def _asarray1d(arr, copy=False): """Ensure 1D array for one array. """ if copy: return asarray(arr).flatten() else: return asarray(arr).ravel()
def meshgrid(x,y): """ For vectors x, y with lengths Nx=len(x) and Ny=len(y), return X, Y where X and Y are (Ny, Nx) shaped arrays with the elements of x and y repeated to fill the matrix EG, [X, Y] = meshgrid([1,2,3], [4,5,6,7]) X = 1 2 3 1 2 3 1 2 3 1 2 3 Y = 4 4 4 5 5 5 6 6 6 7 7 7 """ x = asarray(x) y = asarray(y) numRows, numCols = len(y), len(x) # yes, reversed x = x.reshape(1,numCols) X = x.repeat(numRows, axis=0) y = y.reshape(numRows,1) Y = y.repeat(numCols, axis=1) return X, Y
def apply_along_axis(func1d,axis,arr,*args): """ 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 = asarray(arr) 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 = zeros(nd,'O') indlist = range(nd) indlist.remove(axis) i[axis] = slice(None,None) outshape = asarray(arr.shape).take(indlist) i.put(indlist, ind) res = func1d(arr[tuple(i.tolist())],*args) # if res is a number, then we have a smaller output array if isscalar(res): outarr = zeros(outshape,asarray(res).dtype) outarr[tuple(ind)] = res Ntot = 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) outarr[tuple(ind)] = res k += 1 return outarr else: Ntot = product(outshape) holdshape = outshape outshape = list(arr.shape) outshape[axis] = len(res) outarr = zeros(outshape,asarray(res).dtype) outarr[tuple(i.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) res = func1d(arr[tuple(i.tolist())],*args) outarr[tuple(i.tolist())] = res k += 1 return outarr
def __gmmEm__(self): self.mean = kmeans2(self.data, self.K)[0] self.c = asarray([1.0/self.K]*self.K) self.covm = asarray([identity(self.K)]*self.K) self.p = ndarray((self.N,self.K),dtype='float32') while self.it > 0: self.it -=1 self.__calculateP__() #self.__Estep__() self.__Mstep__()
def polyval(p, x): """ Evaluate a polynomial at specific values. If p is of length N, this function returns the value: p[0]*(x**N-1) + p[1]*(x**N-2) + ... + p[N-2]*x + p[N-1] If x is a sequence then p(x) will be returned for all elements of x. If x is another polynomial then the composite polynomial p(x) will be returned. Parameters ---------- p : {array_like, poly1d} 1D array of polynomial coefficients from highest degree to zero or an instance of poly1d. x : {array_like, poly1d} A number, a 1D array of numbers, or an instance of poly1d. Returns ------- values : {ndarray, poly1d} If either p or x is an instance of poly1d, then an instance of poly1d is returned, otherwise a 1D array is returned. In the case where x is a poly1d, the result is the composition of the two polynomials, i.e., substitution is used. See Also -------- poly1d: A polynomial class. Notes ----- Horner's method is used to evaluate the polynomial. Even so, for polynomials of high degree the values may be inaccurate due to rounding errors. Use carefully. Examples -------- >>> np.polyval([3,0,1], 5) # 3 * 5**2 + 0 * 5**1 + 1 76 """ p = NX.asarray(p) if isinstance(x, poly1d): y = 0 else: x = NX.asarray(x) y = NX.zeros_like(x) for i in range(len(p)): y = x * y + p[i] return y
def apply_over_axes(func, a, axes): """Apply a function repeatedly over multiple axes, keeping the same shape for the resulting array. func is called as res = func(a, axis). The result is assumed to be either the same shape as a or have one less dimension. This call is repeated for each axis in the axes sequence. """ val = asarray(a) N = a.ndim if array(axes).ndim == 0: axes = (axes,) for axis in axes: if axis < 0: axis = N + axis args = (val, axis) res = func(*args) if res.ndim == val.ndim: val = res else: res = expand_dims(res,axis) if res.ndim == val.ndim: val = res else: raise ValueError, "function is not returning"\ " an array of correct shape" return val
def __getitem__(self, indx): reset_full = True if isinstance(indx, Date): indx = self.find_dates(indx) reset_full = False elif numeric.asarray(indx).dtype.kind == 'O': try: indx = self.find_dates(indx) except AttributeError: pass r = ndarray.__getitem__(self, indx) if isinstance(r, (generic, int)): return Date(self.freq, value=r) elif hasattr(r, 'size') and r.size == 1: # need to check if it has a size attribute for situations # like when the datearray is the data for a maskedarray # or some other subclass of ndarray with wierd getitem # behaviour return Date(self.freq, value=r.item()) else: if hasattr(r, '_cachedinfo'): _cache = r._cachedinfo _cache.update(dict([(k,_cache[k][indx]) for k in ('toobj', 'tostr', 'toord') if _cache[k] is not None])) _cache['steps'] = None if reset_full: _cache['full'] = None _cache['hasdups'] = None return r
def guess_freq(dates): """Tries to estimate the frequency of a list of dates, by checking the steps between consecutive dates The steps should be in days. Returns a frequency code (alpha character).""" ddif = numeric.asarray(numpy.diff(dates)) ddif.sort() if ddif.size == 0: fcode = _c.FR_UND elif ddif[0] == ddif[-1] == 1.: fcode = _c.FR_DAY elif (ddif[0] == 1.) and (ddif[-1] == 3.): fcode = _c.FR_BUS elif (ddif[0] > 3.) and (ddif[-1] == 7.): fcode = _c.FR_WK elif (ddif[0] >= 28.) and (ddif[-1] <= 31.): fcode = _c.FR_MTH elif (ddif[0] >= 90.) and (ddif[-1] <= 92.): fcode = _c.FR_QTR elif (ddif[0] >= 365.) and (ddif[-1] <= 366.): fcode = _c.FR_ANN elif numpy.abs(24.*ddif[0] - 1) <= 1e-5 and \ numpy.abs(24.*ddif[-1] - 1) <= 1e-5: fcode = _c.FR_HR elif numpy.abs(1440.*ddif[0] - 1) <= 1e-5 and \ numpy.abs(1440.*ddif[-1] - 1) <= 1e-5: fcode = _c.FR_MIN elif numpy.abs(86400.*ddif[0] - 1) <= 1e-5 and \ numpy.abs(86400.*ddif[-1] - 1) <= 1e-5: fcode = _c.FR_SEC else: warnings.warn("Unable to estimate the frequency! %.3f<>%.3f" %\ (ddif[0], ddif[-1])) fcode = _c.FR_UND return fcode
def iscomplexobj(x): """ Check for a complex type or an array of complex numbers. The type of the input is checked, not the value. Even if the input has an imaginary part equal to zero, `iscomplexobj` evaluates to True. Parameters ---------- x : any The input can be of any type and shape. Returns ------- iscomplexobj : bool The return value, True if `x` is of a complex type or has at least one complex element. See Also -------- isrealobj, iscomplex Examples -------- >>> np.iscomplexobj(1) False >>> np.iscomplexobj(1+0j) True >>> np.iscomplexobj([3, 1+0j, True]) True """ return issubclass(asarray(x).dtype.type, _nx.complexfloating)
def isrealobj(x): """ Return True if x is a not complex type or an array of complex numbers. The type of the input is checked, not the value. So even if the input has an imaginary part equal to zero, `isrealobj` evaluates to False if the data type is complex. Parameters ---------- x : any The input can be of any type and shape. Returns ------- y : bool The return value, False if `x` is of a complex type. See Also -------- iscomplexobj, isreal Examples -------- >>> np.isrealobj(1) True >>> np.isrealobj(1+0j) False >>> np.isrealobj([3, 1+0j, True]) False """ return not issubclass(asarray(x).dtype.type, _nx.complexfloating)
def ix_(*args): """ Construct an open mesh from multiple sequences. This function takes n 1-d sequences and returns n outputs with n dimensions each such that the shape is 1 in all but one dimension and the dimension with the non-unit shape value cycles through all n dimensions. Using ix_() one can quickly construct index arrays that will index the cross product. a[ix_([1,3,7],[2,5,8])] returns the array a[1,2] a[1,5] a[1,8] a[3,2] a[3,5] a[3,8] a[7,2] a[7,5] a[7,8] """ out = [] nd = len(args) baseshape = [1]*nd for k in range(nd): new = _nx.asarray(args[k]) if (new.ndim != 1): raise ValueError, "Cross index must be 1 dimensional" if issubclass(new.dtype.type, _nx.bool_): new = new.nonzero()[0] baseshape[k] = len(new) new = new.reshape(tuple(baseshape)) out.append(new) baseshape[k] = 1 return tuple(out)
def asfarray(a, dtype=_nx.float_): """ Return an array converted to a float type. Parameters ---------- a : array_like The input array. dtype : str or dtype object, optional Float type code to coerce input array `a`. If `dtype` is one of the 'int' dtypes, it is replaced with float64. Returns ------- out : ndarray The input `a` as a float ndarray. Examples -------- >>> np.asfarray([2, 3]) array([2., 3.]) >>> np.asfarray([2, 3], dtype='float') array([2., 3.]) >>> np.asfarray([2, 3], dtype='int8') array([2., 3.]) """ if not _nx.issubdtype(dtype, _nx.inexact): dtype = _nx.float_ return asarray(a, dtype=dtype)
def atleast_3d(*arys): """ Force a sequence of arrays to each be at least 3D. Description: Force an array each be at least 3D. If the array is 0D or 1D, the array is converted to a single 1xNx1 array of values where N is the orginal length of the array. If the array is 2D, the array is converted to a single MxNx1 array of values where MxN is the orginal shape of the array. Otherwise, the array is unaltered. Arguments: arys -- arrays to be converted to 3 or more dimensional array. Returns: input array converted to at least 3D array. """ res = [] for ary in arys: ary = asarray(ary) if len(ary.shape) == 0: result = ary.reshape(1,1,1) elif len(ary.shape) == 1: result = ary[newaxis,:,newaxis] elif len(ary.shape) == 2: result = ary[:,:,newaxis] else: result = ary res.append(result) if len(res) == 1: return res[0] else: return res
def isneginf(x, y=None): """ Return True where x is -infinity, and False otherwise. Parameters ---------- x : array_like The input array. y : array_like A boolean array with the same shape as `x` to store the result. Returns ------- y : ndarray A boolean array where y[i] = True only if x[i] = -Inf. See Also -------- isposinf, isfinite Examples -------- >>> np.isneginf([-np.inf, 0., np.inf]) array([ True, False, False], dtype=bool) """ if y is None: x = nx.asarray(x) y = nx.empty(x.shape, dtype=nx.bool_) nx.logical_and(nx.isinf(x), nx.signbit(x), y) return y
def _fix_int_lt_zero(x): """Convert `x` to double if it has real, negative components. Otherwise, output is just the array version of the input (via asarray). Parameters ---------- x : array_like Returns ------- array Examples -------- >>> _fix_int_lt_zero([1,2]) array([1, 2]) >>> _fix_int_lt_zero([-1,2]) array([-1., 2.]) """ x = asarray(x) if any(isreal(x) & (x < 0)): x = x * 1.0 return x
def diagflat(v,k=0): """Return a 2D array whose k'th diagonal is a flattened v and all other elements are zero. Examples -------- >>> diagflat([[1,2],[3,4]]]) array([[1, 0, 0, 0], [0, 2, 0, 0], [0, 0, 3, 0], [0, 0, 0, 4]]) >>> diagflat([1,2], 1) array([[0, 1, 0], [0, 0, 2], [0, 0, 0]]) """ try: wrap = v.__array_wrap__ except AttributeError: wrap = None v = asarray(v).ravel() s = len(v) n = s + abs(k) res = zeros((n,n), v.dtype) if (k>=0): i = arange(0,n-k) fi = i+k+i*n else: i = arange(0,n+k) fi = i+(i-k)*n res.flat[fi] = v if not wrap: return res return wrap(res)
def diag(v, k=0): """ returns a copy of the the k-th diagonal if v is a 2-d array or returns a 2-d array with v as the k-th diagonal if v is a 1-d array. """ v = asarray(v) s = v.shape if len(s)==1: n = s[0]+abs(k) res = zeros((n,n), v.dtype) if (k>=0): i = arange(0,n-k) fi = i+k+i*n else: i = arange(0,n+k) fi = i+(i-k)*n res.flat[fi] = v return res elif len(s)==2: N1,N2 = s if k >= 0: M = min(N1,N2-k) i = arange(0,M) fi = i+k+i*N2 else: M = min(N1+k,N2) i = arange(0,M) fi = i + (i-k)*N2 return v.flat[fi] else: raise ValueError, "Input must be 1- or 2-d."
def mintypecode(typechars,typeset='GDFgdf',default='d'): """ Return a minimum data type character from typeset that handles all typechars given The returned type character must be the smallest size such that an array of the returned type can handle the data from an array of type t for each t in typechars (or if typechars is an array, then its dtype.char). If the typechars does not intersect with the typeset, then default is returned. If t in typechars is not a string then t=asarray(t).dtype.char is applied. """ typecodes = [(type(t) is type('') and t) or asarray(t).dtype.char\ for t in typechars] intersection = [t for t in typecodes if t in typeset] if not intersection: return default if 'F' in intersection and 'd' in intersection: return 'D' l = [] for t in intersection: i = _typecodes_by_elsize.index(t) l.append((i,t)) l.sort() return l[0][1]
def vander(x, N=None): """ Generate a Van der Monde matrix. The columns of the output matrix are decreasing powers of the input vector. Specifically, the i-th output column is the input vector to the power of ``N - i - 1``. Such a matrix with a geometric progression in each row is named Van Der Monde, or Vandermonde matrix, from Alexandre-Theophile Vandermonde. Parameters ---------- x : array_like 1-D input array. N : int, optional Order of (number of columns in) the output. If `N` is not specified, a square array is returned (``N = len(x)``). Returns ------- out : ndarray Van der Monde matrix of order `N`. The first column is ``x^(N-1)``, the second ``x^(N-2)`` and so forth. References ---------- .. [1] Wikipedia, "Vandermonde matrix", http://en.wikipedia.org/wiki/Vandermonde_matrix Examples -------- >>> x = np.array([1, 2, 3, 5]) >>> N = 3 >>> np.vander(x, N) array([[ 1, 1, 1], [ 4, 2, 1], [ 9, 3, 1], [25, 5, 1]]) >>> np.column_stack([x**(N-1-i) for i in range(N)]) array([[ 1, 1, 1], [ 4, 2, 1], [ 9, 3, 1], [25, 5, 1]]) >>> x = np.array([1, 2, 3, 5]) >>> np.vander(x) array([[ 1, 1, 1, 1], [ 8, 4, 2, 1], [ 27, 9, 3, 1], [125, 25, 5, 1]]) """ x = asarray(x) if N is None: N = len(x) X = ones((len(x), N), x.dtype) for i in range(N - 1): X[:, i] = x ** (N - i - 1) return X
def _fix_real_abs_gt_1(x): """Convert `x` to complex if it has real components x_i with abs(x_i)>1. Otherwise, output is just the array version of the input (via asarray). Parameters ---------- x : array_like Returns ------- array Examples -------- >>> np.lib.scimath._fix_real_abs_gt_1([0,1]) array([0, 1]) >>> np.lib.scimath._fix_real_abs_gt_1([0,2]) array([ 0.+0.j, 2.+0.j]) """ x = asarray(x) if any(isreal(x) & (abs(x)>1)): x = _tocomplex(x) return x
def _fix_real_lt_zero(x): """Convert `x` to complex if it has real, negative components. Otherwise, output is just the array version of the input (via asarray). Parameters ---------- x : array_like Returns ------- array Examples -------- >>> np.lib.scimath._fix_real_lt_zero([1,2]) array([1, 2]) >>> np.lib.scimath._fix_real_lt_zero([-1,2]) array([-1.+0.j, 2.+0.j]) """ x = asarray(x) if any(isreal(x) & (x<0)): x = _tocomplex(x) return x
def polyint(p, m=1, k=None): """Return the mth analytical integral of the polynomial p. If k is None, then zero-valued constants of integration are used. otherwise, k should be a list of length m (or a scalar if m=1) to represent the constants of integration to use for each integration (starting with k[0]) """ m = int(m) if m < 0: raise ValueError, "Order of integral must be positive (see polyder)" if k is None: k = NX.zeros(m, float) k = atleast_1d(k) if len(k) == 1 and m > 1: k = k[0]*NX.ones(m, float) if len(k) < m: raise ValueError, \ "k must be a scalar or a rank-1 array of length 1 or >m." if m == 0: return p else: truepoly = isinstance(p, poly1d) p = NX.asarray(p) y = NX.zeros(len(p)+1, float) y[:-1] = p*1.0/NX.arange(len(p), 0, -1) y[-1] = k[0] val = polyint(y, m-1, k=k[1:]) if truepoly: val = poly1d(val) return val
def asarray_chkfinite(a): """Like asarray, but check that no NaNs or Infs are present. """ a = asarray(a) if (a.dtype.char in typecodes['AllFloat']) \ and (_nx.isnan(a).any() or _nx.isinf(a).any()): raise ValueError, "array must not contain infs or NaNs" return a
def expand_dims(a, axis): """Expand the shape of a by including newaxis before given axis. """ a = asarray(a) shape = a.shape if axis < 0: axis = axis + len(shape) + 1 return a.reshape(shape[:axis] + (1,) + shape[axis:])
def ix_(*args): """ Construct an open mesh from multiple sequences. This function takes N 1-D sequences and returns N outputs with N dimensions each, such that the shape is 1 in all but one dimension and the dimension with the non-unit shape value cycles through all N dimensions. Using `ix_` one can quickly construct index arrays that will index the cross product. ``a[np.ix_([1,3],[2,5])]`` returns the array ``[[a[1,2] a[1,5]], [a[3,2] a[3,5]]]``. Parameters ---------- args : 1-D sequences Returns ------- out : tuple of ndarrays N arrays with N dimensions each, with N the number of input sequences. Together these arrays form an open mesh. See Also -------- ogrid, mgrid, meshgrid Examples -------- >>> a = np.arange(10).reshape(2, 5) >>> a array([[0, 1, 2, 3, 4], [5, 6, 7, 8, 9]]) >>> ixgrid = np.ix_([0,1], [2,4]) >>> ixgrid (array([[0], [1]]), array([[2, 4]])) >>> ixgrid[0].shape, ixgrid[1].shape ((2, 1), (1, 2)) >>> a[ixgrid] array([[2, 4], [7, 9]]) """ out = [] nd = len(args) baseshape = [1] * nd for k in range(nd): new = _nx.asarray(args[k]) if new.ndim != 1: raise ValueError("Cross index must be 1 dimensional") if issubclass(new.dtype.type, _nx.bool_): new = new.nonzero()[0] baseshape[k] = len(new) new = new.reshape(tuple(baseshape)) out.append(new) baseshape[k] = 1 return tuple(out)
def ix_(*args): """ Construct an open mesh from multiple sequences. This function takes N 1-D sequences and returns N outputs with N dimensions each, such that the shape is 1 in all but one dimension and the dimension with the non-unit shape value cycles through all N dimensions. Using `ix_` one can quickly construct index arrays that will index the cross product. ``a[np.ix_([1,3],[2,5])]`` returns the array ``[[a[1,2] a[1,5]], [a[3,2] a[3,5]]]``. Parameters ---------- args : 1-D sequences Returns ------- out : tuple of ndarrays N arrays with N dimensions each, with N the number of input sequences. Together these arrays form an open mesh. See Also -------- ogrid, mgrid, meshgrid Examples -------- >>> a = np.arange(10).reshape(2, 5) >>> a array([[0, 1, 2, 3, 4], [5, 6, 7, 8, 9]]) >>> ixgrid = np.ix_([0,1], [2,4]) >>> ixgrid (array([[0], [1]]), array([[2, 4]])) >>> ixgrid[0].shape, ixgrid[1].shape ((2, 1), (1, 2)) >>> a[ixgrid] array([[2, 4], [7, 9]]) """ out = [] nd = len(args) for k, new in enumerate(args): new = asarray(new) if new.ndim != 1: raise ValueError("Cross index must be 1 dimensional") if new.size == 0: # Explicitly type empty arrays to avoid float default new = new.astype(_nx.intp) if issubdtype(new.dtype, _nx.bool_): new, = new.nonzero() new = new.reshape((1,)*k + (new.size,) + (1,)*(nd-k-1)) out.append(new) return tuple(out)
def expand_dims(a, axis): """ Expand the shape of an array. Insert a new axis, corresponding to a given position in the array shape. Parameters ---------- a : array_like Input array. axis : int Position (amongst axes) where new axis is to be inserted. Returns ------- res : ndarray Output array. The number of dimensions is one greater than that of the input array. See Also -------- squeeze : The inverse operation, removing singleton dimensions reshape : Insert, remove, and combine dimensions, and resize existing ones doc.indexing, atleast_1d, atleast_2d, atleast_3d Examples -------- >>> x = np.array([1,2]) >>> x.shape (2,) The following is equivalent to ``x[np.newaxis,:]`` or ``x[np.newaxis]``: >>> y = np.expand_dims(x, axis=0) >>> y array([[1, 2]]) >>> y.shape (1, 2) >>> y = np.expand_dims(x, axis=1) # Equivalent to x[:,newaxis] >>> y array([[1], [2]]) >>> y.shape (2, 1) Note that some examples may use ``None`` instead of ``np.newaxis``. These are the same objects: >>> np.newaxis is None True """ a = asarray(a) shape = a.shape axis = normalize_axis_index(axis, a.ndim + 1) return a.reshape(shape[:axis] + (1,) + shape[axis:])
def diagflat(v, k=0): """ Create a two-dimensional array with the flattened input as a diagonal. Parameters ---------- v : array_like Input data, which is flattened and set as the `k`-th diagonal of the output. k : int, optional Diagonal to set; 0, the default, corresponds to the "main" diagonal, a positive (negative) `k` giving the number of the diagonal above (below) the main. Returns ------- out : ndarray The 2-D output array. See Also -------- diag : MATLAB work-alike for 1-D and 2-D arrays. diagonal : Return specified diagonals. trace : Sum along diagonals. Examples -------- >>> np.diagflat([[1,2], [3,4]]) array([[1, 0, 0, 0], [0, 2, 0, 0], [0, 0, 3, 0], [0, 0, 0, 4]]) >>> np.diagflat([1,2], 1) array([[0, 1, 0], [0, 0, 2], [0, 0, 0]]) """ try: wrap = v.__array_wrap__ except AttributeError: wrap = None v = asarray(v).ravel() s = len(v) n = s + abs(k) res = zeros((n,n), v.dtype) if (k >= 0): i = arange(0,n-k) fi = i+k+i*n else: i = arange(0,n+k) fi = i+(i-k)*n res.flat[fi] = v if not wrap: return res return wrap(res)
def diag(v, k=0): """ Extract a diagonal or construct a diagonal array. Parameters ---------- v : array_like If `v` is a 2-dimensional array, return a copy of its `k`-th diagonal. If `v` is a 1-dimensional array, return a 2-dimensional array with `v` on the `k`-th diagonal. k : int, optional Diagonal in question. The defaults is 0. Examples -------- >>> x = np.arange(9).reshape((3,3)) >>> x array([[0, 1, 2], [3, 4, 5], [6, 7, 8]]) >>> np.diag(x) array([0, 4, 8]) >>> np.diag(np.diag(x)) array([[0, 0, 0], [0, 4, 0], [0, 0, 8]]) """ v = asarray(v) s = v.shape if len(s)==1: n = s[0]+abs(k) res = zeros((n,n), v.dtype) if (k>=0): i = arange(0,n-k) fi = i+k+i*n else: i = arange(0,n+k) fi = i+(i-k)*n res.flat[fi] = v return res elif len(s)==2: N1,N2 = s if k >= 0: M = min(N1,N2-k) i = arange(0,M) fi = i+k+i*N2 else: M = min(N1+k,N2) i = arange(0,M) fi = i + (i-k)*N2 return v.flat[fi] else: raise ValueError, "Input must be 1- or 2-d."
def mintypecode(typechars,typeset='GDFgdf',default='d'): """ Return the character for the minimum-size type to which given types can be safely cast. The returned type character must represent the smallest size dtype such that an array of the returned type can handle the data from an array of all types in `typechars` (or if `typechars` is an array, then its dtype.char). Parameters ---------- typechars : list of str or array_like If a list of strings, each string should represent a dtype. If array_like, the character representation of the array dtype is used. typeset : str or list of str, optional The set of characters that the returned character is chosen from. The default set is 'GDFgdf'. default : str, optional The default character, this is returned if none of the characters in `typechars` matches a character in `typeset`. Returns ------- typechar : str The character representing the minimum-size type that was found. See Also -------- dtype, sctype2char, maximum_sctype Examples -------- >>> np.mintypecode(['d', 'f', 'S']) 'd' >>> x = np.array([1.1, 2-3.j]) >>> np.mintypecode(x) 'D' >>> np.mintypecode('abceh', default='G') 'G' """ typecodes = [(isinstance(t, str) and t) or asarray(t).dtype.char for t in typechars] intersection = [t for t in typecodes if t in typeset] if not intersection: return default if 'F' in intersection and 'd' in intersection: return 'D' l = [] for t in intersection: i = _typecodes_by_elsize.index(t) l.append((i, t)) l.sort() return l[0][1]
def poly(seq_of_zeros): """ Find the coefficients of a polynomial with the given sequence of roots. Returns the coefficients of the polynomial whose leading coefficient is one for the given sequence of zeros (multiple roots must be included in the sequence as many times as their multiplicity; see Examples). A square matrix (or array, which will be treated as a matrix) can also be given, in which case the coefficients of the characteristic polynomial of the matrix are returned. Parameters ---------- seq_of_zeros : array_like, shape (N,) or (N, N) A sequence of polynomial roots, or a square array or matrix object. Returns ------- c : ndarray 1D array of polynomial coefficients from highest to lowest degree: ``c[0] * x**(N) + c[1] * x**(N-1) + ... + c[N-1] * x + c[N]`` where c[0] always equals 1. Raises ------ ValueError If input is the wrong shape (the input must be a 1-D or square 2-D array). See Also -------- polyval : Evaluate a polynomial at a point. roots : Return the roots of a polynomial. polyfit : Least squares polynomial fit. poly1d : A one-dimensional polynomial class. Notes ----- Specifying the roots of a polynomial still leaves one degree of freedom, typically represented by an undetermined leading coefficient. [1]_ In the case of this function, that coefficient - the first one in the returned array - is always taken as one. (If for some reason you have one other point, the only automatic way presently to leverage that information is to use ``polyfit``.) The characteristic polynomial, :math:`p_a(t)`, of an `n`-by-`n` matrix **A** is given by :math:`p_a(t) = \\mathrm{det}(t\\, \\mathbf{I} - \\mathbf{A})`, where **I** is the `n`-by-`n` identity matrix. [2]_ References ---------- .. [1] M. Sullivan and M. Sullivan, III, "Algebra and Trignometry, Enhanced With Graphing Utilities," Prentice-Hall, pg. 318, 1996. .. [2] G. Strang, "Linear Algebra and Its Applications, 2nd Edition," Academic Press, pg. 182, 1980. Examples -------- Given a sequence of a polynomial's zeros: >>> np.poly((0, 0, 0)) # Multiple root example array([1, 0, 0, 0]) The line above represents z**3 + 0*z**2 + 0*z + 0. >>> np.poly((-1./2, 0, 1./2)) array([ 1. , 0. , -0.25, 0. ]) The line above represents z**3 - z/4 >>> np.poly((np.random.random(1.)[0], 0, np.random.random(1.)[0])) array([ 1. , -0.77086955, 0.08618131, 0. ]) #random Given a square array object: >>> P = np.array([[0, 1./3], [-1./2, 0]]) >>> np.poly(P) array([ 1. , 0. , 0.16666667]) Or a square matrix object: >>> np.poly(np.matrix(P)) array([ 1. , 0. , 0.16666667]) Note how in all cases the leading coefficient is always 1. """ seq_of_zeros = atleast_1d(seq_of_zeros) sh = seq_of_zeros.shape if len(sh) == 2 and sh[0] == sh[1] and sh[0] != 0: seq_of_zeros = eigvals(seq_of_zeros) elif len(sh) == 1: pass else: raise ValueError("input must be 1d or square 2d array.") if len(seq_of_zeros) == 0: return 1.0 a = [1] for k in range(len(seq_of_zeros)): a = NX.convolve(a, [1, -seq_of_zeros[k]], mode='full') if issubclass(a.dtype.type, NX.complexfloating): # if complex roots are all complex conjugates, the roots are real. roots = NX.asarray(seq_of_zeros, complex) pos_roots = sort_complex(NX.compress(roots.imag > 0, roots)) neg_roots = NX.conjugate( sort_complex(NX.compress(roots.imag < 0, roots))) if (len(pos_roots) == len(neg_roots) and NX.alltrue(neg_roots == pos_roots)): a = a.real.copy() return a
def _fix_real_lt_zero(x): x = asarray(x) if any(isreal(x) & (x<0)): x = _tocomplex(x) return x
def polyder(p, m=1): """ Return the derivative of the specified order of a polynomial. Parameters ---------- p : poly1d or sequence Polynomial to differentiate. A sequence is interpreted as polynomial coefficients, see `poly1d`. m : int, optional Order of differentiation (default: 1) Returns ------- der : poly1d A new polynomial representing the derivative. See Also -------- polyint : Anti-derivative of a polynomial. poly1d : Class for one-dimensional polynomials. Examples -------- The derivative of the polynomial :math:`x^3 + x^2 + x^1 + 1` is: >>> p = np.poly1d([1,1,1,1]) >>> p2 = np.polyder(p) >>> p2 poly1d([3, 2, 1]) which evaluates to: >>> p2(2.) 17.0 We can verify this, approximating the derivative with ``(f(x + h) - f(x))/h``: >>> (p(2. + 0.001) - p(2.)) / 0.001 17.007000999997857 The fourth-order derivative of a 3rd-order polynomial is zero: >>> np.polyder(p, 2) poly1d([6, 2]) >>> np.polyder(p, 3) poly1d([6]) >>> np.polyder(p, 4) poly1d([ 0.]) """ m = int(m) if m < 0: raise ValueError("Order of derivative must be positive (see polyint)") truepoly = isinstance(p, poly1d) p = NX.asarray(p) n = len(p) - 1 y = p[:-1] * NX.arange(n, 0, -1) if m == 0: val = p else: val = polyder(y, m - 1) if truepoly: val = poly1d(val) return val
def _fix_real_abs_gt_1(x): x = asarray(x) if any(isreal(x) & (abs(x)>1)): x = _tocomplex(x) return x
def expand_dims(a, axis): """ Expand the shape of an array. Insert a new axis that will appear at the `axis` position in the expanded array shape. .. note:: Previous to NumPy 1.13.0, neither ``axis < -a.ndim - 1`` nor ``axis > a.ndim`` raised errors or put the new axis where documented. Those axis values are now deprecated and will raise an AxisError in the future. Parameters ---------- a : array_like Input array. axis : int Position in the expanded axes where the new axis is placed. Returns ------- res : ndarray View of `a` with the number of dimensions increased by one. See Also -------- squeeze : The inverse operation, removing singleton dimensions reshape : Insert, remove, and combine dimensions, and resize existing ones doc.indexing, atleast_1d, atleast_2d, atleast_3d Examples -------- >>> x = np.array([1,2]) >>> x.shape (2,) The following is equivalent to ``x[np.newaxis,:]`` or ``x[np.newaxis]``: >>> y = np.expand_dims(x, axis=0) >>> y array([[1, 2]]) >>> y.shape (1, 2) >>> y = np.expand_dims(x, axis=1) # Equivalent to x[:,np.newaxis] >>> y array([[1], [2]]) >>> y.shape (2, 1) Note that some examples may use ``None`` instead of ``np.newaxis``. These are the same objects: >>> np.newaxis is None True """ if isinstance(a, matrix): a = asarray(a) else: a = asanyarray(a) shape = a.shape if axis > a.ndim or axis < -a.ndim - 1: # 2017-05-17, 1.13.0 warnings.warn( "Both axis > a.ndim and axis < -a.ndim - 1 are " "deprecated and will raise an AxisError in the future.", DeprecationWarning, stacklevel=2) # When the deprecation period expires, delete this if block, if axis < 0: axis = axis + a.ndim + 1 # and uncomment the following line. # axis = normalize_axis_index(axis, a.ndim + 1) return a.reshape(shape[:axis] + (1, ) + shape[axis:])
def poly(seq_of_zeros): """ Return polynomial coefficients given a sequence of roots. Calculate the coefficients of a polynomial given the zeros of the polynomial. If a square matrix is given, then the coefficients for characteristic equation of the matrix, defined by :math:`\\mathrm{det}(\\mathbf{A} - \\lambda \\mathbf{I})`, are returned. Parameters ---------- seq_of_zeros : ndarray A sequence of polynomial roots or a square matrix. Returns ------- coefs : ndarray A sequence of polynomial coefficients representing the polynomial :math:`\\mathrm{coefs}[0] x^{n-1} + \\mathrm{coefs}[1] x^{n-2} + ... + \\mathrm{coefs}[2] x + \\mathrm{coefs}[n]` See Also -------- numpy.poly1d : A one-dimensional polynomial class. numpy.roots : Return the roots of the polynomial coefficients in p numpy.polyfit : Least squares polynomial fit Examples -------- Given a sequence of polynomial zeros, >>> b = np.roots([1, 3, 1, 5, 6]) >>> np.poly(b) array([ 1., 3., 1., 5., 6.]) Given a square matrix, >>> P = np.array([[19, 3], [-2, 26]]) >>> np.poly(P) array([ 1., -45., 500.]) """ seq_of_zeros = atleast_1d(seq_of_zeros) sh = seq_of_zeros.shape if len(sh) == 2 and sh[0] == sh[1]: seq_of_zeros = _eigvals(seq_of_zeros) elif len(sh) ==1: pass else: raise ValueError, "input must be 1d or square 2d array." if len(seq_of_zeros) == 0: return 1.0 a = [1] for k in range(len(seq_of_zeros)): a = NX.convolve(a, [1, -seq_of_zeros[k]], mode='full') if issubclass(a.dtype.type, NX.complexfloating): # if complex roots are all complex conjugates, the roots are real. roots = NX.asarray(seq_of_zeros, complex) pos_roots = sort_complex(NX.compress(roots.imag > 0, roots)) neg_roots = NX.conjugate(sort_complex( NX.compress(roots.imag < 0,roots))) if (len(pos_roots) == len(neg_roots) and NX.alltrue(neg_roots == pos_roots)): a = a.real.copy() return a
def diag(v, k=0): """ Extract a diagonal or construct a diagonal array. Parameters ---------- v : array_like If `v` is a 2-D array, return a copy of its `k`-th diagonal. If `v` is a 1-D array, return a 2-D array with `v` on the `k`-th diagonal. k : int, optional Diagonal in question. The default is 0. Use `k>0` for diagonals above the main diagonal, and `k<0` for diagonals below the main diagonal. Returns ------- out : ndarray The extracted diagonal or constructed diagonal array. See Also -------- diagonal : Return specified diagonals. diagflat : Create a 2-D array with the flattened input as a diagonal. trace : Sum along diagonals. triu : Upper triangle of an array. tril : Lower triange of an array. Examples -------- >>> x = np.arange(9).reshape((3,3)) >>> x array([[0, 1, 2], [3, 4, 5], [6, 7, 8]]) >>> np.diag(x) array([0, 4, 8]) >>> np.diag(x, k=1) array([1, 5]) >>> np.diag(x, k=-1) array([3, 7]) >>> np.diag(np.diag(x)) array([[0, 0, 0], [0, 4, 0], [0, 0, 8]]) """ v = asarray(v) s = v.shape if len(s) == 1: n = s[0] + abs(k) res = zeros((n, n), v.dtype) if k >= 0: i = k else: i = (-k) * n res[:n - k].flat[i::n + 1] = v return res elif len(s) == 2: if k >= s[1]: return empty(0, dtype=v.dtype) if v.flags.f_contiguous: # faster slicing v, k, s = v.T, -k, s[::-1] if k >= 0: i = k else: i = (-k) * s[1] return v[:s[1] - k].flat[i::s[1] + 1] else: raise ValueError, "Input must be 1- or 2-d."
def __init__(self, arr): self.iter = asarray(arr).flat
def ix_(*args): """ Construct an open mesh from multiple sequences. This function takes N 1-D sequences and returns N outputs with N dimensions each, such that the shape is 1 in all but one dimension and the dimension with the non-unit shape value cycles through all N dimensions. Using `ix_` one can quickly construct index arrays that will index the cross product. ``a[np.ix_([1,3],[2,5])]`` returns the array ``[[a[1,2] a[1,5]], [a[3,2] a[3,5]]]``. Parameters ---------- args : 1-D sequences Each sequence should be of integer or boolean type. Boolean sequences will be interpreted as boolean masks for the corresponding dimension (equivalent to passing in ``np.nonzero(boolean_sequence)``). Returns ------- out : tuple of ndarrays N arrays with N dimensions each, with N the number of input sequences. Together these arrays form an open mesh. See Also -------- ogrid, mgrid, meshgrid Examples -------- >>> a = np.arange(10).reshape(2, 5) >>> a array([[0, 1, 2, 3, 4], [5, 6, 7, 8, 9]]) >>> ixgrid = np.ix_([0, 1], [2, 4]) >>> ixgrid (array([[0], [1]]), array([[2, 4]])) >>> ixgrid[0].shape, ixgrid[1].shape ((2, 1), (1, 2)) >>> a[ixgrid] array([[2, 4], [7, 9]]) >>> ixgrid = np.ix_([True, True], [2, 4]) >>> a[ixgrid] array([[2, 4], [7, 9]]) >>> ixgrid = np.ix_([True, True], [False, False, True, False, True]) >>> a[ixgrid] array([[2, 4], [7, 9]]) """ out = [] nd = len(args) for k, new in enumerate(args): new = asarray(new) if new.ndim != 1: raise ValueError("Cross index must be 1 dimensional") if new.size == 0: # Explicitly type empty arrays to avoid float default new = new.astype(_nx.intp) if issubdtype(new.dtype, _nx.bool_): new, = new.nonzero() new = new.reshape((1, ) * k + (new.size, ) + (1, ) * (nd - k - 1)) out.append(new) return tuple(out)
def diag(v, k=0): """ Extract a diagonal or construct a diagonal array. See the more detailed documentation for ``numpy.diagonal`` if you use this function to extract a diagonal and wish to write to the resulting array; whether it returns a copy or a view depends on what version of numpy you are using. Parameters ---------- v : array_like If `v` is a 2-D array, return a copy of its `k`-th diagonal. If `v` is a 1-D array, return a 2-D array with `v` on the `k`-th diagonal. k : int, optional Diagonal in question. The default is 0. Use `k>0` for diagonals above the main diagonal, and `k<0` for diagonals below the main diagonal. Returns ------- out : ndarray The extracted diagonal or constructed diagonal array. See Also -------- diagonal : Return specified diagonals. diagflat : Create a 2-D array with the flattened input as a diagonal. trace : Sum along diagonals. triu : Upper triangle of an array. tril : Lower triange of an array. Examples -------- >>> x = np.arange(9).reshape((3,3)) >>> x array([[0, 1, 2], [3, 4, 5], [6, 7, 8]]) >>> np.diag(x) array([0, 4, 8]) >>> np.diag(x, k=1) array([1, 5]) >>> np.diag(x, k=-1) array([3, 7]) >>> np.diag(np.diag(x)) array([[0, 0, 0], [0, 4, 0], [0, 0, 8]]) """ v = asarray(v) s = v.shape if len(s) == 1: n = s[0]+abs(k) res = zeros((n,n), v.dtype) if k >= 0: i = k else: i = (-k) * n res[:n-k].flat[i::n+1] = v return res elif len(s) == 2: return v.diagonal(k) else: raise ValueError("Input must be 1- or 2-d.")
def histogram2d(x, y, bins=10, range=None, normed=False, weights=None): """ Compute the bi-dimensional histogram of two data samples. Parameters ---------- x : array_like, shape (N,) An array containing the x coordinates of the points to be histogrammed. y : array_like, shape (N,) An array containing the y coordinates of the points to be histogrammed. bins : int or [int, int] or array_like or [array, array], optional The bin specification: * If int, the number of bins for the two dimensions (nx=ny=bins). * If [int, int], the number of bins in each dimension (nx, ny = bins). * If array_like, the bin edges for the two dimensions (x_edges=y_edges=bins). * If [array, array], the bin edges in each dimension (x_edges, y_edges = bins). range : array_like, shape(2,2), optional The leftmost and rightmost edges of the bins along each dimension (if not specified explicitly in the `bins` parameters): ``[[xmin, xmax], [ymin, ymax]]``. All values outside of this range will be considered outliers and not tallied in the histogram. normed : bool, optional If False, returns the number of samples in each bin. If True, returns the bin density ``bin_count / sample_count / bin_area``. weights : array_like, shape(N,), optional An array of values ``w_i`` weighing each sample ``(x_i, y_i)``. Weights are normalized to 1 if `normed` is True. If `normed` is False, the values of the returned histogram are equal to the sum of the weights belonging to the samples falling into each bin. Returns ------- H : ndarray, shape(nx, ny) The bi-dimensional histogram of samples `x` and `y`. Values in `x` are histogrammed along the first dimension and values in `y` are histogrammed along the second dimension. xedges : ndarray, shape(nx,) The bin edges along the first dimension. yedges : ndarray, shape(ny,) The bin edges along the second dimension. See Also -------- histogram : 1D histogram histogramdd : Multidimensional histogram Notes ----- When `normed` is True, then the returned histogram is the sample density, defined such that the sum over bins of the product ``bin_value * bin_area`` is 1. Please note that the histogram does not follow the Cartesian convention where `x` values are on the abscissa and `y` values on the ordinate axis. Rather, `x` is histogrammed along the first dimension of the array (vertical), and `y` along the second dimension of the array (horizontal). This ensures compatibility with `histogramdd`. Examples -------- >>> import matplotlib as mpl >>> import matplotlib.pyplot as plt Construct a 2D-histogram with variable bin width. First define the bin edges: >>> xedges = [0, 1, 1.5, 3, 5] >>> yedges = [0, 2, 3, 4, 6] Next we create a histogram H with random bin content: >>> x = np.random.normal(3, 1, 100) >>> y = np.random.normal(1, 1, 100) >>> H, xedges, yedges = np.histogram2d(y, x, bins=(xedges, yedges)) Or we fill the histogram H with a determined bin content: >>> H = np.ones((4, 4)).cumsum().reshape(4, 4) >>> print H[::-1] # This shows the bin content in the order as plotted [[ 13. 14. 15. 16.] [ 9. 10. 11. 12.] [ 5. 6. 7. 8.] [ 1. 2. 3. 4.]] Imshow can only do an equidistant representation of bins: >>> fig = plt.figure(figsize=(7, 3)) >>> ax = fig.add_subplot(131) >>> ax.set_title('imshow:\nequidistant') >>> im = plt.imshow(H, interpolation='nearest', origin='low', extent=[xedges[0], xedges[-1], yedges[0], yedges[-1]]) pcolormesh can displaying exact bin edges: >>> ax = fig.add_subplot(132) >>> ax.set_title('pcolormesh:\nexact bin edges') >>> X, Y = np.meshgrid(xedges, yedges) >>> ax.pcolormesh(X, Y, H) >>> ax.set_aspect('equal') NonUniformImage displays exact bin edges with interpolation: >>> ax = fig.add_subplot(133) >>> ax.set_title('NonUniformImage:\ninterpolated') >>> im = mpl.image.NonUniformImage(ax, interpolation='bilinear') >>> xcenters = xedges[:-1] + 0.5 * (xedges[1:] - xedges[:-1]) >>> ycenters = yedges[:-1] + 0.5 * (yedges[1:] - yedges[:-1]) >>> im.set_data(xcenters, ycenters, H) >>> ax.images.append(im) >>> ax.set_xlim(xedges[0], xedges[-1]) >>> ax.set_ylim(yedges[0], yedges[-1]) >>> ax.set_aspect('equal') >>> plt.show() """ from numpy import histogramdd try: N = len(bins) except TypeError: N = 1 if N != 1 and N != 2: xedges = yedges = asarray(bins, float) bins = [xedges, yedges] hist, edges = histogramdd([x, y], bins, range, normed, weights) return hist, edges[0], edges[1]
def histogram2d(x, y, bins=10, range=None, normed=None, weights=None, density=None): """ Compute the bi-dimensional histogram of two data samples. Parameters ---------- x : array_like, shape (N,) An array containing the x coordinates of the points to be histogrammed. y : array_like, shape (N,) An array containing the y coordinates of the points to be histogrammed. bins : int or array_like or [int, int] or [array, array], optional The bin specification: * If int, the number of bins for the two dimensions (nx=ny=bins). * If array_like, the bin edges for the two dimensions (x_edges=y_edges=bins). * If [int, int], the number of bins in each dimension (nx, ny = bins). * If [array, array], the bin edges in each dimension (x_edges, y_edges = bins). * A combination [int, array] or [array, int], where int is the number of bins and array is the bin edges. range : array_like, shape(2,2), optional The leftmost and rightmost edges of the bins along each dimension (if not specified explicitly in the `bins` parameters): ``[[xmin, xmax], [ymin, ymax]]``. All values outside of this range will be considered outliers and not tallied in the histogram. density : bool, optional If False, the default, returns the number of samples in each bin. If True, returns the probability *density* function at the bin, ``bin_count / sample_count / bin_area``. normed : bool, optional An alias for the density argument that behaves identically. To avoid confusion with the broken normed argument to `histogram`, `density` should be preferred. weights : array_like, shape(N,), optional An array of values ``w_i`` weighing each sample ``(x_i, y_i)``. Weights are normalized to 1 if `normed` is True. If `normed` is False, the values of the returned histogram are equal to the sum of the weights belonging to the samples falling into each bin. Returns ------- H : ndarray, shape(nx, ny) The bi-dimensional histogram of samples `x` and `y`. Values in `x` are histogrammed along the first dimension and values in `y` are histogrammed along the second dimension. xedges : ndarray, shape(nx+1,) The bin edges along the first dimension. yedges : ndarray, shape(ny+1,) The bin edges along the second dimension. See Also -------- histogram : 1D histogram histogramdd : Multidimensional histogram Notes ----- When `normed` is True, then the returned histogram is the sample density, defined such that the sum over bins of the product ``bin_value * bin_area`` is 1. Please note that the histogram does not follow the Cartesian convention where `x` values are on the abscissa and `y` values on the ordinate axis. Rather, `x` is histogrammed along the first dimension of the array (vertical), and `y` along the second dimension of the array (horizontal). This ensures compatibility with `histogramdd`. Examples -------- >>> from matplotlib.image import NonUniformImage >>> import matplotlib.pyplot as plt Construct a 2-D histogram with variable bin width. First define the bin edges: >>> xedges = [0, 1, 3, 5] >>> yedges = [0, 2, 3, 4, 6] Next we create a histogram H with random bin content: >>> x = np.random.normal(2, 1, 100) >>> y = np.random.normal(1, 1, 100) >>> H, xedges, yedges = np.histogram2d(x, y, bins=(xedges, yedges)) >>> H = H.T # Let each row list bins with common y range. :func:`imshow <matplotlib.pyplot.imshow>` can only display square bins: >>> fig = plt.figure(figsize=(7, 3)) >>> ax = fig.add_subplot(131, title='imshow: square bins') >>> plt.imshow(H, interpolation='nearest', origin='low', ... extent=[xedges[0], xedges[-1], yedges[0], yedges[-1]]) <matplotlib.image.AxesImage object at 0x...> :func:`pcolormesh <matplotlib.pyplot.pcolormesh>` can display actual edges: >>> ax = fig.add_subplot(132, title='pcolormesh: actual edges', ... aspect='equal') >>> X, Y = np.meshgrid(xedges, yedges) >>> ax.pcolormesh(X, Y, H) <matplotlib.collections.QuadMesh object at 0x...> :class:`NonUniformImage <matplotlib.image.NonUniformImage>` can be used to display actual bin edges with interpolation: >>> ax = fig.add_subplot(133, title='NonUniformImage: interpolated', ... aspect='equal', xlim=xedges[[0, -1]], ylim=yedges[[0, -1]]) >>> im = NonUniformImage(ax, interpolation='bilinear') >>> xcenters = (xedges[:-1] + xedges[1:]) / 2 >>> ycenters = (yedges[:-1] + yedges[1:]) / 2 >>> im.set_data(xcenters, ycenters, H) >>> ax.images.append(im) >>> plt.show() """ from numpy import histogramdd try: N = len(bins) except TypeError: N = 1 if N != 1 and N != 2: xedges = yedges = asarray(bins) bins = [xedges, yedges] hist, edges = histogramdd([x, y], bins, range, normed, weights, density) return hist, edges[0], edges[1]
def vander(x, N=None, increasing=False): """ Generate a Vandermonde matrix. The columns of the output matrix are powers of the input vector. The order of the powers is determined by the `increasing` boolean argument. Specifically, when `increasing` is False, the `i`-th output column is the input vector raised element-wise to the power of ``N - i - 1``. Such a matrix with a geometric progression in each row is named for Alexandre- Theophile Vandermonde. Parameters ---------- x : array_like 1-D input array. N : int, optional Number of columns in the output. If `N` is not specified, a square array is returned (``N = len(x)``). increasing : bool, optional Order of the powers of the columns. If True, the powers increase from left to right, if False (the default) they are reversed. .. versionadded:: 1.9.0 Returns ------- out : ndarray Vandermonde matrix. If `increasing` is False, the first column is ``x^(N-1)``, the second ``x^(N-2)`` and so forth. If `increasing` is True, the columns are ``x^0, x^1, ..., x^(N-1)``. See Also -------- polynomial.polynomial.polyvander Examples -------- >>> x = np.array([1, 2, 3, 5]) >>> N = 3 >>> np.vander(x, N) array([[ 1, 1, 1], [ 4, 2, 1], [ 9, 3, 1], [25, 5, 1]]) >>> np.column_stack([x**(N-1-i) for i in range(N)]) array([[ 1, 1, 1], [ 4, 2, 1], [ 9, 3, 1], [25, 5, 1]]) >>> x = np.array([1, 2, 3, 5]) >>> np.vander(x) array([[ 1, 1, 1, 1], [ 8, 4, 2, 1], [ 27, 9, 3, 1], [125, 25, 5, 1]]) >>> np.vander(x, increasing=True) array([[ 1, 1, 1, 1], [ 1, 2, 4, 8], [ 1, 3, 9, 27], [ 1, 5, 25, 125]]) The determinant of a square Vandermonde matrix is the product of the differences between the values of the input vector: >>> np.linalg.det(np.vander(x)) 48.000000000000043 # may vary >>> (5-3)*(5-2)*(5-1)*(3-2)*(3-1)*(2-1) 48 """ x = asarray(x) if x.ndim != 1: raise ValueError("x must be a one-dimensional array or sequence.") if N is None: N = len(x) v = empty((len(x), N), dtype=promote_types(x.dtype, int)) tmp = v[:, ::-1] if not increasing else v if N > 0: tmp[:, 0] = 1 if N > 1: tmp[:, 1:] = x[:, None] multiply.accumulate(tmp[:, 1:], out=tmp[:, 1:], axis=1) return v
def vander(x, N=None): """ Generate a Van der Monde matrix. The columns of the output matrix are decreasing powers of the input vector. Specifically, the `i`-th output column is the input vector raised element-wise to the power of ``N - i - 1``. Such a matrix with a geometric progression in each row is named for Alexandre-Theophile Vandermonde. Parameters ---------- x : array_like 1-D input array. N : int, optional Order of (number of columns in) the output. If `N` is not specified, a square array is returned (``N = len(x)``). Returns ------- out : ndarray Van der Monde matrix of order `N`. The first column is ``x^(N-1)``, the second ``x^(N-2)`` and so forth. Examples -------- >>> x = np.array([1, 2, 3, 5]) >>> N = 3 >>> np.vander(x, N) array([[ 1, 1, 1], [ 4, 2, 1], [ 9, 3, 1], [25, 5, 1]]) >>> np.column_stack([x**(N-1-i) for i in range(N)]) array([[ 1, 1, 1], [ 4, 2, 1], [ 9, 3, 1], [25, 5, 1]]) >>> x = np.array([1, 2, 3, 5]) >>> np.vander(x) array([[ 1, 1, 1, 1], [ 8, 4, 2, 1], [ 27, 9, 3, 1], [125, 25, 5, 1]]) The determinant of a square Vandermonde matrix is the product of the differences between the values of the input vector: >>> np.linalg.det(np.vander(x)) 48.000000000000043 >>> (5-3)*(5-2)*(5-1)*(3-2)*(3-1)*(2-1) 48 """ x = asarray(x) if N is None: N=len(x) X = ones( (len(x),N), x.dtype) for i in range(N - 1): X[:,i] = x**(N - i - 1) return X
def polyint(p, m=1, k=None): """ Return an antiderivative (indefinite integral) of a polynomial. The returned order `m` antiderivative `P` of polynomial `p` satisfies :math:`\\frac{d^m}{dx^m}P(x) = p(x)` and is defined up to `m - 1` integration constants `k`. The constants determine the low-order polynomial part .. math:: \\frac{k_{m-1}}{0!} x^0 + \\ldots + \\frac{k_0}{(m-1)!}x^{m-1} of `P` so that :math:`P^{(j)}(0) = k_{m-j-1}`. Parameters ---------- p : {array_like, poly1d} Polynomial to differentiate. A sequence is interpreted as polynomial coefficients, see `poly1d`. m : int, optional Order of the antiderivative. (Default: 1) k : {None, list of `m` scalars, scalar}, optional Integration constants. They are given in the order of integration: those corresponding to highest-order terms come first. If ``None`` (default), all constants are assumed to be zero. If `m = 1`, a single scalar can be given instead of a list. See Also -------- polyder : derivative of a polynomial poly1d.integ : equivalent method Examples -------- The defining property of the antiderivative: >>> p = np.poly1d([1,1,1]) >>> P = np.polyint(p) >>> P poly1d([ 0.33333333, 0.5 , 1. , 0. ]) >>> np.polyder(P) == p True The integration constants default to zero, but can be specified: >>> P = np.polyint(p, 3) >>> P(0) 0.0 >>> np.polyder(P)(0) 0.0 >>> np.polyder(P, 2)(0) 0.0 >>> P = np.polyint(p, 3, k=[6,5,3]) >>> P poly1d([ 0.01666667, 0.04166667, 0.16666667, 3. , 5. , 3. ]) Note that 3 = 6 / 2!, and that the constants are given in the order of integrations. Constant of the highest-order polynomial term comes first: >>> np.polyder(P, 2)(0) 6.0 >>> np.polyder(P, 1)(0) 5.0 >>> P(0) 3.0 """ m = int(m) if m < 0: raise ValueError("Order of integral must be positive (see polyder)") if k is None: k = NX.zeros(m, float) k = atleast_1d(k) if len(k) == 1 and m > 1: k = k[0] * NX.ones(m, float) if len(k) < m: raise ValueError( "k must be a scalar or a rank-1 array of length 1 or >m.") truepoly = isinstance(p, poly1d) p = NX.asarray(p) if m == 0: if truepoly: return poly1d(p) return p else: # Note: this must work also with object and integer arrays y = NX.concatenate((p.__truediv__(NX.arange(len(p), 0, -1)), [k[0]])) val = polyint(y, m - 1, k=k[1:]) if truepoly: return poly1d(val) return val
def apply_over_axes(func, a, axes): """ Apply a function repeatedly over multiple axes. `func` is called as `res = func(a, axis)`, where `axis` is the first element of `axes`. The result `res` of the function call must have either the same dimensions as `a` or one less dimension. If `res` has one less dimension than `a`, a dimension is inserted before `axis`. The call to `func` is then repeated for each axis in `axes`, with `res` as the first argument. Parameters ---------- func : function This function must take two arguments, `func(a, axis)`. a : array_like Input array. axes : array_like Axes over which `func` is applied; the elements must be integers. Returns ------- apply_over_axis : ndarray The output array. The number of dimensions is the same as `a`, but the shape can be different. This depends on whether `func` changes the shape of its output with respect to its input. See Also -------- apply_along_axis : Apply a function to 1-D slices of an array along the given axis. Notes ------ This function is equivalent to tuple axis arguments to reorderable ufuncs with keepdims=True. Tuple axis arguments to ufuncs have been availabe since version 1.7.0. Examples -------- >>> a = np.arange(24).reshape(2,3,4) >>> a array([[[ 0, 1, 2, 3], [ 4, 5, 6, 7], [ 8, 9, 10, 11]], [[12, 13, 14, 15], [16, 17, 18, 19], [20, 21, 22, 23]]]) Sum over axes 0 and 2. The result has same number of dimensions as the original array: >>> np.apply_over_axes(np.sum, a, [0,2]) array([[[ 60], [ 92], [124]]]) Tuple axis arguments to ufuncs are equivalent: >>> np.sum(a, axis=(0,2), keepdims=True) array([[[ 60], [ 92], [124]]]) """ val = asarray(a) N = a.ndim if array(axes).ndim == 0: axes = (axes,) for axis in axes: if axis < 0: axis = N + axis args = (val, axis) res = func(*args) if res.ndim == val.ndim: val = res else: res = expand_dims(res, axis) if res.ndim == val.ndim: val = res else: raise ValueError("function is not returning " "an array of the correct shape") return val
def __array__(self, t=None): if t: return NX.asarray(self.coeffs, t) else: return NX.asarray(self.coeffs)
def polyfit(x, y, deg, rcond=None, full=False): """Least squares polynomial fit. Required arguments x -- vector of sample points y -- vector or 2D array of values to fit deg -- degree of the fitting polynomial Keyword arguments rcond -- relative condition number of the fit (default len(x)*eps) full -- return full diagnostic output (default False) Returns full == False -- coefficients full == True -- coefficients, residuals, rank, singular values, rcond. Warns RankWarning -- if rank is reduced and not full output 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 n=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 Method: 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 is a len(x) x N+1 matrix, p is a N+1 length vector, and y is a len(x) x 1 vector 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 is preferred and that is the method used 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 may also be set to a value smaller than its default, but this may result in bad fits. 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. The warnings can be turned off by: >>> import numpy >>> import warnings >>> warnings.simplefilter('ignore',numpy.RankWarning) 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 basic problem is that the powers x**n are generally a poor basis for the functions on the sample interval with the result that the Vandermonde matrix is ill conditioned and computation of the polynomial values is sensitive to coefficient error. The quality of the resulting fit should be checked against the data whenever the condition number is large, as 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. See also polyval """ order = int(deg) + 1 x = NX.asarray(x) + 0.0 y = NX.asarray(y) + 0.0 # check arguments. if deg < 0: raise ValueError, "expected deg >= 0" if x.ndim != 1 or x.size == 0: raise TypeError, "expected non-empty vector for x" if y.ndim < 1 or y.ndim > 2: raise TypeError, "expected 1D or 2D array for y" if x.shape[0] != y.shape[0]: raise TypeError, "expected x and y to have same length" # set rcond if rcond is None: xtype = x.dtype if xtype == NX.single or xtype == NX.csingle: rcond = len(x) * _single_eps else: rcond = len(x) * _double_eps # scale x to improve condition number scale = abs(x).max() if scale != 0: x /= scale # solve least squares equation for powers of x v = vander(x, order) c, resids, rank, s = _lstsq(v, y, rcond) # warn on rank reduction, which indicates an ill conditioned matrix if rank != order and not full: msg = "Polyfit may be poorly conditioned" warnings.warn(msg, RankWarning) # scale returned coefficients if scale != 0: c /= vander([scale], order)[0] if full: return c, resids, rank, s, rcond else: return c
print("") print("Error for degree 5 is",1-(sum(pow(y-y2,2)))/sum(pow(y-np.mean(y),2))) print("") plt.scatter(x, y) plt.scatter(x, y2) plt.show() #to find the cofficients for degree 7 def Regression_degree_7(x,y): vanderX=np.vander(x,8) #get vandemonde matrix a = np.linalg.inv(np.transpose(vanderX).dot(vanderX)).dot(np.transpose(vanderX).dot(y)) # get the cofficients y2 = find_y2(a, x) #get the expected y print("Cofficients for degree 7 are :", a) print("Error for degree 7 is ",1-(sum(pow(y-y2,2)))/sum(pow(y-np.mean(y),2))) plt.scatter(x, y) plt.scatter(x, y2) plt.show() if __name__ == "__main__": data = scipy.io.loadmat("data.mat") x = (data['x']).ravel() #change the shape to 1-D y = (data['y']).ravel() x = NX.asarray(x) + 0.0 y = NX.asarray(y) + 0.0 Regression_degree_1(x,y) Regression_degree_3(x, y) Regression_degree_5(x, y) Regression_degree_7(x, y)
def polyfit(x, y, deg, rcond=None, full=False, w=None, cov=False): """ Least squares polynomial fit. .. note:: This forms part of the old polynomial API. Since version 1.4, the new polynomial API defined in `numpy.polynomial` is preferred. A summary of the differences can be found in the :doc:`transition guide </reference/routines.polynomials>`. Fit a polynomial ``p(x) = p[0] * x**deg + ... + p[deg]`` of degree `deg` to points `(x, y)`. Returns a vector of coefficients `p` that minimises the squared error in the order `deg`, `deg-1`, ... `0`. The `Polynomial.fit <numpy.polynomial.polynomial.Polynomial.fit>` class method is recommended for new code as it is more stable numerically. See the documentation of the method for more information. Parameters ---------- x : array_like, shape (M,) x-coordinates of the M sample points ``(x[i], y[i])``. y : array_like, shape (M,) or (M, K) y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column. deg : int Degree of the fitting polynomial rcond : float, optional Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases. full : bool, optional Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned. w : array_like, shape (M,), optional Weights. If not None, the weight ``w[i]`` applies to the unsquared residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are chosen so that the errors of the products ``w[i]*y[i]`` all have the same variance. When using inverse-variance weighting, use ``w[i] = 1/sigma(y[i])``. The default value is None. cov : bool or str, optional If given and not `False`, return not just the estimate but also its covariance matrix. By default, the covariance are scaled by chi2/dof, where dof = M - (deg + 1), i.e., the weights are presumed to be unreliable except in a relative sense and everything is scaled such that the reduced chi2 is unity. This scaling is omitted if ``cov='unscaled'``, as is relevant for the case that the weights are w = 1/sigma, with sigma known to be a reliable estimate of the uncertainty. Returns ------- p : ndarray, shape (deg + 1,) or (deg + 1, K) Polynomial coefficients, highest power first. If `y` was 2-D, the coefficients for `k`-th data set are in ``p[:,k]``. residuals, rank, singular_values, rcond Present only if `full` = True. Residuals is sum of squared residuals of the least-squares fit, the effective rank of the scaled Vandermonde coefficient matrix, its singular values, and the specified value of `rcond`. For more details, see `linalg.lstsq`. V : ndarray, shape (M,M) or (M,M,K) Present only if `full` = False and `cov`=True. The covariance matrix of the polynomial coefficient estimates. The diagonal of this matrix are the variance estimates for each coefficient. If y is a 2-D array, then the covariance matrix for the `k`-th data set are in ``V[:,:,k]`` Warns ----- RankWarning The rank of the coefficient matrix in the least-squares fit is deficient. The warning is only raised if `full` = False. The warnings can be turned off by >>> import warnings >>> warnings.simplefilter('ignore', np.RankWarning) See Also -------- polyval : Compute polynomial values. linalg.lstsq : Computes a least-squares fit. scipy.interpolate.UnivariateSpline : Computes spline fits. Notes ----- The solution minimizes the squared error .. math :: E = \\sum_{j=0}^k |p(x_j) - y_j|^2 in the equations:: x[0]**n * p[0] + ... + x[0] * p[n-1] + p[n] = y[0] x[1]**n * p[0] + ... + x[1] * p[n-1] + p[n] = y[1] ... x[k]**n * p[0] + ... + x[k] * p[n-1] + p[n] = y[k] The coefficient matrix of the coefficients `p` is a Vandermonde matrix. `polyfit` issues a `RankWarning` when the least-squares fit is badly conditioned. This implies that the best fit is not well-defined due to numerical error. The results may be improved by lowering the polynomial degree or by replacing `x` by `x` - `x`.mean(). The `rcond` parameter can also be set to a value smaller than its default, but the resulting fit may be spurious: including contributions from the small singular values can add numerical noise to the result. Note that fitting polynomial coefficients is inherently badly conditioned when the degree of the polynomial is large or the interval of sample points is badly centered. The quality of the fit should always be checked in these cases. When polynomial fits are not satisfactory, splines may be a good alternative. References ---------- .. [1] Wikipedia, "Curve fitting", https://en.wikipedia.org/wiki/Curve_fitting .. [2] Wikipedia, "Polynomial interpolation", https://en.wikipedia.org/wiki/Polynomial_interpolation Examples -------- >>> import warnings >>> x = np.array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0]) >>> y = np.array([0.0, 0.8, 0.9, 0.1, -0.8, -1.0]) >>> z = np.polyfit(x, y, 3) >>> z array([ 0.08703704, -0.81349206, 1.69312169, -0.03968254]) # may vary It is convenient to use `poly1d` objects for dealing with polynomials: >>> p = np.poly1d(z) >>> p(0.5) 0.6143849206349179 # may vary >>> p(3.5) -0.34732142857143039 # may vary >>> p(10) 22.579365079365115 # may vary High-order polynomials may oscillate wildly: >>> with warnings.catch_warnings(): ... warnings.simplefilter('ignore', np.RankWarning) ... p30 = np.poly1d(np.polyfit(x, y, 30)) ... >>> p30(4) -0.80000000000000204 # may vary >>> p30(5) -0.99999999999999445 # may vary >>> p30(4.5) -0.10547061179440398 # may vary Illustration: >>> import matplotlib.pyplot as plt >>> xp = np.linspace(-2, 6, 100) >>> _ = plt.plot(x, y, '.', xp, p(xp), '-', xp, p30(xp), '--') >>> plt.ylim(-2,2) (-2, 2) >>> plt.show() """ order = int(deg) + 1 x = NX.asarray(x) + 0.0 y = NX.asarray(y) + 0.0 # check arguments. if deg < 0: raise ValueError("expected deg >= 0") if x.ndim != 1: raise TypeError("expected 1D vector for x") if x.size == 0: raise TypeError("expected non-empty vector for x") if y.ndim < 1 or y.ndim > 2: raise TypeError("expected 1D or 2D array for y") if x.shape[0] != y.shape[0]: raise TypeError("expected x and y to have same length") # set rcond if rcond is None: rcond = len(x) * finfo(x.dtype).eps # set up least squares equation for powers of x lhs = vander(x, order) rhs = y # apply weighting if w is not None: w = NX.asarray(w) + 0.0 if w.ndim != 1: raise TypeError("expected a 1-d array for weights") if w.shape[0] != y.shape[0]: raise TypeError("expected w and y to have the same length") lhs *= w[:, NX.newaxis] if rhs.ndim == 2: rhs *= w[:, NX.newaxis] else: rhs *= w # scale lhs to improve condition number and solve scale = NX.sqrt((lhs * lhs).sum(axis=0)) lhs /= scale c, resids, rank, s = lstsq(lhs, rhs, rcond) c = (c.T / scale).T # broadcast scale coefficients # warn on rank reduction, which indicates an ill conditioned matrix if rank != order and not full: msg = "Polyfit may be poorly conditioned" warnings.warn(msg, RankWarning, stacklevel=4) if full: return c, resids, rank, s, rcond elif cov: Vbase = inv(dot(lhs.T, lhs)) Vbase /= NX.outer(scale, scale) if cov == "unscaled": fac = 1 else: if len(x) <= order: raise ValueError("the number of data points must exceed order " "to scale the covariance matrix") # note, this used to be: fac = resids / (len(x) - order - 2.0) # it was deciced that the "- 2" (originally justified by "Bayesian # uncertainty analysis") is not was the user expects # (see gh-11196 and gh-11197) fac = resids / (len(x) - order) if y.ndim == 1: return c, Vbase * fac else: return c, Vbase[:, :, NX.newaxis] * fac else: return c
def expand_dims(a, axis): """ Expand the shape of an array. Insert a new axis that will appear at the `axis` position in the expanded array shape. Parameters ---------- a : array_like Input array. axis : int or tuple of ints Position in the expanded axes where the new axis (or axes) is placed. .. deprecated:: 1.13.0 Passing an axis where ``axis > a.ndim`` will be treated as ``axis == a.ndim``, and passing ``axis < -a.ndim - 1`` will be treated as ``axis == 0``. This behavior is deprecated. .. versionchanged:: 1.18.0 A tuple of axes is now supported. Out of range axes as described above are now forbidden and raise an `AxisError`. Returns ------- result : ndarray View of `a` with the number of dimensions increased. See Also -------- squeeze : The inverse operation, removing singleton dimensions reshape : Insert, remove, and combine dimensions, and resize existing ones doc.indexing, atleast_1d, atleast_2d, atleast_3d Examples -------- >>> x = np.array([1, 2]) >>> x.shape (2,) The following is equivalent to ``x[np.newaxis, :]`` or ``x[np.newaxis]``: >>> y = np.expand_dims(x, axis=0) >>> y array([[1, 2]]) >>> y.shape (1, 2) The following is equivalent to ``x[:, np.newaxis]``: >>> y = np.expand_dims(x, axis=1) >>> y array([[1], [2]]) >>> y.shape (2, 1) ``axis`` may also be a tuple: >>> y = np.expand_dims(x, axis=(0, 1)) >>> y array([[[1, 2]]]) >>> y = np.expand_dims(x, axis=(2, 0)) >>> y array([[[1], [2]]]) Note that some examples may use ``None`` instead of ``np.newaxis``. These are the same objects: >>> np.newaxis is None True """ if isinstance(a, matrix): a = asarray(a) else: a = asanyarray(a) if type(axis) not in (tuple, list): axis = (axis, ) out_ndim = len(axis) + a.ndim axis = normalize_axis_tuple(axis, out_ndim) shape_it = iter(a.shape) shape = [1 if ax in axis else next(shape_it) for ax in range(out_ndim)] return a.reshape(shape)
def histogram2d(x, y, bins=10, range=None, normed=False, weights=None): """ Compute the bi-dimensional histogram of two data samples. Parameters ---------- x : array_like, shape(N,) A sequence of values to be histogrammed along the first dimension. y : array_like, shape(M,) A sequence of values to be histogrammed along the second dimension. bins : int or [int, int] or array_like or [array, array], optional The bin specification: * If int, the number of bins for the two dimensions (nx=ny=bins). * If [int, int], the number of bins in each dimension (nx, ny = bins). * If array_like, the bin edges for the two dimensions (x_edges=y_edges=bins). * If [array, array], the bin edges in each dimension (x_edges, y_edges = bins). range : array_like, shape(2,2), optional The leftmost and rightmost edges of the bins along each dimension (if not specified explicitly in the `bins` parameters): ``[[xmin, xmax], [ymin, ymax]]``. All values outside of this range will be considered outliers and not tallied in the histogram. normed : bool, optional If False, returns the number of samples in each bin. If True, returns the bin density, i.e. the bin count divided by the bin area. weights : array_like, shape(N,), optional An array of values ``w_i`` weighing each sample ``(x_i, y_i)``. Weights are normalized to 1 if `normed` is True. If `normed` is False, the values of the returned histogram are equal to the sum of the weights belonging to the samples falling into each bin. Returns ------- H : ndarray, shape(nx, ny) The bi-dimensional histogram of samples `x` and `y`. Values in `x` are histogrammed along the first dimension and values in `y` are histogrammed along the second dimension. xedges : ndarray, shape(nx,) The bin edges along the first dimension. yedges : ndarray, shape(ny,) The bin edges along the second dimension. See Also -------- histogram: 1D histogram histogramdd: Multidimensional histogram Notes ----- When `normed` is True, then the returned histogram is the sample density, defined such that: .. math:: \\sum_{i=0}^{nx-1} \\sum_{j=0}^{ny-1} H_{i,j} \\Delta x_i \\Delta y_j = 1 where `H` is the histogram array and :math:`\\Delta x_i \\Delta y_i` the area of bin `{i,j}`. Please note that the histogram does not follow the Cartesian convention where `x` values are on the abcissa and `y` values on the ordinate axis. Rather, `x` is histogrammed along the first dimension of the array (vertical), and `y` along the second dimension of the array (horizontal). This ensures compatibility with `histogramdd`. Examples -------- >>> x, y = np.random.randn(2, 100) >>> H, xedges, yedges = np.histogram2d(x, y, bins=(5, 8)) >>> H.shape, xedges.shape, yedges.shape ((5, 8), (6,), (9,)) We can now use the Matplotlib to visualize this 2-dimensional histogram: >>> extent = [yedges[0], yedges[-1], xedges[-1], xedges[0]] >>> import matplotlib.pyplot as plt >>> plt.imshow(H, extent=extent, interpolation='nearest') <matplotlib.image.AxesImage object at ...> >>> plt.colorbar() <matplotlib.colorbar.Colorbar instance at ...> >>> plt.show() """ from numpy import histogramdd try: N = len(bins) except TypeError: N = 1 if N != 1 and N != 2: xedges = yedges = asarray(bins, float) bins = [xedges, yedges] hist, edges = histogramdd([x,y], bins, range, normed, weights) return hist, edges[0], edges[1]
def fromarrays(arrayList, dtype=None, shape=None, formats=None, names=None, titles=None, aligned=False, byteorder=None): """ create a record array from a (flat) list of arrays >>> x1=N.array([1,2,3,4]) >>> x2=N.array(['a','dd','xyz','12']) >>> x3=N.array([1.1,2,3,4]) >>> r = fromarrays([x1,x2,x3],names='a,b,c') >>> print r[1] (2, 'dd', 2.0) >>> x1[1]=34 >>> r.a array([1, 2, 3, 4]) """ arrayList = [sb.asarray(x) for x in arrayList] if shape is None or shape == 0: shape = arrayList[0].shape if isinstance(shape, int): shape = (shape, ) if formats is None and dtype is None: # go through each object in the list to see if it is an ndarray # and determine the formats. formats = '' for obj in arrayList: if not isinstance(obj, ndarray): raise ValueError, "item in the array list must be an ndarray." formats += _typestr[obj.dtype.type] if issubclass(obj.dtype.type, nt.flexible): formats += ` obj.itemsize ` formats += ',' formats = formats[:-1] if dtype is not None: descr = sb.dtype(dtype) _names = descr.names else: parsed = format_parser(formats, names, titles, aligned, byteorder) _names = parsed._names descr = parsed._descr # Determine shape from data-type. if len(descr) != len(arrayList): raise ValueError, "mismatch between the number of fields "\ "and the number of arrays" d0 = descr[0].shape nn = len(d0) if nn > 0: shape = shape[:-nn] for k, obj in enumerate(arrayList): nn = len(descr[k].shape) testshape = obj.shape[:len(obj.shape) - nn] if testshape != shape: raise ValueError, "array-shape mismatch in array %d" % k _array = recarray(shape, descr) # populate the record array (makes a copy) for i in range(len(arrayList)): _array[_names[i]] = arrayList[i] return _array
def isrealobj(x): """Return True if x is not a complex type. Unlike isreal(x), complex(3.0) is considered a complex object. """ return not issubclass(asarray(x).dtype.type, _nx.complexfloating)
def poly(seq_of_zeros): """ Find the coefficients of a polynomial with the given sequence of roots. .. note:: This forms part of the old polynomial API. Since version 1.4, the new polynomial API defined in `numpy.polynomial` is preferred. A summary of the differences can be found in the :doc:`transition guide </reference/routines.polynomials>`. Returns the coefficients of the polynomial whose leading coefficient is one for the given sequence of zeros (multiple roots must be included in the sequence as many times as their multiplicity; see Examples). A square matrix (or array, which will be treated as a matrix) can also be given, in which case the coefficients of the characteristic polynomial of the matrix are returned. Parameters ---------- seq_of_zeros : array_like, shape (N,) or (N, N) A sequence of polynomial roots, or a square array or matrix object. Returns ------- c : ndarray 1D array of polynomial coefficients from highest to lowest degree: ``c[0] * x**(N) + c[1] * x**(N-1) + ... + c[N-1] * x + c[N]`` where c[0] always equals 1. Raises ------ ValueError If input is the wrong shape (the input must be a 1-D or square 2-D array). See Also -------- polyval : Compute polynomial values. roots : Return the roots of a polynomial. polyfit : Least squares polynomial fit. poly1d : A one-dimensional polynomial class. Notes ----- Specifying the roots of a polynomial still leaves one degree of freedom, typically represented by an undetermined leading coefficient. [1]_ In the case of this function, that coefficient - the first one in the returned array - is always taken as one. (If for some reason you have one other point, the only automatic way presently to leverage that information is to use ``polyfit``.) The characteristic polynomial, :math:`p_a(t)`, of an `n`-by-`n` matrix **A** is given by :math:`p_a(t) = \\mathrm{det}(t\\, \\mathbf{I} - \\mathbf{A})`, where **I** is the `n`-by-`n` identity matrix. [2]_ References ---------- .. [1] M. Sullivan and M. Sullivan, III, "Algebra and Trignometry, Enhanced With Graphing Utilities," Prentice-Hall, pg. 318, 1996. .. [2] G. Strang, "Linear Algebra and Its Applications, 2nd Edition," Academic Press, pg. 182, 1980. Examples -------- Given a sequence of a polynomial's zeros: >>> np.poly((0, 0, 0)) # Multiple root example array([1., 0., 0., 0.]) The line above represents z**3 + 0*z**2 + 0*z + 0. >>> np.poly((-1./2, 0, 1./2)) array([ 1. , 0. , -0.25, 0. ]) The line above represents z**3 - z/4 >>> np.poly((np.random.random(1)[0], 0, np.random.random(1)[0])) array([ 1. , -0.77086955, 0.08618131, 0. ]) # random Given a square array object: >>> P = np.array([[0, 1./3], [-1./2, 0]]) >>> np.poly(P) array([1. , 0. , 0.16666667]) Note how in all cases the leading coefficient is always 1. """ seq_of_zeros = atleast_1d(seq_of_zeros) sh = seq_of_zeros.shape if len(sh) == 2 and sh[0] == sh[1] and sh[0] != 0: seq_of_zeros = eigvals(seq_of_zeros) elif len(sh) == 1: dt = seq_of_zeros.dtype # Let object arrays slip through, e.g. for arbitrary precision if dt != object: seq_of_zeros = seq_of_zeros.astype(mintypecode(dt.char)) else: raise ValueError("input must be 1d or non-empty square 2d array.") if len(seq_of_zeros) == 0: return 1.0 dt = seq_of_zeros.dtype a = ones((1, ), dtype=dt) for zero in seq_of_zeros: a = NX.convolve(a, array([1, -zero], dtype=dt), mode='full') if issubclass(a.dtype.type, NX.complexfloating): # if complex roots are all complex conjugates, the roots are real. roots = NX.asarray(seq_of_zeros, complex) if NX.all(NX.sort(roots) == NX.sort(roots.conjugate())): a = a.real.copy() return a
def polyfit(x, y, deg, rcond=None, full=False, w=None, cov=False): """ Least squares polynomial fit. Fit a polynomial ``p(x) = p[0] * x**deg + ... + p[deg]`` of degree `deg` to points `(x, y)`. Returns a vector of coefficients `p` that minimises the squared error. Parameters ---------- x : array_like, shape (M,) x-coordinates of the M sample points ``(x[i], y[i])``. y : array_like, shape (M,) or (M, K) y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column. deg : int Degree of the fitting polynomial rcond : float, optional Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases. full : bool, optional Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned. w : array_like, shape (M,), optional weights to apply to the y-coordinates of the sample points. cov : bool, optional Return the estimate and the covariance matrix of the estimate If full is True, then cov is not returned. Returns ------- p : ndarray, shape (M,) or (M, K) Polynomial coefficients, highest power first. If `y` was 2-D, the coefficients for `k`-th data set are in ``p[:,k]``. residuals, rank, singular_values, rcond : present only if `full` = True Residuals of the least-squares fit, the effective rank of the scaled Vandermonde coefficient matrix, its singular values, and the specified value of `rcond`. For more details, see `linalg.lstsq`. V : ndaray, shape (M,M) or (M,M,K) : present only if `full` = False and `cov`=True The covariance matrix of the polynomial coefficient estimates. The diagonal of this matrix are the variance estimates for each coefficient. If y is a 2-d array, then the covariance matrix for the `k`-th data set are in ``V[:,:,k]`` Warns ----- RankWarning The rank of the coefficient matrix in the least-squares fit is deficient. The warning is only raised if `full` = False. The warnings can be turned off by >>> import warnings >>> warnings.simplefilter('ignore', np.RankWarning) See Also -------- polyval : Computes polynomial values. linalg.lstsq : Computes a least-squares fit. scipy.interpolate.UnivariateSpline : Computes spline fits. Notes ----- The solution minimizes the squared error .. math :: E = \\sum_{j=0}^k |p(x_j) - y_j|^2 in the equations:: x[0]**n * p[n] + ... + x[0] * p[1] + p[0] = y[0] x[1]**n * p[n] + ... + x[1] * p[1] + p[0] = y[1] ... x[k]**n * p[n] + ... + x[k] * p[1] + p[0] = y[k] The coefficient matrix of the coefficients `p` is a Vandermonde matrix. `polyfit` issues a `RankWarning` when the least-squares fit is badly conditioned. This implies that the best fit is not well-defined due to numerical error. The results may be improved by lowering the polynomial degree or by replacing `x` by `x` - `x`.mean(). The `rcond` parameter can also be set to a value smaller than its default, but the resulting fit may be spurious: including contributions from the small singular values can add numerical noise to the result. Note that fitting polynomial coefficients is inherently badly conditioned when the degree of the polynomial is large or the interval of sample points is badly centered. The quality of the fit should always be checked in these cases. When polynomial fits are not satisfactory, splines may be a good alternative. References ---------- .. [1] Wikipedia, "Curve fitting", http://en.wikipedia.org/wiki/Curve_fitting .. [2] Wikipedia, "Polynomial interpolation", http://en.wikipedia.org/wiki/Polynomial_interpolation Examples -------- >>> x = np.array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0]) >>> y = np.array([0.0, 0.8, 0.9, 0.1, -0.8, -1.0]) >>> z = np.polyfit(x, y, 3) >>> z array([ 0.08703704, -0.81349206, 1.69312169, -0.03968254]) It is convenient to use `poly1d` objects for dealing with polynomials: >>> p = np.poly1d(z) >>> p(0.5) 0.6143849206349179 >>> p(3.5) -0.34732142857143039 >>> p(10) 22.579365079365115 High-order polynomials may oscillate wildly: >>> p30 = np.poly1d(np.polyfit(x, y, 30)) /... RankWarning: Polyfit may be poorly conditioned... >>> p30(4) -0.80000000000000204 >>> p30(5) -0.99999999999999445 >>> p30(4.5) -0.10547061179440398 Illustration: >>> import matplotlib.pyplot as plt >>> xp = np.linspace(-2, 6, 100) >>> plt.plot(x, y, '.', xp, p(xp), '-', xp, p30(xp), '--') [<matplotlib.lines.Line2D object at 0x...>, <matplotlib.lines.Line2D object at 0x...>, <matplotlib.lines.Line2D object at 0x...>] >>> plt.ylim(-2,2) (-2, 2) >>> plt.show() """ order = int(deg) + 1 x = NX.asarray(x) + 0.0 y = NX.asarray(y) + 0.0 # check arguments. if deg < 0: raise ValueError("expected deg >= 0") if x.ndim != 1: raise TypeError("expected 1D vector for x") if x.size == 0: raise TypeError("expected non-empty vector for x") if y.ndim < 1 or y.ndim > 2: raise TypeError("expected 1D or 2D array for y") if x.shape[0] != y.shape[0]: raise TypeError("expected x and y to have same length") # set rcond if rcond is None: rcond = len(x) * finfo(x.dtype).eps # set up least squares equation for powers of x lhs = vander(x, order) rhs = y # apply weighting if w is not None: w = NX.asarray(w) + 0.0 if w.ndim != 1: raise TypeError("expected a 1-d array for weights") if w.shape[0] != y.shape[0]: raise TypeError("expected w and y to have the same length") lhs *= w[:, NX.newaxis] if rhs.ndim == 2: rhs *= w[:, NX.newaxis] else: rhs *= w # scale lhs to improve condition number and solve scale = NX.sqrt((lhs * lhs).sum(axis=0)) lhs /= scale c, resids, rank, s = lstsq(lhs, rhs, rcond) c = (c.T / scale).T # broadcast scale coefficients # warn on rank reduction, which indicates an ill conditioned matrix if rank != order and not full: msg = "Polyfit may be poorly conditioned" warnings.warn(msg, RankWarning) if full: return c, resids, rank, s, rcond elif cov: Vbase = inv(dot(lhs.T, lhs)) Vbase /= NX.outer(scale, scale) # Some literature ignores the extra -2.0 factor in the denominator, but # it is included here because the covariance of Multivariate Student-T # (which is implied by a Bayesian uncertainty analysis) includes it. # Plus, it gives a slightly more conservative estimate of uncertainty. fac = resids / (len(x) - order - 2.0) if y.ndim == 1: return c, Vbase * fac else: return c, Vbase[:, :, NX.newaxis] * fac else: return c
def polyval(p, x): """ Evaluate a polynomial at specific values. If `p` is of length N, this function returns the value: ``p[0]*x**(N-1) + p[1]*x**(N-2) + ... + p[N-2]*x + p[N-1]`` If `x` is a sequence, then `p(x)` is returned for each element of `x`. If `x` is another polynomial then the composite polynomial `p(x(t))` is returned. Parameters ---------- p : array_like or poly1d object 1D array of polynomial coefficients (including coefficients equal to zero) from highest degree to the constant term, or an instance of poly1d. x : array_like or poly1d object A number, a 1D array of numbers, or an instance of poly1d, "at" which to evaluate `p`. Returns ------- values : ndarray or poly1d If `x` is a poly1d instance, the result is the composition of the two polynomials, i.e., `x` is "substituted" in `p` and the simplified result is returned. In addition, the type of `x` - array_like or poly1d - governs the type of the output: `x` array_like => `values` array_like, `x` a poly1d object => `values` is also. See Also -------- poly1d: A polynomial class. Notes ----- Horner's scheme [1]_ is used to evaluate the polynomial. Even so, for polynomials of high degree the values may be inaccurate due to rounding errors. Use carefully. References ---------- .. [1] I. N. Bronshtein, K. A. Semendyayev, and K. A. Hirsch (Eng. trans. Ed.), *Handbook of Mathematics*, New York, Van Nostrand Reinhold Co., 1985, pg. 720. Examples -------- >>> np.polyval([3,0,1], 5) # 3 * 5**2 + 0 * 5**1 + 1 76 >>> np.polyval([3,0,1], np.poly1d(5)) poly1d([ 76.]) >>> np.polyval(np.poly1d([3,0,1]), 5) 76 >>> np.polyval(np.poly1d([3,0,1]), np.poly1d(5)) poly1d([ 76.]) """ p = NX.asarray(p) if isinstance(x, poly1d): y = 0 else: x = NX.asarray(x) y = NX.zeros_like(x) for i in range(len(p)): y = x * y + p[i] return y
def apply_along_axis(func1d, axis, arr, *args, **kwargs): """ Apply a function to 1-D slices along the given axis. Execute `func1d(a, *args)` where `func1d` operates on 1-D arrays and `a` is a 1-D slice of `arr` along `axis`. Parameters ---------- func1d : function This function should accept 1-D arrays. It is applied to 1-D slices of `arr` along the specified axis. axis : integer Axis along which `arr` is sliced. arr : ndarray Input array. args : any Additional arguments to `func1d`. kwargs : any Additional named arguments to `func1d`. .. versionadded:: 1.9.0 Returns ------- apply_along_axis : ndarray The output array. The shape of `outarr` is identical to the shape of `arr`, except along the `axis` dimension, where the length of `outarr` is equal to the size of the return value of `func1d`. If `func1d` returns a scalar `outarr` will have one fewer dimensions than `arr`. See Also -------- apply_over_axes : Apply a function repeatedly over multiple axes. Examples -------- >>> def my_func(a): ... \"\"\"Average first and last element of a 1-D array\"\"\" ... return (a[0] + a[-1]) * 0.5 >>> b = np.array([[1,2,3], [4,5,6], [7,8,9]]) >>> np.apply_along_axis(my_func, 0, b) array([ 4., 5., 6.]) >>> np.apply_along_axis(my_func, 1, b) array([ 2., 5., 8.]) For a function that doesn't return a scalar, the number of dimensions in `outarr` is the same as `arr`. >>> b = np.array([[8,1,7], [4,3,9], [5,2,6]]) >>> np.apply_along_axis(sorted, 1, b) array([[1, 7, 8], [3, 4, 9], [2, 5, 6]]) """ arr = asarray(arr) 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 = zeros(nd, 'O') indlist = list(range(nd)) indlist.remove(axis) i[axis] = slice(None, None) outshape = asarray(arr.shape).take(indlist) i.put(indlist, ind) res = func1d(arr[tuple(i.tolist())], *args, **kwargs) # if res is a number, then we have a smaller output array if isscalar(res): outarr = zeros(outshape, asarray(res).dtype) outarr[tuple(ind)] = res Ntot = 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 k += 1 return outarr else: Ntot = product(outshape) holdshape = outshape outshape = list(arr.shape) outshape[axis] = len(res) outarr = zeros(outshape, asarray(res).dtype) outarr[tuple(i.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) res = func1d(arr[tuple(i.tolist())], *args, **kwargs) outarr[tuple(i.tolist())] = res k += 1 return outarr
def _fix_int_lt_zero(x): x = asarray(x) if any(isreal(x) & (x < 0)): x = x * 1.0 return x
def iscomplexobj(x): """Return True if x is a complex type or an array of complex numbers. Unlike iscomplex(x), complex(3.0) is considered a complex object. """ return issubclass(asarray(x).dtype.type, _nx.complexfloating)