B ʾZm@sFddlmZmZmZddlZddlZddlZddlZddlZddl m Z m Z m Z m Z ddlmZmZmZmZddlmZmZmZmZdej_ejddkrddZnd dZd Ze eZd Ze eZd d Z ddZ!ddZ"ddZ#ddZ$ddZ%ddZ&ddZ'ddZ(ddZ)d d!Z*d"d#Z+d$d%Z,d&d'Z-d(d)Z.d*d+Z/d,d-Z0d.d/Z1d0d1Z2e d2Z3d3d4Z4e d5Z5d6d7Z6e d8Z7d9d:Z8e d;Z9dd?Z;d@dAZdFdGZ?e dHZ@dIdJZAdKdLZBdMdNZCdOdPZDdQdRZEdSdTZFdUdVZGdWZHdXdYZIdZd[ZJd\d]ZKd^d_ZLeMd`krBddlNZNeNOdS)a)divisionabsolute_importprint_functionN)NumpyDocString FunctionDocClassDoc ParseError)SphinxDocStringSphinxClassDocSphinxFunctionDocget_doc_object) assert_equal assert_raisesassert_list_equal assert_truecCs|S)N)srrsrcCs t|dS)NZunicode_escape)Zunicode)rrrrrsa numpy.multivariate_normal(mean, cov, shape=None, spam=None) Draw values from a multivariate normal distribution with specified mean and covariance. The multivariate normal or Gaussian distribution is a generalisation of the one-dimensional normal distribution to higher dimensions. Parameters ---------- mean : (N,) ndarray Mean of the N-dimensional distribution. .. math:: (1+2+3)/3 cov : (N, N) ndarray Covariance matrix of the distribution. shape : tuple of ints Given a shape of, for example, (m,n,k), m*n*k samples are generated, and packed in an m-by-n-by-k arrangement. Because each sample is N-dimensional, the output shape is (m,n,k,N). Returns ------- out : ndarray The drawn samples, arranged according to `shape`. If the shape given is (m,n,...), then the shape of `out` is (m,n,...,N). In other words, each entry ``out[i,j,...,:]`` is an N-dimensional value drawn from the distribution. list of str This is not a real return value. It exists to test anonymous return values. no_description Other Parameters ---------------- spam : parrot A parrot off its mortal coil. Raises ------ RuntimeError Some error Warns ----- RuntimeWarning Some warning Warnings -------- Certain warnings apply. Notes ----- Instead of specifying the full covariance matrix, popular approximations include: - Spherical covariance (`cov` is a multiple of the identity matrix) - Diagonal covariance (`cov` has non-negative elements only on the diagonal) This geometrical property can be seen in two dimensions by plotting generated data-points: >>> mean = [0,0] >>> cov = [[1,0],[0,100]] # diagonal covariance, points lie on x or y-axis >>> x,y = multivariate_normal(mean,cov,5000).T >>> plt.plot(x,y,'x'); plt.axis('equal'); plt.show() Note that the covariance matrix must be symmetric and non-negative definite. References ---------- .. [1] A. Papoulis, "Probability, Random Variables, and Stochastic Processes," 3rd ed., McGraw-Hill Companies, 1991 .. [2] R.O. Duda, P.E. Hart, and D.G. Stork, "Pattern Classification," 2nd ed., Wiley, 2001. See Also -------- some, other, funcs otherfunc : relationship Examples -------- >>> mean = (1,2) >>> cov = [[1,0],[1,0]] >>> x = multivariate_normal(mean,cov,(3,3)) >>> print x.shape (3, 3, 2) The following is probably true, given that 0.6 is roughly twice the standard deviation: >>> print list( (x[0,0,:] - mean) < 0.6 ) [True, True] .. index:: random :refguide: random;distributions, random;gauss z Test generator Yields ------ a : int The number of apples. b : int The number of bananas. int The number of unknowns. cCs(tddsttdds$tdS)N Signatureznumpy.multivariate_normal(z spam=None))doc startswithAssertionErrorendswithrrrrtest_signaturesrcCs0tdddsttddds,tdS)NSummaryrz Draw valuesz covariance.)rrrrrrrr test_summarysrcCstdddstdS)NzExtended SummaryrzThe multivariate normal)rrrrrrrtest_extended_summarysrcCs~tttddtddtdDdddgtdd\}}}t|d |d d s^ttdd d d d ksztdS)N ParametersrcSsg|]\}}}|qSrr).0n_rrr sz#test_parameters..ZmeanZcovshapez(N, N) ndarrayrzCovariance matrixrz (1+2+3)/3)r lenrrr)argarg_typedescrrrtest_parameterss  r+cCs^tttddtddtdDdgtdd\}}}t|d|ddsZtdS) NzOther Parametersr&cSsg|]\}}}|qSrr)r!r"r#rrrr$sz)test_other_parameters..spamrZparrotzA parrot off its mortal coil)r r'rrr)r(r)r*rrrtest_other_parameterss  r-cCstttddtdd\}}}t|dt|d|ddsJt|dds\ttdd \}}}t|d t|d |dd st|dd sttdd\}}}t|dt|d d |rtdS)NZReturnsrroutZndarrayzThe drawn samplesrz distribution.r&z list of strzThis is not a realzanonymous return values.Zno_description)r r'rrrrjoinstrip)r(r)r*rrr test_returnss      r3cCstd}tt|ddddg}xZt||D]L\\}}}\}}}t||t|||ddsft|d|s,tq,WdS)NZYieldsr)aintzapples.)br5zbananas.)r5r/z unknowns.rz The number of) doc_yieldsr r'ziprrr)Zsectiontruthr(r)r*Zarg_Z arg_type_endrrr test_yieldss   r;cCsd}ttt|dS)Nz Test having returns and yields. Returns ------- int The number of apples. Yields ------ a : int The number of apples. b : int The number of bananas. )r ValueErrorr)doc_textrrrtest_returnyieldsr>c Csd}ttt|Gdddt}dd}y t|Wn>tk rr}z tdt|kp^dt|kWdd}~XYnXy t|Wn>tk r}z tdt|kpdt|kWdd}~XYnXdS) Nzy Test having a section Notes twice Notes ----- See the next note for more information Notes ----- That should break... c@s eZdZdZddZddZdS)z!test_section_twice..Dummyz Dummy class. Notes ----- First note. Notes ----- Second note. cSsdS)zSpam Spam spam.Nr)selfr4r6rrrr, sz&test_section_twice..Dummy.spamcSsdS)zCheese No cheese.Nr)r?cdrrrhamsz%test_section_twice..Dummy.hamN)__name__ __module__ __qualname____doc__r,rBrrrrDummys rGcSsdS)z Dummy function. Notes ----- First note. Notes ----- Second note. Nr)r(rrr dummy_funcs z&test_section_twice..dummy_funcz!test_section_twice..Dummyztest_docscrape.Dummyzfunction dummy_func)rr<robjectr rstrr )r=rGrHerrrtest_section_twices      rLcCsBtdddsttddds,ttttdddS)NZNotesrZInsteadrz definite.)rrrrr r'rrrr test_notes0srNcCs0tdddsttddds,tdS)NZ Referencesrz..rz2001.)rrrrrrrrtest_references6srOcCs0tdddsttddds,tdS)NZExamplesrz>>>rzTrue])rrrrrrrr test_examples;srPcCs>ttdddtttddtttddddS)NindexdefaultZrandomr0Zrefguide)r rr'rrrr test_index@srScCs.tdd|}tdd|}tdd|}|S)z1Remove leading, trailing and multiple blank linesz^\s*\nr/z\n\s*$z\n\s*\nz\n\n)resub)rrrr_strip_blank_linesFsrVcCsRt|}t|}ddt|dD}ddt|dD}t||dS)NcSsg|] }|qSr)rstrip)r!lrrrr$Qsz(line_by_line_compare.. cSsg|] }|qSr)rW)r!rXrrrr$Rs)textwrapdedentrVsplitr)r4r6rrrline_by_line_compareNs   r]cCstttddS)Na numpy.multivariate_normal(mean, cov, shape=None, spam=None) Draw values from a multivariate normal distribution with specified mean and covariance. The multivariate normal or Gaussian distribution is a generalisation of the one-dimensional normal distribution to higher dimensions. Parameters ---------- mean : (N,) ndarray Mean of the N-dimensional distribution. .. math:: (1+2+3)/3 cov : (N, N) ndarray Covariance matrix of the distribution. shape : tuple of ints Given a shape of, for example, (m,n,k), m*n*k samples are generated, and packed in an m-by-n-by-k arrangement. Because each sample is N-dimensional, the output shape is (m,n,k,N). Returns ------- out : ndarray The drawn samples, arranged according to `shape`. If the shape given is (m,n,...), then the shape of `out` is (m,n,...,N). In other words, each entry ``out[i,j,...,:]`` is an N-dimensional value drawn from the distribution. list of str This is not a real return value. It exists to test anonymous return values. no_description Other Parameters ---------------- spam : parrot A parrot off its mortal coil. Raises ------ RuntimeError Some error Warns ----- RuntimeWarning Some warning Warnings -------- Certain warnings apply. See Also -------- `some`_, `other`_, `funcs`_ `otherfunc`_ relationship Notes ----- Instead of specifying the full covariance matrix, popular approximations include: - Spherical covariance (`cov` is a multiple of the identity matrix) - Diagonal covariance (`cov` has non-negative elements only on the diagonal) This geometrical property can be seen in two dimensions by plotting generated data-points: >>> mean = [0,0] >>> cov = [[1,0],[0,100]] # diagonal covariance, points lie on x or y-axis >>> x,y = multivariate_normal(mean,cov,5000).T >>> plt.plot(x,y,'x'); plt.axis('equal'); plt.show() Note that the covariance matrix must be symmetric and non-negative definite. References ---------- .. [1] A. Papoulis, "Probability, Random Variables, and Stochastic Processes," 3rd ed., McGraw-Hill Companies, 1991 .. [2] R.O. Duda, P.E. Hart, and D.G. Stork, "Pattern Classification," 2nd ed., Wiley, 2001. Examples -------- >>> mean = (1,2) >>> cov = [[1,0],[1,0]] >>> x = multivariate_normal(mean,cov,(3,3)) >>> print x.shape (3, 3, 2) The following is probably true, given that 0.6 is roughly twice the standard deviation: >>> print list( (x[0,0,:] - mean) < 0.6 ) [True, True] .. index:: random :refguide: random;distributions, random;gauss)r]rJrrrrrtest_strVskr^cCstttddS)NzTest generator Yields ------ a : int The number of apples. b : int The number of bananas. int The number of unknowns. .. index:: )r]rJr7rrrrtest_yield_strs r_cCstt}tt|ddS)Na .. index:: random single: random;distributions, random;gauss Draw values from a multivariate normal distribution with specified mean and covariance. The multivariate normal or Gaussian distribution is a generalisation of the one-dimensional normal distribution to higher dimensions. :Parameters: **mean** : (N,) ndarray Mean of the N-dimensional distribution. .. math:: (1+2+3)/3 **cov** : (N, N) ndarray Covariance matrix of the distribution. **shape** : tuple of ints Given a shape of, for example, (m,n,k), m*n*k samples are generated, and packed in an m-by-n-by-k arrangement. Because each sample is N-dimensional, the output shape is (m,n,k,N). :Returns: **out** : ndarray The drawn samples, arranged according to `shape`. If the shape given is (m,n,...), then the shape of `out` is (m,n,...,N). In other words, each entry ``out[i,j,...,:]`` is an N-dimensional value drawn from the distribution. **list of str** This is not a real return value. It exists to test anonymous return values. **no_description** .. :Other Parameters: **spam** : parrot A parrot off its mortal coil. :Raises: **RuntimeError** Some error :Warns: **RuntimeWarning** Some warning .. warning:: Certain warnings apply. .. seealso:: :obj:`some`, :obj:`other`, :obj:`funcs` :obj:`otherfunc` relationship .. rubric:: Notes Instead of specifying the full covariance matrix, popular approximations include: - Spherical covariance (`cov` is a multiple of the identity matrix) - Diagonal covariance (`cov` has non-negative elements only on the diagonal) This geometrical property can be seen in two dimensions by plotting generated data-points: >>> mean = [0,0] >>> cov = [[1,0],[0,100]] # diagonal covariance, points lie on x or y-axis >>> x,y = multivariate_normal(mean,cov,5000).T >>> plt.plot(x,y,'x'); plt.axis('equal'); plt.show() Note that the covariance matrix must be symmetric and non-negative definite. .. rubric:: References .. [1] A. Papoulis, "Probability, Random Variables, and Stochastic Processes," 3rd ed., McGraw-Hill Companies, 1991 .. [2] R.O. Duda, P.E. Hart, and D.G. Stork, "Pattern Classification," 2nd ed., Wiley, 2001. .. only:: latex [1]_, [2]_ .. rubric:: Examples >>> mean = (1,2) >>> cov = [[1,0],[1,0]] >>> x = multivariate_normal(mean,cov,(3,3)) >>> print x.shape (3, 3, 2) The following is probably true, given that 0.6 is roughly twice the standard deviation: >>> print list( (x[0,0,:] - mean) < 0.6 ) [True, True] )r doc_txtr]rJ) sphinx_docrrrtest_sphinx_strssrbcCstt}tt|ddS)NzTest generator :Yields: **a** : int The number of apples. **b** : int The number of bananas. **int** The number of unknowns. )r doc_yields_txtr]rJ)rarrrtest_sphinx_yields_strPs rda Returns array of indices of the maximum values of along the given axis. 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