"""Masked arrays add-ons. A collection of utilities for maskedarray :author: Pierre Gerard-Marchant :contact: pierregm_at_uga_dot_edu :version: $Id: extras.py 3473 2007-10-29 15:18:13Z jarrod.millman $ """ __author__ = "Pierre GF Gerard-Marchant ($Author: jarrod.millman $)" __version__ = '1.0' __revision__ = "$Revision: 3473 $" __date__ = '$Date: 2007-10-29 17:18:13 +0200 (Mon, 29 Oct 2007) $' __all__ = ['apply_along_axis', 'atleast_1d', 'atleast_2d', 'atleast_3d', 'average', 'column_stack','compress_cols','compress_rowcols', 'compress_rows', 'count_masked', 'corrcoef', 'cov', 'diagflat', 'dot','dstack', 'ediff1d','expand_dims', 'flatnotmasked_contiguous','flatnotmasked_edges', 'hsplit','hstack', 'mask_cols','mask_rowcols','mask_rows','masked_all','masked_all_like', 'median','mr_', 'notmasked_contiguous','notmasked_edges', 'polyfit', 'row_stack', 'vander','vstack', ] from itertools import groupby import warnings import core as ma from core import MaskedArray, MAError, add, array, asarray, concatenate, count,\ filled, getmask, getmaskarray, masked, masked_array, mask_or, nomask, ones,\ sort, zeros #from core import * import numpy as np from numpy import ndarray, array as nxarray import numpy.core.umath as umath from numpy.lib.index_tricks import AxisConcatenator from numpy.lib.polynomial import _lstsq #............................................................................... def issequence(seq): """Is seq a sequence (ndarray, list or tuple)?""" if isinstance(seq, ndarray): return True elif isinstance(seq, tuple): return True elif isinstance(seq, list): return True return False def count_masked(arr, axis=None): """Count the number of masked elements along the given axis. Parameters ---------- axis : int, optional Axis along which to count. If None (default), a flattened version of the array is used. """ m = getmaskarray(arr) return m.sum(axis) def masked_all(shape, dtype=float): """Return an empty masked array of the given shape and dtype, where all the data are masked. Parameters ---------- dtype : dtype, optional Data type of the output. """ a = masked_array(np.empty(shape, dtype), mask=np.ones(shape, bool)) return a def masked_all_like(arr): """Return an empty masked array of the same shape and dtype as the array `a`, where all the data are masked. """ a = masked_array(np.empty_like(arr), mask=np.ones(arr.shape, bool)) return a #####-------------------------------------------------------------------------- #---- --- Standard functions --- #####-------------------------------------------------------------------------- class _fromnxfunction: """Defines a wrapper to adapt numpy functions to masked arrays.""" def __init__(self, funcname): self.__name__ = funcname self.__doc__ = self.getdoc() def getdoc(self): "Retrieves the __doc__ string from the function." return getattr(np, self.__name__).__doc__ +\ "*Notes*:\n (The function is applied to both the _data and the _mask, if any.)" def __call__(self, *args, **params): func = getattr(np, self.__name__) if len(args)==1: x = args[0] if isinstance(x, ndarray): _d = func(np.asarray(x), **params) _m = func(getmaskarray(x), **params) return masked_array(_d, mask=_m) elif isinstance(x, tuple) or isinstance(x, list): _d = func(tuple([np.asarray(a) for a in x]), **params) _m = func(tuple([getmaskarray(a) for a in x]), **params) return masked_array(_d, mask=_m) else: arrays = [] args = list(args) while len(args)>0 and issequence(args[0]): arrays.append(args.pop(0)) res = [] for x in arrays: _d = func(np.asarray(x), *args, **params) _m = func(getmaskarray(x), *args, **params) res.append(masked_array(_d, mask=_m)) return res atleast_1d = _fromnxfunction('atleast_1d') atleast_2d = _fromnxfunction('atleast_2d') atleast_3d = _fromnxfunction('atleast_3d') vstack = row_stack = _fromnxfunction('vstack') hstack = _fromnxfunction('hstack') column_stack = _fromnxfunction('column_stack') dstack = _fromnxfunction('dstack') hsplit = _fromnxfunction('hsplit') diagflat = _fromnxfunction('diagflat') def expand_dims(a, axis): """Expands the shape of a by including newaxis before axis. """ if not isinstance(a, MaskedArray): return np.expand_dims(a, axis) elif getmask(a) is nomask: return np.expand_dims(a, axis).view(MaskedArray) m = getmaskarray(a) return masked_array(np.expand_dims(a, axis), mask=np.expand_dims(m, axis)) #####-------------------------------------------------------------------------- #---- #####-------------------------------------------------------------------------- def flatten_inplace(seq): """Flatten a sequence in place.""" k = 0 while (k != len(seq)): while hasattr(seq[k],'__iter__'): seq[k:(k+1)] = seq[k] k += 1 return seq def apply_along_axis(func1d, axis, arr, *args, **kwargs): """Execute func1d(arr[i],*args) where func1d takes 1-D arrays and arr is an N-d array. i varies so as to apply the function along the given axis for each 1-d subarray in arr. """ arr = array(arr, copy=False, subok=True) nd = arr.ndim if axis < 0: axis += nd if (axis >= nd): raise ValueError("axis must be less than arr.ndim; axis=%d, rank=%d." % (axis,nd)) ind = [0]*(nd-1) i = np.zeros(nd,'O') indlist = range(nd) indlist.remove(axis) i[axis] = slice(None,None) outshape = np.asarray(arr.shape).take(indlist) i.put(indlist, ind) j = i.copy() res = func1d(arr[tuple(i.tolist())], *args, **kwargs) # if res is a number, then we have a smaller output array asscalar = np.isscalar(res) if not asscalar: try: len(res) except TypeError: asscalar = True # Note: we shouldn't set the dtype of the output from the first result... #...so we force the type to object, and build a list of dtypes #...we'll just take the largest, to avoid some downcasting dtypes = [] if asscalar: dtypes.append(np.asarray(res).dtype) outarr = zeros(outshape, object) outarr[tuple(ind)] = res Ntot = np.product(outshape) k = 1 while k < Ntot: # increment the index ind[-1] += 1 n = -1 while (ind[n] >= outshape[n]) and (n > (1-nd)): ind[n-1] += 1 ind[n] = 0 n -= 1 i.put(indlist, ind) res = func1d(arr[tuple(i.tolist())], *args, **kwargs) outarr[tuple(ind)] = res dtypes.append(asarray(res).dtype) k += 1 else: res = array(res, copy=False, subok=True) j = i.copy() j[axis] = ([slice(None, None)] * res.ndim) j.put(indlist, ind) Ntot = np.product(outshape) holdshape = outshape outshape = list(arr.shape) outshape[axis] = res.shape dtypes.append(asarray(res).dtype) outshape = flatten_inplace(outshape) outarr = zeros(outshape, object) outarr[tuple(flatten_inplace(j.tolist()))] = res k = 1 while k < Ntot: # increment the index ind[-1] += 1 n = -1 while (ind[n] >= holdshape[n]) and (n > (1-nd)): ind[n-1] += 1 ind[n] = 0 n -= 1 i.put(indlist, ind) j.put(indlist, ind) res = func1d(arr[tuple(i.tolist())], *args, **kwargs) outarr[tuple(flatten_inplace(j.tolist()))] = res dtypes.append(asarray(res).dtype) k += 1 max_dtypes = np.dtype(np.asarray(dtypes).max()) if not hasattr(arr, '_mask'): result = np.asarray(outarr, dtype=max_dtypes) else: result = asarray(outarr, dtype=max_dtypes) result.fill_value = ma.default_fill_value(result) return result def average(a, axis=None, weights=None, returned=False): """Average the array over the given axis. Parameters ---------- axis : {None,int}, optional Axis along which to perform the operation. If None, applies to a flattened version of the array. weights : {None, sequence}, optional Sequence of weights. The weights must have the shape of a, or be 1D with length the size of a along the given axis. If no weights are given, weights are assumed to be 1. returned : {False, True}, optional Flag indicating whether a tuple (result, sum of weights/counts) should be returned as output (True), or just the result (False). """ a = asarray(a) mask = a.mask ash = a.shape if ash == (): ash = (1,) if axis is None: if mask is nomask: if weights is None: n = a.sum(axis=None) d = float(a.size) else: w = filled(weights, 0.0).ravel() n = umath.add.reduce(a._data.ravel() * w) d = umath.add.reduce(w) del w else: if weights is None: n = a.filled(0).sum(axis=None) d = umath.add.reduce((-mask).ravel().astype(int)) else: w = array(filled(weights, 0.0), float, mask=mask).ravel() n = add.reduce(a.ravel() * w) d = add.reduce(w) del w else: if mask is nomask: if weights is None: d = ash[axis] * 1.0 n = add.reduce(a._data, axis, dtype=float) else: w = filled(weights, 0.0) wsh = w.shape if wsh == (): wsh = (1,) if wsh == ash: w = np.array(w, float, copy=0) n = add.reduce(a*w, axis) d = add.reduce(w, axis) del w elif wsh == (ash[axis],): ni = ash[axis] r = [None]*len(ash) r[axis] = slice(None, None, 1) w = eval ("w["+ repr(tuple(r)) + "] * ones(ash, float)") n = add.reduce(a*w, axis, dtype=float) d = add.reduce(w, axis, dtype=float) del w, r else: raise ValueError, 'average: weights wrong shape.' else: if weights is None: n = add.reduce(a, axis, dtype=float) d = umath.add.reduce((-mask), axis=axis, dtype=float) else: w = filled(weights, 0.0) wsh = w.shape if wsh == (): wsh = (1,) if wsh == ash: w = array(w, dtype=float, mask=mask, copy=0) n = add.reduce(a*w, axis, dtype=float) d = add.reduce(w, axis, dtype=float) elif wsh == (ash[axis],): ni = ash[axis] r = [None]*len(ash) r[axis] = slice(None, None, 1) w = eval ("w["+ repr(tuple(r)) + \ "] * masked_array(ones(ash, float), mask)") n = add.reduce(a*w, axis, dtype=float) d = add.reduce(w, axis, dtype=float) else: raise ValueError, 'average: weights wrong shape.' del w if n is masked or d is masked: return masked result = n/d del n if isinstance(result, MaskedArray): if ((axis is None) or (axis==0 and a.ndim == 1)) and \ (result.mask is nomask): result = result._data if returned: if not isinstance(d, MaskedArray): d = masked_array(d) if isinstance(d, ndarray) and (not d.shape == result.shape): d = ones(result.shape, dtype=float) * d if returned: return result, d else: return result def median(a, axis=None, out=None, overwrite_input=False): """Compute the median along the specified axis. Returns the median of the array elements. Parameters ---------- a : array-like Input array or object that can be converted to an array axis : {None, int}, optional Axis along which the medians are computed. The default (axis=None) is to compute the median along a flattened version of the array. out : {None, ndarray}, optional Alternative output array in which to place the result. It must have the same shape and buffer length as the expected output but the type will be cast if necessary. overwrite_input : {False, True}, optional If True, then allow use of memory of input array (a) for calculations. The input array will be modified by the call to median. This will save memory when you do not need to preserve the contents of the input array. Treat the input as undefined, but it will probably be fully or partially sorted. Default is False. Note that, if overwrite_input is true, and the input is not already an ndarray, an error will be raised. Returns ------- median : ndarray. A new array holding the result is returned unless out is specified, in which case a reference to out is returned. Return datatype is float64 for ints and floats smaller than float64, or the input datatype otherwise. See Also ------- mean Notes ----- Given a vector V with N non masked values, the median of V is the middle value of a sorted copy of V (Vs) - i.e. Vs[(N-1)/2], when N is odd, or {Vs[N/2 - 1] + Vs[N/2]}/2. when N is even. """ def _median1D(data): counts = filled(count(data, axis),0) (idx, rmd) = divmod(counts, 2) if rmd: choice = slice(idx, idx+1) else: choice = slice(idx-1, idx+1) return data[choice].mean(0) # if overwrite_input: if axis is None: asorted = a.ravel() asorted.sort() else: a.sort(axis=axis) asorted = a else: asorted = sort(a, axis=axis) if axis is None: result = _median1D(asorted) else: result = apply_along_axis(_median1D, axis, asorted) if out is not None: out = result return result #.............................................................................. def compress_rowcols(x, axis=None): """Suppress the rows and/or columns of a 2D array that contains masked values. The suppression behavior is selected with the `axis`parameter. - If axis is None, rows and columns are suppressed. - If axis is 0, only rows are suppressed. - If axis is 1 or -1, only columns are suppressed. Parameters ---------- axis : int, optional Axis along which to perform the operation. If None, applies to a flattened version of the array. Returns ------- compressed_array : an ndarray. """ x = asarray(x) if x.ndim != 2: raise NotImplementedError, "compress2d works for 2D arrays only." m = getmask(x) # Nothing is masked: return x if m is nomask or not m.any(): return x._data # All is masked: return empty if m.all(): return nxarray([]) # Builds a list of rows/columns indices (idxr, idxc) = (range(len(x)), range(x.shape[1])) masked = m.nonzero() if not axis: for i in np.unique(masked[0]): idxr.remove(i) if axis in [None, 1, -1]: for j in np.unique(masked[1]): idxc.remove(j) return x._data[idxr][:,idxc] def compress_rows(a): """Suppress whole rows of a 2D array that contain masked values. """ return compress_rowcols(a, 0) def compress_cols(a): """Suppress whole columnss of a 2D array that contain masked values. """ return compress_rowcols(a, 1) def mask_rowcols(a, axis=None): """Mask whole rows and/or columns of a 2D array that contain masked values. The masking behavior is selected with the `axis`parameter. - If axis is None, rows and columns are masked. - If axis is 0, only rows are masked. - If axis is 1 or -1, only columns are masked. Parameters ---------- axis : int, optional Axis along which to perform the operation. If None, applies to a flattened version of the array. Returns ------- a *pure* ndarray. """ a = asarray(a) if a.ndim != 2: raise NotImplementedError, "compress2d works for 2D arrays only." m = getmask(a) # Nothing is masked: return a if m is nomask or not m.any(): return a maskedval = m.nonzero() a._mask = a._mask.copy() if not axis: a[np.unique(maskedval[0])] = masked if axis in [None, 1, -1]: a[:,np.unique(maskedval[1])] = masked return a def mask_rows(a, axis=None): """Mask whole rows of a 2D array that contain masked values. Parameters ---------- axis : int, optional Axis along which to perform the operation. If None, applies to a flattened version of the array. """ return mask_rowcols(a, 0) def mask_cols(a, axis=None): """Mask whole columns of a 2D array that contain masked values. Parameters ---------- axis : int, optional Axis along which to perform the operation. If None, applies to a flattened version of the array. """ return mask_rowcols(a, 1) def dot(a,b, strict=False): """Return the dot product of two 2D masked arrays a and b. Like the generic numpy equivalent, the product sum is over the last dimension of a and the second-to-last dimension of b. If strict is True, masked values are propagated: if a masked value appears in a row or column, the whole row or column is considered masked. Parameters ---------- strict : {boolean} Whether masked data are propagated (True) or set to 0 for the computation. Notes ----- The first argument is not conjugated. """ #!!!: Works only with 2D arrays. There should be a way to get it to run with higher dimension if strict and (a.ndim == 2) and (b.ndim == 2): a = mask_rows(a) b = mask_cols(b) # d = np.dot(filled(a, 0), filled(b, 0)) # am = (~getmaskarray(a)) bm = (~getmaskarray(b)) m = ~np.dot(am, bm) return masked_array(d, mask=m) #............................................................................... def ediff1d(array, to_end=None, to_begin=None): """Return the differences between consecutive elements of an array, possibly with prefixed and/or appended values. Parameters ---------- array : {array} Input array, will be flattened before the difference is taken. to_end : {number}, optional If provided, this number will be tacked onto the end of the returned differences. to_begin : {number}, optional If provided, this number will be taked onto the beginning of the returned differences. Returns ------- ed : {array} The differences. Loosely, this will be (ary[1:] - ary[:-1]). """ a = masked_array(array, copy=True) if a.ndim > 1: a.reshape((a.size,)) (d, m, n) = (a._data, a._mask, a.size-1) dd = d[1:]-d[:-1] if m is nomask: dm = nomask else: dm = m[1:]-m[:-1] # if to_end is not None: to_end = asarray(to_end) nend = to_end.size if to_begin is not None: to_begin = asarray(to_begin) nbegin = to_begin.size r_data = np.empty((n+nend+nbegin,), dtype=a.dtype) r_mask = np.zeros((n+nend+nbegin,), dtype=bool) r_data[:nbegin] = to_begin._data r_mask[:nbegin] = to_begin._mask r_data[nbegin:-nend] = dd r_mask[nbegin:-nend] = dm else: r_data = np.empty((n+nend,), dtype=a.dtype) r_mask = np.zeros((n+nend,), dtype=bool) r_data[:-nend] = dd r_mask[:-nend] = dm r_data[-nend:] = to_end._data r_mask[-nend:] = to_end._mask # elif to_begin is not None: to_begin = asarray(to_begin) nbegin = to_begin.size r_data = np.empty((n+nbegin,), dtype=a.dtype) r_mask = np.zeros((n+nbegin,), dtype=bool) r_data[:nbegin] = to_begin._data r_mask[:nbegin] = to_begin._mask r_data[nbegin:] = dd r_mask[nbegin:] = dm # else: r_data = dd r_mask = dm return masked_array(r_data, mask=r_mask) def _covhelper(x, y=None, rowvar=True, allow_masked=True): """ Private function for the computation of covariance and correlation coefficients. """ x = ma.array(x, ndmin=2, copy=True, dtype=float) xmask = ma.getmaskarray(x) # Quick exit if we can't process masked data if not allow_masked and xmask.any(): raise ValueError("Cannot process masked data...") # if x.shape[0] == 1: rowvar = True # Make sure that rowvar is either 0 or 1 rowvar = int(bool(rowvar)) axis = 1-rowvar if rowvar: tup = (slice(None), None) else: tup = (None, slice(None)) # if y is None: xnotmask = np.logical_not(xmask).astype(int) else: y = array(y, copy=False, ndmin=2, dtype=float) ymask = ma.getmaskarray(y) if not allow_masked and ymask.any(): raise ValueError("Cannot process masked data...") if xmask.any() or ymask.any(): if y.shape == x.shape: # Define some common mask common_mask = np.logical_or(xmask, ymask) if common_mask is not nomask: x.unshare_mask() y.unshare_mask() xmask = x._mask = y._mask = ymask = common_mask x = ma.concatenate((x,y),axis) xnotmask = np.logical_not(np.concatenate((xmask, ymask), axis)).astype(int) x -= x.mean(axis=rowvar)[tup] return (x, xnotmask, rowvar) def cov(x, y=None, rowvar=True, bias=False, allow_masked=True): """Estimates the covariance matrix. Normalization is by (N-1) where N is the number of observations (unbiased estimate). If bias is True then normalization is by N. By default, masked values are recognized as such. If x and y have the same shape, a common mask is allocated: if x[i,j] is masked, then y[i,j] will also be masked. Setting `allow_masked` to False will raise an exception if values are missing in either of the input arrays. Parameters ---------- x : array-like Input data. If x is a 1D array, returns the variance. If x is a 2D array, returns the covariance matrix. y : {None, array-like}, optional Optional set of variables. rowvar : {False, True} optional If rowvar is true, then each row is a variable with observations in columns. If rowvar is False, each column is a variable and the observations are in the rows. bias : {False, True} optional Whether to use a biased (True) or unbiased (False) estimate of the covariance. If bias is True, then the normalization is by N, the number of observations. Otherwise, the normalization is by (N-1). allow_masked : {True, False} optional If True, masked values are propagated pair-wise: if a value is masked in x, the corresponding value is masked in y. If False, raises a ValueError exception when some values are missing. Raises ------ ValueError: Raised if some values are missing and allow_masked is False. """ (x, xnotmask, rowvar) = _covhelper(x, y, rowvar, allow_masked) if not rowvar: fact = np.dot(xnotmask.T, xnotmask)*1. - (1 - bool(bias)) result = (dot(x.T, x.conj(), strict=False) / fact).squeeze() else: fact = np.dot(xnotmask, xnotmask.T)*1. - (1 - bool(bias)) result = (dot(x, x.T.conj(), strict=False) / fact).squeeze() return result def corrcoef(x, y=None, rowvar=True, bias=False, allow_masked=True): """The correlation coefficients formed from the array x, where the rows are the observations, and the columns are variables. corrcoef(x,y) where x and y are 1d arrays is the same as corrcoef(transpose([x,y])) Parameters ---------- x : ndarray Input data. If x is a 1D array, returns the variance. If x is a 2D array, returns the covariance matrix. y : {None, ndarray} optional Optional set of variables. rowvar : {False, True} optional If True, then each row is a variable with observations in columns. If False, each column is a variable and the observations are in the rows. bias : {False, True} optional Whether to use a biased (True) or unbiased (False) estimate of the covariance. If True, then the normalization is by N, the number of non-missing observations. Otherwise, the normalization is by (N-1). allow_masked : {True, False} optional If True, masked values are propagated pair-wise: if a value is masked in x, the corresponding value is masked in y. If False, raises an exception. See Also -------- cov """ # Get the data (x, xnotmask, rowvar) = _covhelper(x, y, rowvar, allow_masked) # Compute the covariance matrix if not rowvar: fact = np.dot(xnotmask.T, xnotmask)*1. - (1 - bool(bias)) c = (dot(x.T, x.conj(), strict=False) / fact).squeeze() else: fact = np.dot(xnotmask, xnotmask.T)*1. - (1 - bool(bias)) c = (dot(x, x.T.conj(), strict=False) / fact).squeeze() # Check whether we have a scalar try: diag = ma.diagonal(c) except ValueError: return 1 # if xnotmask.all(): _denom = ma.sqrt(ma.multiply.outer(diag, diag)) else: _denom = diagflat(diag) n = x.shape[1-rowvar] if rowvar: for i in range(n-1): for j in range(i+1,n): _x = mask_cols(vstack((x[i], x[j]))).var(axis=1, ddof=1-bias) _denom[i,j] = _denom[j,i] = ma.sqrt(ma.multiply.reduce(_x)) else: for i in range(n-1): for j in range(i+1,n): _x = mask_cols(vstack((x[:,i], x[:,j]))).var(axis=1, ddof=1-bias) _denom[i,j] = _denom[j,i] = ma.sqrt(ma.multiply.reduce(_x)) return c/_denom #####-------------------------------------------------------------------------- #---- --- Concatenation helpers --- #####-------------------------------------------------------------------------- class MAxisConcatenator(AxisConcatenator): """Translate slice objects to concatenation along an axis. """ def __init__(self, axis=0): AxisConcatenator.__init__(self, axis, matrix=False) def __getitem__(self,key): if isinstance(key, str): raise MAError, "Unavailable for masked array." if type(key) is not tuple: key = (key,) objs = [] scalars = [] final_dtypedescr = None for k in range(len(key)): scalar = False if type(key[k]) is slice: step = key[k].step start = key[k].start stop = key[k].stop if start is None: start = 0 if step is None: step = 1 if type(step) is type(1j): size = int(abs(step)) newobj = np.linspace(start, stop, num=size) else: newobj = np.arange(start, stop, step) elif type(key[k]) is str: if (key[k] in 'rc'): self.matrix = True self.col = (key[k] == 'c') continue try: self.axis = int(key[k]) continue except (ValueError, TypeError): raise ValueError, "Unknown special directive" elif type(key[k]) in np.ScalarType: newobj = asarray([key[k]]) scalars.append(k) scalar = True else: newobj = key[k] objs.append(newobj) if isinstance(newobj, ndarray) and not scalar: if final_dtypedescr is None: final_dtypedescr = newobj.dtype elif newobj.dtype > final_dtypedescr: final_dtypedescr = newobj.dtype if final_dtypedescr is not None: for k in scalars: objs[k] = objs[k].astype(final_dtypedescr) res = concatenate(tuple(objs),axis=self.axis) return self._retval(res) class mr_class(MAxisConcatenator): """Translate slice objects to concatenation along the first axis. For example: >>> np.ma.mr_[np.ma.array([1,2,3]), 0, 0, np.ma.array([4,5,6])] array([1, 2, 3, 0, 0, 4, 5, 6]) """ def __init__(self): MAxisConcatenator.__init__(self, 0) mr_ = mr_class() #####-------------------------------------------------------------------------- #---- Find unmasked data --- #####-------------------------------------------------------------------------- def flatnotmasked_edges(a): """Find the indices of the first and last not masked values in a 1D masked array. If all values are masked, returns None. """ m = getmask(a) if m is nomask or not np.any(m): return [0,-1] unmasked = np.flatnonzero(~m) if len(unmasked) > 0: return unmasked[[0,-1]] else: return None def notmasked_edges(a, axis=None): """Find the indices of the first and last not masked values along the given axis in a masked array. If all values are masked, return None. Otherwise, return a list of 2 tuples, corresponding to the indices of the first and last unmasked values respectively. Parameters ---------- axis : int, optional Axis along which to perform the operation. If None, applies to a flattened version of the array. """ a = asarray(a) if axis is None or a.ndim == 1: return flatnotmasked_edges(a) m = getmask(a) idx = array(np.indices(a.shape), mask=np.asarray([m]*a.ndim)) return [tuple([idx[i].min(axis).compressed() for i in range(a.ndim)]), tuple([idx[i].max(axis).compressed() for i in range(a.ndim)]),] def flatnotmasked_contiguous(a): """Find contiguous unmasked data in a flattened masked array. Return a sorted sequence of slices (start index, end index). """ m = getmask(a) if m is nomask: return (a.size, [0,-1]) unmasked = np.flatnonzero(~m) if len(unmasked) == 0: return None result = [] for k, group in groupby(enumerate(unmasked), lambda (i,x):i-x): tmp = np.array([g[1] for g in group], int) # result.append((tmp.size, tuple(tmp[[0,-1]]))) result.append( slice(tmp[0], tmp[-1]) ) result.sort() return result def notmasked_contiguous(a, axis=None): """Find contiguous unmasked data in a masked array along the given axis. Parameters ---------- axis : int, optional Axis along which to perform the operation. If None, applies to a flattened version of the array. Returns ------- A sorted sequence of slices (start index, end index). Notes ----- Only accepts 2D arrays at most. """ a = asarray(a) nd = a.ndim if nd > 2: raise NotImplementedError,"Currently limited to atmost 2D array." if axis is None or nd == 1: return flatnotmasked_contiguous(a) # result = [] # other = (axis+1)%2 idx = [0,0] idx[axis] = slice(None, None) # for i in range(a.shape[other]): idx[other] = i result.append( flatnotmasked_contiguous(a[idx]) ) return result #####-------------------------------------------------------------------------- #---- Polynomial fit --- #####-------------------------------------------------------------------------- def vander(x, n=None): """%s Notes ----- Masked values in x will result in rows of zeros. """ _vander = np.vander(x, n) m = getmask(x) if m is not nomask: _vander[m] = 0 return _vander def polyfit(x, y, deg, rcond=None, full=False): """ Least squares polynomial fit. Do a best fit polynomial of degree 'deg' of 'x' to 'y'. Return value is a vector of polynomial coefficients [pk ... p1 p0]. Eg, for ``deg = 2``:: p2*x0^2 + p1*x0 + p0 = y1 p2*x1^2 + p1*x1 + p0 = y1 p2*x2^2 + p1*x2 + p0 = y2 ..... p2*xk^2 + p1*xk + p0 = yk Parameters ---------- x : array_like 1D vector of sample points. y : array_like 1D vector or 2D array of values to fit. The values should run down the columns in the 2D case. deg : integer Degree of the fitting polynomial rcond: {None, float}, optional Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The defaul value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases. full : {False, boolean}, optional Switch determining nature of return value. When it is False just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned. Returns ------- coefficients, [residuals, rank, singular_values, rcond] : variable When full=False, only the coefficients are returned, running down the appropriate colume when y is a 2D array. When full=True, the rank of the scaled Vandermonde matrix, its effective rank in light of the rcond value, its singular values, and the specified value of rcond are also returned. Warns ----- RankWarning : if rank is reduced and not full output The warnings can be turned off by: >>> import warnings >>> warnings.simplefilter('ignore',np.RankWarning) See Also -------- polyval : computes polynomial values. Notes ----- If X is a the Vandermonde Matrix computed from x (see http://mathworld.wolfram.com/VandermondeMatrix.html), then the polynomial least squares solution is given by the 'p' in X*p = y where X.shape is a matrix of dimensions (len(x), deg + 1), p is a vector of dimensions (deg + 1, 1), and y is a vector of dimensions (len(x), 1). This equation can be solved as p = (XT*X)^-1 * XT * y where XT is the transpose of X and -1 denotes the inverse. However, this method is susceptible to rounding errors and generally the singular value decomposition of the matrix X is preferred and that is what is done here. The singular value method takes a paramenter, 'rcond', which sets a limit on the relative size of the smallest singular value to be used in solving the equation. This may result in lowering the rank of the Vandermonde matrix, in which case a RankWarning is issued. If polyfit issues a RankWarning, try a fit of lower degree or replace x by x - x.mean(), both of which will generally improve the condition number. The routine already normalizes the vector x by its maximum absolute value to help in this regard. The rcond parameter can be set to a value smaller than its default, but the resulting fit may be spurious. The current default value of rcond is len(x)*eps, where eps is the relative precision of the floating type being used, generally around 1e-7 and 2e-16 for IEEE single and double precision respectively. This value of rcond is fairly conservative but works pretty well when x - x.mean() is used in place of x. DISCLAIMER: Power series fits are full of pitfalls for the unwary once the degree of the fit becomes large or the interval of sample points is badly centered. The problem is that the powers x**n are generally a poor basis for the polynomial functions on the sample interval, resulting in a Vandermonde matrix is ill conditioned and coefficients sensitive to rounding erros. The computation of the polynomial values will also sensitive to rounding errors. Consequently, the quality of the polynomial fit should be checked against the data whenever the condition number is large. The quality of polynomial fits *can not* be taken for granted. If all you want to do is draw a smooth curve through the y values and polyfit is not doing the job, try centering the sample range or look into scipy.interpolate, which includes some nice spline fitting functions that may be of use. For more info, see http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html, but note that the k's and n's in the superscripts and subscripts on that page. The linear algebra is correct, however. Notes ----- Any masked values in x is propagated in y, and vice-versa. """ order = int(deg) + 1 x = asarray(x) mx = getmask(x) y = asarray(y) if y.ndim == 1: m = mask_or(mx, getmask(y)) elif y.ndim == 2: y = mask_rows(y) my = getmask(y) if my is not nomask: m = mask_or(mx, my[:,0]) else: m = mx else: raise TypeError,"Expected a 1D or 2D array for y!" if m is not nomask: x[m] = y[m] = masked # Set rcond if rcond is None : rcond = len(x)*np.finfo(x.dtype).eps # Scale x to improve condition number scale = abs(x).max() if scale != 0 : x = x / scale # solve least squares equation for powers of x v = vander(x, order) c, resids, rank, s = _lstsq(v, y.filled(0), rcond) # warn on rank reduction, which indicates an ill conditioned matrix if rank != order and not full: warnings.warn("Polyfit may be poorly conditioned", np.RankWarning) # scale returned coefficients if scale != 0 : if c.ndim == 1 : c /= np.vander([scale], order)[0] else : c /= np.vander([scale], order).T if full : return c, resids, rank, s, rcond else : return c ################################################################################