DŽg  ddlZddlZddlZddlZddlZddlZddlZddl m cm Z ddl mZmZddl mZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZddl m!Z!m"Z"m#Z#m$Z$m%Z%m&Z&m'Z'm(Z(m)Z)m*Z*m+Z+m,Z,m-Z-ddl.m/Z/m0Z0m1Z1m2Z2m3Z3m4Z4ddl5m6Z6ddl7m8Z8ddl9m:Z:m;Z;mZ?m@ZAdd lBmCZCdd lDmEZEdd lFmGZGmHZHejejd ZJgdZKeLeLddeLddeLddeLddeLddeLddeLddeLddeLddeLd deLd!deLd"d#eLd$d% ZMdd&ZNeJeNdd(ZOdd)ZPeJePdd*ZQeEd d+ZRd,ZSddd-d.ZTeJeTdejd-d0ZVeEd dd1ZWd2ZXeJeXd3ZYdd4ZZeJeZdd5Z[dd6Z\eJe\dd7Z]ddd8d9Z^eJe^dd'd8d:Z_dd;Z`eJe`d'dZbeJebdd?Z>dd@ZceJecddAZddddBdCZeeJeeddDe!zdBdEZfdFZgeJegdGZhddHZieJeiddIZjdJZkeJekdKZldLZmeJemdMZnddOZodPZpdQjepZrdRjerZsdSjesZtdTjetZudUZvdVZwdWZxdXZy ddYZzdZZ{eEd Gd[d\Z| ddd]d^Z}eJe} ddd]d_Z~ddd]d`ZeJeddNejejfdd]daZeEd dbZeEd dcZeEd ddZeEd deZgdfZgdgZdhZdiZdjZdkZeJedlZeEd dmZdnZeJedoZddpZ ddqZeJeddrZddsZ ddddtduZeJe ddddtdvZ ddddtdwZeJe ddddtdxZ ddyZdzZd{Zd|ed}efd~ZdZddZdZdZdZdZ ddej(dej(dej(dededej(f dZdZ ddej(dej(defdZddZeJeddZeEd ddZdddddZeJedNd/dddZddZeJeddZddZeJeddZddZeJeddZddZeJeddZy)N) transpose overrides)ones zeros_likearange concatenatearrayasarray asanyarrayemptyndarraytakedotwhereintpintegerisscalarabsolute) piaddarctan2 frompyfunccos less_equalsqrtsinmodexp not_equalsubtractminimum)ravelnonzero partitionmeananysum) typecodes)diag)_placebincountnormalize_axis_index _monotonicityinterpinterp_complex)_array_converter) set_module) histogram histogramddnumpy)module)&select piecewise trim_zeroscopyiterable percentilediffgradientangleunwrap sort_complexfliprot90extractplace vectorizeasarray_chkfiniteaverager+digitizecovcorrcoefmediansinchamminghanningbartlettblackmankaiser trapezoidtrapzi0meshgriddeleteinsertappendr.quantilect||SN) _inverted_cdfn quantiless e/home/mcse/projects/flask_80/flask-venv/lib/python3.12/site-packages/numpy/lib/_function_base_impl.pyraFs }Q /J)get_virtual_index fix_gammac||zdz SNr]s r`raraJsI /Brbc:t|jdd|dk(S)N??rshape default_valueconditioned_valuer_get_gamma_maskrmgamma_s r`raraKs?++!1* $rbct||Sr[)_closest_observationr]s r`raraRs/CADM0Orbct||ddS)Nrrg_compute_virtual_indexr]s r`raraXq)Q2rbc|Sr[rhrrs r`raraZ5rbct||ddS)Nrkrxr]s r`rara]sq)S#6rbc|Sr[rhrrs r`rara_r|rbct||ddSNrrxr]s r`rarabrzrbc|Sr[rhrrs r`raradr|rbc|dz |zSrfrhr]s r`raraksA/Brbc|Sr[rhrrs r`raralr|rbct||ddS)NgUUUUUU?rxr]s r`raraoq)Wg>rbc|Sr[rhrrs r`raraqr|rbct||ddS)Ng?rxr]s r`raratrrbc|Sr[rhrrs r`raravr|rbcrtj|dz |zjtjSrf)npfloorastyperr]s r`rarazs+rxx Ui 0!!'rbcrtj|dz |zjtjSrf)rceilrrr]s r`raras+rww Ui 0!!'rbcvdtj|dz |ztj|dz |zzzS)Nrkrg)rrrr]s r`raras:s!a%9,-''1q5I-./00rbc@t|jdd|dzdk(S)Nrkrgrrlrprsindexs r`raras#++ !)q. ("rbcrtj|dz |zjtjSrf)raroundrrr]s r`raras+ryy Ui 0!!'rb) inverted_cdfaveraged_inverted_cdfclosest_observationinterpolated_inverted_cdfhazenweibulllinearmedian_unbiasednormal_unbiasedlowerhighermidpointnearestc|fSr[rh)mkaxess r`_rot90_dispatcherr 4Krbrgct|}t|dk7r tdt|}|d|dk(s!t |d|dz |j k(r td|d|j k\s8|d|j ks%|d|j k\s|d|j kr%tdj ||j |dz}|dk(r|ddS|dk(rtt||d|dStd|j }||d||dc||d<||d<|dk(rtt||d|Stt|||dS) a Rotate an array by 90 degrees in the plane specified by axes. Rotation direction is from the first towards the second axis. This means for a 2D array with the default `k` and `axes`, the rotation will be counterclockwise. Parameters ---------- m : array_like Array of two or more dimensions. k : integer Number of times the array is rotated by 90 degrees. axes : (2,) array_like The array is rotated in the plane defined by the axes. Axes must be different. .. versionadded:: 1.12.0 Returns ------- y : ndarray A rotated view of `m`. See Also -------- flip : Reverse the order of elements in an array along the given axis. fliplr : Flip an array horizontally. flipud : Flip an array vertically. Notes ----- ``rot90(m, k=1, axes=(1,0))`` is the reverse of ``rot90(m, k=1, axes=(0,1))`` ``rot90(m, k=1, axes=(1,0))`` is equivalent to ``rot90(m, k=-1, axes=(0,1))`` Examples -------- >>> m = np.array([[1,2],[3,4]], int) >>> m array([[1, 2], [3, 4]]) >>> np.rot90(m) array([[2, 4], [1, 3]]) >>> np.rot90(m, 2) array([[4, 3], [2, 1]]) >>> m = np.arange(8).reshape((2,2,2)) >>> np.rot90(m, 1, (1,2)) array([[[1, 3], [0, 2]], [[5, 7], [4, 6]]]) zlen(axes) must be 2.rrgzAxes must be different.z*Axes={} out of range for array of ndim={}.N) tuplelen ValueErrorr rndimformatrArr)rrr axes_lists r`rBrBsx ;D 4yA~/001 A Aw$q'Xd1gQ&78AFFB233 Q166T!Ww. 7aff Q166' 1E VD!&& !# #FAAvt AvDDG$d1g..q!&&!I09$q'0B09$q'0B-YtAw47+ Avaa)955Ia+T!W55rbc|fSr[rh)raxiss r`_flip_dispatcherrrrbcht|ds t|}|)tjdddf|jz}||St j ||j}tjddg|jz}|D]}tjddd||<t|}||S)a Reverse the order of elements in an array along the given axis. The shape of the array is preserved, but the elements are reordered. .. versionadded:: 1.12.0 Parameters ---------- m : array_like Input array. axis : None or int or tuple of ints, optional Axis or axes along which to flip over. The default, axis=None, will flip over all of the axes of the input array. If axis is negative it counts from the last to the first axis. If axis is a tuple of ints, flipping is performed on all of the axes specified in the tuple. .. versionchanged:: 1.15.0 None and tuples of axes are supported Returns ------- out : array_like A view of `m` with the entries of axis reversed. Since a view is returned, this operation is done in constant time. See Also -------- flipud : Flip an array vertically (axis=0). fliplr : Flip an array horizontally (axis=1). Notes ----- flip(m, 0) is equivalent to flipud(m). flip(m, 1) is equivalent to fliplr(m). flip(m, n) corresponds to ``m[...,::-1,...]`` with ``::-1`` at position n. flip(m) corresponds to ``m[::-1,::-1,...,::-1]`` with ``::-1`` at all positions. flip(m, (0, 1)) corresponds to ``m[::-1,::-1,...]`` with ``::-1`` at position 0 and position 1. Examples -------- >>> A = np.arange(8).reshape((2,2,2)) >>> A array([[[0, 1], [2, 3]], [[4, 5], [6, 7]]]) >>> np.flip(A, 0) array([[[4, 5], [6, 7]], [[0, 1], [2, 3]]]) >>> np.flip(A, 1) array([[[2, 3], [0, 1]], [[6, 7], [4, 5]]]) >>> np.flip(A) array([[[7, 6], [5, 4]], [[3, 2], [1, 0]]]) >>> np.flip(A, (0, 2)) array([[[5, 4], [7, 6]], [[1, 0], [3, 2]]]) >>> A = np.random.randn(3,4,5) >>> np.all(np.flip(A,2) == A[:,:,::-1,...]) True rN)hasattrr rs_r_nxnormalize_axis_tupler)rrindexeraxs r`rArAsb 1f  AJ |552;.166) W: ''aff5558*qvv% &B%%"+GBK &. W:rbc: t|y#t$rYywxYw)a Check whether or not an object can be iterated over. Parameters ---------- y : object Input object. Returns ------- b : bool Return ``True`` if the object has an iterator method or is a sequence and ``False`` otherwise. Examples -------- >>> np.iterable([1, 2, 3]) True >>> np.iterable(2) False Notes ----- In most cases, the results of ``np.iterable(obj)`` are consistent with ``isinstance(obj, collections.abc.Iterable)``. One notable exception is the treatment of 0-dimensional arrays:: >>> from collections.abc import Iterable >>> a = np.array(1.0) # 0-dimensional numpy array >>> isinstance(a, Iterable) True >>> np.iterable(a) False FT)iter TypeError)ys r`r:r:Ws(L Q  s  ctj|}j|jk7r td|jt fdDk7r t d|j tj}|jt fdtjD}|S)zEValidate weights array. We assume, weights is not None. z;Axis must be specified when shapes of a and weights differ.c3<K|]}j|ywr[rm).0ras r` z%_weights_are_valid..s9baggbk9szIShape of weights must be consistent with shape of a along specified axis.c34K|]\}}|vr|ndyw)rgNrh)rrsrs r`rz%_weights_are_valid..s* A$)B')Dja!7 A) rr rmrrrrargsortreshape enumerate)weightsrrwgts `` r`_weights_are_validrs -- C ww#)) <  999D99 934 4 mmBJJt,-kk% A-6qww-? AAB Jrb)keepdimsc ||fSr[rh)rrrreturnedrs r`_average_dispatcherrs w<rbFctj|}|"tj||jd}|tj uri}nd|i}|\|j |fi|}tj|}|jj|j|jz }nt|||} t|jjtjtjfr,tj|j| jd} n*tj|j| j} | jd || d|}tj |dk(r t#dtj$|| | j|fi||z x}}|rK|j&|j&k7r.tj(||j&j+}||fS|S) a Compute the weighted average along the specified axis. Parameters ---------- a : array_like Array containing data to be averaged. If `a` is not an array, a conversion is attempted. axis : None or int or tuple of ints, optional Axis or axes along which to average `a`. The default, `axis=None`, will average over all of the elements of the input array. If axis is negative it counts from the last to the first axis. .. versionadded:: 1.7.0 If axis is a tuple of ints, averaging is performed on all of the axes specified in the tuple instead of a single axis or all the axes as before. weights : array_like, optional An array of weights associated with the values in `a`. Each value in `a` contributes to the average according to its associated weight. The array of weights must be the same shape as `a` if no axis is specified, otherwise the weights must have dimensions and shape consistent with `a` along the specified axis. If `weights=None`, then all data in `a` are assumed to have a weight equal to one. The calculation is:: avg = sum(a * weights) / sum(weights) where the sum is over all included elements. The only constraint on the values of `weights` is that `sum(weights)` must not be 0. returned : bool, optional Default is `False`. If `True`, the tuple (`average`, `sum_of_weights`) is returned, otherwise only the average is returned. If `weights=None`, `sum_of_weights` is equivalent to the number of elements over which the average is taken. keepdims : bool, optional If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original `a`. *Note:* `keepdims` will not work with instances of `numpy.matrix` or other classes whose methods do not support `keepdims`. .. versionadded:: 1.23.0 Returns ------- retval, [sum_of_weights] : array_type or double Return the average along the specified axis. When `returned` is `True`, return a tuple with the average as the first element and the sum of the weights as the second element. `sum_of_weights` is of the same type as `retval`. The result dtype follows a general pattern. If `weights` is None, the result dtype will be that of `a` , or ``float64`` if `a` is integral. Otherwise, if `weights` is not None and `a` is non- integral, the result type will be the type of lowest precision capable of representing values of both `a` and `weights`. If `a` happens to be integral, the previous rules still applies but the result dtype will at least be ``float64``. Raises ------ ZeroDivisionError When all weights along axis are zero. See `numpy.ma.average` for a version robust to this type of error. TypeError When `weights` does not have the same shape as `a`, and `axis=None`. ValueError When `weights` does not have dimensions and shape consistent with `a` along specified `axis`. See Also -------- mean ma.average : average for masked arrays -- useful if your data contains "missing" values numpy.result_type : Returns the type that results from applying the numpy type promotion rules to the arguments. Examples -------- >>> data = np.arange(1, 5) >>> data array([1, 2, 3, 4]) >>> np.average(data) 2.5 >>> np.average(np.arange(1, 11), weights=np.arange(10, 0, -1)) 4.0 >>> data = np.arange(6).reshape((3, 2)) >>> data array([[0, 1], [2, 3], [4, 5]]) >>> np.average(data, axis=1, weights=[1./4, 3./4]) array([0.75, 2.75, 4.75]) >>> np.average(data, weights=[1./4, 3./4]) Traceback (most recent call last): ... TypeError: Axis must be specified when shapes of a and weights differ. With ``keepdims=True``, the following result has shape (3, 1). >>> np.average(data, axis=1, keepdims=True) array([[0.5], [2.5], [4.5]]) >>> data = np.arange(8).reshape((2, 2, 2)) >>> data array([[[0, 1], [2, 3]], [[4, 5], [6, 7]]]) >>> np.average(data, axis=(0, 1), weights=[[1./4, 3./4], [1., 1./2]]) array([3.4, 4.4]) >>> np.average(data, axis=0, weights=[[1./4, 3./4], [1., 1./2]]) Traceback (most recent call last): ... ValueError: Shape of weights must be consistent with shape of a along specified axis. rargnamerrrrf8rdtyperz(Weights sum to zero, can't be normalizedrrh)rr rrr_NoValuer%rtypesizer issubclassrbool result_typer'r&ZeroDivisionErrormultiplyrm broadcast_tor9) rrrrr keepdims_kwavg avg_as_arraysclr result_dtypes r`rGrGs~ aA ''afffE2;; !8, affT)[)}}S)   %%aff\->->&>? ADA aggllRZZ$9 :>>!''399dCL>>!''399=LcggC4|C{C 66#* #:< <2R[[C ,..1c$G:EGILM Ml 99 ** *//#|'9'9:??ACCx rbct|||}|jjtdvr.t j |j s td|S)aConvert the input to an array, checking for NaNs or Infs. Parameters ---------- a : array_like Input data, in any form that can be converted to an array. This includes lists, lists of tuples, tuples, tuples of tuples, tuples of lists and ndarrays. Success requires no NaNs or Infs. dtype : data-type, optional By default, the data-type is inferred from the input data. order : {'C', 'F', 'A', 'K'}, optional Memory layout. 'A' and 'K' depend on the order of input array a. 'C' row-major (C-style), 'F' column-major (Fortran-style) memory representation. 'A' (any) means 'F' if `a` is Fortran contiguous, 'C' otherwise 'K' (keep) preserve input order Defaults to 'C'. Returns ------- out : ndarray Array interpretation of `a`. No copy is performed if the input is already an ndarray. If `a` is a subclass of ndarray, a base class ndarray is returned. Raises ------ ValueError Raises ValueError if `a` contains NaN (Not a Number) or Inf (Infinity). See Also -------- asarray : Create and array. asanyarray : Similar function which passes through subclasses. ascontiguousarray : Convert input to a contiguous array. asfortranarray : Convert input to an ndarray with column-major memory order. fromiter : Create an array from an iterator. fromfunction : Construct an array by executing a function on grid positions. Examples -------- Convert a list into an array. If all elements are finite ``asarray_chkfinite`` is identical to ``asarray``. >>> a = [1, 2] >>> np.asarray_chkfinite(a, dtype=float) array([1., 2.]) Raises ValueError if array_like contains Nans or Infs. >>> a = [1, 2, np.inf] >>> try: ... np.asarray_chkfinite(a) ... except ValueError: ... print('ValueError') ... ValueError )rorderAllFloatz#array must not contain infs or NaNs)r rcharr(risfiniteallr)rrrs r`rFrFHsP~ e,Aww||y,,R[[^5G5G5I 13 3 Hrbc/XK|tj|r |Ed{yy7wr[)rr:)xcondlistfunclistargskws r`_piecewise_dispatcherrs( G {{8s *(*cxt|}t|}t|s(t|dtt fs|j dk7r|g}t|t}t|}||dz k(r8tj|dd}tj||gd}|dz }n$||k7rtdj|||dzt|}t||D]S\} } t| t j"j$s| || <0|| } | j&dkDsE| | g|i||| <U|S)a Evaluate a piecewise-defined function. Given a set of conditions and corresponding functions, evaluate each function on the input data wherever its condition is true. Parameters ---------- x : ndarray or scalar The input domain. condlist : list of bool arrays or bool scalars Each boolean array corresponds to a function in `funclist`. Wherever `condlist[i]` is True, `funclist[i](x)` is used as the output value. Each boolean array in `condlist` selects a piece of `x`, and should therefore be of the same shape as `x`. The length of `condlist` must correspond to that of `funclist`. If one extra function is given, i.e. if ``len(funclist) == len(condlist) + 1``, then that extra function is the default value, used wherever all conditions are false. funclist : list of callables, f(x,*args,**kw), or scalars Each function is evaluated over `x` wherever its corresponding condition is True. It should take a 1d array as input and give an 1d array or a scalar value as output. If, instead of a callable, a scalar is provided then a constant function (``lambda x: scalar``) is assumed. args : tuple, optional Any further arguments given to `piecewise` are passed to the functions upon execution, i.e., if called ``piecewise(..., ..., 1, 'a')``, then each function is called as ``f(x, 1, 'a')``. kw : dict, optional Keyword arguments used in calling `piecewise` are passed to the functions upon execution, i.e., if called ``piecewise(..., ..., alpha=1)``, then each function is called as ``f(x, alpha=1)``. Returns ------- out : ndarray The output is the same shape and type as x and is found by calling the functions in `funclist` on the appropriate portions of `x`, as defined by the boolean arrays in `condlist`. Portions not covered by any condition have a default value of 0. See Also -------- choose, select, where Notes ----- This is similar to choose or select, except that functions are evaluated on elements of `x` that satisfy the corresponding condition from `condlist`. The result is:: |-- |funclist[0](x[condlist[0]]) out = |funclist[1](x[condlist[1]]) |... |funclist[n2](x[condlist[n2]]) |-- Examples -------- Define the signum function, which is -1 for ``x < 0`` and +1 for ``x >= 0``. >>> x = np.linspace(-2.5, 2.5, 6) >>> np.piecewise(x, [x < 0, x >= 0], [-1, 1]) array([-1., -1., -1., 1., 1., 1.]) Define the absolute value, which is ``-x`` for ``x <0`` and ``x`` for ``x >= 0``. >>> np.piecewise(x, [x < 0, x >= 0], [lambda x: -x, lambda x: x]) array([2.5, 1.5, 0.5, 0.5, 1.5, 2.5]) Apply the same function to a scalar value. >>> y = -2 >>> np.piecewise(y, [y < 0, y >= 0], [lambda x: -x, lambda x: x]) array(2) rrrgT)rrrz>8X"6Q? Q b J VAq!A#   1 A(H-2 d$  8 89AdGT7Dyy1}t1d1b1$ 2 Hrbc#<K|Ed{|Ed{y77wr[rh)r choicelistdefaults r`_select_dispatcherrs!s c$t|t|k7r tdt|dk(r td|Dcgc]5}t|ttt fvr|nt j|7}}|jt|ttt fvr|nt j| t j|}t j|}t j|}t|D]E\}}|jjt jus-tdj||dj dk(r|dj"} n)t j|d|ddj"} t j$| |d|} |ddd}|ddd}t'||D]\}}t j(| || | Scc}w#t$r}d|}t|dd}~wwxYw) a Return an array drawn from elements in choicelist, depending on conditions. Parameters ---------- condlist : list of bool ndarrays The list of conditions which determine from which array in `choicelist` the output elements are taken. When multiple conditions are satisfied, the first one encountered in `condlist` is used. choicelist : list of ndarrays The list of arrays from which the output elements are taken. It has to be of the same length as `condlist`. default : scalar, optional The element inserted in `output` when all conditions evaluate to False. Returns ------- output : ndarray The output at position m is the m-th element of the array in `choicelist` where the m-th element of the corresponding array in `condlist` is True. See Also -------- where : Return elements from one of two arrays depending on condition. take, choose, compress, diag, diagonal Examples -------- Beginning with an array of integers from 0 to 5 (inclusive), elements less than ``3`` are negated, elements greater than ``3`` are squared, and elements not meeting either of these conditions (exactly ``3``) are replaced with a `default` value of ``42``. >>> x = np.arange(6) >>> condlist = [x<3, x>3] >>> choicelist = [x, x**2] >>> np.select(condlist, choicelist, 42) array([ 0, 1, 2, 42, 16, 25]) When multiple conditions are satisfied, the first one encountered in `condlist` is used. >>> condlist = [x<=4, x>3] >>> choicelist = [x, x**2] >>> np.select(condlist, choicelist, 55) array([ 0, 1, 2, 3, 4, 25]) z7list of cases must be same length as list of conditionsrz3select with an empty condition list is not possiblez9Choicelist and default value do not have a common dtype: Nz7invalid entry {} in condlist: should be boolean ndarrayrr)rrrintfloatcomplexrr rXrrbroadcast_arraysrrrrrrmfullrcopyto) rrrchoiceremsgir result_shaperesults r`r6r6sh 8}J' EG G 8}NOO!" v,3w"77RZZ=OO"J"g3w2G!Gg::g.0' +""H-H$$j1JX&U4 ::??"'' )IPPQRSU UU !}Q{(( **8A; 1 FqIOO WW\:b>5 9F BFF#J"~HJ1.  &&-. MQ" 'I!Mn$&'s:G+?G00 H9H  Hc|fSr[rhrrsuboks r`_copy_dispatcherr}rrbc t|||dS)a Return an array copy of the given object. Parameters ---------- a : array_like Input data. order : {'C', 'F', 'A', 'K'}, optional Controls the memory layout of the copy. 'C' means C-order, 'F' means F-order, 'A' means 'F' if `a` is Fortran contiguous, 'C' otherwise. 'K' means match the layout of `a` as closely as possible. (Note that this function and :meth:`ndarray.copy` are very similar, but have different default values for their order= arguments.) subok : bool, optional If True, then sub-classes will be passed-through, otherwise the returned array will be forced to be a base-class array (defaults to False). .. versionadded:: 1.19.0 Returns ------- arr : ndarray Array interpretation of `a`. See Also -------- ndarray.copy : Preferred method for creating an array copy Notes ----- This is equivalent to: >>> np.array(a, copy=True) #doctest: +SKIP The copy made of the data is shallow, i.e., for arrays with object dtype, the new array will point to the same objects. See Examples from `ndarray.copy`. Examples -------- Create an array x, with a reference y and a copy z: >>> x = np.array([1, 2, 3]) >>> y = x >>> z = np.copy(x) Note that, when we modify x, y changes, but not z: >>> x[0] = 10 >>> x[0] == y[0] True >>> x[0] == z[0] False Note that, np.copy clears previously set WRITEABLE=False flag. >>> a = np.array([1, 2, 3]) >>> a.flags["WRITEABLE"] = False >>> b = np.copy(a) >>> b.flags["WRITEABLE"] True >>> b[0] = 3 >>> b array([3, 2, 3]) T)rrr9)r rs r`r9r9sH %u4 88rb)r edge_orderc',K||Ed{y7wr[rh)frrvarargss r`_gradient_dispatcherr s Gs c( tj|}|j}|tt |}nt j ||}t|}t|}|dk(rdg|z}nA|dk(r"tj|ddk(r||z}n||k(r t|}t|D]\} } tj| } | jdk(r+| jdk7r tdt| |j|| k7r tdtj| jtjr| jtj } tj"| } | | dk(j%r| d} | || <n t'd|dkDr td g} t)dg|z} t)dg|z}t)dg|z}t)dg|z}|j}|j*tj,urAtj|j.j1d d }|j3|}n|j*tj4urnxtj|tj6rnStj|tjr|jtj }tj }t9||D]p\}}|j||dzkr td tj:|| }tj|dk(}t)dd| |<t)dd||<t)dd||<t)dd||<|r0|t||t|z d|zz |t| <n|dd}|dd}| |||zzz }||z ||zz }||||zzz }tj<|t> }d||<|x|_ x|_ |_ ||t|z||t|zz||t|zz|t| <|dk(rd| |<d||<d||<|r|n|d}|t||t|z |z |t| <d| |<d||<d||<|r|n|d}|t||t|z |z |t| <n3d| |<d||<d||<d||<|rd|z }d|z }d|z }n3|d}|d}d|z|z |||zzz }||z||zz }| |||zzz }||t|z||t|zz||t|zz|t| <d| |<d||<d||<d||<|rd|z }d|z }d|z }n2|d}|d}||||zzz }||z ||zz }d|z|z|||zzz }||t|z||t|zz||t|zz|t| <| jA|t)d| |<t)d||<t)d||<t)d||<s|dk(r| dSt| S)a Return the gradient of an N-dimensional array. The gradient is computed using second order accurate central differences in the interior points and either first or second order accurate one-sides (forward or backwards) differences at the boundaries. The returned gradient hence has the same shape as the input array. Parameters ---------- f : array_like An N-dimensional array containing samples of a scalar function. varargs : list of scalar or array, optional Spacing between f values. Default unitary spacing for all dimensions. Spacing can be specified using: 1. single scalar to specify a sample distance for all dimensions. 2. N scalars to specify a constant sample distance for each dimension. i.e. `dx`, `dy`, `dz`, ... 3. N arrays to specify the coordinates of the values along each dimension of F. The length of the array must match the size of the corresponding dimension 4. Any combination of N scalars/arrays with the meaning of 2. and 3. If `axis` is given, the number of varargs must equal the number of axes. Default: 1. edge_order : {1, 2}, optional Gradient is calculated using N-th order accurate differences at the boundaries. Default: 1. .. versionadded:: 1.9.1 axis : None or int or tuple of ints, optional Gradient is calculated only along the given axis or axes The default (axis = None) is to calculate the gradient for all the axes of the input array. axis may be negative, in which case it counts from the last to the first axis. .. versionadded:: 1.11.0 Returns ------- gradient : ndarray or list of ndarray A list of ndarrays (or a single ndarray if there is only one dimension) corresponding to the derivatives of f with respect to each dimension. Each derivative has the same shape as f. Examples -------- >>> f = np.array([1, 2, 4, 7, 11, 16], dtype=float) >>> np.gradient(f) array([1. , 1.5, 2.5, 3.5, 4.5, 5. ]) >>> np.gradient(f, 2) array([0.5 , 0.75, 1.25, 1.75, 2.25, 2.5 ]) Spacing can be also specified with an array that represents the coordinates of the values F along the dimensions. For instance a uniform spacing: >>> x = np.arange(f.size) >>> np.gradient(f, x) array([1. , 1.5, 2.5, 3.5, 4.5, 5. ]) Or a non uniform one: >>> x = np.array([0., 1., 1.5, 3.5, 4., 6.], dtype=float) >>> np.gradient(f, x) array([1. , 3. , 3.5, 6.7, 6.9, 2.5]) For two dimensional arrays, the return will be two arrays ordered by axis. In this example the first array stands for the gradient in rows and the second one in columns direction: >>> np.gradient(np.array([[1, 2, 6], [3, 4, 5]], dtype=float)) [array([[ 2., 2., -1.], [ 2., 2., -1.]]), array([[1. , 2.5, 4. ], [1. , 1. , 1. ]])] In this example the spacing is also specified: uniform for axis=0 and non uniform for axis=1 >>> dx = 2. >>> y = [1., 1.5, 3.5] >>> np.gradient(np.array([[1, 2, 6], [3, 4, 5]], dtype=float), dx, y) [array([[ 1. , 1. , -0.5], [ 1. , 1. , -0.5]]), array([[2. , 2. , 2. ], [2. , 1.7, 0.5]])] It is possible to specify how boundaries are treated using `edge_order` >>> x = np.array([0, 1, 2, 3, 4]) >>> f = x**2 >>> np.gradient(f, edge_order=1) array([1., 2., 4., 6., 7.]) >>> np.gradient(f, edge_order=2) array([0., 2., 4., 6., 8.]) The `axis` keyword can be used to specify a subset of axes of which the gradient is calculated >>> np.gradient(np.array([[1, 2, 6], [3, 4, 5]], dtype=float), axis=0) array([[ 2., 2., -1.], [ 2., 2., -1.]]) Notes ----- Assuming that :math:`f\in C^{3}` (i.e., :math:`f` has at least 3 continuous derivatives) and let :math:`h_{*}` be a non-homogeneous stepsize, we minimize the "consistency error" :math:`\eta_{i}` between the true gradient and its estimate from a linear combination of the neighboring grid-points: .. math:: \eta_{i} = f_{i}^{\left(1\right)} - \left[ \alpha f\left(x_{i}\right) + \beta f\left(x_{i} + h_{d}\right) + \gamma f\left(x_{i}-h_{s}\right) \right] By substituting :math:`f(x_{i} + h_{d})` and :math:`f(x_{i} - h_{s})` with their Taylor series expansion, this translates into solving the following the linear system: .. math:: \left\{ \begin{array}{r} \alpha+\beta+\gamma=0 \\ \beta h_{d}-\gamma h_{s}=1 \\ \beta h_{d}^{2}+\gamma h_{s}^{2}=0 \end{array} \right. The resulting approximation of :math:`f_{i}^{(1)}` is the following: .. math:: \hat f_{i}^{(1)} = \frac{ h_{s}^{2}f\left(x_{i} + h_{d}\right) + \left(h_{d}^{2} - h_{s}^{2}\right)f\left(x_{i}\right) - h_{d}^{2}f\left(x_{i}-h_{s}\right)} { h_{s}h_{d}\left(h_{d} + h_{s}\right)} + \mathcal{O}\left(\frac{h_{d}h_{s}^{2} + h_{s}h_{d}^{2}}{h_{d} + h_{s}}\right) It is worth noting that if :math:`h_{s}=h_{d}` (i.e., data are evenly spaced) we find the standard second order approximation: .. math:: \hat f_{i}^{(1)}= \frac{f\left(x_{i+1}\right) - f\left(x_{i-1}\right)}{2h} + \mathcal{O}\left(h^{2}\right) With a similar procedure the forward/backward approximations used for boundaries can be derived. References ---------- .. [1] Quarteroni A., Sacco R., Saleri F. (2007) Numerical Mathematics (Texts in Applied Mathematics). New York: Springer. .. [2] Durran D. R. (1999) Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. New York: Springer. .. [3] Fornberg B. (1988) Generation of Finite Difference Formulas on Arbitrarily Spaced Grids, Mathematics of Computation 51, no. 184 : 699-706. `PDF `_. Nrrjrgz&distances must be either scalars or 1dzGwhen 1d, distances must match the length of the corresponding dimensionzinvalid number of argumentsrz)'edge_order' greater than 2 not supporteddatetime timedeltazlShape of array too small to calculate a numerical gradient, at least (edge_order + 1) elements are required.rrr @ggrkgg?)!rr rrrangerrrrrrrm issubdtyperrrfloat64r<rrslicer datetime64namereplaceview timedelta64inexactr empty_likerr rX)rrrrNrlen_axesr^dxr distancesdiffxoutvalsslice1slice2slice3slice4otypeax_dxoutuniform_spacingdx1dx2rbcrmdx_0dx_ns r`r=r=s^ aA A |U1X''a04yH G AAvUX  aBGGGAJ'1, x  h ']%bM LAy i0I~~"1$ !IJJ9~a!11 "MNN}}Y__bjj9&,,RZZ8 GGI&Eq!&&(aBqE% (566A~DEE GDk]1_FDk]1_FDk]1_FDk]1_F GGE zzR]]"++J DE FF5M r~~ % ubjj ) == +$A 4}Z# e 774=:> )CD DmmAU+''%.A-Q|t T2t Q|t Q~t "#E&M"2QuV}5E"E"u*!UCf "+C)CsSy)*AssSy)AscCi()AGGAS)EE$K*/ /AG /ag!"QuV}%5!5AeFmrbc |||fSr[rh)rr^rprependrXs r`_diff_dispatcherrG(s w rbrcT|dk(r|S|dkrtdt|zt|}|j}|dk(r tdt ||}g}|t j urnt j|}|jdk(r9t|j}d||<t j|t|}|j||j||t j urnt j|}|jdk(r9t|j}d||<t j|t|}|j|t|dkDrt j||}tdg|z}tdg|z} tdd||<tdd| |<t|}t| } |jt j k(rt"nt$} t'|D]} | |||| }|S)a0 Calculate the n-th discrete difference along the given axis. The first difference is given by ``out[i] = a[i+1] - a[i]`` along the given axis, higher differences are calculated by using `diff` recursively. Parameters ---------- a : array_like Input array n : int, optional The number of times values are differenced. If zero, the input is returned as-is. axis : int, optional The axis along which the difference is taken, default is the last axis. prepend, append : array_like, optional Values to prepend or append to `a` along axis prior to performing the difference. Scalar values are expanded to arrays with length 1 in the direction of axis and the shape of the input array in along all other axes. Otherwise the dimension and shape must match `a` except along axis. .. versionadded:: 1.16.0 Returns ------- diff : ndarray The n-th differences. The shape of the output is the same as `a` except along `axis` where the dimension is smaller by `n`. The type of the output is the same as the type of the difference between any two elements of `a`. This is the same as the type of `a` in most cases. A notable exception is `datetime64`, which results in a `timedelta64` output array. See Also -------- gradient, ediff1d, cumsum Notes ----- Type is preserved for boolean arrays, so the result will contain `False` when consecutive elements are the same and `True` when they differ. For unsigned integer arrays, the results will also be unsigned. This should not be surprising, as the result is consistent with calculating the difference directly: >>> u8_arr = np.array([1, 0], dtype=np.uint8) >>> np.diff(u8_arr) array([255], dtype=uint8) >>> u8_arr[1,...] - u8_arr[0,...] 255 If this is not desirable, then the array should be cast to a larger integer type first: >>> i16_arr = u8_arr.astype(np.int16) >>> np.diff(i16_arr) array([-1], dtype=int16) Examples -------- >>> x = np.array([1, 2, 4, 7, 0]) >>> np.diff(x) array([ 1, 2, 3, -7]) >>> np.diff(x, n=2) array([ 1, 1, -10]) >>> x = np.array([[1, 3, 6, 10], [0, 5, 6, 8]]) >>> np.diff(x) array([[2, 3, 4], [5, 1, 2]]) >>> np.diff(x, axis=0) array([[-1, 2, 0, -2]]) >>> x = np.arange('1066-10-13', '1066-10-16', dtype=np.datetime64) >>> np.diff(x) array([1, 1], dtype='timedelta64[D]') rz#order must be non-negative but got z4diff requires input that is at least one dimensionalrgNr)rreprr rr,rrrrmrrrXrrr)rrrr r&) rr^rrFrXndcombinedrmr7r8oprts r`r<r<,sj Av1u 1DG ;= = 1 A B QwOPP b )DHbkk!--( <<1 MEE$KooguU|F4Lr?F4L 6]F 6]Fgg(hB 1X% qy!F) $% Hrbc |||fSr[rh)rxpfpleftrightperiods r`_interp_dispatcherrSs r2;rbcJtj|}tj|rt}tj}nt }tj }|>|dk(r tdt|}d}d}tj|tj }tj|tj }tj||}|jdk7s|jdk7r td|jd|jdk7r td||z}||z}tj|}||}||}tj|dd|z ||dd|zf}tj|dd||ddf}||||||S) a One-dimensional linear interpolation for monotonically increasing sample points. Returns the one-dimensional piecewise linear interpolant to a function with given discrete data points (`xp`, `fp`), evaluated at `x`. Parameters ---------- x : array_like The x-coordinates at which to evaluate the interpolated values. xp : 1-D sequence of floats The x-coordinates of the data points, must be increasing if argument `period` is not specified. Otherwise, `xp` is internally sorted after normalizing the periodic boundaries with ``xp = xp % period``. fp : 1-D sequence of float or complex The y-coordinates of the data points, same length as `xp`. left : optional float or complex corresponding to fp Value to return for `x < xp[0]`, default is `fp[0]`. right : optional float or complex corresponding to fp Value to return for `x > xp[-1]`, default is `fp[-1]`. period : None or float, optional A period for the x-coordinates. This parameter allows the proper interpolation of angular x-coordinates. Parameters `left` and `right` are ignored if `period` is specified. .. versionadded:: 1.10.0 Returns ------- y : float or complex (corresponding to fp) or ndarray The interpolated values, same shape as `x`. Raises ------ ValueError If `xp` and `fp` have different length If `xp` or `fp` are not 1-D sequences If `period == 0` See Also -------- scipy.interpolate Warnings -------- The x-coordinate sequence is expected to be increasing, but this is not explicitly enforced. However, if the sequence `xp` is non-increasing, interpolation results are meaningless. Note that, since NaN is unsortable, `xp` also cannot contain NaNs. A simple check for `xp` being strictly increasing is:: np.all(np.diff(xp) > 0) Examples -------- >>> xp = [1, 2, 3] >>> fp = [3, 2, 0] >>> np.interp(2.5, xp, fp) 1.0 >>> np.interp([0, 1, 1.5, 2.72, 3.14], xp, fp) array([3. , 3. , 2.5 , 0.56, 0. ]) >>> UNDEF = -99.0 >>> np.interp(3.14, xp, fp, right=UNDEF) -99.0 Plot an interpolant to the sine function: >>> x = np.linspace(0, 2*np.pi, 10) >>> y = np.sin(x) >>> xvals = np.linspace(0, 2*np.pi, 50) >>> yinterp = np.interp(xvals, x, y) >>> import matplotlib.pyplot as plt >>> plt.plot(x, y, 'o') [] >>> plt.plot(xvals, yinterp, '-x') [] >>> plt.show() Interpolation with periodic x-coordinates: >>> x = [-180, -170, -185, 185, -10, -5, 0, 365] >>> xp = [190, -190, 350, -350] >>> fp = [5, 10, 3, 4] >>> np.interp(x, xp, fp, period=360) array([7.5 , 5. , 8.75, 6.25, 3. , 3.25, 3.5 , 3.75]) Complex interpolation: >>> x = [1.5, 4.0] >>> xp = [2,3,5] >>> fp = [1.0j, 0, 2+3j] >>> np.interp(x, xp, fp) array([0.+1.j , 1.+1.5j]) Nrzperiod must be a non-zero valuerrgz!Data points must be 1-D sequencesz$fp and xp are not of the same lengthr) rr iscomplexobjcompiled_interp_complex complex128compiled_interpr(rabsrrmrr) rrNrOrPrQrR interp_func input_dtypeasort_xps r`r.r.sqR BB r- mm % jj   Q;>? ?V JJq + ZZ"** - ZZ+ . 77a<277a<@A A 88A;"((1+ %CD D J &[::b> \ \ ^^RWV^RAa@ A ^^RWb"Qq'2 3 q"b$ ..rbc|fSr[rh)zdegs r`_angle_dispatcherr`Brrbct|}t|jjtj r|j }|j}nd}|}t||}|r |dtz z}|S)a Return the angle of the complex argument. Parameters ---------- z : array_like A complex number or sequence of complex numbers. deg : bool, optional Return angle in degrees if True, radians if False (default). Returns ------- angle : ndarray or scalar The counterclockwise angle from the positive real axis on the complex plane in the range ``(-pi, pi]``, with dtype as numpy.float64. .. versionchanged:: 1.16.0 This function works on subclasses of ndarray like `ma.array`. See Also -------- arctan2 absolute Notes ----- This function passes the imaginary and real parts of the argument to `arctan2` to compute the result; consequently, it follows the convention of `arctan2` when the magnitude of the argument is zero. See example. Examples -------- >>> np.angle([1.0, 1.0j, 1+1j]) # in radians array([ 0. , 1.57079633, 0.78539816]) # may vary >>> np.angle(1+1j, deg=True) # in degrees 45.0 >>> np.angle([0., -0., complex(0., -0.), complex(-0., -0.)]) # convention array([ 0. , 3.14159265, -0. , -3.14159265]) r) r rrrrcomplexfloatingimagrealrr)r^r_zimagzrealrs r`r>r>FscT 1 A!'',, 3 34uA  SV  Hrb)rRc|fSr[rh)pdiscontrrRs r`_unwrap_dispatcherrk~rrbrcnt|}|j}t||}||dz }tddg|z}tdd||<t |}t j ||}tj|tjrt|d\}} | dk(} n|dz }d} | } t|| z || z} | r!tj| || | k(|dkDz| |z } tj| dt||kt|d|}||| j|z||<|S) a Unwrap by taking the complement of large deltas with respect to the period. This unwraps a signal `p` by changing elements which have an absolute difference from their predecessor of more than ``max(discont, period/2)`` to their `period`-complementary values. For the default case where `period` is :math:`2\pi` and `discont` is :math:`\pi`, this unwraps a radian phase `p` such that adjacent differences are never greater than :math:`\pi` by adding :math:`2k\pi` for some integer :math:`k`. Parameters ---------- p : array_like Input array. discont : float, optional Maximum discontinuity between values, default is ``period/2``. Values below ``period/2`` are treated as if they were ``period/2``. To have an effect different from the default, `discont` should be larger than ``period/2``. axis : int, optional Axis along which unwrap will operate, default is the last axis. period : float, optional Size of the range over which the input wraps. By default, it is ``2 pi``. .. versionadded:: 1.21.0 Returns ------- out : ndarray Output array. See Also -------- rad2deg, deg2rad Notes ----- If the discontinuity in `p` is smaller than ``period/2``, but larger than `discont`, no unwrapping is done because taking the complement would only make the discontinuity larger. Examples -------- >>> phase = np.linspace(0, np.pi, num=5) >>> phase[3:] += np.pi >>> phase array([ 0. , 0.78539816, 1.57079633, 5.49778714, 6.28318531]) # may vary >>> np.unwrap(phase) array([ 0. , 0.78539816, 1.57079633, -0.78539816, 0. ]) # may vary >>> np.unwrap([0, 1, 2, -1, 0], period=4) array([0, 1, 2, 3, 4]) >>> np.unwrap([ 1, 2, 3, 4, 5, 6, 1, 2, 3], period=6) array([1, 2, 3, 4, 5, 6, 7, 8, 9]) >>> np.unwrap([2, 3, 4, 5, 2, 3, 4, 5], period=4) array([2, 3, 4, 5, 6, 7, 8, 9]) >>> phase_deg = np.mod(np.linspace(0 ,720, 19), 360) - 180 >>> np.unwrap(phase_deg, period=360) array([-180., -140., -100., -60., -20., 20., 60., 100., 140., 180., 220., 260., 300., 340., 380., 420., 460., 500., 540.]) rNrrgrTr )r9r)r rr<r)rrrrr'rdivmodrrrYr cumsum)rirjrrRrJddr7r interval_highremboundary_ambiguous interval_lowddmod ph_correctups r`r?r?s>D  A B ad B(D$  #FD>F4L 6]F NN2v &E ~~eS[[)#FA. s AX !!>L \!6 *\ 9E 5-</BF; =JJJz1CGg$56 qt5 )B6Z..t44BvJ Irbc|fSr[rh)rs r` _sort_complexrxrrbcdt|d}|jt|jjt j sd|jjdvr|jdS|jjdk(r|jdS|jdS|S)a Sort a complex array using the real part first, then the imaginary part. Parameters ---------- a : array_like Input array Returns ------- out : complex ndarray Always returns a sorted complex array. Examples -------- >>> np.sort_complex([5, 3, 6, 2, 1]) array([1.+0.j, 2.+0.j, 3.+0.j, 5.+0.j, 6.+0.j]) >>> np.sort_complex([1 + 2j, 2 - 1j, 3 - 2j, 3 - 3j, 3 + 5j]) array([1.+2.j, 2.-1.j, 3.-3.j, 3.-2.j, 3.+5.j]) Tr9bhBHFgGD) r sortrrrrrcrr)rrAs r`r@r@s0 adAFFH aggllC$7$7 8 77<<6 !88C= WW\\S 88C= 88C= rbc|fSr[rh)filttrims r` _trim_zerosr s 7Nrbcd}|j}d|vr|D]}|dk7rn|dz}t|}d|vr|dddD]}|dk7rn|dz }|||S)a/ Trim the leading and/or trailing zeros from a 1-D array or sequence. Parameters ---------- filt : 1-D array or sequence Input array. trim : str, optional A string with 'f' representing trim from front and 'b' to trim from back. Default is 'fb', trim zeros from both front and back of the array. Returns ------- trimmed : 1-D array or sequence The result of trimming the input. The input data type is preserved. Examples -------- >>> a = np.array((0, 0, 0, 1, 2, 3, 0, 2, 1, 0)) >>> np.trim_zeros(a) array([1, 2, 3, 0, 2, 1]) >>> np.trim_zeros(a, 'b') array([0, 0, 0, ..., 0, 2, 1]) The input data type is preserved, list/tuple in means list/tuple out. >>> np.trim_zeros([0, 1, 2, 0]) [1, 2] rr|rrgBNr)upperr)rrfirstrlasts r`r8r8sF E ::>> arr = np.arange(12).reshape((3, 4)) >>> arr array([[ 0, 1, 2, 3], [ 4, 5, 6, 7], [ 8, 9, 10, 11]]) >>> condition = np.mod(arr, 3)==0 >>> condition array([[ True, False, False, True], [False, False, True, False], [False, True, False, False]]) >>> np.extract(condition, arr) array([0, 3, 6, 9]) If `condition` is boolean: >>> arr[condition] array([0, 3, 6, 9]) r)rrr"r#rs r`rCrCHs)b 88E#Ji(8 9! < ==rbc |||fSr[rhrmaskrs r`_place_dispatcherr|s t rbct|||S)a Change elements of an array based on conditional and input values. Similar to ``np.copyto(arr, vals, where=mask)``, the difference is that `place` uses the first N elements of `vals`, where N is the number of True values in `mask`, while `copyto` uses the elements where `mask` is True. Note that `extract` does the exact opposite of `place`. Parameters ---------- arr : ndarray Array to put data into. mask : array_like Boolean mask array. Must have the same size as `a`. vals : 1-D sequence Values to put into `a`. Only the first N elements are used, where N is the number of True values in `mask`. If `vals` is smaller than N, it will be repeated, and if elements of `a` are to be masked, this sequence must be non-empty. See Also -------- copyto, put, take, extract Examples -------- >>> arr = np.arange(6).reshape(2, 3) >>> np.place(arr, arr>2, [44, 55]) >>> arr array([[ 0, 1, 2], [44, 55, 44]]) )r*rs r`rDrDsJ #tT ""rbTctjdtd|tj}|r|j d|zn|j d|z|j y)az Display a message on a device. .. deprecated:: 2.0 Use your own printing function instead. Parameters ---------- mesg : str Message to display. device : object Device to write message. If None, defaults to ``sys.stdout`` which is very similar to ``print``. `device` needs to have ``write()`` and ``flush()`` methods. linefeed : bool, optional Option whether to print a line feed or not. Defaults to True. Raises ------ AttributeError If `device` does not have a ``write()`` or ``flush()`` method. Examples -------- Besides ``sys.stdout``, a file-like object can also be used as it has both required methods: >>> from io import StringIO >>> buf = StringIO() >>> np.disp('"Display" in a file', device=buf) >>> buf.getvalue() '"Display" in a file\n' zW`disp` is deprecated, use your own printing function instead. (deprecated in NumPy 2.0)r stacklevelNz%s z%s)warningswarnDeprecationWarningsysstdoutwriteflush)mesgdevicelinefeeds r`disprs]J MM $  ~ Vd]# TD[! LLN rbz\w+z(?:{0:}(?:,{0:})*)?z\({}\)z{0:}(?:,{0:})*z ^{0:}->{0:}$ctjdd|}tjt|st dj |t d|jdDS)as Parse string signatures for a generalized universal function. Arguments --------- signature : string Generalized universal function signature, e.g., ``(m,n),(n,p)->(m,p)`` for ``np.matmul``. Returns ------- Tuple of input and output core dimensions parsed from the signature, each of the form List[Tuple[str, ...]]. z\s+z not a valid gufunc signature: {}c 3K|]M}tjt|Dcgc]%}ttjt|'c}Oycc}wwr[)refindall _ARGUMENTr_DIMENSION_NAME)rarg_listargs r`rz*_parse_gufunc_signature..sK8 ZZ 8<> ?C89>8>s!A*A Az->)rsubmatch _SIGNATURErrrsplit) signatures r`_parse_gufunc_signaturersdvr9-I 88J * . 5 5i @B B 8!*!68 88rbc  |syt|}|j|krtd|j|fz|j| d}t ||D].\}}||vr |||k7std|d|d|||||<0y)aO Incrementally check and update core dimension sizes for a single argument. Arguments --------- dim_sizes : Dict[str, int] Sizes of existing core dimensions. Will be updated in-place. arg : ndarray Argument to examine. core_dims : Tuple[str, ...] Core dimensions for this argument. NzR%d-dimensional argument does not have enough dimensions for all core dimensions %rz%inconsistent size for core dimension z: z vs )rrrrmr) dim_sizesr core_dims num_core_dims core_shapedimrs r`_update_dim_sizesrs  NM xx- 4xx# $% % M>?+JJ/" T ) y~% D)C.233"IcN"rbc`g}i}t||D]r\}}t||||jt|z }tj j jd|jd|}|j|tt j jj|}||fS)a Parse broadcast and core dimensions for vectorize with a signature. Arguments --------- args : Tuple[ndarray, ...] Tuple of input arguments to examine. input_core_dims : List[Tuple[str, ...]] List of core dimensions corresponding to each input. Returns ------- broadcast_shape : Tuple[int, ...] Common shape to broadcast all non-core dimensions to. dim_sizes : Dict[str, int] Common sizes for named core dimensions. rN) rrrrrlib stride_tricks as_stridedrmrX_stride_tricks_impl_broadcast_shape) rinput_core_dimsbroadcast_argsrrrr dummy_arraybroadcast_shapes r`_parse_input_dimensionsr"s$NIdO4+Y)S)4xx#i.(ff**55a5D9IJ k* + ff00AA O I %%rbcX|Dcgc]}|tfd|Dzc}Scc}w)z=Helper for calculating broadcast shapes with core dimensions.c3(K|] }| ywr[rh)rrrs r`rz$_calculate_shapes..Cs#HsIcN#H)r)rrlist_of_core_dimsrs ` r`_calculate_shapesrAs5/ 0 e#Hi#HH H 00 0s'ct|||}|dgt|z}|tdt||D}|Stdt|||D}|S)z/Helper for creating output arrays in vectorize.Nc3PK|]\}}tj|| yw)rmrN)rr )rrmrs r`rz!_create_arrays..Ns,@'5%xxe599@s$&c3TK|] \}}}tj|||"ywr)rr0)rrrmrs r`rz!_create_arrays..Qs.8/65%}}V5FF8s&()rrrr)rrrdtypesresultsshapesarrayss r`_create_arraysrGs~ ;L MF ~#f+%@+.vv+>@@ M8gvv688 Mrbc:|jdvr |jS|S)NSU)rrs r`_get_vectorize_dtyperWs zzTzz Lrbc\eZdZdZej dddddfdZdZdZdZ dZ d Z d Z y) rEa vectorize(pyfunc=np._NoValue, otypes=None, doc=None, excluded=None, cache=False, signature=None) Returns an object that acts like pyfunc, but takes arrays as input. Define a vectorized function which takes a nested sequence of objects or numpy arrays as inputs and returns a single numpy array or a tuple of numpy arrays. The vectorized function evaluates `pyfunc` over successive tuples of the input arrays like the python map function, except it uses the broadcasting rules of numpy. The data type of the output of `vectorized` is determined by calling the function with the first element of the input. This can be avoided by specifying the `otypes` argument. Parameters ---------- pyfunc : callable, optional A python function or method. Can be omitted to produce a decorator with keyword arguments. otypes : str or list of dtypes, optional The output data type. It must be specified as either a string of typecode characters or a list of data type specifiers. There should be one data type specifier for each output. doc : str, optional The docstring for the function. If None, the docstring will be the ``pyfunc.__doc__``. excluded : set, optional Set of strings or integers representing the positional or keyword arguments for which the function will not be vectorized. These will be passed directly to `pyfunc` unmodified. .. versionadded:: 1.7.0 cache : bool, optional If `True`, then cache the first function call that determines the number of outputs if `otypes` is not provided. .. versionadded:: 1.7.0 signature : string, optional Generalized universal function signature, e.g., ``(m,n),(n)->(m)`` for vectorized matrix-vector multiplication. If provided, ``pyfunc`` will be called with (and expected to return) arrays with shapes given by the size of corresponding core dimensions. By default, ``pyfunc`` is assumed to take scalars as input and output. .. versionadded:: 1.12.0 Returns ------- out : callable A vectorized function if ``pyfunc`` was provided, a decorator otherwise. See Also -------- frompyfunc : Takes an arbitrary Python function and returns a ufunc Notes ----- The `vectorize` function is provided primarily for convenience, not for performance. The implementation is essentially a for loop. If `otypes` is not specified, then a call to the function with the first argument will be used to determine the number of outputs. The results of this call will be cached if `cache` is `True` to prevent calling the function twice. However, to implement the cache, the original function must be wrapped which will slow down subsequent calls, so only do this if your function is expensive. The new keyword argument interface and `excluded` argument support further degrades performance. References ---------- .. [1] :doc:`/reference/c-api/generalized-ufuncs` Examples -------- >>> def myfunc(a, b): ... "Return a-b if a>b, otherwise return a+b" ... if a > b: ... return a - b ... else: ... return a + b >>> vfunc = np.vectorize(myfunc) >>> vfunc([1, 2, 3, 4], 2) array([3, 4, 1, 2]) The docstring is taken from the input function to `vectorize` unless it is specified: >>> vfunc.__doc__ 'Return a-b if a>b, otherwise return a+b' >>> vfunc = np.vectorize(myfunc, doc='Vectorized `myfunc`') >>> vfunc.__doc__ 'Vectorized `myfunc`' The output type is determined by evaluating the first element of the input, unless it is specified: >>> out = vfunc([1, 2, 3, 4], 2) >>> type(out[0]) >>> vfunc = np.vectorize(myfunc, otypes=[float]) >>> out = vfunc([1, 2, 3, 4], 2) >>> type(out[0]) The `excluded` argument can be used to prevent vectorizing over certain arguments. This can be useful for array-like arguments of a fixed length such as the coefficients for a polynomial as in `polyval`: >>> def mypolyval(p, x): ... _p = list(p) ... res = _p.pop(0) ... while _p: ... res = res*x + _p.pop(0) ... return res >>> vpolyval = np.vectorize(mypolyval, excluded=['p']) >>> vpolyval(p=[1, 2, 3], x=[0, 1]) array([3, 6]) Positional arguments may also be excluded by specifying their position: >>> vpolyval.excluded.add(0) >>> vpolyval([1, 2, 3], x=[0, 1]) array([3, 6]) The `signature` argument allows for vectorizing functions that act on non-scalar arrays of fixed length. For example, you can use it for a vectorized calculation of Pearson correlation coefficient and its p-value: >>> import scipy.stats >>> pearsonr = np.vectorize(scipy.stats.pearsonr, ... signature='(n),(n)->(),()') >>> pearsonr([[0, 1, 2, 3]], [[1, 2, 3, 4], [4, 3, 2, 1]]) (array([ 1., -1.]), array([ 0., 0.])) Or for a vectorized convolution: >>> convolve = np.vectorize(np.convolve, signature='(n),(m)->(k)') >>> convolve(np.eye(4), [1, 2, 1]) array([[1., 2., 1., 0., 0., 0.], [0., 1., 2., 1., 0., 0.], [0., 0., 1., 2., 1., 0.], [0., 0., 0., 1., 2., 1.]]) Decorator syntax is supported. The decorator can be called as a function to provide keyword arguments: >>> @np.vectorize ... def identity(x): ... return x ... >>> identity([0, 1, 2]) array([0, 1, 2]) >>> @np.vectorize(otypes=[float]) ... def as_float(x): ... return x ... >>> as_float([0, 1, 2]) array([0., 1., 2.]) NFc|tjk7rt|sd}d}t||z||_||_||_|tjk7rt|dr|j|_i|_ d|_ ||_ |t|dr|j|_ n||_ t|tr!|D]} | tdvstd| nDt!|r,|D cgc] } t#t%j&| "}} n | td||_| t+}t+||_|t/||_yd|_ycc} w)NzWhen used as a decorator, zonly accepts keyword arguments.__name____doc__AllzInvalid otype specified: zInvalid otype specification)rrcallablerpyfunccacherrr_ufunc_docrrstrr(rr:rrrotypessetexcludedr_in_and_out_core_dims) selfrrdocrrrpart1part2rrs r`__init__zvectorize.__init__ sQ bkk !HV,<1E5EEEM* *  " R[[ WVZ%@"OODM   ;7695!>>DLDI fc " Ny//$d%LMM Nf BHIQ*399Q<8IFI  :; ;   uHH   )@)KD &)-D &Js2%E%c|j|_||_|j|j|_y|j|_yr[)rrrr)rrrkwargss r` _init_stage_2zvectorize._init_stage_22 s4  99 !>>DL99DLrbc j}s|sj}|}nt|}Dcgc] }||vs| c} t|Dcgc] }||vs| c} t |  fd} Dcgc]}|| }}|j Dcgc]}| c}j ||Scc}wcc}wcc}wcc}w)z Return arrays with the results of `pyfunc` broadcast (vectorized) over `args` and `kwargs` not in `excluded`. c tD] \}}|||<jt|tdjiSr[)rupdaterrr)vargs_n_iindsrnamesrthe_argss r`rz'vectorize._call_as_normal..funcM s['o-FB#(9HRL- c%s4yz):;<"t{{H777rbrr)rrrr&rextend_vectorize_call) rrrrrrnargsrrrrrs ` ` @@@r`_call_as_normalzvectorize._call_as_normal: s ==h;;DE IE"(?BBh,>R?E!&uD281CBDDDzH 8 8 )--"T"X-E- LLu5&*5 6##E#::@D.5s" B>B> CC; C C c|jtjur|j|i||S|j|i|Sr[)rrrrr)rrrs r`__call__zvectorize.__call__X sD ;;"++ % D   / /K#t##T4V44rbc X |s td|j|j}t|}t|j}|jus||jvrt ||}nd}|jur|jj ||}||fS|Dcgc] }t|}}tjd|Dr td|Dcgc]}|jd}}|} |jr | g fd} n} t| tr t| }nd}| f} djt|D cgc]$} t| | j j"&c} }t | t||}||fScc}wcc}wcc} w) zReturn (ufunc, otypes).zargs can not be emptyNc3:K|]}|jdk(ywrN)rrrs r`rz2vectorize._get_ufunc_and_otypes..~ s:cCHHM:s?cannot call `vectorize` on size 0 inputs unless `otypes` is setrc2rjS|Sr[)pop)r_cachers r`_funcz.vectorize._get_ufunc_and_otypes.._func s%zz|+#U|+rbrgr)rrrrrr setdefaultr builtinsr&flatrrrjoinr&rr) rrrrninnoutufuncrinputsoutputsr_krs ` @r`_get_ufunc_and_otypeszvectorize._get_ufunc_and_otypes_ s45 5 ;; "[[Fd)Ct{{#D4;;&#T[[*@"4d3t{{" ..sE:Xf}K-11SGCL1D1||:T:: ":;;.22cchhqk2F2FmG zz!, '5)7|"*WW(-d 5"$&gbk288==56F uc$i6Ef}K2 305s&F%F")F'c |j|j||}|S|s |}|S|j||\}}|Dcgc]}t|t}}||}|j dk(rt||d}|St t||D cgc]\} } t| | c} } }|Scc}wcc} } w)z1Vectorized call to `func` over positional `args`.rrrgr)r_vectorize_call_with_signaturerr objectr rr) rrrresr rrr r rts r`rzvectorize._vectorize_call s >> %55dDAC  &C !66Dt6LME6<@@aj&1@F@VnGzzQ q : ),Wf)=?!%A(3?@ A?s B;C c |j\}}t|t|k7r#tdt|dt|td|D}t ||\}t ||}t ||Dcgc]\}}tj||d}}}d} |j} t|} tj|D]|fd|D} t| tr t| nd} | | k7rtd | d| | dk(r| f} | 0t | |D]\}}t||t||| | } t | | D] \}}||< | D| td tj fd |Dr td t||| } | dk(r| d S| Scc}}w)z;Vectorized call over positional arguments with a signature.z/wrong number of positional arguments: expected z, got c32K|]}t|ywr[)r rs r`rz;vectorize._vectorize_call_with_signature.. s5Z_5sTrNc3(K|] }| ywr[rh)rrrs r`rz;vectorize._vectorize_call_with_signature.. s8CSZ8rrgz.wrong number of outputs from pyfunc: expected rc34K|]}|D]}|v ywr[rh)rdimsrrs r`rz;vectorize._vectorize_call_with_signature.. s1- $'+- # y0-0-rzYcannot call `vectorize` with a signature including new output dimensions on size 0 inputsr)rrrrrrrrrrndindexrrrrrr&)rrrroutput_core_dimsr input_shapesrrmr rr r n_resultsrroutputrrs @@r`rz(vectorize._vectorize_call_with_signature s%,0,F,F)) t9O, ,"?3SY@A A555%< /&#"())8: #&dL"9;CU$7;;#$ZZ1 'E8489G(27E(BG Iy  Y())qy"*),W6F)GD%FI%iCD)))967L#&gw"7 ' &u  ') '. ?~ ":;;||-(8--!"*++%_i%5v?G"QYwqz3G3Y;s "G) r __module__ __qualname__rrrrrrrrrrrhrbr`rErE]sCfN!kk$Det*.X%;<5CJ*:4rbrErc||||fSr[rh)rrrowvarbiasddoffweightsaweightsrs r`_cov_dispatcherr( s q(H %%rbc@||t|k7r tdtj|}|jdkDr td|/tj|}|jdkDr td|L|%tj |tj }n%tj ||tj }t|d|}|s|jddk7r |j}|jddk(r%tjgjddS|Ht|dd| }|s|jddk7r |j}tj||fd }| |dk(rd}nd}d} |tj|t }tj|tj|k(s td |jdkDr t!d |jd|jdk7r t!dt#|dkr td|} |tj|t }|jdkDr t!d|jd|jdk7r t!dt#|dkr td| |} n| |z} t%|d| d\} } | d} | |jd|z } n'|dk(r| } n|| |z } n| |t'| |zz| z z } | dkrt)j*dt,dd} || dddfz}| |j} n|| zj} t/|| j1}|tj2d| z}|j5S)a' Estimate a covariance matrix, given data and weights. Covariance indicates the level to which two variables vary together. If we examine N-dimensional samples, :math:`X = [x_1, x_2, ... x_N]^T`, then the covariance matrix element :math:`C_{ij}` is the covariance of :math:`x_i` and :math:`x_j`. The element :math:`C_{ii}` is the variance of :math:`x_i`. See the notes for an outline of the algorithm. Parameters ---------- m : array_like A 1-D or 2-D array containing multiple variables and observations. Each row of `m` represents a variable, and each column a single observation of all those variables. Also see `rowvar` below. y : array_like, optional An additional set of variables and observations. `y` has the same form as that of `m`. rowvar : bool, optional If `rowvar` is True (default), then each row represents a variable, with observations in the columns. Otherwise, the relationship is transposed: each column represents a variable, while the rows contain observations. bias : bool, optional Default normalization (False) is by ``(N - 1)``, where ``N`` is the number of observations given (unbiased estimate). If `bias` is True, then normalization is by ``N``. These values can be overridden by using the keyword ``ddof`` in numpy versions >= 1.5. ddof : int, optional If not ``None`` the default value implied by `bias` is overridden. Note that ``ddof=1`` will return the unbiased estimate, even if both `fweights` and `aweights` are specified, and ``ddof=0`` will return the simple average. See the notes for the details. The default value is ``None``. .. versionadded:: 1.5 fweights : array_like, int, optional 1-D array of integer frequency weights; the number of times each observation vector should be repeated. .. versionadded:: 1.10 aweights : array_like, optional 1-D array of observation vector weights. These relative weights are typically large for observations considered "important" and smaller for observations considered less "important". If ``ddof=0`` the array of weights can be used to assign probabilities to observation vectors. .. versionadded:: 1.10 dtype : data-type, optional Data-type of the result. By default, the return data-type will have at least `numpy.float64` precision. .. versionadded:: 1.20 Returns ------- out : ndarray The covariance matrix of the variables. See Also -------- corrcoef : Normalized covariance matrix Notes ----- Assume that the observations are in the columns of the observation array `m` and let ``f = fweights`` and ``a = aweights`` for brevity. The steps to compute the weighted covariance are as follows:: >>> m = np.arange(10, dtype=np.float64) >>> f = np.arange(10) * 2 >>> a = np.arange(10) ** 2. >>> ddof = 1 >>> w = f * a >>> v1 = np.sum(w) >>> v2 = np.sum(w * a) >>> m -= np.sum(m * w, axis=None, keepdims=True) / v1 >>> cov = np.dot(m * w, m.T) * v1 / (v1**2 - ddof * v2) Note that when ``a == 1``, the normalization factor ``v1 / (v1**2 - ddof * v2)`` goes over to ``1 / (np.sum(f) - ddof)`` as it should. Examples -------- Consider two variables, :math:`x_0` and :math:`x_1`, which correlate perfectly, but in opposite directions: >>> x = np.array([[0, 2], [1, 1], [2, 0]]).T >>> x array([[0, 1, 2], [2, 1, 0]]) Note how :math:`x_0` increases while :math:`x_1` decreases. The covariance matrix shows this clearly: >>> np.cov(x) array([[ 1., -1.], [-1., 1.]]) Note that element :math:`C_{0,1}`, which shows the correlation between :math:`x_0` and :math:`x_1`, is negative. Further, note how `x` and `y` are combined: >>> x = [-2.1, -1, 4.3] >>> y = [3, 1.1, 0.12] >>> X = np.stack((x, y), axis=0) >>> np.cov(X) array([[11.71 , -4.286 ], # may vary [-4.286 , 2.144133]]) >>> np.cov(x, y) array([[11.71 , -4.286 ], # may vary [-4.286 , 2.144133]]) >>> np.cov(x) array(11.71) Nzddof must be integerrzm has more than 2 dimensionszy has more than 2 dimensions)ndminrrrgr9r*rrrzfweights must be integerz'cannot handle multidimensional fweightsz,incompatible numbers of samples and fweightszfweights cannot be negativez'cannot handle multidimensional aweightsz,incompatible numbers of samples and aweightszaweights cannot be negativeT)rrrz!Degrees of freedom <= 0 for slicerr)r rrr rrr(r rmTrrr rrr RuntimeErrorr&rGr'rrRuntimeWarningrconj true_dividesqueeze)rrr#r$r%r&r'rXwrw_sumfactX_TrBs r`rIrI s|x DCI- "$ $ 1 Avvz788} JJqM 66A:;< < } 9NN1bjj1ENN1a4E aq&A aggajAo CCwwqzQxx|##Aq))} !$au 5!''!*/A NNAq6 * | 19DD A::he4vvh"))H"556*, , ==1 9; ; >>!  *>@ @ x!| -/ / ::he4 ==1 9; ; >>!  *>@ @ x!| -/ / 9A MAA=JC !HE ywwqzD    t|tC( O+E11 qy 9$ 4QWAyccsgg AsxxzA4  A 99;rbc ||fSr[rh)rrr#r$r%rs r`_corrcoef_dispatcherr8 s q6Mrbc|tjus|tjurtjdtdt ||||} t |}t|j}||dddfz}||dddfz}tj|jdd|jtj|r-tj|jdd|j|S#t$r||z cYSwxYw) ai Return Pearson product-moment correlation coefficients. Please refer to the documentation for `cov` for more detail. The relationship between the correlation coefficient matrix, `R`, and the covariance matrix, `C`, is .. math:: R_{ij} = \frac{ C_{ij} } { \sqrt{ C_{ii} C_{jj} } } The values of `R` are between -1 and 1, inclusive. Parameters ---------- x : array_like A 1-D or 2-D array containing multiple variables and observations. Each row of `x` represents a variable, and each column a single observation of all those variables. Also see `rowvar` below. y : array_like, optional An additional set of variables and observations. `y` has the same shape as `x`. rowvar : bool, optional If `rowvar` is True (default), then each row represents a variable, with observations in the columns. Otherwise, the relationship is transposed: each column represents a variable, while the rows contain observations. bias : _NoValue, optional Has no effect, do not use. .. deprecated:: 1.10.0 ddof : _NoValue, optional Has no effect, do not use. .. deprecated:: 1.10.0 dtype : data-type, optional Data-type of the result. By default, the return data-type will have at least `numpy.float64` precision. .. versionadded:: 1.20 Returns ------- R : ndarray The correlation coefficient matrix of the variables. See Also -------- cov : Covariance matrix Notes ----- Due to floating point rounding the resulting array may not be Hermitian, the diagonal elements may not be 1, and the elements may not satisfy the inequality abs(a) <= 1. The real and imaginary parts are clipped to the interval [-1, 1] in an attempt to improve on that situation but is not much help in the complex case. This function accepts but discards arguments `bias` and `ddof`. This is for backwards compatibility with previous versions of this function. These arguments had no effect on the return values of the function and can be safely ignored in this and previous versions of numpy. Examples -------- In this example we generate two random arrays, ``xarr`` and ``yarr``, and compute the row-wise and column-wise Pearson correlation coefficients, ``R``. Since ``rowvar`` is true by default, we first find the row-wise Pearson correlation coefficients between the variables of ``xarr``. >>> import numpy as np >>> rng = np.random.default_rng(seed=42) >>> xarr = rng.random((3, 3)) >>> xarr array([[0.77395605, 0.43887844, 0.85859792], [0.69736803, 0.09417735, 0.97562235], [0.7611397 , 0.78606431, 0.12811363]]) >>> R1 = np.corrcoef(xarr) >>> R1 array([[ 1. , 0.99256089, -0.68080986], [ 0.99256089, 1. , -0.76492172], [-0.68080986, -0.76492172, 1. ]]) If we add another set of variables and observations ``yarr``, we can compute the row-wise Pearson correlation coefficients between the variables in ``xarr`` and ``yarr``. >>> yarr = rng.random((3, 3)) >>> yarr array([[0.45038594, 0.37079802, 0.92676499], [0.64386512, 0.82276161, 0.4434142 ], [0.22723872, 0.55458479, 0.06381726]]) >>> R2 = np.corrcoef(xarr, yarr) >>> R2 array([[ 1. , 0.99256089, -0.68080986, 0.75008178, -0.934284 , -0.99004057], [ 0.99256089, 1. , -0.76492172, 0.82502011, -0.97074098, -0.99981569], [-0.68080986, -0.76492172, 1. , -0.99507202, 0.89721355, 0.77714685], [ 0.75008178, 0.82502011, -0.99507202, 1. , -0.93657855, -0.83571711], [-0.934284 , -0.97074098, 0.89721355, -0.93657855, 1. , 0.97517215], [-0.99004057, -0.99981569, 0.77714685, -0.83571711, 0.97517215, 1. ]]) Finally if we use the option ``rowvar=False``, the columns are now being treated as the variables and we will find the column-wise Pearson correlation coefficients between variables in ``xarr`` and ``yarr``. >>> R3 = np.corrcoef(xarr, yarr, rowvar=False) >>> R3 array([[ 1. , 0.77598074, -0.47458546, -0.75078643, -0.9665554 , 0.22423734], [ 0.77598074, 1. , -0.92346708, -0.99923895, -0.58826587, -0.44069024], [-0.47458546, -0.92346708, 1. , 0.93773029, 0.23297648, 0.75137473], [-0.75078643, -0.99923895, 0.93773029, 1. , 0.55627469, 0.47536961], [-0.9665554 , -0.58826587, 0.23297648, 0.55627469, 1. , -0.46666491], [ 0.22423734, -0.44069024, 0.75137473, 0.47536961, -0.46666491, 1. ]]) z/bias and ddof have no effect and are deprecatedrrrNrrgr=) rrrrrrIr)rrrecliprUrd) rrr#r$r%rrBdstddevs r`rJrJ s@ 2;;$bkk"9 G(Q 8 Aq&&A G !&&\F4AaA GGAFFBqvv& q A166* H 1u s C::D  D cDtjd|g}|d}|dkrtg|jS|dk(rtd|jSt d|z |d}ddt t |z|dz z zzdt dt z|z|dz z zzS) a Return the Blackman window. The Blackman window is a taper formed by using the first three terms of a summation of cosines. It was designed to have close to the minimal leakage possible. It is close to optimal, only slightly worse than a Kaiser window. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. Returns ------- out : ndarray The window, with the maximum value normalized to one (the value one appears only if the number of samples is odd). See Also -------- bartlett, hamming, hanning, kaiser Notes ----- The Blackman window is defined as .. math:: w(n) = 0.42 - 0.5 \cos(2\pi n/M) + 0.08 \cos(4\pi n/M) Most references to the Blackman window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. It is also known as an apodization (which means "removing the foot", i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function. It is known as a "near optimal" tapering function, almost as good (by some measures) as the kaiser window. References ---------- Blackman, R.B. and Tukey, J.W., (1958) The measurement of power spectra, Dover Publications, New York. Oppenheim, A.V., and R.W. Schafer. Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice-Hall, 1999, pp. 468-471. Examples -------- >>> import matplotlib.pyplot as plt >>> np.blackman(12) array([-1.38777878e-17, 3.26064346e-02, 1.59903635e-01, # may vary 4.14397981e-01, 7.36045180e-01, 9.67046769e-01, 9.67046769e-01, 7.36045180e-01, 4.14397981e-01, 1.59903635e-01, 3.26064346e-02, -1.38777878e-17]) Plot the window and the frequency response. .. plot:: :include-source: import matplotlib.pyplot as plt from numpy.fft import fft, fftshift window = np.blackman(51) plt.plot(window) plt.title("Blackman window") plt.ylabel("Amplitude") plt.xlabel("Sample") plt.show() # doctest: +SKIP plt.figure() A = fft(window, 2048) / 25.5 mag = np.abs(fftshift(A)) freq = np.linspace(-0.5, 0.5, len(A)) with np.errstate(divide='ignore', invalid='ignore'): response = 20 * np.log10(mag) response = np.clip(response, -100, 100) plt.plot(freq, response) plt.title("Frequency response of Blackman window") plt.ylabel("Magnitude [dB]") plt.xlabel("Normalized frequency [cycles per sample]") plt.axis('tight') plt.show() rrgrrgzG?rkg{Gz?r$rr rrrrrMvaluesr^s r`rPrPw spXXsAh Fq A1uRv||,,AvAV\\**qsAqA #c"Q$!*o% %SR1Q3-@(@ @@rbctjd|g}|d}|dkrtg|jS|dk(rtd|jSt d|z |d}t t |dd||dz z zd||dz z z S)a Return the Bartlett window. The Bartlett window is very similar to a triangular window, except that the end points are at zero. It is often used in signal processing for tapering a signal, without generating too much ripple in the frequency domain. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. Returns ------- out : array The triangular window, with the maximum value normalized to one (the value one appears only if the number of samples is odd), with the first and last samples equal to zero. See Also -------- blackman, hamming, hanning, kaiser Notes ----- The Bartlett window is defined as .. math:: w(n) = \frac{2}{M-1} \left( \frac{M-1}{2} - \left|n - \frac{M-1}{2}\right| \right) Most references to the Bartlett window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. Note that convolution with this window produces linear interpolation. It is also known as an apodization (which means "removing the foot", i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function. The Fourier transform of the Bartlett window is the product of two sinc functions. Note the excellent discussion in Kanasewich [2]_. References ---------- .. [1] M.S. Bartlett, "Periodogram Analysis and Continuous Spectra", Biometrika 37, 1-16, 1950. .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The University of Alberta Press, 1975, pp. 109-110. .. [3] A.V. Oppenheim and R.W. Schafer, "Discrete-Time Signal Processing", Prentice-Hall, 1999, pp. 468-471. .. [4] Wikipedia, "Window function", https://en.wikipedia.org/wiki/Window_function .. [5] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, "Numerical Recipes", Cambridge University Press, 1986, page 429. Examples -------- >>> import matplotlib.pyplot as plt >>> np.bartlett(12) array([ 0. , 0.18181818, 0.36363636, 0.54545455, 0.72727273, # may vary 0.90909091, 0.90909091, 0.72727273, 0.54545455, 0.36363636, 0.18181818, 0. ]) Plot the window and its frequency response (requires SciPy and matplotlib). .. plot:: :include-source: import matplotlib.pyplot as plt from numpy.fft import fft, fftshift window = np.bartlett(51) plt.plot(window) plt.title("Bartlett window") plt.ylabel("Amplitude") plt.xlabel("Sample") plt.show() plt.figure() A = fft(window, 2048) / 25.5 mag = np.abs(fftshift(A)) freq = np.linspace(-0.5, 0.5, len(A)) with np.errstate(divide='ignore', invalid='ignore'): response = 20 * np.log10(mag) response = np.clip(response, -100, 100) plt.plot(freq, response) plt.title("Frequency response of Bartlett window") plt.ylabel("Magnitude [dB]") plt.xlabel("Normalized frequency [cycles per sample]") plt.axis('tight') plt.show() rrgrrr)rr rrrrrr@s r`rOrO s~XXsAh Fq A1uRv||,,AvAV\\**qsAqA Aq!1q!A#w;AqsG <>> np.hanning(12) array([0. , 0.07937323, 0.29229249, 0.57115742, 0.82743037, 0.97974649, 0.97974649, 0.82743037, 0.57115742, 0.29229249, 0.07937323, 0. ]) Plot the window and its frequency response. .. plot:: :include-source: import matplotlib.pyplot as plt from numpy.fft import fft, fftshift window = np.hanning(51) plt.plot(window) plt.title("Hann window") plt.ylabel("Amplitude") plt.xlabel("Sample") plt.show() plt.figure() A = fft(window, 2048) / 25.5 mag = np.abs(fftshift(A)) freq = np.linspace(-0.5, 0.5, len(A)) with np.errstate(divide='ignore', invalid='ignore'): response = 20 * np.log10(mag) response = np.clip(response, -100, 100) plt.plot(freq, response) plt.title("Frequency response of the Hann window") plt.ylabel("Magnitude [dB]") plt.xlabel("Normalized frequency [cycles per sample]") plt.axis('tight') plt.show() rrgrrrkr?r@s r`rNrND stXXsAh Fq A1uRv||,,AvAV\\**qsAqA SAqs_$ $$rbctjd|g}|d}|dkrtg|jS|dk(rtd|jSt d|z |d}ddt t |z|dz z zzS)a Return the Hamming window. The Hamming window is a taper formed by using a weighted cosine. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. Returns ------- out : ndarray The window, with the maximum value normalized to one (the value one appears only if the number of samples is odd). See Also -------- bartlett, blackman, hanning, kaiser Notes ----- The Hamming window is defined as .. math:: w(n) = 0.54 - 0.46\cos\left(\frac{2\pi{n}}{M-1}\right) \qquad 0 \leq n \leq M-1 The Hamming was named for R. W. Hamming, an associate of J. W. Tukey and is described in Blackman and Tukey. It was recommended for smoothing the truncated autocovariance function in the time domain. Most references to the Hamming window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. It is also known as an apodization (which means "removing the foot", i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function. References ---------- .. [1] Blackman, R.B. and Tukey, J.W., (1958) The measurement of power spectra, Dover Publications, New York. .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The University of Alberta Press, 1975, pp. 109-110. .. [3] Wikipedia, "Window function", https://en.wikipedia.org/wiki/Window_function .. [4] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, "Numerical Recipes", Cambridge University Press, 1986, page 425. Examples -------- >>> np.hamming(12) array([ 0.08 , 0.15302337, 0.34890909, 0.60546483, 0.84123594, # may vary 0.98136677, 0.98136677, 0.84123594, 0.60546483, 0.34890909, 0.15302337, 0.08 ]) Plot the window and the frequency response. .. plot:: :include-source: import matplotlib.pyplot as plt from numpy.fft import fft, fftshift window = np.hamming(51) plt.plot(window) plt.title("Hamming window") plt.ylabel("Amplitude") plt.xlabel("Sample") plt.show() plt.figure() A = fft(window, 2048) / 25.5 mag = np.abs(fftshift(A)) freq = np.linspace(-0.5, 0.5, len(A)) response = 20 * np.log10(mag) response = np.clip(response, -100, 100) plt.plot(freq, response) plt.title("Frequency response of Hamming window") plt.ylabel("Magnitude [dB]") plt.xlabel("Normalized frequency [cycles per sample]") plt.axis('tight') plt.show() rrgrrgHzG?gq= ףp?r?r@s r`rMrM snXXsAh Fq A1uRv||,,AvAV\\**qsAqA $s2a41:& &&rb)g4!\Tg}b3g0 Kg5dMv;p>g"c쑾g$>g'doҾgY(X?>gZY&+g|t(?gRBguZ?gI ^qga?g!Ng-Ί>?g-4pKgw?gWӿg*5N?)gT`g0fFVg!g["d,->gmրVX>gna>g+A>gRx?gI墌k?g b?cx|d}d}tdt|D]}|}|}||z|z ||z}d|z zS)Nrrrgrk)r&r)rrb0b1rb2s r`_chbevlrJK sZ aB B 1c$i !   rTBYa ! R=rbcFt|t|dz dz tzS)Nr$r)rrJ_i0Ars r`_i0_1rNW s q6GAcE!GT* **rbc^t|td|z dz tzt|z S)Ng@@r$)rrJ_i0BrrMs r`_i0_2rQ[ s) q6GDFSL$/ /$q' 99rbc|fSr[rhrMs r`_i0_dispatcherrS_ rrbc0tj|}|jjdk(r t d|jjdk7r|j t }tj|}t||dkgttgS)a Modified Bessel function of the first kind, order 0. Usually denoted :math:`I_0`. Parameters ---------- x : array_like of float Argument of the Bessel function. Returns ------- out : ndarray, shape = x.shape, dtype = float The modified Bessel function evaluated at each of the elements of `x`. See Also -------- scipy.special.i0, scipy.special.iv, scipy.special.ive Notes ----- The scipy implementation is recommended over this function: it is a proper ufunc written in C, and more than an order of magnitude faster. We use the algorithm published by Clenshaw [1]_ and referenced by Abramowitz and Stegun [2]_, for which the function domain is partitioned into the two intervals [0,8] and (8,inf), and Chebyshev polynomial expansions are employed in each interval. Relative error on the domain [0,30] using IEEE arithmetic is documented [3]_ as having a peak of 5.8e-16 with an rms of 1.4e-16 (n = 30000). References ---------- .. [1] C. W. Clenshaw, "Chebyshev series for mathematical functions", in *National Physical Laboratory Mathematical Tables*, vol. 5, London: Her Majesty's Stationery Office, 1962. .. [2] M. Abramowitz and I. A. Stegun, *Handbook of Mathematical Functions*, 10th printing, New York: Dover, 1964, pp. 379. https://personal.math.ubc.ca/~cbm/aands/page_379.htm .. [3] https://metacpan.org/pod/distribution/Math-Cephes/lib/Math/Cephes.pod#i0:-Modified-Bessel-function-of-order-zero Examples -------- >>> np.i0(0.) array(1.0) >>> np.i0([0, 1, 2, 3]) array([1. , 1.26606588, 2.2795853 , 4.88079259]) rBz#i0 not supported for complex valuesrg @) rr rkindrrr rYr7rNrQrMs r`rTrTc srf aAww||s=>>ww||s HHUO q A Qc UEN 33rbctjd||g}|d}|d}|dk(r!tjd|jSt d|}|dz dz }t |t d||z |z dzz zt |z S)a Return the Kaiser window. The Kaiser window is a taper formed by using a Bessel function. Parameters ---------- M : int Number of points in the output window. If zero or less, an empty array is returned. beta : float Shape parameter for window. Returns ------- out : array The window, with the maximum value normalized to one (the value one appears only if the number of samples is odd). See Also -------- bartlett, blackman, hamming, hanning Notes ----- The Kaiser window is defined as .. math:: w(n) = I_0\left( \beta \sqrt{1-\frac{4n^2}{(M-1)^2}} \right)/I_0(\beta) with .. math:: \quad -\frac{M-1}{2} \leq n \leq \frac{M-1}{2}, where :math:`I_0` is the modified zeroth-order Bessel function. The Kaiser was named for Jim Kaiser, who discovered a simple approximation to the DPSS window based on Bessel functions. The Kaiser window is a very good approximation to the Digital Prolate Spheroidal Sequence, or Slepian window, which is the transform which maximizes the energy in the main lobe of the window relative to total energy. The Kaiser can approximate many other windows by varying the beta parameter. ==== ======================= beta Window shape ==== ======================= 0 Rectangular 5 Similar to a Hamming 6 Similar to a Hanning 8.6 Similar to a Blackman ==== ======================= A beta value of 14 is probably a good starting point. Note that as beta gets large, the window narrows, and so the number of samples needs to be large enough to sample the increasingly narrow spike, otherwise NaNs will get returned. Most references to the Kaiser window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. It is also known as an apodization (which means "removing the foot", i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function. References ---------- .. [1] J. F. Kaiser, "Digital Filters" - Ch 7 in "Systems analysis by digital computer", Editors: F.F. Kuo and J.F. Kaiser, p 218-285. John Wiley and Sons, New York, (1966). .. [2] E.R. Kanasewich, "Time Sequence Analysis in Geophysics", The University of Alberta Press, 1975, pp. 177-178. .. [3] Wikipedia, "Window function", https://en.wikipedia.org/wiki/Window_function Examples -------- >>> import matplotlib.pyplot as plt >>> np.kaiser(12, 14) array([7.72686684e-06, 3.46009194e-03, 4.65200189e-02, # may vary 2.29737120e-01, 5.99885316e-01, 9.45674898e-01, 9.45674898e-01, 5.99885316e-01, 2.29737120e-01, 4.65200189e-02, 3.46009194e-03, 7.72686684e-06]) Plot the window and the frequency response. .. plot:: :include-source: import matplotlib.pyplot as plt from numpy.fft import fft, fftshift window = np.kaiser(51, 14) plt.plot(window) plt.title("Kaiser window") plt.ylabel("Amplitude") plt.xlabel("Sample") plt.show() plt.figure() A = fft(window, 2048) / 25.5 mag = np.abs(fftshift(A)) freq = np.linspace(-0.5, 0.5, len(A)) response = 20 * np.log10(mag) response = np.clip(response, -100, 100) plt.plot(freq, response) plt.title("Frequency response of Kaiser window") plt.ylabel("Magnitude [dB]") plt.xlabel("Normalized frequency [cycles per sample]") plt.axis('tight') plt.show() rrgrrrr$)rr rrrrTr)rAbetarBr^alphas r`rQrQ snXXsAtn %Fq A !9DAvwwq --q! A qS#IE dT!agu_s2233 4RX ==rbc|fSr[rhrMs r`_sinc_dispatcherrZ#rrbcvtj|}tt|dk(d|z}t ||z S)a Return the normalized sinc function. The sinc function is equal to :math:`\sin(\pi x)/(\pi x)` for any argument :math:`x\ne 0`. ``sinc(0)`` takes the limit value 1, making ``sinc`` not only everywhere continuous but also infinitely differentiable. .. note:: Note the normalization factor of ``pi`` used in the definition. This is the most commonly used definition in signal processing. Use ``sinc(x / np.pi)`` to obtain the unnormalized sinc function :math:`\sin(x)/x` that is more common in mathematics. Parameters ---------- x : ndarray Array (possibly multi-dimensional) of values for which to calculate ``sinc(x)``. Returns ------- out : ndarray ``sinc(x)``, which has the same shape as the input. Notes ----- The name sinc is short for "sine cardinal" or "sinus cardinalis". The sinc function is used in various signal processing applications, including in anti-aliasing, in the construction of a Lanczos resampling filter, and in interpolation. For bandlimited interpolation of discrete-time signals, the ideal interpolation kernel is proportional to the sinc function. References ---------- .. [1] Weisstein, Eric W. "Sinc Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SincFunction.html .. [2] Wikipedia, "Sinc function", https://en.wikipedia.org/wiki/Sinc_function Examples -------- >>> import matplotlib.pyplot as plt >>> x = np.linspace(-4, 4, 41) >>> np.sinc(x) array([-3.89804309e-17, -4.92362781e-02, -8.40918587e-02, # may vary -8.90384387e-02, -5.84680802e-02, 3.89804309e-17, 6.68206631e-02, 1.16434881e-01, 1.26137788e-01, 8.50444803e-02, -3.89804309e-17, -1.03943254e-01, -1.89206682e-01, -2.16236208e-01, -1.55914881e-01, 3.89804309e-17, 2.33872321e-01, 5.04551152e-01, 7.56826729e-01, 9.35489284e-01, 1.00000000e+00, 9.35489284e-01, 7.56826729e-01, 5.04551152e-01, 2.33872321e-01, 3.89804309e-17, -1.55914881e-01, -2.16236208e-01, -1.89206682e-01, -1.03943254e-01, -3.89804309e-17, 8.50444803e-02, 1.26137788e-01, 1.16434881e-01, 6.68206631e-02, 3.89804309e-17, -5.84680802e-02, -8.90384387e-02, -8.40918587e-02, -4.92362781e-02, -3.89804309e-17]) >>> plt.plot(x, np.sinc(x)) [] >>> plt.title("Sinc Function") Text(0.5, 1.0, 'Sinc Function') >>> plt.ylabel("Amplitude") Text(0, 0.5, 'Amplitude') >>> plt.xlabel("X") Text(0.5, 0, 'X') >>> plt.show() rg#B ;)rr rrr)rrs r`rLrL's7X aA U167A &&A q6!8Orbc L tj|}|jdd |jdd}|tjurd}|j} t j | |r/|-t fdt|D}|tf|z|d<t dk(r d|d<ntt|t z }t|}tt|D]\} } |j| | }|j|j d|dz}d |d<n|r|d |z}|tf|z|d<||fi|} ||S|rA tj"f|z} nt fd t|D} | tf| z} | S) a- Internal Function. Call `func` with `a` as first argument swapping the axes to use extended axis on functions that don't support it natively. Returns result and a.shape with axis dims set to 1. Parameters ---------- a : array_like Input array or object that can be converted to an array. func : callable Reduction function capable of receiving a single axis argument. It is called with `a` as first argument followed by `kwargs`. kwargs : keyword arguments additional keyword arguments to pass to `func`. Returns ------- result : tuple Result of func(a, **kwargs) and a.shape with axis dims set to 1 which can be used to reshape the result to the same shape a ufunc with keepdims=True would produce. rNr=Fc3@K|]}|vrdn tdywr)r)rrrs r`rz_ureduce..s'"H89dAd 3"Hsrgrrrrc3\K|]#}|vrtjn td%ywr[)rnewaxisr)r^s r`rz_ureduce..s,$ 4i U4[8$s),)rr getrrrrrr&Ellipsisrrrsortedswapaxesrrmrb)rrrrr=rJ index_outkeepnkeeprrrindex_rrs @r`_ureducerlxs4 aA ::fd #D **UD !C2;; B ''b1 !"H=B2Y"HH #XL9$< =u t9>!!WF6NuRy>CI-DIE!&,/ %1JJq!$ % !''&5/E12AF6N !BJ #XL9$< =u  Q&A   <zznr)G$r$$G xlW$ % Hrbc ||fSr[rhrrr=overwrite_inputrs r`_median_dispatcherrps s8Orbc,t|t||||S)a 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 : {int, sequence of int, None}, optional Axis or axes along which the medians are computed. The default, axis=None, will compute the median along a flattened version of the array. .. versionadded:: 1.9.0 If a sequence of axes, the array is first flattened along the given axes, then the median is computed along the resulting flattened axis. out : 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 (of the output) will be cast if necessary. overwrite_input : bool, 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. If `overwrite_input` is ``True`` and `a` is not already an `ndarray`, an error will be raised. keepdims : bool, optional If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original `arr`. .. versionadded:: 1.9.0 Returns ------- median : ndarray A new array holding the result. If the input contains integers or floats smaller than ``float64``, then the output data-type is ``np.float64``. Otherwise, the data-type of the output is the same as that of the input. If `out` is specified, that array is returned instead. See Also -------- mean, percentile Notes ----- Given a vector ``V`` of length ``N``, the median of ``V`` is the middle value of a sorted copy of ``V``, ``V_sorted`` - i e., ``V_sorted[(N-1)/2]``, when ``N`` is odd, and the average of the two middle values of ``V_sorted`` when ``N`` is even. Examples -------- >>> a = np.array([[10, 7, 4], [3, 2, 1]]) >>> a array([[10, 7, 4], [ 3, 2, 1]]) >>> np.median(a) np.float64(3.5) >>> np.median(a, axis=0) array([6.5, 4.5, 2.5]) >>> np.median(a, axis=1) array([7., 2.]) >>> np.median(a, axis=(0, 1)) np.float64(3.5) >>> m = np.median(a, axis=0) >>> out = np.zeros_like(m) >>> np.median(a, axis=0, out=m) array([6.5, 4.5, 2.5]) >>> m array([6.5, 4.5, 2.5]) >>> b = a.copy() >>> np.median(b, axis=1, overwrite_input=True) array([7., 2.]) >>> assert not np.all(a==b) >>> b = a.copy() >>> np.median(b, axis=None, overwrite_input=True) np.float64(3.5) >>> assert not np.all(a==b) )rrrr=ro)rl_medianrns r`rKrKst AGhTs$3 55rbctj|}| |j}n|j|}|dzdk(r |dz}|dz |g}n |dz dzg}tj|j tj xs|j jdv}|r|jd|r:|"|j}|j|n$|j|||}nt|||}|jdk(r|jS|d}tdg|jz} |j|dz} |j|dzdk(rt| | dz| |<nt| dz | dz| |<t| } t|| ||} |r0|dkDr+tj j"j%|| |} | S) NrrrgMmrrrhrr=)rr rrmr'rr/rUrXr"r$itemr)rrr%r _utils_impl_median_nancheck) rrr=roszszhkth supports_nanspartrrrouts r`rrrr(s aA | VV WWT] Av{AgQwnQ1}oMM!''2::6N!'',,$:NM 2 <779D NN3  KK$K 'DCd+ zzRyy{ |T{mdii'G JJt  !E zz$!q eU1W- eAguQw/ GnG W Dc 2Davv!!224tD Krb)r interpolationc||||fSr[rh rqrr=romethodrrrs r`_percentile_dispatcherr` q#w rbc f| t||d}tj|}|jjdk(r t dtj ||jjdk(r|jjdnd}t|}t|s td|k|dk7rd|d } t| |"tj||jd }t||| }tj|d kr tdt||||||||S)a|* Compute the q-th percentile of the data along the specified axis. Returns the q-th percentile(s) of the array elements. Parameters ---------- a : array_like of real numbers Input array or object that can be converted to an array. q : array_like of float Percentage or sequence of percentages for the percentiles to compute. Values must be between 0 and 100 inclusive. axis : {int, tuple of int, None}, optional Axis or axes along which the percentiles are computed. The default is to compute the percentile(s) along a flattened version of the array. .. versionchanged:: 1.9.0 A tuple of axes is supported out : 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 (of the output) will be cast if necessary. overwrite_input : bool, optional If True, then allow the input array `a` to be modified by intermediate calculations, to save memory. In this case, the contents of the input `a` after this function completes is undefined. method : str, optional This parameter specifies the method to use for estimating the percentile. There are many different methods, some unique to NumPy. See the notes for explanation. The options sorted by their R type as summarized in the H&F paper [1]_ are: 1. 'inverted_cdf' 2. 'averaged_inverted_cdf' 3. 'closest_observation' 4. 'interpolated_inverted_cdf' 5. 'hazen' 6. 'weibull' 7. 'linear' (default) 8. 'median_unbiased' 9. 'normal_unbiased' The first three methods are discontinuous. NumPy further defines the following discontinuous variations of the default 'linear' (7.) option: * 'lower' * 'higher', * 'midpoint' * 'nearest' .. versionchanged:: 1.22.0 This argument was previously called "interpolation" and only offered the "linear" default and last four options. keepdims : bool, optional If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original array `a`. .. versionadded:: 1.9.0 weights : array_like, optional An array of weights associated with the values in `a`. Each value in `a` contributes to the percentile according to its associated weight. The weights array can either be 1-D (in which case its length must be the size of `a` along the given axis) or of the same shape as `a`. If `weights=None`, then all data in `a` are assumed to have a weight equal to one. Only `method="inverted_cdf"` supports weights. See the notes for more details. .. versionadded:: 2.0.0 interpolation : str, optional Deprecated name for the method keyword argument. .. deprecated:: 1.22.0 Returns ------- percentile : scalar or ndarray If `q` is a single percentile and `axis=None`, then the result is a scalar. If multiple percentiles are given, first axis of the result corresponds to the percentiles. The other axes are the axes that remain after the reduction of `a`. If the input contains integers or floats smaller than ``float64``, the output data-type is ``float64``. Otherwise, the output data-type is the same as that of the input. If `out` is specified, that array is returned instead. See Also -------- mean median : equivalent to ``percentile(..., 50)`` nanpercentile quantile : equivalent to percentile, except q in the range [0, 1]. Notes ----- In general, the percentile at percentage level :math:`q` of a cumulative distribution function :math:`F(y)=P(Y \leq y)` with probability measure :math:`P` is defined as any number :math:`x` that fulfills the *coverage conditions* .. math:: P(Y < x) \leq q/100 \quad\text{and} \quad P(Y \leq x) \geq q/100 with random variable :math:`Y\sim P`. Sample percentiles, the result of ``percentile``, provide nonparametric estimation of the underlying population counterparts, represented by the unknown :math:`F`, given a data vector ``a`` of length ``n``. One type of estimators arises when one considers :math:`F` as the empirical distribution function of the data, i.e. :math:`F(y) = \frac{1}{n} \sum_i 1_{a_i \leq y}`. Then, different methods correspond to different choices of :math:`x` that fulfill the above inequalities. Methods that follow this approach are ``inverted_cdf`` and ``averaged_inverted_cdf``. A more general way to define sample percentile estimators is as follows. The empirical q-percentile of ``a`` is the ``n * q/100``-th value of the way from the minimum to the maximum in a sorted copy of ``a``. The values and distances of the two nearest neighbors as well as the `method` parameter will determine the percentile if the normalized ranking does not match the location of ``n * q/100`` exactly. This function is the same as the median if ``q=50``, the same as the minimum if ``q=0`` and the same as the maximum if ``q=100``. The optional `method` parameter specifies the method to use when the desired percentile lies between two indexes ``i`` and ``j = i + 1``. In that case, we first determine ``i + g``, a virtual index that lies between ``i`` and ``j``, where ``i`` is the floor and ``g`` is the fractional part of the index. The final result is, then, an interpolation of ``a[i]`` and ``a[j]`` based on ``g``. During the computation of ``g``, ``i`` and ``j`` are modified using correction constants ``alpha`` and ``beta`` whose choices depend on the ``method`` used. Finally, note that since Python uses 0-based indexing, the code subtracts another 1 from the index internally. The following formula determines the virtual index ``i + g``, the location of the percentile in the sorted sample: .. math:: i + g = (q / 100) * ( n - alpha - beta + 1 ) + alpha The different methods then work as follows inverted_cdf: method 1 of H&F [1]_. This method gives discontinuous results: * if g > 0 ; then take j * if g = 0 ; then take i averaged_inverted_cdf: method 2 of H&F [1]_. This method gives discontinuous results: * if g > 0 ; then take j * if g = 0 ; then average between bounds closest_observation: method 3 of H&F [1]_. This method gives discontinuous results: * if g > 0 ; then take j * if g = 0 and index is odd ; then take j * if g = 0 and index is even ; then take i interpolated_inverted_cdf: method 4 of H&F [1]_. This method gives continuous results using: * alpha = 0 * beta = 1 hazen: method 5 of H&F [1]_. This method gives continuous results using: * alpha = 1/2 * beta = 1/2 weibull: method 6 of H&F [1]_. This method gives continuous results using: * alpha = 0 * beta = 0 linear: method 7 of H&F [1]_. This method gives continuous results using: * alpha = 1 * beta = 1 median_unbiased: method 8 of H&F [1]_. This method is probably the best method if the sample distribution function is unknown (see reference). This method gives continuous results using: * alpha = 1/3 * beta = 1/3 normal_unbiased: method 9 of H&F [1]_. This method is probably the best method if the sample distribution function is known to be normal. This method gives continuous results using: * alpha = 3/8 * beta = 3/8 lower: NumPy method kept for backwards compatibility. Takes ``i`` as the interpolation point. higher: NumPy method kept for backwards compatibility. Takes ``j`` as the interpolation point. nearest: NumPy method kept for backwards compatibility. Takes ``i`` or ``j``, whichever is nearest. midpoint: NumPy method kept for backwards compatibility. Uses ``(i + j) / 2``. For weighted percentiles, the above coverage conditions still hold. The empirical cumulative distribution is simply replaced by its weighted version, i.e. :math:`P(Y \leq t) = \frac{1}{\sum_i w_i} \sum_i w_i 1_{x_i \leq t}`. Only ``method="inverted_cdf"`` supports weights. Examples -------- >>> a = np.array([[10, 7, 4], [3, 2, 1]]) >>> a array([[10, 7, 4], [ 3, 2, 1]]) >>> np.percentile(a, 50) 3.5 >>> np.percentile(a, 50, axis=0) array([6.5, 4.5, 2.5]) >>> np.percentile(a, 50, axis=1) array([7., 2.]) >>> np.percentile(a, 50, axis=1, keepdims=True) array([[7.], [2.]]) >>> m = np.percentile(a, 50, axis=0) >>> out = np.zeros_like(m) >>> np.percentile(a, 50, axis=0, out=out) array([6.5, 4.5, 2.5]) >>> m array([6.5, 4.5, 2.5]) >>> b = a.copy() >>> np.percentile(b, 50, axis=1, overwrite_input=True) array([7., 2.]) >>> assert not np.all(a == b) The different methods can be visualized graphically: .. plot:: import matplotlib.pyplot as plt a = np.arange(4) p = np.linspace(0, 100, 6001) ax = plt.gca() lines = [ ('linear', '-', 'C0'), ('inverted_cdf', ':', 'C1'), # Almost the same as `inverted_cdf`: ('averaged_inverted_cdf', '-.', 'C1'), ('closest_observation', ':', 'C2'), ('interpolated_inverted_cdf', '--', 'C1'), ('hazen', '--', 'C3'), ('weibull', '-.', 'C4'), ('median_unbiased', '--', 'C5'), ('normal_unbiased', '-.', 'C6'), ] for method, style, color in lines: ax.plot( p, np.percentile(a, p, method=method), label=method, linestyle=style, color=color) ax.set( title='Percentiles for different methods and data: ' + str(a), xlabel='Percentile', ylabel='Estimated percentile value', yticks=a) ax.legend(bbox_to_anchor=(1.03, 1)) plt.tight_layout() plt.show() References ---------- .. [1] R. J. Hyndman and Y. Fan, "Sample quantiles in statistical packages," The American Statistician, 50(4), pp. 361-365, 1996 r;rB"a must be an array of real numbersrdz)Percentiles must be in the range [0, 100]r2Only method 'inverted_cdf' supports weights. Got: .rrrrWeights must be non-negative.)_check_interpolation_as_methodrr rrUrr0r_quantile_is_validrrrrrr&_quantile_unchecked rrrr=rorrrrrs r`r;r;fs(~  / M<1 aAww||s<== qqww||s/B!'',,s+LA1 A a DEE ^ #!(!%CS/ !  ++D!&&&ID$WE 66'A+ <= =  1dC&(G EErbc||||fSr[rhrs r`_quantile_dispatcherrrrbc | t||d}tj|}|jjdk(r t dt |ttfr;|jjdk(r"tj||j}ntj|}t|s td|k|dk7rd|d } t| |"tj||jd }t||| }tj|d kr tdt!||||||||S)a50 Compute the q-th quantile of the data along the specified axis. .. versionadded:: 1.15.0 Parameters ---------- a : array_like of real numbers Input array or object that can be converted to an array. q : array_like of float Probability or sequence of probabilities for the quantiles to compute. Values must be between 0 and 1 inclusive. axis : {int, tuple of int, None}, optional Axis or axes along which the quantiles are computed. The default is to compute the quantile(s) along a flattened version of the array. out : 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 (of the output) will be cast if necessary. overwrite_input : bool, optional If True, then allow the input array `a` to be modified by intermediate calculations, to save memory. In this case, the contents of the input `a` after this function completes is undefined. method : str, optional This parameter specifies the method to use for estimating the quantile. There are many different methods, some unique to NumPy. See the notes for explanation. The options sorted by their R type as summarized in the H&F paper [1]_ are: 1. 'inverted_cdf' 2. 'averaged_inverted_cdf' 3. 'closest_observation' 4. 'interpolated_inverted_cdf' 5. 'hazen' 6. 'weibull' 7. 'linear' (default) 8. 'median_unbiased' 9. 'normal_unbiased' The first three methods are discontinuous. NumPy further defines the following discontinuous variations of the default 'linear' (7.) option: * 'lower' * 'higher', * 'midpoint' * 'nearest' .. versionchanged:: 1.22.0 This argument was previously called "interpolation" and only offered the "linear" default and last four options. keepdims : bool, optional If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original array `a`. weights : array_like, optional An array of weights associated with the values in `a`. Each value in `a` contributes to the quantile according to its associated weight. The weights array can either be 1-D (in which case its length must be the size of `a` along the given axis) or of the same shape as `a`. If `weights=None`, then all data in `a` are assumed to have a weight equal to one. Only `method="inverted_cdf"` supports weights. See the notes for more details. .. versionadded:: 2.0.0 interpolation : str, optional Deprecated name for the method keyword argument. .. deprecated:: 1.22.0 Returns ------- quantile : scalar or ndarray If `q` is a single probability and `axis=None`, then the result is a scalar. If multiple probability levels are given, first axis of the result corresponds to the quantiles. The other axes are the axes that remain after the reduction of `a`. If the input contains integers or floats smaller than ``float64``, the output data-type is ``float64``. Otherwise, the output data-type is the same as that of the input. If `out` is specified, that array is returned instead. See Also -------- mean percentile : equivalent to quantile, but with q in the range [0, 100]. median : equivalent to ``quantile(..., 0.5)`` nanquantile Notes ----- Given a sample `a` from an underlying distribution, `quantile` provides a nonparametric estimate of the inverse cumulative distribution function. By default, this is done by interpolating between adjacent elements in ``y``, a sorted copy of `a`:: (1-g)*y[j] + g*y[j+1] where the index ``j`` and coefficient ``g`` are the integral and fractional components of ``q * (n-1)``, and ``n`` is the number of elements in the sample. This is a special case of Equation 1 of H&F [1]_. More generally, - ``j = (q*n + m - 1) // 1``, and - ``g = (q*n + m - 1) % 1``, where ``m`` may be defined according to several different conventions. The preferred convention may be selected using the ``method`` parameter: =============================== =============== =============== ``method`` number in H&F ``m`` =============================== =============== =============== ``interpolated_inverted_cdf`` 4 ``0`` ``hazen`` 5 ``1/2`` ``weibull`` 6 ``q`` ``linear`` (default) 7 ``1 - q`` ``median_unbiased`` 8 ``q/3 + 1/3`` ``normal_unbiased`` 9 ``q/4 + 3/8`` =============================== =============== =============== Note that indices ``j`` and ``j + 1`` are clipped to the range ``0`` to ``n - 1`` when the results of the formula would be outside the allowed range of non-negative indices. The ``- 1`` in the formulas for ``j`` and ``g`` accounts for Python's 0-based indexing. The table above includes only the estimators from H&F that are continuous functions of probability `q` (estimators 4-9). NumPy also provides the three discontinuous estimators from H&F (estimators 1-3), where ``j`` is defined as above, ``m`` is defined as follows, and ``g`` is a function of the real-valued ``index = q*n + m - 1`` and ``j``. 1. ``inverted_cdf``: ``m = 0`` and ``g = int(index - j > 0)`` 2. ``averaged_inverted_cdf``: ``m = 0`` and ``g = (1 + int(index - j > 0)) / 2`` 3. ``closest_observation``: ``m = -1/2`` and ``g = 1 - int((index == j) & (j%2 == 1))`` For backward compatibility with previous versions of NumPy, `quantile` provides four additional discontinuous estimators. Like ``method='linear'``, all have ``m = 1 - q`` so that ``j = q*(n-1) // 1``, but ``g`` is defined as follows. - ``lower``: ``g = 0`` - ``midpoint``: ``g = 0.5`` - ``higher``: ``g = 1`` - ``nearest``: ``g = (q*(n-1) % 1) > 0.5`` **Weighted quantiles:** More formally, the quantile at probability level :math:`q` of a cumulative distribution function :math:`F(y)=P(Y \leq y)` with probability measure :math:`P` is defined as any number :math:`x` that fulfills the *coverage conditions* .. math:: P(Y < x) \leq q \quad\text{and}\quad P(Y \leq x) \geq q with random variable :math:`Y\sim P`. Sample quantiles, the result of ``quantile``, provide nonparametric estimation of the underlying population counterparts, represented by the unknown :math:`F`, given a data vector ``a`` of length ``n``. One type of estimators arises when one considers :math:`F` as the empirical distribution function of the data, i.e. :math:`F(y) = \frac{1}{n} \sum_i 1_{a_i \leq y}`. Then, different methods correspond to different choices of :math:`x` that fulfill the above inequalities. Methods that follow this approach are ``inverted_cdf`` and ``averaged_inverted_cdf``. A more general way to define sample quantile estimators is as follows. The empirical q-quantile of ``a`` is the ``n * q``-th value of the way from the minimum to the maximum in a sorted copy of ``a``. The values and distances of the two nearest neighbors as well as the `method` parameter will determine the quantile if the normalized ranking does not match the location of ``n * q`` exactly. This function is the same as the median if ``q=0.5``, the same as the minimum if ``q=0.0`` and the same as the maximum if ``q=1.0``. The optional `method` parameter specifies the method to use when the desired quantile lies between two indexes ``i`` and ``j = i + 1``. In that case, we first determine ``i + g``, a virtual index that lies between ``i`` and ``j``, where ``i`` is the floor and ``g`` is the fractional part of the index. The final result is, then, an interpolation of ``a[i]`` and ``a[j]`` based on ``g``. During the computation of ``g``, ``i`` and ``j`` are modified using correction constants ``alpha`` and ``beta`` whose choices depend on the ``method`` used. Finally, note that since Python uses 0-based indexing, the code subtracts another 1 from the index internally. The following formula determines the virtual index ``i + g``, the location of the quantile in the sorted sample: .. math:: i + g = q * ( n - alpha - beta + 1 ) + alpha The different methods then work as follows inverted_cdf: method 1 of H&F [1]_. This method gives discontinuous results: * if g > 0 ; then take j * if g = 0 ; then take i averaged_inverted_cdf: method 2 of H&F [1]_. This method gives discontinuous results: * if g > 0 ; then take j * if g = 0 ; then average between bounds closest_observation: method 3 of H&F [1]_. This method gives discontinuous results: * if g > 0 ; then take j * if g = 0 and index is odd ; then take j * if g = 0 and index is even ; then take i interpolated_inverted_cdf: method 4 of H&F [1]_. This method gives continuous results using: * alpha = 0 * beta = 1 hazen: method 5 of H&F [1]_. This method gives continuous results using: * alpha = 1/2 * beta = 1/2 weibull: method 6 of H&F [1]_. This method gives continuous results using: * alpha = 0 * beta = 0 linear: method 7 of H&F [1]_. This method gives continuous results using: * alpha = 1 * beta = 1 median_unbiased: method 8 of H&F [1]_. This method is probably the best method if the sample distribution function is unknown (see reference). This method gives continuous results using: * alpha = 1/3 * beta = 1/3 normal_unbiased: method 9 of H&F [1]_. This method is probably the best method if the sample distribution function is known to be normal. This method gives continuous results using: * alpha = 3/8 * beta = 3/8 lower: NumPy method kept for backwards compatibility. Takes ``i`` as the interpolation point. higher: NumPy method kept for backwards compatibility. Takes ``j`` as the interpolation point. nearest: NumPy method kept for backwards compatibility. Takes ``i`` or ``j``, whichever is nearest. midpoint: NumPy method kept for backwards compatibility. Uses ``(i + j) / 2``. **Weighted quantiles:** For weighted quantiles, the above coverage conditions still hold. The empirical cumulative distribution is simply replaced by its weighted version, i.e. :math:`P(Y \leq t) = \frac{1}{\sum_i w_i} \sum_i w_i 1_{x_i \leq t}`. Only ``method="inverted_cdf"`` supports weights. Examples -------- >>> a = np.array([[10, 7, 4], [3, 2, 1]]) >>> a array([[10, 7, 4], [ 3, 2, 1]]) >>> np.quantile(a, 0.5) 3.5 >>> np.quantile(a, 0.5, axis=0) array([6.5, 4.5, 2.5]) >>> np.quantile(a, 0.5, axis=1) array([7., 2.]) >>> np.quantile(a, 0.5, axis=1, keepdims=True) array([[7.], [2.]]) >>> m = np.quantile(a, 0.5, axis=0) >>> out = np.zeros_like(m) >>> np.quantile(a, 0.5, axis=0, out=out) array([6.5, 4.5, 2.5]) >>> m array([6.5, 4.5, 2.5]) >>> b = a.copy() >>> np.quantile(b, 0.5, axis=1, overwrite_input=True) array([7., 2.]) >>> assert not np.all(a == b) See also `numpy.percentile` for a visualization of most methods. References ---------- .. [1] R. J. Hyndman and Y. Fan, "Sample quantiles in statistical packages," The American Statistician, 50(4), pp. 361-365, 1996 rYrBrrrz%Quantiles must be in the range [0, 1]rrrrrrrr)rrr rrUrrr r rrrrrrr&rrs r`rYrYs0d  / M:/ aAww||s<==!c5\"qww||s': MM!177 + MM!  a @AA ^ #!(!%CS/ !  ++D!&&&ID$WE 66'A+ <= =  1dC&(G EErbc 2t|t||||||| S)z.Assumes that q is in [0, 1], and is an ndarray)rrrrrr=ror)rl_quantile_ureduce_func)rrrr=rorrrs r`rr;s+ A/#%$3! ##rbc|jdk(r=|jdkr.t|jD]}d||cxkrdkryyy|jdk\r|j dksyy)Nrg rrjFrT)rrr&minmax)rrs r`rrOspvv{qvv{qvv A1Q4&3&'  1 A rbcftjd|dtd|dk7r td|S)Nz!the `interpolation=` argument to z was renamed to `method=`, which has additional options. Users of the modes 'nearest', 'lower', 'higher', or 'midpoint' are encouraged to review the method they used. (Deprecated NumPy 1.22)rrrzjYou shall not pass both `method` and `interpolation`! (`interpolation` is Deprecated in favor of `method`))rrrr)rrfnames r`rr[sM MM +E73" " q * CD D rbrXrWc0||z||d|z |z zzzdz S)a0 Compute the floating point indexes of an array for the linear interpolation of quantiles. n : array_like The sample sizes. quantiles : array_like The quantiles values. alpha : float A constant used to correct the index computed. beta : float A constant used to correct the index computed. alpha and beta values depend on the chosen method (see quantile documentation) Reference: Hyndman&Fan paper "Sample Quantiles in Statistical Packages", DOI: 10.1080/00031305.1996.10473566 rgrh)r^r_rXrWs r`ryryls3( y= IUT!12 2    rbctj||z }|d||}tj||jS)a Compute gamma (a.k.a 'm' or 'weight') for the linear interpolation of quantiles. virtual_indexes : array_like The indexes where the percentile is supposed to be found in the sorted sample. previous_indexes : array_like The floor values of virtual_indexes. interpolation : dict The interpolation method chosen, which may have a specific rule modifying gamma. gamma is usually the fractional part of virtual_indexes but can be modified by the interpolation method. rdr)rr r)virtual_indexesprevious_indexesrrss r` _get_gammarsD" MM/,<< =E F;  7E ==o&;&; <.s!eqjRXXe_q5HA5M%Nrbrgrkrr^r_ gamma_funs r`rvrvs*OI /Y!0Cc0I09 ;;rbc,d}t||zdz |S)Nc |dk(Srrhrrs r`raz_inverted_cdf..s %1*rbrgrrs r`r\r\s"-I /Y!0C09 ;;rbrrrrroreturnc2|jdkDr td|r.|'d}|j}|dn|j}n@|}|}n;|'d}|j}|dn|j}n|j }|}t ||||||} | S)Nrzq must be a scalar or 1dr)r_rrr=r)rrr"flattenr9 _quantile) rrrrr=rorrrrs r`rrs vvz344 <D'')C!/$w}}CCC <D))+C!/$w/@C&&(CC s!" $" $F MrbcBtjtj|}tj|dz}||dz k\}|jr d||<d||<|dk}|jr d||<d||<tj|j tj r/tj|}|jr d||<d||<|jtj}|jtj}||fS)a Get the valid indexes of arr neighbouring virtual_indexes. Note This is a companion function to linear interpolation of Quantiles Returns ------- (previous_indexes, next_indexes): Tuple A Tuple of virtual_indexes neighbouring indexes rgrr) rr rr&r'rr/isnanrr)rrvalid_values_countr next_indexesindexes_above_boundsindexes_below_boundsvirtual_indexes_nanss r` _get_indexesrs}}RXXo%>?==!1A!56L*.@1.DD!13-.-/ )**Q.!12-.-. )* }}SYY +!xx8  # # %57 1 213L- .'..rww7&&rww/L \ ))rbrr_c  tj|}|j||dk7rtj||d}tj|j tj xs|j jdv}| t|}|d}tj|}|dd} n7tj|j tj} |d k(xr| } | r|rF|jt|jd gfd tj |d } n<|j|jd tj"d t$} t'||d|} nEt)||\} }|jtj*tjdd g| j|jfd |rtj |d } nd} || }||}t-|| |}|jd|j.dz zz}|j1|}t3||||} nftj|}|dk7rtj||d}tj4|dd}tj6||d }|j|jk(rtj6||d }n|j1d |df}|rtj |d } ntj"d t$} |j9dtj:}||d z}j jdk(r|j=j }fd}|jdd}j.dkDrj|z}|tj>||} n/|j|k7rd|d|jd}t||} |jdd}tj@|D]E}||tjBddf|z|tjBddf|z| d|z<G| jdk(r2| j tj dk(r| jE} tjF| r3| j.dk(r ||d } | StjH| |d | | S#t$r$t|dtjdwxYw)!a  Private function that doesn't support extended axis or keepdims. These methods are extended to this function using _ureduce See nanpercentile for parameter usage It computes the quantiles of the array for the given axis. A linear interpolation is performed based on the `interpolation`. By default, the method is "linear" where alpha == beta == 1 which performs the 7th method of Hyndman&Fan. With "median_unbiased" we get alpha == beta == 1/3 thus the 8th method of Hyndman&Fan. r) destinationrtNz$ is not a valid method. Use one of: rcrdTrrr)r.Frrurgrgr:stable)rrU.rrcrtj|d}t|dz }t||d}|S)NrPsidergrr)r searchsortedr!r)rcdfindicesrr_ values_counts r` find_cdf_1dz_quantile..find_cdf_1ds;ooc96BGg|a'78G#wQ/FMrbrz%Wrong shape of argument 'out', shape=z is required; got shape=r).rhOr )%rr rmmoveaxisr'rr/rU_QuantileMethodsKeyErrorrkeysrr$rr"rr rrruniquerrrrrtake_along_axisrnr(rr0rrrvr&r)rr_rrr=rr| method_propsrsupports_integersint_virtual_indicesslices_having_nansrrrrrrsr index_arrayrrr_shaperNkkkrs ` @r`rr!s* -- C99T?L qykk#t3 cii,F $0F  8+F3L <,':;L%'XXe4%@"#QC@F-9#:I:F.H * l MM "..1b'*:*@*@*B*6*<*<*>*,-.   %'XXc'l%;"%)"+,H|$D0@,OE*004388a<3HHLMM,/E8!F--( 19kk'4Q?Gjj18<   k: ==CII %((+AFGoob)+s*:;G !##g,!7 "$%t!< nn!2::n6 s7| ??  3 &**Y__-C ))AB- >>A oo/G ;]]3g6FyyG#>wiH..1ii[; o%FYYqr]**R. B"-BEE!"JO$c"%%*r/&:#F6B;   <<2 &,,"((3-"?[[]F vv ! ;;!  WF M IIfc'l2D E MQ 8*@#((*+-.37 8 8s  T-U c ||fSr[rhrrr3rs r`_trapezoid_dispatcherrs q6Mrbc t|}||}nft|}|jdk(r?t|}dg|jz}|jd||<|j |}n t||}|j}t dg|z}t dg|z}t dd||<t dd||< ||t ||t |zzdz j|} | S#t$retj|}tj|}tj||t ||t |zzdz |} Y| SwxYw)a- Integrate along the given axis using the composite trapezoidal rule. If `x` is provided, the integration happens in sequence along its elements - they are not sorted. Integrate `y` (`x`) along each 1d slice on the given axis, compute :math:`\int y(x) dx`. When `x` is specified, this integrates along the parametric curve, computing :math:`\int_t y(t) dt = \int_t y(t) \left.\frac{dx}{dt}\right|_{x=x(t)} dt`. .. versionadded:: 2.0.0 Parameters ---------- y : array_like Input array to integrate. x : array_like, optional The sample points corresponding to the `y` values. If `x` is None, the sample points are assumed to be evenly spaced `dx` apart. The default is None. dx : scalar, optional The spacing between sample points when `x` is None. The default is 1. axis : int, optional The axis along which to integrate. Returns ------- trapezoid : float or ndarray Definite integral of `y` = n-dimensional array as approximated along a single axis by the trapezoidal rule. If `y` is a 1-dimensional array, then the result is a float. If `n` is greater than 1, then the result is an `n`-1 dimensional array. See Also -------- sum, cumsum Notes ----- Image [2]_ illustrates trapezoidal rule -- y-axis locations of points will be taken from `y` array, by default x-axis distances between points will be 1.0, alternatively they can be provided with `x` array or with `dx` scalar. Return value will be equal to combined area under the red lines. References ---------- .. [1] Wikipedia page: https://en.wikipedia.org/wiki/Trapezoidal_rule .. [2] Illustration image: https://en.wikipedia.org/wiki/File:Composite_trapezoidal_rule_illustration.png Examples -------- Use the trapezoidal rule on evenly spaced points: >>> np.trapezoid([1, 2, 3]) 4.0 The spacing between sample points can be selected by either the ``x`` or ``dx`` arguments: >>> np.trapezoid([1, 2, 3], x=[4, 6, 8]) 8.0 >>> np.trapezoid([1, 2, 3], dx=2) 8.0 Using a decreasing ``x`` corresponds to integrating in reverse: >>> np.trapezoid([1, 2, 3], x=[8, 6, 4]) -8.0 More generally ``x`` is used to integrate along a parametric curve. We can estimate the integral :math:`\int_0^1 x^2 = 1/3` using: >>> x = np.linspace(0, 1, num=50) >>> y = x**2 >>> np.trapezoid(y, x) 0.33340274885464394 Or estimate the area of a circle, noting we repeat the sample which closes the curve: >>> theta = np.linspace(0, 2 * np.pi, num=1000, endpoint=True) >>> np.trapezoid(np.cos(theta), x=np.sin(theta)) 3.141571941375841 ``np.trapezoid`` can be applied along a specified axis to do multiple computations in one call: >>> a = np.arange(6).reshape(2, 3) >>> a array([[0, 1, 2], [3, 4, 5]]) >>> np.trapezoid(a, axis=0) array([1.5, 2.5, 3.5]) >>> np.trapezoid(a, axis=1) array([2., 8.]) Nrgrrrr$) r rr<rmrr)rr'rrr rreduce) rrr3rr<rmrJr7r8rets r`rRrRsWR 1 Ay  qM 66Q;QACJE''!*E$K % AQT"A BDk]2 FDk]2 FD>F4Lr?F4LLAeFm$qv'7783>CCDI J L JJqM JJqMjjaf .qv/??@DdK J Ls2C44A*E"!E"cXtjdtdt||||S)z `trapz` is deprecated in NumPy 2.0. Please use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`. zs`trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.rr)rr3r)rrrrRrs r`rSrS[s/ MM @  Q! ..rb)r9sparseindexingc|Sr[rh)r9rrxis r`_meshgrid_dispatcherrms Irbxyc t|}|dvr tdd|z}t|Dcgc]8\}}tj|j |d|dz||dzdz:}}}|dk(r%|dkDr d|d dz|d _d |d dz|d_|stj|d d i}|rtd|D}|Scc}}w)a Return a tuple of coordinate matrices from coordinate vectors. Make N-D coordinate arrays for vectorized evaluations of N-D scalar/vector fields over N-D grids, given one-dimensional coordinate arrays x1, x2,..., xn. .. versionchanged:: 1.9 1-D and 0-D cases are allowed. Parameters ---------- x1, x2,..., xn : array_like 1-D arrays representing the coordinates of a grid. indexing : {'xy', 'ij'}, optional Cartesian ('xy', default) or matrix ('ij') indexing of output. See Notes for more details. .. versionadded:: 1.7.0 sparse : bool, optional If True the shape of the returned coordinate array for dimension *i* is reduced from ``(N1, ..., Ni, ... Nn)`` to ``(1, ..., 1, Ni, 1, ..., 1)``. These sparse coordinate grids are intended to be use with :ref:`basics.broadcasting`. When all coordinates are used in an expression, broadcasting still leads to a fully-dimensonal result array. Default is False. .. versionadded:: 1.7.0 copy : bool, optional If False, a view into the original arrays are returned in order to conserve memory. Default is True. Please note that ``sparse=False, copy=False`` will likely return non-contiguous arrays. Furthermore, more than one element of a broadcast array may refer to a single memory location. If you need to write to the arrays, make copies first. .. versionadded:: 1.7.0 Returns ------- X1, X2,..., XN : tuple of ndarrays For vectors `x1`, `x2`,..., `xn` with lengths ``Ni=len(xi)``, returns ``(N1, N2, N3,..., Nn)`` shaped arrays if indexing='ij' or ``(N2, N1, N3,..., Nn)`` shaped arrays if indexing='xy' with the elements of `xi` repeated to fill the matrix along the first dimension for `x1`, the second for `x2` and so on. Notes ----- This function supports both indexing conventions through the indexing keyword argument. Giving the string 'ij' returns a meshgrid with matrix indexing, while 'xy' returns a meshgrid with Cartesian indexing. In the 2-D case with inputs of length M and N, the outputs are of shape (N, M) for 'xy' indexing and (M, N) for 'ij' indexing. In the 3-D case with inputs of length M, N and P, outputs are of shape (N, M, P) for 'xy' indexing and (M, N, P) for 'ij' indexing. The difference is illustrated by the following code snippet:: xv, yv = np.meshgrid(x, y, indexing='ij') for i in range(nx): for j in range(ny): # treat xv[i,j], yv[i,j] xv, yv = np.meshgrid(x, y, indexing='xy') for i in range(nx): for j in range(ny): # treat xv[j,i], yv[j,i] In the 1-D and 0-D case, the indexing and sparse keywords have no effect. See Also -------- mgrid : Construct a multi-dimensional "meshgrid" using indexing notation. ogrid : Construct an open multi-dimensional "meshgrid" using indexing notation. :ref:`how-to-index` Examples -------- >>> nx, ny = (3, 2) >>> x = np.linspace(0, 1, nx) >>> y = np.linspace(0, 1, ny) >>> xv, yv = np.meshgrid(x, y) >>> xv array([[0. , 0.5, 1. ], [0. , 0.5, 1. ]]) >>> yv array([[0., 0., 0.], [1., 1., 1.]]) The result of `meshgrid` is a coordinate grid: >>> import matplotlib.pyplot as plt >>> plt.plot(xv, yv, marker='o', color='k', linestyle='none') >>> plt.show() You can create sparse output arrays to save memory and computation time. >>> xv, yv = np.meshgrid(x, y, sparse=True) >>> xv array([[0. , 0.5, 1. ]]) >>> yv array([[0.], [1.]]) `meshgrid` is very useful to evaluate functions on a grid. If the function depends on all coordinates, both dense and sparse outputs can be used. >>> x = np.linspace(-5, 5, 101) >>> y = np.linspace(-5, 5, 101) >>> # full coordinate arrays >>> xx, yy = np.meshgrid(x, y) >>> zz = np.sqrt(xx**2 + yy**2) >>> xx.shape, yy.shape, zz.shape ((101, 101), (101, 101), (101, 101)) >>> # sparse coordinate arrays >>> xs, ys = np.meshgrid(x, y, sparse=True) >>> zs = np.sqrt(xs**2 + ys**2) >>> xs.shape, ys.shape, zs.shape ((1, 101), (101, 1), (101, 101)) >>> np.array_equal(zz, zs) True >>> h = plt.contourf(x, y, zs) >>> plt.axis('scaled') >>> plt.colorbar() >>> plt.show() )rijz.Valid values for `indexing` are 'xy' and 'ij'.rNr_rgr)rgrrr)rrgrTc3<K|]}|jywr[rz)rrs r`rzmeshgrid.. s0Aqvvx0s) rrrrr rrmrr) r9rrrrs0rrrs r`rUrUrsJ r7D|# <> > B#B-)!QmmA&&r"1v~1q56 'BC)F)4D1H!BqrF*q !BqrF*q  $$f9D9 000 M)s=Cc ||fSr[rh)robjrs r`_delete_dispatcherrs :rbct|}|jd\}|j}|jjrdnd}|'|dk7r|j }|j}|dz }n t ||}tdg|z}|j|}t|j}t|tr|j|\} } } t| | | } t| } | dkr#|j|j|d S| dkr| } | d } | ddz} ||xx| zcc<t!||j"|}| dk(rn)td| ||<|t%||t%|<| |k(rnJt| | z d||<tdg|z}t| d||<|t%||t%|<| dk(rn|t'| | z t( }d|d| | z | <t| | | z ||<tdg|z}t| | ||<|t%|}|||<|t%||t%|<|j|d St|t*t,frt|t(sd }nd}|}t/j0|}|j2dk(r0t|t.j4s|j7t8}n9|j2dk(r*|j"j:d vr|j=}d }|r|| ks||k\rt?d|||fz|dkr||z }||xxdzcc<t!||j"|}td|||<|t%||t%|<t|d||<tdg|z}t|dzd||<|t%||t%|<nk|j"t(k(r.|j|fk7rtAdjC||}nt'|t( }d||f<|||<|t%|}|j|d S)a Return a new array with sub-arrays along an axis deleted. For a one dimensional array, this returns those entries not returned by `arr[obj]`. Parameters ---------- arr : array_like Input array. obj : slice, int or array of ints Indicate indices of sub-arrays to remove along the specified axis. .. versionchanged:: 1.19.0 Boolean indices are now treated as a mask of elements to remove, rather than being cast to the integers 0 and 1. axis : int, optional The axis along which to delete the subarray defined by `obj`. If `axis` is None, `obj` is applied to the flattened array. Returns ------- out : ndarray A copy of `arr` with the elements specified by `obj` removed. Note that `delete` does not occur in-place. If `axis` is None, `out` is a flattened array. See Also -------- insert : Insert elements into an array. append : Append elements at the end of an array. Notes ----- Often it is preferable to use a boolean mask. For example: >>> arr = np.arange(12) + 1 >>> mask = np.ones(len(arr), dtype=bool) >>> mask[[0,2,4]] = False >>> result = arr[mask,...] Is equivalent to ``np.delete(arr, [0,2,4], axis=0)``, but allows further use of `mask`. Examples -------- >>> arr = np.array([[1,2,3,4], [5,6,7,8], [9,10,11,12]]) >>> arr array([[ 1, 2, 3, 4], [ 5, 6, 7, 8], [ 9, 10, 11, 12]]) >>> np.delete(arr, 1, 0) array([[ 1, 2, 3, 4], [ 9, 10, 11, 12]]) >>> np.delete(arr, np.s_[::2], 1) array([[ 2, 4], [ 6, 8], [10, 12]]) >>> np.delete(arr, [1,3,5], None) array([ 1, 3, 5, 7, 8, 9, 10, 11, 12]) Frr|CNrgr)r to_scalarrrTuiz2index %i is out of bounds for axis %i with size %iz\boolean array argument obj to delete must be one dimensional and match the axis length of {})"r0 as_arraysrflagsfncr"r,r)rmrrrr&rwrapr9r rrrrr rrr rr rrrUrv IndexErrorrr)rrrconvrarrorderslobjr1newshapestartstopstepxrnumtodelnewslobj2rh single_value_objs r`rVrVsAB C D >>> &DC 88DiimmsH | 19))+Cxxax#D$/ 4[M$ E $ACIIH#uKKNtT 5$ %r7 q=99SXXHX59G G !85DrFEa519D("Hcii2 A: e,E$K #E%L 1Ce  19 X t4E$KDk]4'F t,F4L #E&M 2Ce  19 U $/D%*D!$u*T! "tH}5E$KDk]4'F -F4LeFm$CF4L #E&M 2Ce yyy..#W~&z#t/D  jjo 88q=D"**!=**T"C XX]syy~~5((*CL 1"Hq $N+, , !G 1HC!Hcii2D#&d e -E%LC&d +t#SUD)t f .E%L 99 yyQD  "006q ;; 4D&DDJd %, 99SE9 **rbc |||fSr[rh)rrrBrs r`_insert_dispatcherrs f rbct|}|jd\}|j}|jjrdnd}|'|dk7r|j }|j}|dz }n t ||}tdg|z}|j|}t|j} t|trt|j|dti} nttj|} | j t"k(r2t%j&dt(d | j+t} n| jdkDr t-d | j.dk(r[| j1} | | ks| |kDrt3d |d |d|| dkr| |z } t|d|j|j }| jdk(rtj4|d|}|j|} | |xx| z cc<t7| |j |} td| ||<|t9|| t9|<t| | | z||<|| t9|<t| | zd||<tdg|z}t| d||<|t9|| t9|<|j;| dS| j.dk(r/t|tj<s| j+t} | | dkxx|z cc<t?| } | jAd}| |xxtj| z cc<| |xx| z cc<tC| |t"}d|| <t7| |j |} tdg|z}| ||<|||<|| t9|<|| t9|<|j;| dS)a Insert values along the given axis before the given indices. Parameters ---------- arr : array_like Input array. obj : int, slice or sequence of ints Object that defines the index or indices before which `values` is inserted. .. versionadded:: 1.8.0 Support for multiple insertions when `obj` is a single scalar or a sequence with one element (similar to calling insert multiple times). values : array_like Values to insert into `arr`. If the type of `values` is different from that of `arr`, `values` is converted to the type of `arr`. `values` should be shaped so that ``arr[...,obj,...] = values`` is legal. axis : int, optional Axis along which to insert `values`. If `axis` is None then `arr` is flattened first. Returns ------- out : ndarray A copy of `arr` with `values` inserted. Note that `insert` does not occur in-place: a new array is returned. If `axis` is None, `out` is a flattened array. See Also -------- append : Append elements at the end of an array. concatenate : Join a sequence of arrays along an existing axis. delete : Delete elements from an array. Notes ----- Note that for higher dimensional inserts ``obj=0`` behaves very different from ``obj=[0]`` just like ``arr[:,0,:] = values`` is different from ``arr[:,[0],:] = values``. Examples -------- >>> a = np.array([[1, 1], [2, 2], [3, 3]]) >>> a array([[1, 1], [2, 2], [3, 3]]) >>> np.insert(a, 1, 5) array([1, 5, 1, ..., 2, 3, 3]) >>> np.insert(a, 1, 5, axis=1) array([[1, 5, 1], [2, 5, 2], [3, 5, 3]]) Difference between sequence and scalars: >>> np.insert(a, [1], [[1],[2],[3]], axis=1) array([[1, 1, 1], [2, 2, 2], [3, 3, 3]]) >>> np.array_equal(np.insert(a, 1, [1, 2, 3], axis=1), ... np.insert(a, [1], [[1],[2],[3]], axis=1)) True >>> b = a.flatten() >>> b array([1, 1, 2, 2, 3, 3]) >>> np.insert(b, [2, 2], [5, 6]) array([1, 1, 5, ..., 2, 3, 3]) >>> np.insert(b, slice(2, 4), [5, 6]) array([1, 1, 5, ..., 2, 3, 3]) >>> np.insert(b, [2, 2], [7.13, False]) # type casting array([1, 1, 7, ..., 2, 3, 3]) >>> x = np.arange(8).reshape(2, 4) >>> idx = (1, 3) >>> np.insert(x, idx, 999, axis=1) array([[ 0, 999, 1, 2, 999, 3], [ 4, 999, 5, 6, 999, 7]]) Frr|rNrgrzrin the future insert will treat boolean arrays and array-likes as a boolean index instead of casting it to integerrrzDindex array argument obj to insert must be one dimensional or scalarzindex z is out of bounds for axis z with size rr+r mergesort)rUr)"r0rrrrr"r,r)rmrrrrrrr rrrr FutureWarningrrrrvrrr rrr rrr)rrrBrrrrrr1rrrnumnewrrrold_masks r`rWrWsr C D >>> &DC 88DiimmsH | 19))+Cxxax#D$/ 4[M$ E $ACIIH#u#++a.55((3- ==D  MM(Q 8nnT*G \\A  ||q  A2:vcU*EdVL**+./ / AI QJEvD J <<1 [[D1Fd#& Hcii2D%(d e -E%LE5<0d "E%LE&L$/d +%UD)t f .E%Lyyy..  :c2::#>..& GaKA \F OOO -E ENbii''N TNfNHTN$/HHW #))X .CDk]4 FE$KF4LCe Cf  99SE9 **rbc ||fSr[rhrrBrs r`_append_dispatcherrs =rbct|}|9|jdk7r|j}t|}|jdz }t||f|S)a Append values to the end of an array. Parameters ---------- arr : array_like Values are appended to a copy of this array. values : array_like These values are appended to a copy of `arr`. It must be of the correct shape (the same shape as `arr`, excluding `axis`). If `axis` is not specified, `values` can be any shape and will be flattened before use. axis : int, optional The axis along which `values` are appended. If `axis` is not given, both `arr` and `values` are flattened before use. Returns ------- append : ndarray A copy of `arr` with `values` appended to `axis`. Note that `append` does not occur in-place: a new array is allocated and filled. If `axis` is None, `out` is a flattened array. See Also -------- insert : Insert elements into an array. delete : Delete elements from an array. Examples -------- >>> np.append([1, 2, 3], [[4, 5, 6], [7, 8, 9]]) array([1, 2, 3, ..., 7, 8, 9]) When `axis` is specified, `values` must have the correct shape. >>> np.append([[1, 2, 3], [4, 5, 6]], [[7, 8, 9]], axis=0) array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) >>> np.append([[1, 2, 3], [4, 5, 6]], [7, 8, 9], axis=0) Traceback (most recent call last): ... ValueError: all the input arrays must have same number of dimensions, but the array at index 0 has 2 dimension(s) and the array at index 1 has 1 dimension(s) rgr)r rr"rr s r`rXrXsPb S/C | 88q=))+Cvxxz V}4 00rbc ||fSr[rh)rbinsrQs r`_digitize_dispatcherrs t9rbctj|}tj|}tj|jtj r t dt|}|dk(r td|rdnd}|dk(r*t|tj|ddd||z Stj|||S) aE Return the indices of the bins to which each value in input array belongs. ========= ============= ============================ `right` order of bins returned index `i` satisfies ========= ============= ============================ ``False`` increasing ``bins[i-1] <= x < bins[i]`` ``True`` increasing ``bins[i-1] < x <= bins[i]`` ``False`` decreasing ``bins[i-1] > x >= bins[i]`` ``True`` decreasing ``bins[i-1] >= x > bins[i]`` ========= ============= ============================ If values in `x` are beyond the bounds of `bins`, 0 or ``len(bins)`` is returned as appropriate. Parameters ---------- x : array_like Input array to be binned. Prior to NumPy 1.10.0, this array had to be 1-dimensional, but can now have any shape. bins : array_like Array of bins. It has to be 1-dimensional and monotonic. right : bool, optional Indicating whether the intervals include the right or the left bin edge. Default behavior is (right==False) indicating that the interval does not include the right edge. The left bin end is open in this case, i.e., bins[i-1] <= x < bins[i] is the default behavior for monotonically increasing bins. Returns ------- indices : ndarray of ints Output array of indices, of same shape as `x`. Raises ------ ValueError If `bins` is not monotonic. TypeError If the type of the input is complex. See Also -------- bincount, histogram, unique, searchsorted Notes ----- If values in `x` are such that they fall outside the bin range, attempting to index `bins` with the indices that `digitize` returns will result in an IndexError. .. versionadded:: 1.10.0 `numpy.digitize` is implemented in terms of `numpy.searchsorted`. This means that a binary search is used to bin the values, which scales much better for larger number of bins than the previous linear search. It also removes the requirement for the input array to be 1-dimensional. For monotonically *increasing* `bins`, the following are equivalent:: np.digitize(x, bins, right=True) np.searchsorted(bins, x, side='left') Note that as the order of the arguments are reversed, the side must be too. The `searchsorted` call is marginally faster, as it does not do any monotonicity checks. Perhaps more importantly, it supports all dtypes. Examples -------- >>> x = np.array([0.2, 6.4, 3.0, 1.6]) >>> bins = np.array([0.0, 1.0, 2.5, 4.0, 10.0]) >>> inds = np.digitize(x, bins) >>> inds array([1, 4, 3, 2]) >>> for n in range(x.size): ... print(bins[inds[n]-1], "<=", x[n], "<", bins[inds[n]]) ... 0.0 <= 0.2 < 1.0 4.0 <= 6.4 < 10.0 2.5 <= 3.0 < 4.0 1.0 <= 1.6 < 2.5 >>> x = np.array([1.2, 10.0, 12.4, 15.5, 20.]) >>> bins = np.array([0, 5, 10, 15, 20]) >>> np.digitize(x,bins,right=True) array([1, 2, 3, 4, 4]) >>> np.digitize(x,bins,right=False) array([1, 3, 3, 4, 5]) zx may not be complexrz3bins must be monotonically increasing or decreasingrPrQrNr) rr rr'rrcrr-rrr)rrrQmonors r`rHrHsv AA ;;t D }}QWWc112.//  D qyNOO6D rz4y3++D2JEEEad33rb)NN)rg)rrgr[)NNN)NNFr`)KF)NNNN)F)Nr)fb)NT)NNNNNN)NTFNNN)NNFF)NNNNN)NNFrF)NNFrFN)NNFr)rrNN)Nrjr)rcollections.abcr functoolsrrrr4rnumpy._core.numeric_corenumericr numpy._corerrrrrrr r r r r rrrrrrrnumpy._core.umathrrrrrrrrrrrr r!numpy._core.fromnumericr"r#r$r%r&r'numpy._core.numerictypesr(numpy.lib._twodim_base_implr)numpy._core.multiarrayr*r+r,r-r.rXr/rVnumpy._core._multiarray_umathr0 numpy._utilsr1numpy.lib._histograms_implr2r3partialarray_function_dispatch__all__dictrrrBrrAr:rrrrGrFrr7rr6rr9r r=rGr<rSr`r>rkr?rxr@rr8rrCrrDrrr_CORE_DIMENSION_LISTr_ARGUMENT_LISTrrrrrrrrEr(rIr8rJrPrOrNrMrLrPrJrNrQrSrTrQrZrLrlrprKrrrr;rrYrrrr ryrrrqrrvr\r rrrrrrRrSrrUrrVrrWrrXrrHrhrbr`r+s !!,/,;#>,)++ %%g7 6JBO #3( 7(  3(  B( ?( ?( 1  1  0"  1 WPf*+X6,X6v)*Z+Zz G))X2!% ,-b[[b.bJ GB B J./u 0u p +,f-fR)*C9+C9P,0D -.#U/Up  )*b"++bkkA +A H+,I/-I/X*+4 ,4 nT+,\qt\-\~'!(!H%1&1h,-0>.0>f*+$#,$#N4 p,33OD   1 2 !((3  " "> 2 82"D&>0     GU4U4U4p =A,0&;?& )@DZ $Z*Zz# -.t"++BKKU U /U p G_A_AD Gf=f=R Ga%a%H G^'^'FB: +:(84)84z G~>~>B)*M+M`J \@D +,Z5-Z5z5pGK15 BF)- /0$ YE!YE1YEx EI/3 @D'+ -." nEnE/nEf " (-'!& $#( "  U 2=00 ;;  %$ 88$ 88$$ $  $XX$N *L  n XXn88nnb./A0AH G//"$(t -.EDZ/Zz+,o+-o+d+,s+-s+l+,61-61r-.k4/k4rb