_base_present([X]) # Fill in the values of the distance matrix. _distance_wrap.to_squareform_from_vector_wrap(M, X) # Return the distance matrix. M = M + M.transpose() return M elif len(s) == 2: if s[0] != s[1]: raise ValueError('The matrix argument must be square.') if checks: is_valid_dm(X, throw=True, name='X') # One-side of the dimensions is set here. d = s[0] if d <= 1: return np.array([], dtype=np.double) # Create a vector. v = np.zeros(((d * (d - 1) / 2),), dtype=np.double) # Since the C code does not support striding using strides. # The dimensions are used instead. [X] = _copy_arrays_if_base_present([X]) # Convert the vector to squareform. _distance_wrap.to_vector_from_squareform_wrap(X, v) return v else: raise ValueError(('The first argument must be one or two dimensional ' 'array. A %d-dimensional array is not ' 'permitted') % len(s)) def is_valid_dm(D, tol=0.0, throw=False, name="D", warning=False): """ Returns True if the variable D passed is a valid distance matrix. Distance matrices must be 2-dimensional numpy arrays containing doubles. They must have a zero-diagonal, and they must be symmetric. Parameters ---------- D : ndarray The candidate object to test for validity. tol : double The distance matrix should be symmetric. tol is the maximum difference between the :math:`ij`th entry and the :math:`ji`th entry for the distance metric to be considered symmetric. throw : bool An exception is thrown if the distance matrix passed is not valid. name : string the name of the variable to checked. This is useful if throw is set to ``True`` so the offending variable can be identified in the exception message when an exception is thrown. warning : bool Instead of throwing an exception, a warning message is raised. Returns ------- Returns ``True`` if the variable ``D`` passed is a valid distance matrix. Small numerical differences in ``D`` and ``D.T`` and non-zeroness of the diagonal are ignored if they are within the tolerance specified by ``tol``. """ D = np.asarray(D, order='c') valid = True try: s = D.shape if D.dtype != np.double: if name: raise TypeError(('Distance matrix \'%s\' must contain doubles ' '(double).') % name) else: raise TypeError('Distance matrix must contain doubles ' '(double).') if len(D.shape) != 2: if name: raise ValueError(('Distance matrix \'%s\' must have shape=2 ' '(i.e. be two-dimensional).') % name) else: raise ValueError('Distance matrix must have shape=2 (i.e. ' 'be two-dimensional).') if tol == 0.0: if not (D == D.T).all(): if name: raise ValueError(('Distance matrix \'%s\' must be ' 'symmetric.') % name) else: raise ValueError('Distance matrix must be symmetric.') if not (D[xrange(0, s[0]), xrange(0, s[0])] == 0).all(): if name: raise ValueError(('Distance matrix \'%s\' diagonal must ' 'be zero.') % name) else: raise ValueError('Distance matrix diagonal must be zero.') else: if not (D - D.T <= tol).all(): if name: raise ValueError(('Distance matrix \'%s\' must be ' 'symmetric within tolerance %d.') % (name, tol)) else: raise ValueError('Distance matrix must be symmetric within' ' tolerance %5.5f.' % tol) if not (D[xrange(0, s[0]), xrange(0, s[0])] <= tol).all(): if name: raise ValueError(('Distance matrix \'%s\' diagonal must be' ' close to zero within tolerance %5.5f.') % (name, tol)) else: raise ValueError(('Distance matrix \'%s\' diagonal must be' ' close to zero within tolerance %5.5f.') % tol) except Exception, e: if throw: raise if warning: warnings.warn(str(e)) valid = False return valid def is_valid_y(y, warning=False, throw=False, name=None): r""" Returns ``True`` if the variable ``y`` passed is a valid condensed distance matrix. Condensed distance matrices must be 1-dimensional numpy arrays containing doubles. Their length must be a binomial coefficient :math:`{n \choose 2}` for some positive integer n. Parameters ---------- y : ndarray The condensed distance matrix. warning : bool, optional Invokes a warning if the variable passed is not a valid condensed distance matrix. The warning message explains why the distance matrix is not valid. 'name' is used when referencing the offending variable. throws : throw, optional Throws an exception if the variable passed is not a valid condensed distance matrix. name : bool, optional Used when referencing the offending variable in the warning or exception message. """ y = np.asarray(y, order='c') valid = True try: if type(y) != np.ndarray: if name: raise TypeError(('\'%s\' passed as a condensed distance ' 'matrix is not a numpy array.') % name) else: raise TypeError('Variable is not a numpy array.') if y.dtype != np.double: if name: raise TypeError(('Condensed distance matrix \'%s\' must ' 'contain doubles (double).') % name) else: raise TypeError('Condensed distance matrix must contain ' 'doubles (double).') if len(y.shape) != 1: if name: raise ValueError(('Condensed distance matrix \'%s\' must ' 'have shape=1 (i.e. be one-dimensional).') % name) else: raise ValueError('Condensed distance matrix must have shape=1 ' '(i.e. be one-dimensional).') n = y.shape[0] d = int(np.ceil(np.sqrt(n * 2))) if (d * (d - 1) / 2) != n: if name: raise ValueError(('Length n of condensed distance matrix ' '\'%s\' must be a binomial coefficient, i.e.' 'there must be a k such that ' '(k \choose 2)=n)!') % name) else: raise ValueError('Length n of condensed distance matrix must ' 'be a binomial coefficient, i.e. there must ' 'be a k such that (k \choose 2)=n)!') except Exception, e: if throw: raise if warning: warnings.warn(str(e)) valid = False return valid def num_obs_dm(d): """ Returns the number of original observations that correspond to a square, redundant distance matrix ``D``. Parameters ---------- d : ndarray The target distance matrix. Returns ------- numobs : int The number of observations in the redundant distance matrix. """ d = np.asarray(d, order='c') is_valid_dm(d, tol=np.inf, throw=True, name='d') return d.shape[0] def num_obs_y(Y): """ Returns the number of original observations that correspond to a condensed distance matrix ``Y``. Parameters ---------- Y : ndarray The number of original observations in the condensed observation ``Y``. Returns ------- n : int The number of observations in the condensed distance matrix passed. """ Y = np.asarray(Y, order='c') is_valid_y(Y, throw=True, name='Y') k = Y.shape[0] if k == 0: raise ValueError("The number of observations cannot be determined on " "an empty distance matrix.") d = int(np.ceil(np.sqrt(k * 2))) if (d * (d - 1) / 2) != k: raise ValueError("Invalid condensed distance matrix passed. Must be " "some k where k=(n choose 2) for some n >= 2.") return d def cdist(XA, XB, metric='euclidean', p=2, V=None, VI=None, w=None): r""" Computes distance between each pair of observation vectors in the Cartesian product of two collections of vectors. ``XA`` is a :math:`m_A` by :math:`n` array while ``XB`` is a :math:`m_B` by :math:`n` array. A :math:`m_A` by :math:`m_B` array is returned. An exception is thrown if ``XA`` and ``XB`` do not have the same number of columns. A rectangular distance matrix ``Y`` is returned. For each :math:`i` and :math:`j`, the metric ``dist(u=XA[i], v=XB[j])`` is computed and stored in the :math:`ij` th entry. The following are common calling conventions: 1. ``Y = cdist(XA, XB, 'euclidean')`` Computes the distance between :math:`m` points using Euclidean distance (2-norm) as the distance metric between the points. The points are arranged as :math:`m` :math:`n`-dimensional row vectors in the matrix X. 2. ``Y = cdist(XA, XB, 'minkowski', p)`` Computes the distances using the Minkowski distance :math:`||u-v||_p` (:math:`p`-norm) where :math:`p \geq 1`. 3. ``Y = cdist(XA, XB, 'cityblock')`` Computes the city block or Manhattan distance between the points. 4. ``Y = cdist(XA, XB, 'seuclidean', V=None)`` Computes the standardized Euclidean distance. The standardized Euclidean distance between two n-vectors ``u`` and ``v`` is .. math:: \sqrt{\sum {(u_i-v_i)^2 / V[x_i]}}. V is the variance vector; V[i] is the variance computed over all the i'th components of the points. If not passed, it is automatically computed. 5. ``Y = cdist(XA, XB, 'sqeuclidean')`` Computes the squared Euclidean distance :math:`||u-v||_2^2` between the vectors. 6. ``Y = cdist(XA, XB, 'cosine')`` Computes the cosine distance between vectors u and v, .. math:: \frac{1 - uv^T} {{|u|}_2 {|v|}_2} where :math:`|*|_2` is the 2-norm of its argument *. 7. ``Y = cdist(XA, XB, 'correlation')`` Computes the correlation distance between vectors u and v. This is .. math:: \frac{1 - (u - n{|u|}_1){(v - n{|v|}_1)}^T} {{|(u - n{|u|}_1)|}_2 {|(v - n{|v|}_1)|}^T} where :math:`|*|_1` is the Manhattan (or 1-norm) of its argument, and :math:`n` is the common dimensionality of the vectors. 8. ``Y = cdist(XA, XB, 'hamming')`` Computes the normalized Hamming distance, or the proportion of those vector elements between two n-vectors ``u`` and ``v`` which disagree. To save memory, the matrix ``X`` can be of type boolean. 9. ``Y = cdist(XA, XB, 'jaccard')`` Computes the Jaccard distance between the points. Given two vectors, ``u`` and ``v``, the Jaccard distance is the proportion of those elements ``u[i]`` and ``v[i]`` that disagree where at least one of them is non-zero. 10. ``Y = cdist(XA, XB, 'chebyshev')`` Computes the Chebyshev distance between the points. The Chebyshev distance between two n-vectors ``u`` and ``v`` is the maximum norm-1 distance between their respective elements. More precisely, the distance is given by .. math:: d(u,v) = \max_i {|u_i-v_i|}. 11. ``Y = cdist(XA, XB, 'canberra')`` Computes the Canberra distance between the points. The Canberra distance between two points ``u`` and ``v`` is .. math:: d(u,v) = \sum_u \frac{|u_i-v_i|} {(|u_i|+|v_i|)} 12. ``Y = cdist(XA, XB, 'braycurtis')`` Computes the Bray-Curtis distance between the points. The Bray-Curtis distance between two points ``u`` and ``v`` is .. math:: d(u,v) = \frac{\sum_i (u_i-v_i)} {\sum_i (u_i+v_i)} 13. ``Y = cdist(XA, XB, 'mahalanobis', VI=None)`` Computes the Mahalanobis distance between the points. The Mahalanobis distance between two points ``u`` and ``v`` is :math:`(u-v)(1/V)(u-v)^T` where :math:`(1/V)` (the ``VI`` variable) is the inverse covariance. If ``VI`` is not None, ``VI`` will be used as the inverse covariance matrix. 14. ``Y = cdist(XA, XB, 'yule')`` Computes the Yule distance between the boolean vectors. (see yule function documentation) 15. ``Y = cdist(XA, XB, 'matching')`` Computes the matching distance between the boolean vectors. (see matching function documentation) 16. ``Y = cdist(XA, XB, 'dice')`` Computes the Dice distance between the boolean vectors. (see dice function documentation) 17. ``Y = cdist(XA, XB, 'kulsinski')`` Computes the Kulsinski distance between the boolean vectors. (see kulsinski function documentation) 18. ``Y = cdist(XA, XB, 'rogerstanimoto')`` Computes the Rogers-Tanimoto distance between the boolean vectors. (see rogerstanimoto function documentation) 19. ``Y = cdist(XA, XB, 'russellrao')`` Computes the Russell-Rao distance between the boolean vectors. (see russellrao function documentation) 20. ``Y = cdist(XA, XB, 'sokalmichener')`` Computes the Sokal-Michener distance between the boolean vectors. (see sokalmichener function documentation) 21. ``Y = cdist(XA, XB, 'sokalsneath')`` Computes the Sokal-Sneath distance between the vectors. (see sokalsneath function documentation) 22. ``Y = cdist(XA, XB, 'wminkowski')`` Computes the weighted Minkowski distance between the vectors. (see sokalsneath function documentation) 23. ``Y = cdist(XA, XB, f)`` Computes the distance between all pairs of vectors in X using the user supplied 2-arity function f. For example, Euclidean distance between the vectors could be computed as follows:: dm = cdist(XA, XB, (lambda u, v: np.sqrt(((u-v)*(u-v).T).sum()))) Note that you should avoid passing a reference to one of the distance functions defined in this library. For example,:: dm = cdist(XA, XB, sokalsneath) would calculate the pair-wise distances between the vectors in X using the Python function sokalsneath. This would result in sokalsneath being called :math:`{n \choose 2}` times, which is inefficient. Instead, the optimized C version is more efficient, and we call it using the following syntax.:: dm = cdist(XA, XB, 'sokalsneath') Parameters ---------- XA : ndarray An :math:`m_A` by :math:`n` array of :math:`m_A` original observations in an :math:`n`-dimensional space. XB : ndarray An :math:`m_B` by :math:`n` array of :math:`m_B` original observations in an :math:`n`-dimensional space. metric : string or function The distance metric to use. The distance function can be 'braycurtis', 'canberra', 'chebyshev', 'cityblock', 'correlation', 'cosine', 'dice', 'euclidean', 'hamming', 'jaccard', 'kulsinski', 'mahalanobis', 'matching', 'minkowski', 'rogerstanimoto', 'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath', 'sqeuclidean', 'wminkowski', 'yule'. w : ndarray The weight vector (for weighted Minkowski). p : double The p-norm to apply (for Minkowski, weighted and unweighted) V : ndarray The variance vector (for standardized Euclidean). VI : ndarray The inverse of the covariance matrix (for Mahalanobis). Returns ------- Y : ndarray A :math:`m_A` by :math:`m_B` distance matrix. """ # 21. Y = cdist(XA, XB, 'test_Y') # # Computes the distance between all pairs of vectors in X # using the distance metric Y but with a more succint, # verifiable, but less efficient implementation. XA = np.asarray(XA, order='c') XB = np.asarray(XB, order='c') #if np.issubsctype(X, np.floating) and not np.issubsctype(X, np.double): # raise TypeError('Floating point arrays must be 64-bit (got %r).' % # (X.dtype.type,)) # The C code doesn't do striding. [XA] = _copy_arrays_if_base_present([_convert_to_double(XA)]) [XB] = _copy_arrays_if_base_present([_convert_to_double(XB)]) s = XA.shape sB = XB.shape if len(s) != 2: raise ValueError('XA must be a 2-dimensional array.') if len(sB) != 2: raise ValueError('XB must be a 2-dimensional array.') if s[1] != sB[1]: raise ValueError('XA and XB must have the same number of columns ' '(i.e. feature dimension.)') mA = s[0] mB = sB[0] n = s[1] dm = np.zeros((mA, mB), dtype=np.double) if callable(metric): if metric == minkowski: for i in xrange(0, mA): for j in xrange(0, mB): dm[i, j] = minkowski(XA[i, :], XB[j, :], p) elif metric == wminkowski: for i in xrange(0, mA): for j in xrange(0, mB): dm[i, j] = wminkowski(XA[i, :], XB[j, :], p, w) elif metric == seuclidean: for i in xrange(0, mA): for j in xrange(0, mB): dm[i, j] = seuclidean(XA[i, :], XB[j, :], V) elif metric == mahalanobis: for i in xrange(0, mA): for j in xrange(0, mB): dm[i, j] = mahalanobis(XA[i, :], XB[j, :], V) else: for i in xrange(0, mA): for j in xrange(0, mB): dm[i, j] = metric(XA[i, :], XB[j, :]) elif isinstance(metric, basestring): mstr = metric.lower() #if XA.dtype != np.double and \ # (mstr != 'hamming' and mstr != 'jaccard'): # TypeError('A double array must be passed.') if mstr in set(['euclidean', 'euclid', 'eu', 'e']): _distance_wrap.cdist_euclidean_wrap(_convert_to_double(XA), _convert_to_double(XB), dm) elif mstr in set(['sqeuclidean', 'sqe', 'sqeuclid']): _distance_wrap.cdist_euclidean_wrap(_convert_to_double(XA), _convert_to_double(XB), dm) dm **= 2.0 elif mstr in set(['cityblock', 'cblock', 'cb', 'c']): _distance_wrap.cdist_city_block_wrap(_convert_to_double(XA), _convert_to_double(XB), dm) elif mstr in set(['hamming', 'hamm', 'ha', 'h']): if XA.dtype == np.bool: _distance_wrap.cdist_hamming_bool_wrap(_convert_to_bool(XA), _convert_to_bool(XB), dm) else: _distance_wrap.cdist_hamming_wrap(_convert_to_double(XA), _convert_to_double(XB), dm) elif mstr in set(['jaccard', 'jacc', 'ja', 'j']): if XA.dtype == np.bool: _distance_wrap.cdist_jaccard_bool_wrap(_convert_to_bool(XA), _convert_to_bool(XB), dm) else: _distance_wrap.cdist_jaccard_wrap(_convert_to_double(XA), _convert_to_double(XB), dm) elif mstr in set(['chebychev', 'chebyshev', 'cheby', 'cheb', 'ch']): _distance_wrap.cdist_chebyshev_wrap(_convert_to_double(XA), _convert_to_double(XB), dm) elif mstr in set(['minkowski', 'mi', 'm', 'pnorm']): _distance_wrap.cdist_minkowski_wrap(_convert_to_double(XA), _convert_to_double(XB), dm, p) elif mstr in set(['wminkowski', 'wmi', 'wm', 'wpnorm']): _distance_wrap.cdist_weighted_minkowski_wrap(_convert_to_double(XA), _convert_to_double(XB), dm, p, _convert_to_double(w)) elif mstr in set(['seuclidean', 'se', 's']): if V is not None: V = np.asarray(V, order='c') if type(V) != np.ndarray: raise TypeError('Variance vector V must be a numpy array') if V.dtype != np.double: raise TypeError('Variance vector V must contain doubles.') if len(V.shape) != 1: raise ValueError('Variance vector V must be ' 'one-dimensional.') if V.shape[0] != n: raise ValueError('Variance vector V must be of the same ' 'dimension as the vectors on which the ' 'distances are computed.') # The C code doesn't do striding. [VV] = _copy_arrays_if_base_present([_convert_to_double(V)]) else: X = np.vstack([XA, XB]) VV = np.var(X, axis=0, ddof=1) X = None del X _distance_wrap.cdist_seuclidean_wrap(_convert_to_double(XA), _convert_to_double(XB), VV, dm) # Need to test whether vectorized cosine works better. # Find out: Is there a dot subtraction operator so I can # subtract matrices in a similar way to multiplying them? # Need to get rid of as much unnecessary C code as possible. elif mstr in set(['cosine', 'cos']): normsA = np.sqrt(np.sum(XA * XA, axis=1)) normsB = np.sqrt(np.sum(XB * XB, axis=1)) _distance_wrap.cdist_cosine_wrap(_convert_to_double(XA), _convert_to_double(XB), dm, normsA, normsB) elif mstr in set(['correlation', 'co']): XA2 = XA - XA.mean(1)[:, np.newaxis] XB2 = XB - XB.mean(1)[:, np.newaxis] #X2 = X - np.matlib.repmat(np.mean(X, axis=1).reshape(m, 1), 1, n) normsA = np.sqrt(np.sum(XA2 * XA2, axis=1)) normsB = np.sqrt(np.sum(XB2 * XB2, axis=1)) _distance_wrap.cdist_cosine_wrap(_convert_to_double(XA2), _convert_to_double(XB2), _convert_to_double(dm), _convert_to_double(normsA), _convert_to_double(normsB)) elif mstr in set(['mahalanobis', 'mahal', 'mah']): if VI is not None: VI = _convert_to_double(np.asarray(VI, order='c')) if type(VI) != np.ndarray: raise TypeError('VI must be a numpy array.') if VI.dtype != np.double: raise TypeError('The array must contain 64-bit floats.') [VI] = _copy_arrays_if_base_present([VI]) else: X = np.vstack([XA, XB]) V = np.cov(X.T) X = None del X VI = _convert_to_double(np.linalg.inv(V).T.copy()) # (u-v)V^(-1)(u-v)^T _distance_wrap.cdist_mahalanobis_wrap(_convert_to_double(XA), _convert_to_double(XB), VI, dm) elif mstr == 'canberra': _distance_wrap.cdist_canberra_wrap(_convert_to_double(XA), _convert_to_double(XB), dm) elif mstr == 'braycurtis': _distance_wrap.cdist_bray_curtis_wrap(_convert_to_double(XA), _convert_to_double(XB), dm) elif mstr == 'yule': _distance_wrap.cdist_yule_bool_wrap(_convert_to_bool(XA), _convert_to_bool(XB), dm) elif mstr == 'matching': _distance_wrap.cdist_matching_bool_wrap(_convert_to_bool(XA), _convert_to_bool(XB), dm) elif mstr == 'kulsinski': _distance_wrap.cdist_kulsinski_bool_wrap(_convert_to_bool(XA), _convert_to_bool(XB), dm) elif mstr == 'dice': _distance_wrap.cdist_dice_bool_wrap(_convert_to_bool(XA), _convert_to_bool(XB), dm) elif mstr == 'rogerstanimoto': _distance_wrap.cdist_rogerstanimoto_bool_wrap(_convert_to_bool(XA), _convert_to_bool(XB), dm) elif mstr == 'russellrao': _distance_wrap.cdist_russellrao_bool_wrap(_convert_to_bool(XA), _convert_to_bool(XB), dm) elif mstr == 'sokalmichener': _distance_wrap.cdist_sokalmichener_bool_wrap(_convert_to_bool(XA), _convert_to_bool(XB), dm) elif mstr == 'sokalsneath': _distance_wrap.cdist_sokalsneath_bool_wrap(_convert_to_bool(XA), _convert_to_bool(XB), dm) elif metric == 'test_euclidean': dm = cdist(XA, XB, euclidean) elif metric == 'test_seuclidean': if V is None: V = np.var(np.vstack([XA, XB]), axis=0, ddof=1) else: V = np.asarray(V, order='c') dm = cdist(XA, XB, lambda u, v: seuclidean(u, v, V)) elif metric == 'test_sqeuclidean': dm = cdist(XA, XB, lambda u, v: sqeuclidean(u, v)) elif metric == 'test_braycurtis': dm = cdist(XA, XB, braycurtis) elif metric == 'test_mahalanobis': if VI is None: X = np.vstack([XA, XB]) V = np.cov(X.T) VI = np.linalg.inv(V) X = None del X else: VI = np.asarray(VI, order='c') [VI] = _copy_arrays_if_base_present([VI]) # (u-v)V^(-1)(u-v)^T dm = cdist(XA, XB, (lambda u, v: mahalanobis(u, v, VI))) elif metric == 'test_canberra': dm = cdist(XA, XB, canberra) elif metric == 'test_cityblock': dm = cdist(XA, XB, cityblock) elif metric == 'test_minkowski': dm = cdist(XA, XB, minkowski, p=p) elif metric == 'test_wminkowski': dm = cdist(XA, XB, wminkowski, p=p, w=w) elif metric == 'test_cosine': dm = cdist(XA, XB, cosine) elif metric == 'test_correlation': dm = cdist(XA, XB, correlation) elif metric == 'test_hamming': dm = cdist(XA, XB, hamming) elif metric == 'test_jaccard': dm = cdist(XA, XB, jaccard) elif metric == 'test_chebyshev' or metric == 'test_chebychev': dm = cdist(XA, XB, chebyshev) elif metric == 'test_yule': dm = cdist(XA, XB, yule) elif metric == 'test_matching': dm = cdist(XA, XB, matching) elif metric == 'test_dice': dm = cdist(XA, XB, dice) elif metric == 'test_kulsinski': dm = cdist(XA, XB, kulsinski) elif metric == 'test_rogerstanimoto': dm = cdist(XA, XB, rogerstanimoto) elif metric == 'test_russellrao': dm = cdist(XA, XB, russellrao) elif metric == 'test_sokalsneath': dm = cdist(XA, XB, sokalsneath) elif metric == 'test_sokalmichener': dm = cdist(XA, XB, sokalmichener) else: raise ValueError('Unknown Distance Metric: %s' % mstr) else: raise TypeError('2nd argument metric must be a string identifier ' 'or a function.') return dm