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kf/vf(H(@ Pf(ff(f(H(f(LG\HW01W(d$L\$l$\$l$d$XcH~@L MAiH BH5LwAf.GHf.AVAUATUHSH00L$ f(ȉIfʿPI\$(D$T$) $%HcHI$D$f( $A\$d$ I$LLT$(HLH@HAL$8f(ID$\Ml$Il$ Mt$0AT$(T$$${$$LLl$T$HLAD$Hf(\f(FIct$AXD$HI<$At$HAD$Hq$I$H0L[]A\A]A^DH~@AE1L 7LH A,H5!KH0L[]A\A]A^ff.HtKHcW w9}3BH)HHt@G HH9tHHf/Dw1f.gf.HHt SHH8H;H[AVAUIATUHS+IU@H5JIċELH1tLE1L5J~3ZH5>H=GL ~[]A\f(ff.AWAVAUATUSHHHt$,fAHf.TQAIcE]Gd-ALHD$(McHD$IN<JAJ4J,MHD$^Ht$YIBHD$ H$DMA@HD$H<$8f/c\f.QHL;t$(A^]YAH|$LD$DֹDHA1f(AI@fflHfYXfXL9u܉ЃtDHcH Y XA DL\f(A^DHYMXL9bH|$ EH<$IMLf(1EJ,HAAUH1AHHAuHH1[]A\A]A^A_HH[]A\A]A^A_L AfH 0H5Y<H=HHf[]A\A]A^A_dfDT$蹡Ht$@IIDrufVӀ@+tRH|$HT$H|$HtH|$AwHHcHH\$(HLl$D$D$Ll$8~\$$AGIcLd$8LcHEIL9nLL%LLl$8BDI9uL @-A2H  AH5A-H=-H蝔H|$3D$D$HH[]A\A]A^A_DHD$ HcHHfDL ,A2H ~H5,H=|,茿D$fL n,AfH F!H5,H=D,THܓD$Df.DHHH@Hff.ATISHHHT$0HL$8LD$@LL$Ht7)D$P)L$`)T$p)$)$)$)$)$H$s H;D$HD$HD$ HD$HcCD$ 09+fDs Hcbs HHHcC9}HL$LHǾCH1[A\fDAUIHATIUSH舕Au I}HʼnIcE(9%Au HcAu IEHIcE9}ۍULHHcjAmH1[]A\A]fHtUHH?Ht H]ff.@Ht HGf.H觙HtH1L + Ac2H H5c*ff.HHtH1L AcUH H5*趼̛f.ff/f(vf(f(*f/f/H(kf/f(L$T$#L$f/ %gf(T$uv{f(-\fTIof/kH(YYXfDff(^^XPk^f/ f%Xuw6f(T$d$L$kT$d$L$S@fH(XYjH(ۗf.fH8f(f/sfW-Vff/f(sfW5Dff(f(l$ ]\t$T$(Y\$f(d$詒d$\$t$l$ tT$(Yef(Cf/r1f/sf/ܸwfW%e1f/H8fH'Hf.H׏Hf.f.ATLcSLILHI~L1HÒIHL[A\DHt ffDff.@HGf~ lHHLf(HHfT_f(H9uf/f(tyf(f(fHHDfHf^fYXfXH9u߉tH^YXf.w%Qf(Yff(1Hf(T$茘T$HYff.fH?7GH(fH~%kHLf(HDHfT_f(H9uf/f(f(fHHDfHf^fYXfXH9u߉@tH^YXf.f(QYÅOσtuf(HfHHfDfHf^HH9uȃtHH^H(f(1kf(|1f(Ht$|$\$Ht$|$\$HHtkHtf~btf1fHffHfYXfXH9u݉tHYXf(fDff(f1HH_cython_3_1_6takes no arguments%.200s() %s (%zd given)takes exactly one argumentBad call flags for CyFunctiontakes no keyword arguments%.200s() %s__pyx_capi____loader__loader__file__origin__package__parent__path__submodule_search_locations[objid: %s] %d : %s => %sneeds an argumentkeywords must be stringsendunparsable format string'complex double''signed char''unsigned char''short''unsigned short''int''unsigned int''long''unsigned long''long long''unsigned long long''double''complex long double''bool''char''complex float''float'a structPython objecta pointera string'long double'an integer is requireds#dpdflogpdfpmfbuffer dtypeBuffer not C contiguous.scipy._lib._ccallbackLowLevelCallablePyCapsule_GetPointer failedinvalid callable givencannot import name %Snumpy._core._multiarray_umathnumpy.core._multiarray_umath_ARRAY_API_ARRAY_API is NULL pointernumpy.import_arrayBitGenerator_callback_wrapperexactlyself__pyx_unpickle__URNGtupleExpected %s, got %.200sNo module named '%U'name '%U' is not defined__reduce__wrap_dist__reduce_cython__Memoryview is not initialized_validate_domain__init___validate_qmc_inputat leastat mostset_random_stateu_errorppfppf_hat__setstate_cython___validate_args__cinit__qrvsbuiltinscython_runtime__builtins__does not matchnumpyflatiterbroadcastndarraygenericnumberunsignedintegerinexactcomplexfloatingflexiblecharacterufuncnumpy.random.bit_generatorSeedSequenceSeedlessSequencescipy._lib.messagestreamMessageStreamscipy._cyutilitymemoryview_allocate_bufferarray_cwrappermemoryview_cwrappermemview_sliceslice_memviewslicepybuffer_indexint (__Pyx_memviewslice *)transpose_memslicememoryview_fromsliceget_slice_from_memviewslice_copymemoryview_copymemoryview_copy_from_sliceget_best_orderslice_get_sizefill_contig_strides_arraycopy_data_to_temp_err_extents_err_dimint (PyObject *, PyObject *)_errint (void)_err_no_memorymemoryview_copy_contentsbroadcast_leadingrefcount_copyingrefcount_objects_in_slice_slice_assign_scalar__orig_bases____module____dictoffset____vectorcalloffset____weaklistoffset__func_doc__doc__func_name__name____qualname__func_dict__dict__func_globals__globals__func_closure__closure__func_code__code__func_defaults__defaults____kwdefaults____annotations___is_coroutineunuran_wrappermidpoint_errorsqueeze_hat_ratiohat_areasqueeze_areaintdouble__int__ returned non-int (type %.200s). The ability to return an instance of a strict subclass of int is deprecated, and may be removed in a future version of Python.__int__ returned non-int (type %.200s)%s() got an unexpected keyword argument '%U'base class '%.200s' is not a heap typeextension type '%.200s' has no __dict__ slot, but base type '%.200s' has: either add 'cdef dict __dict__' to the extension type or add '__slots__ = [...]' to the base typeC function %.200s.%.200s has wrong signature (expected %.500s, got %.500s)%.200s does not export expected C function %.200sInterpreter change detected - this module can only be loaded into one interpreter per process.Shared Cython type %.200s is not a type objectShared Cython type %.200s has the wrong size, try recompilingtoo many values to unpack (expected %zd)unbound method %.200S() needs an argument%.200s.%.200s is not a type object%.200s.%.200s size changed, may indicate binary incompatibility. Expected %zd from C header, got %zd from PyObjectUnexpected format string character: '%c'Acquisition count is %d (line %d)%.200s() keywords must be strings%s() got multiple values for keyword argument '%U'invalid vtable found for imported typemultiple bases have vtable conflict: '%.200s' and '%.200s'__annotations__ must be set to a dict object__qualname__ must be set to a string object__name__ must be set to a string object__kwdefaults__ must be set to a dict objectchanges to cyfunction.__kwdefaults__ will not currently affect the values used in function calls__defaults__ must be set to a tuple objectchanges to cyfunction.__defaults__ will not currently affect the values used in function callsfunction's dictionary may not be deletedsetting function's dictionary to a non-dictBuffer dtype mismatch, expected %s%s%s but got %sBuffer dtype mismatch, expected '%s' but got %s in '%s.%s'Expected a dimension of size %zu, got %zuExpected %d dimensions, got %dPython does not define a standard format string size for long double ('g')..Buffer dtype mismatch; next field is at offset %zd but %zd expectedBig-endian buffer not supported on little-endian compilerBuffer acquisition: Expected '{' after 'T'Cannot handle repeated arrays in format stringDoes not understand character buffer dtype format string ('%c')Expected a dimension of size %zu, got %dExpected a comma in format string, got '%c'Expected %d dimension(s), got %dUnexpected end of format string, expected ')'instance exception may not have a separate valuecalling %R should have returned an instance of BaseException, not %Rraise: exception class must be a subclass of BaseExceptionexception causes must derive from BaseExceptioncan't convert negative value to size_tvalue too large to convert to intBuffer has wrong number of dimensions (expected %d, got %d)Item size of buffer (%zu byte%s) does not match size of '%s' (%zu byte%s)Buffer exposes suboffsets but no stridesC-contiguous buffer is not indirect in dimension %dBuffer and memoryview are not contiguous in the same dimension.Buffer is not indirectly contiguous in dimension %d.Buffer is not indirectly accessible in dimension %d.Buffer not compatible with direct access in dimension %d.memviewslice is already initialized!hasattr(): attribute name must be stringcannot fit '%.200s' into an index-sized integer while calling a Python objectNULL result without error in PyObject_CallInvalid scipy.LowLevelCallable signature "%s". Expected one of: %R'%.200s' object is not subscriptable_ARRAY_API is not PyCapsule objectmodule compiled against ABI version 0x%x but this version of numpy is 0x%xmodule was compiled against NumPy C-API version 0x%x (NumPy 1.23) but the running NumPy has C-API version 0x%x. Check the section C-API incompatibility at the Troubleshooting ImportError section at https://numpy.org/devdocs/user/troubleshooting-importerror.html#c-api-incompatibility for indications on how to solve this problem.FATAL: module compiled as unknown endianFATAL: module compiled as little endian, but detected different endianness at runtime../../tmp/build-env-99e64tom/lib/python3.12/site-packages/numpy/__init__.cython-30.pxdscipy/stats/_unuran/unuran_wrapper.pyxscipy.stats._unuran.unuran_wrapper._URNG.get_urng'NoneType' object is not iterablescipy.stats._unuran.unuran_wrapper._pack_distr%.200s() takes %.8s %zd positional argument%.1s (%zd given)free variable '%s' referenced before assignment in enclosing scopescipy.stats._unuran.unuran_wrapper.TransformedDensityRejection.__cinit__._callback_wrapperscipy.stats._unuran.unuran_wrapper.TransformedDensityRejection.squeeze_hat_ratio.__get__scipy.stats._unuran.unuran_wrapper.TransformedDensityRejection.hat_area.__get__scipy.stats._unuran.unuran_wrapper.TransformedDensityRejection.squeeze_area.__get__scipy.stats._unuran.unuran_wrapper.SimpleRatioUniforms.__cinit__._callback_wrapperscipy.stats._unuran.unuran_wrapper.NumericalInversePolynomial.__cinit__._callback_wrapperscipy.stats._unuran.unuran_wrapper.NumericalInversePolynomial.intervals.__get__scipy.stats._unuran.unuran_wrapper.NumericalInverseHermite.__cinit__._callback_wrapperscipy.stats._unuran.unuran_wrapper.NumericalInverseHermite.intervals.__get__scipy.stats._unuran.unuran_wrapper.NumericalInverseHermite.midpoint_error.__get__'NoneType' object is not subscriptablescipy.stats._unuran.unuran_wrapper.__pyx_unpickle__URNG__set_statescipy.stats._unuran.unuran_wrapper.__pyx_unpickle__URNGscipy.stats._unuran.unuran_wrapper.SimpleRatioUniforms._validate_argsargument after ** must be a mapping, not NoneTypescipy.stats._unuran.unuran_wrapper.Method.__reduce__scipy.stats._unuran.unuran_wrapper.NumericalInversePolynomial._validate_argsscipy.stats._unuran.unuran_wrapper._URNG.get_qurngscipy.stats._unuran.unuran_wrapper.get_numpy_rngscipy.stats._unuran.unuran_wrapper._setup_unuranlocal variable '%s' referenced before assignmentjoin() result is too long for a Python stringscipy.stats._unuran.unuran_wrapper._unpack_distscipy.stats._unuran.unuran_wrapper.NumericalInversePolynomial._cdfOut of bounds on buffer access (axis %d)scipy.stats._unuran.unuran_wrapper.NumericalInversePolynomial.cdfscipy.stats._unuran.unuran_wrapper.Method._check_errorcodescipy.stats._unuran.unuran_wrapper.NumericalInversePolynomial.u_errorscipy.stats._unuran.unuran_wrapper._URNG.__reduce_cython__scipy.stats._unuran.unuran_wrapper._URNG._next_qdoublescipy.stats._unuran.unuran_wrapper._unpack_dist.wrap_dist.pdfscipy.stats._unuran.unuran_wrapper._unpack_dist.wrap_dist.logpdfscipy.stats._unuran.unuran_wrapper._validate_domainscipy.stats._unuran.unuran_wrapper._unpack_dist.wrap_dist.pmfscipy.stats._unuran.unuran_wrapper._unpack_dist.wrap_dist.__init__need more than %zd value%.1s to unpackscipy.stats._unuran.unuran_wrapper._unpack_dist.wrap_dist.cdfscipy.stats._unuran.unuran_wrapper._validate_qmc_inputscipy.stats._unuran.unuran_wrapper.Method.set_random_statescipy.stats._unuran.unuran_wrapper._validate_pvscipy.stats._unuran.unuran_wrapper._URNG.__init__scipy.stats._unuran.unuran_wrapper.NumericalInverseHermite.u_errorscipy.stats._unuran.unuran_wrapper.NumericalInversePolynomial.ppfscipy.stats._unuran.unuran_wrapper.NumericalInverseHermite.ppfscipy.stats._unuran.unuran_wrapper.TransformedDensityRejection.ppf_hatscipy.stats._unuran.unuran_wrapper.DiscreteAliasUrn._validate_argsscipy.stats._unuran.unuran_wrapper.Method._rvs_discrscipy.stats._unuran.unuran_wrapper.Method._rvs_contscipy.stats._unuran.unuran_wrapper.Method.rvsscipy.stats._unuran.unuran_wrapper._URNG.__setstate_cython__scipy.stats._unuran.unuran_wrapper.NumericalInverseHermite._validate_argsscipy.stats._unuran.unuran_wrapper.Method._set_rngscipy.stats._unuran.unuran_wrapper.DiscreteGuideTable.__cinit__scipy.stats._unuran.unuran_wrapper.DiscreteAliasUrn.__cinit__scipy.stats._unuran.unuran_wrapper.NumericalInverseHermite.__cinit__scipy.stats._unuran.unuran_wrapper.NumericalInversePolynomial.__cinit__scipy.stats._unuran.unuran_wrapper.TransformedDensityRejection.__cinit__scipy.stats._unuran.unuran_wrapper.DiscreteGuideTable._validate_argsscipy.stats._unuran.unuran_wrapper.SimpleRatioUniforms.__cinit__scipy.stats._unuran.unuran_wrapper.NumericalInverseHermite.qrvsscipy.stats._unuran.unuran_wrapper.NumericalInversePolynomial.qrvsscipy.stats._unuran.unuran_wrapper.DiscreteGuideTable.ppfscipy.stats._unuran.unuran_wrapper.TransformedDensityRejection._validate_argsModule 'unuran_wrapper' has already been imported. Re-initialisation is not supported.scipy.stats._unuran.unuran_wrappercompile time Python version %d.%d of module '%.100s' %s runtime version %d.%dUnable to initialize pickling for %.200sint (struct __pyx_array_obj *)struct __pyx_array_obj *(PyObject *, Py_ssize_t, char *, char const *, char *)PyObject *(PyObject *, int, int, __Pyx_TypeInfo const *)struct __pyx_memoryview_obj *(struct __pyx_memoryview_obj *, PyObject *)int (__Pyx_memviewslice *, Py_ssize_t, Py_ssize_t, Py_ssize_t, int, int, int *, Py_ssize_t, Py_ssize_t, Py_ssize_t, int, int, int, int)char *(Py_buffer *, char *, Py_ssize_t, Py_ssize_t)PyObject *(__Pyx_memviewslice, int, PyObject *(*)(char *), int (*)(char *, PyObject *), int)__Pyx_memviewslice *(struct __pyx_memoryview_obj *, __Pyx_memviewslice *)void (struct __pyx_memoryview_obj *, __Pyx_memviewslice *)PyObject *(struct __pyx_memoryview_obj *)PyObject *(struct __pyx_memoryview_obj *, __Pyx_memviewslice *)char (__Pyx_memviewslice *, int)Py_ssize_t (__Pyx_memviewslice *, int)Py_ssize_t (Py_ssize_t *, Py_ssize_t *, Py_ssize_t, int, char)void *(__Pyx_memviewslice *, __Pyx_memviewslice *, char, int)int (int, Py_ssize_t, Py_ssize_t)int (PyObject *, PyObject *, int)int (__Pyx_memviewslice, __Pyx_memviewslice, int, int, int)void (__Pyx_memviewslice *, int, int)void (__Pyx_memviewslice *, int, int, int)void (char *, Py_ssize_t *, Py_ssize_t *, int, int)void (__Pyx_memviewslice *, int, size_t, void *, int)void (char *, Py_ssize_t *, Py_ssize_t *, int, size_t, void *)__mro_entries__ must return a tuplemetaclass conflict: the metaclass of a derived class must be a (non-strict) subclass of the metaclasses of all its basesinit scipy.stats._unuran.unuran_wrapper_cython_3_1_6.cython_function_or_method_cython_3_1_6._common_types_metatypescipy.stats._unuran.unuran_wrapper.__pyx_scope_struct_3___cinit__scipy.stats._unuran.unuran_wrapper.__pyx_scope_struct_2___cinit__scipy.stats._unuran.unuran_wrapper.__pyx_scope_struct_1___cinit__scipy.stats._unuran.unuran_wrapper.__pyx_scope_struct____cinit__scipy.stats._unuran.unuran_wrapper.DiscreteGuideTable DiscreteGuideTable(dist, *, domain=None, guide_factor=1, random_state=None) Discrete Guide Table method. The Discrete Guide Table method samples from arbitrary, but finite, probability vectors. It uses the probability vector of size :math:`N` or a probability mass function with a finite support to generate random numbers from the distribution. Discrete Guide Table has a very slow set up (linear with the vector length) but provides very fast sampling. Parameters ---------- dist : array_like or object, optional Probability vector (PV) of the distribution. If PV isn't available, an instance of a class with a ``pmf`` method is expected. The signature of the PMF is expected to be: ``def pmf(self, k: int) -> float``. i.e. it should accept a Python integer and return a Python float. domain : int, optional Support of the PMF. If a probability vector (``pv``) is not available, a finite domain must be given. i.e. the PMF must have a finite support. Default is ``None``. When ``None``: * If a ``support`` method is provided by the distribution object `dist`, it is used to set the domain of the distribution. * Otherwise, the support is assumed to be ``(0, len(pv))``. When this parameter is passed in combination with a probability vector, ``domain[0]`` is used to relocate the distribution from ``(0, len(pv))`` to ``(domain[0], domain[0]+len(pv))`` and ``domain[1]`` is ignored. See Notes and tutorial for a more detailed explanation. guide_factor: int, optional Size of the guide table relative to length of PV. Larger guide tables result in faster generation time but require a more expensive setup. Sizes larger than 3 are not recommended. If the relative size is set to 0, sequential search is used. Default is 1. random_state : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional A NumPy random number generator or seed for the underlying NumPy random number generator used to generate the stream of uniform random numbers. If `random_state` is None (or `np.random`), the `numpy.random.RandomState` singleton is used. If `random_state` is an int, a new ``RandomState`` instance is used, seeded with `random_state`. If `random_state` is already a ``Generator`` or ``RandomState`` instance then that instance is used. Notes ----- This method works when either a finite probability vector is available or the PMF of the distribution is available. In case a PMF is only available, the *finite* support (domain) of the PMF must also be given. It is recommended to first obtain the probability vector by evaluating the PMF at each point in the support and then using it instead. DGT samples from arbitrary but finite probability vectors. Random numbers are generated by the inversion method, i.e. 1. Generate a random number U ~ U(0,1). 2. Find smallest integer I such that F(I) = P(X<=I) >= U. Step (2) is the crucial step. Using sequential search requires O(E(X)) comparisons, where E(X) is the expectation of the distribution. Indexed search, however, uses a guide table to jump to some I' <= I near I to find X in constant time. Indeed the expected number of comparisons is reduced to 2, when the guide table has the same size as the probability vector (this is the default). For larger guide tables this number becomes smaller (but is always larger than 1), for smaller tables it becomes larger. For the limit case of table size 1 the algorithm simply does sequential search. On the other hand the setup time for guide table is O(N), where N denotes the length of the probability vector (for size 1 no preprocessing is required). Moreover, for very large guide tables memory effects might even reduce the speed of the algorithm. So we do not recommend to use guide tables that are more than three times larger than the given probability vector. If only a few random numbers have to be generated, (much) smaller table sizes are better. The size of the guide table relative to the length of the given probability vector can be set by the ``guide_factor`` parameter. If a probability vector is given, it must be a 1-dimensional array of non-negative floats without any ``inf`` or ``nan`` values. Also, there must be at least one non-zero entry otherwise an exception is raised. By default, the probability vector is indexed starting at 0. However, this can be changed by passing a ``domain`` parameter. When ``domain`` is given in combination with the PV, it has the effect of relocating the distribution from ``(0, len(pv))`` to ``(domain[0], domain[0] + len(pv))``. ``domain[1]`` is ignored in this case. References ---------- .. [1] UNU.RAN reference manual, Section 5.8.4, "DGT - (Discrete) Guide Table method (indexed search)" https://statmath.wu.ac.at/unuran/doc/unuran.html#DGT .. [2] H.C. Chen and Y. Asau (1974). On generating random variates from an empirical distribution, AIIE Trans. 6, pp. 163-166. Examples -------- >>> from scipy.stats.sampling import DiscreteGuideTable >>> import numpy as np To create a random number generator using a probability vector, use: >>> pv = [0.1, 0.3, 0.6] >>> urng = np.random.default_rng() >>> rng = DiscreteGuideTable(pv, random_state=urng) The RNG has been setup. Now, we can now use the `rvs` method to generate samples from the distribution: >>> rvs = rng.rvs(size=1000) To verify that the random variates follow the given distribution, we can use the chi-squared test (as a measure of goodness-of-fit): >>> from scipy.stats import chisquare >>> _, freqs = np.unique(rvs, return_counts=True) >>> freqs = freqs / np.sum(freqs) >>> freqs array([0.092, 0.355, 0.553]) >>> chisquare(freqs, pv).pvalue 0.9987382966178464 As the p-value is very high, we fail to reject the null hypothesis that the observed frequencies are the same as the expected frequencies. Hence, we can safely assume that the variates have been generated from the given distribution. Note that this just gives the correctness of the algorithm and not the quality of the samples. If a PV is not available, an instance of a class with a PMF method and a finite domain can also be passed. >>> urng = np.random.default_rng() >>> from scipy.stats import binom >>> n, p = 10, 0.2 >>> dist = binom(n, p) >>> rng = DiscreteGuideTable(dist, random_state=urng) Now, we can sample from the distribution using the `rvs` method and also measure the goodness-of-fit of the samples: >>> rvs = rng.rvs(1000) >>> _, freqs = np.unique(rvs, return_counts=True) >>> freqs = freqs / np.sum(freqs) >>> obs_freqs = np.zeros(11) # some frequencies may be zero. >>> obs_freqs[:freqs.size] = freqs >>> pv = [dist.pmf(i) for i in range(0, 11)] >>> pv = np.asarray(pv) / np.sum(pv) >>> chisquare(obs_freqs, pv).pvalue 0.9999999999999989 To check that the samples have been drawn from the correct distribution, we can visualize the histogram of the samples: >>> import matplotlib.pyplot as plt >>> rvs = rng.rvs(1000) >>> fig = plt.figure() >>> ax = fig.add_subplot(111) >>> x = np.arange(0, n+1) >>> fx = dist.pmf(x) >>> fx = fx / fx.sum() >>> ax.plot(x, fx, 'bo', label='true distribution') >>> ax.vlines(x, 0, fx, lw=2) >>> ax.hist(rvs, bins=np.r_[x, n+1]-0.5, density=True, alpha=0.5, ... color='r', label='samples') >>> ax.set_xlabel('x') >>> ax.set_ylabel('PMF(x)') >>> ax.set_title('Discrete Guide Table Samples') >>> plt.legend() >>> plt.show() To set the size of the guide table use the `guide_factor` keyword argument. This sets the size of the guide table relative to the probability vector >>> rng = DiscreteGuideTable(pv, guide_factor=1, random_state=urng) To calculate the PPF of a binomial distribution with :math:`n=4` and :math:`p=0.1`: we can set up a guide table as follows: >>> n, p = 4, 0.1 >>> dist = binom(n, p) >>> rng = DiscreteGuideTable(dist, random_state=42) >>> rng.ppf(0.5) 0.0 scipy.stats._unuran.unuran_wrapper.DiscreteAliasUrn DiscreteAliasUrn(dist, *, domain=None, urn_factor=1, random_state=None) Discrete Alias-Urn Method. This method is used to sample from univariate discrete distributions with a finite domain. It uses the probability vector of size :math:`N` or a probability mass function with a finite support to generate random numbers from the distribution. Parameters ---------- dist : array_like or object, optional Probability vector (PV) of the distribution. If PV isn't available, an instance of a class with a ``pmf`` method is expected. The signature of the PMF is expected to be: ``def pmf(self, k: int) -> float``. i.e. it should accept a Python integer and return a Python float. domain : int, optional Support of the PMF. If a probability vector (``pv``) is not available, a finite domain must be given. i.e. the PMF must have a finite support. Default is ``None``. When ``None``: * If a ``support`` method is provided by the distribution object `dist`, it is used to set the domain of the distribution. * Otherwise, the support is assumed to be ``(0, len(pv))``. When this parameter is passed in combination with a probability vector, ``domain[0]`` is used to relocate the distribution from ``(0, len(pv))`` to ``(domain[0], domain[0]+len(pv))`` and ``domain[1]`` is ignored. See Notes and tutorial for a more detailed explanation. urn_factor : float, optional Size of the urn table *relative* to the size of the probability vector. It must not be less than 1. Larger tables result in faster generation times but require a more expensive setup. Default is 1. random_state : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional A NumPy random number generator or seed for the underlying NumPy random number generator used to generate the stream of uniform random numbers. If `random_state` is None (or `np.random`), the `numpy.random.RandomState` singleton is used. If `random_state` is an int, a new ``RandomState`` instance is used, seeded with `random_state`. If `random_state` is already a ``Generator`` or ``RandomState`` instance then that instance is used. Notes ----- This method works when either a finite probability vector is available or the PMF of the distribution is available. In case a PMF is only available, the *finite* support (domain) of the PMF must also be given. It is recommended to first obtain the probability vector by evaluating the PMF at each point in the support and then using it instead. If a probability vector is given, it must be a 1-dimensional array of non-negative floats without any ``inf`` or ``nan`` values. Also, there must be at least one non-zero entry otherwise an exception is raised. By default, the probability vector is indexed starting at 0. However, this can be changed by passing a ``domain`` parameter. When ``domain`` is given in combination with the PV, it has the effect of relocating the distribution from ``(0, len(pv))`` to ``(domain[0]``, ``domain[0] + len(pv))``. ``domain[1]`` is ignored in this case. The parameter ``urn_factor`` can be increased for faster generation at the cost of increased setup time. This method uses a table for random variate generation. ``urn_factor`` controls the size of this table relative to the size of the probability vector (or width of the support, in case a PV is not available). As this table is computed during setup time, increasing this parameter linearly increases the time required to setup. It is recommended to keep this parameter under 2. References ---------- .. [1] UNU.RAN reference manual, Section 5.8.2, "DAU - (Discrete) Alias-Urn method", http://statmath.wu.ac.at/software/unuran/doc/unuran.html#DAU .. [2] A.J. Walker (1977). An efficient method for generating discrete random variables with general distributions, ACM Trans. Math. Software 3, pp. 253-256. Examples -------- >>> from scipy.stats.sampling import DiscreteAliasUrn >>> import numpy as np To create a random number generator using a probability vector, use: >>> pv = [0.1, 0.3, 0.6] >>> urng = np.random.default_rng() >>> rng = DiscreteAliasUrn(pv, random_state=urng) The RNG has been setup. Now, we can now use the `rvs` method to generate samples from the distribution: >>> rvs = rng.rvs(size=1000) To verify that the random variates follow the given distribution, we can use the chi-squared test (as a measure of goodness-of-fit): >>> from scipy.stats import chisquare >>> _, freqs = np.unique(rvs, return_counts=True) >>> freqs = freqs / np.sum(freqs) >>> freqs array([0.092, 0.292, 0.616]) >>> chisquare(freqs, pv).pvalue 0.9993602047563164 As the p-value is very high, we fail to reject the null hypothesis that the observed frequencies are the same as the expected frequencies. Hence, we can safely assume that the variates have been generated from the given distribution. Note that this just gives the correctness of the algorithm and not the quality of the samples. If a PV is not available, an instance of a class with a PMF method and a finite domain can also be passed. >>> urng = np.random.default_rng() >>> class Binomial: ... def __init__(self, n, p): ... self.n = n ... self.p = p ... def pmf(self, x): ... # note that the pmf doesn't need to be normalized. ... return self.p**x * (1-self.p)**(self.n-x) ... def support(self): ... return (0, self.n) ... >>> n, p = 10, 0.2 >>> dist = Binomial(n, p) >>> rng = DiscreteAliasUrn(dist, random_state=urng) Now, we can sample from the distribution using the `rvs` method and also measure the goodness-of-fit of the samples: >>> rvs = rng.rvs(1000) >>> _, freqs = np.unique(rvs, return_counts=True) >>> freqs = freqs / np.sum(freqs) >>> obs_freqs = np.zeros(11) # some frequencies may be zero. >>> obs_freqs[:freqs.size] = freqs >>> pv = [dist.pmf(i) for i in range(0, 11)] >>> pv = np.asarray(pv) / np.sum(pv) >>> chisquare(obs_freqs, pv).pvalue 0.9999999999999999 To check that the samples have been drawn from the correct distribution, we can visualize the histogram of the samples: >>> import matplotlib.pyplot as plt >>> rvs = rng.rvs(1000) >>> fig = plt.figure() >>> ax = fig.add_subplot(111) >>> x = np.arange(0, n+1) >>> fx = dist.pmf(x) >>> fx = fx / fx.sum() >>> ax.plot(x, fx, 'bo', label='true distribution') >>> ax.vlines(x, 0, fx, lw=2) >>> ax.hist(rvs, bins=np.r_[x, n+1]-0.5, density=True, alpha=0.5, ... color='r', label='samples') >>> ax.set_xlabel('x') >>> ax.set_ylabel('PMF(x)') >>> ax.set_title('Discrete Alias Urn Samples') >>> plt.legend() >>> plt.show() To set the ``urn_factor``, use: >>> rng = DiscreteAliasUrn(pv, urn_factor=2, random_state=urng) This uses a table twice the size of the probability vector to generate random variates from the distribution. scipy.stats._unuran.unuran_wrapper.NumericalInverseHermite NumericalInverseHermite(dist, *, domain=None, order=3, u_resolution=1e-12, construction_points=None, random_state=None) Hermite interpolation based INVersion of CDF (HINV). HINV is a variant of numerical inversion, where the inverse CDF is approximated using Hermite interpolation, i.e., the interval [0,1] is split into several intervals and in each interval the inverse CDF is approximated by polynomials constructed by means of values of the CDF and PDF at interval boundaries. This makes it possible to improve the accuracy by splitting a particular interval without recomputations in unaffected intervals. Three types of splines are implemented: linear, cubic, and quintic interpolation. For linear interpolation only the CDF is required. Cubic interpolation also requires PDF and quintic interpolation PDF and its derivative. These splines have to be computed in a setup step. However, it only works for distributions with bounded domain; for distributions with unbounded domain the tails are chopped off such that the probability for the tail regions is small compared to the given u-resolution. The method is not exact, as it only produces random variates of the approximated distribution. Nevertheless, the maximal numerical error in "u-direction" (i.e. ``|U - CDF(X)|`` where ``X`` is the approximate percentile corresponding to the quantile ``U`` i.e. ``X = approx_ppf(U)``) can be set to the required resolution (within machine precision). Notice that very small values of the u-resolution are possible but may increase the cost for the setup step. Parameters ---------- dist : object An instance of a class with a ``cdf`` and optionally a ``pdf`` and ``dpdf`` method. * ``cdf``: CDF of the distribution. The signature of the CDF is expected to be: ``def cdf(self, x: float) -> float``. i.e. the CDF should accept a Python float and return a Python float. * ``pdf``: PDF of the distribution. This method is optional when ``order=1``. Must have the same signature as the PDF. * ``dpdf``: Derivative of the PDF w.r.t the variate (i.e. ``x``). This method is optional with ``order=1`` or ``order=3``. Must have the same signature as the CDF. domain : list or tuple of length 2, optional The support of the distribution. Default is ``None``. When ``None``: * If a ``support`` method is provided by the distribution object `dist`, it is used to set the domain of the distribution. * Otherwise the support is assumed to be :math:`(-\infty, \infty)`. order : int, default: ``3`` Set order of Hermite interpolation. Valid orders are 1, 3, and 5. Valid orders are 1, 3, and 5. Notice that order greater than 1 requires the density of the distribution, and order greater than 3 even requires the derivative of the density. Using order 1 results for most distributions in a huge number of intervals and is therefore not recommended. If the maximal error in u-direction is very small (say smaller than 1.e-10), order 5 is recommended as it leads to considerably fewer design points, as long there are no poles or heavy tails. u_resolution : float, default: ``1e-12`` Set maximal tolerated u-error. Notice that the resolution of most uniform random number sources is 2-32= 2.3e-10. Thus a value of 1.e-10 leads to an inversion algorithm that could be called exact. For most simulations slightly bigger values for the maximal error are enough as well. Default is 1e-12. construction_points : array_like, optional Set starting construction points (nodes) for Hermite interpolation. As the possible maximal error is only estimated in the setup it may be necessary to set some special design points for computing the Hermite interpolation to guarantee that the maximal u-error can not be bigger than desired. Such points are points where the density is not differentiable or has a local extremum. random_state : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional A NumPy random number generator or seed for the underlying NumPy random number generator used to generate the stream of uniform random numbers. If `random_state` is None (or `np.random`), the `numpy.random.RandomState` singleton is used. If `random_state` is an int, a new ``RandomState`` instance is used, seeded with `random_state`. If `random_state` is already a ``Generator`` or ``RandomState`` instance then that instance is used. Notes ----- `NumericalInverseHermite` approximates the inverse of a continuous statistical distribution's CDF with a Hermite spline. Order of the hermite spline can be specified by passing the `order` parameter. As described in [1]_, it begins by evaluating the distribution's PDF and CDF at a mesh of quantiles ``x`` within the distribution's support. It uses the results to fit a Hermite spline ``H`` such that ``H(p) == x``, where ``p`` is the array of percentiles corresponding with the quantiles ``x``. Therefore, the spline approximates the inverse of the distribution's CDF to machine precision at the percentiles ``p``, but typically, the spline will not be as accurate at the midpoints between the percentile points:: p_mid = (p[:-1] + p[1:])/2 so the mesh of quantiles is refined as needed to reduce the maximum "u-error":: u_error = np.max(np.abs(dist.cdf(H(p_mid)) - p_mid)) below the specified tolerance `u_resolution`. Refinement stops when the required tolerance is achieved or when the number of mesh intervals after the next refinement could exceed the maximum allowed number of intervals, which is 100000. References ---------- .. [1] Hörmann, Wolfgang, and Josef Leydold. "Continuous random variate generation by fast numerical inversion." ACM Transactions on Modeling and Computer Simulation (TOMACS) 13.4 (2003): 347-362. .. [2] UNU.RAN reference manual, Section 5.3.5, "HINV - Hermite interpolation based INVersion of CDF", https://statmath.wu.ac.at/software/unuran/doc/unuran.html#HINV Examples -------- >>> from scipy.stats.sampling import NumericalInverseHermite >>> from scipy.stats import norm, genexpon >>> from scipy.special import ndtr >>> import numpy as np To create a generator to sample from the standard normal distribution, do: >>> class StandardNormal: ... def pdf(self, x): ... return 1/np.sqrt(2*np.pi) * np.exp(-x**2 / 2) ... def cdf(self, x): ... return ndtr(x) ... >>> dist = StandardNormal() >>> urng = np.random.default_rng() >>> rng = NumericalInverseHermite(dist, random_state=urng) The `NumericalInverseHermite` has a method that approximates the PPF of the distribution. >>> rng = NumericalInverseHermite(dist) >>> p = np.linspace(0.01, 0.99, 99) # percentiles from 1% to 99% >>> np.allclose(rng.ppf(p), norm.ppf(p)) True Depending on the implementation of the distribution's random sampling method, the random variates generated may be nearly identical, given the same random state. >>> dist = genexpon(9, 16, 3) >>> rng = NumericalInverseHermite(dist) >>> # `seed` ensures identical random streams are used by each `rvs` method >>> seed = 500072020 >>> rvs1 = dist.rvs(size=100, random_state=np.random.default_rng(seed)) >>> rvs2 = rng.rvs(size=100, random_state=np.random.default_rng(seed)) >>> np.allclose(rvs1, rvs2) True To check that the random variates closely follow the given distribution, we can look at its histogram: >>> import matplotlib.pyplot as plt >>> dist = StandardNormal() >>> rng = NumericalInverseHermite(dist) >>> rvs = rng.rvs(10000) >>> x = np.linspace(rvs.min()-0.1, rvs.max()+0.1, 1000) >>> fx = norm.pdf(x) >>> plt.plot(x, fx, 'r-', lw=2, label='true distribution') >>> plt.hist(rvs, bins=20, density=True, alpha=0.8, label='random variates') >>> plt.xlabel('x') >>> plt.ylabel('PDF(x)') >>> plt.title('Numerical Inverse Hermite Samples') >>> plt.legend() >>> plt.show() Given the derivative of the PDF w.r.t the variate (i.e. ``x``), we can use quintic Hermite interpolation to approximate the PPF by passing the `order` parameter: >>> class StandardNormal: ... def pdf(self, x): ... return 1/np.sqrt(2*np.pi) * np.exp(-x**2 / 2) ... def dpdf(self, x): ... return -1/np.sqrt(2*np.pi) * x * np.exp(-x**2 / 2) ... def cdf(self, x): ... return ndtr(x) ... >>> dist = StandardNormal() >>> urng = np.random.default_rng() >>> rng = NumericalInverseHermite(dist, order=5, random_state=urng) Higher orders result in a fewer number of intervals: >>> rng3 = NumericalInverseHermite(dist, order=3) >>> rng5 = NumericalInverseHermite(dist, order=5) >>> rng3.intervals, rng5.intervals (3000, 522) The u-error can be estimated by calling the `u_error` method. It runs a small Monte-Carlo simulation to estimate the u-error. By default, 100,000 samples are used. This can be changed by passing the `sample_size` argument: >>> rng1 = NumericalInverseHermite(dist, u_resolution=1e-10) >>> rng1.u_error(sample_size=1000000) # uses one million samples UError(max_error=9.53167544892608e-11, mean_absolute_error=2.2450136432146864e-11) This returns a namedtuple which contains the maximum u-error and the mean absolute u-error. The u-error can be reduced by decreasing the u-resolution (maximum allowed u-error): >>> rng2 = NumericalInverseHermite(dist, u_resolution=1e-13) >>> rng2.u_error(sample_size=1000000) UError(max_error=9.32027892364129e-14, mean_absolute_error=1.5194172675685075e-14) Note that this comes at the cost of increased setup time and number of intervals. >>> rng1.intervals 1022 >>> rng2.intervals 5687 >>> from timeit import timeit >>> f = lambda: NumericalInverseHermite(dist, u_resolution=1e-10) >>> timeit(f, number=1) 0.017409582000254886 # may vary >>> f = lambda: NumericalInverseHermite(dist, u_resolution=1e-13) >>> timeit(f, number=1) 0.08671202100003939 # may vary Since the PPF of the normal distribution is available as a special function, we can also check the x-error, i.e. the difference between the approximated PPF and exact PPF:: >>> import matplotlib.pyplot as plt >>> u = np.linspace(0.01, 0.99, 1000) >>> approxppf = rng.ppf(u) >>> exactppf = norm.ppf(u) >>> error = np.abs(exactppf - approxppf) >>> plt.plot(u, error) >>> plt.xlabel('u') >>> plt.ylabel('error') >>> plt.title('Error between exact and approximated PPF (x-error)') >>> plt.show() Get number of nodes (design points) used for Hermite interpolation in the generator object. The number of intervals is the number of nodes minus 1. scipy.stats._unuran.unuran_wrapper.NumericalInversePolynomial NumericalInversePolynomial(dist, *, mode=None, center=None, domain=None, order=5, u_resolution=1e-10, random_state=None) Polynomial interpolation based INVersion of CDF (PINV). PINV is a variant of numerical inversion, where the inverse CDF is approximated using Newton's interpolating formula. The interval ``[0,1]`` is split into several subintervals. In each of these, the inverse CDF is constructed at nodes ``(CDF(x),x)`` for some points ``x`` in this subinterval. If the PDF is given, then the CDF is computed numerically from the given PDF using adaptive Gauss-Lobatto integration with 5 points. Subintervals are split until the requested accuracy goal is reached. The method is not exact, as it only produces random variates of the approximated distribution. Nevertheless, the maximal tolerated approximation error can be set to be the resolution (but, of course, is bounded by the machine precision). We use the u-error ``|U - CDF(X)|`` to measure the error where ``X`` is the approximate percentile corresponding to the quantile ``U`` i.e. ``X = approx_ppf(U)``. We call the maximal tolerated u-error the u-resolution of the algorithm. Both the order of the interpolating polynomial and the u-resolution can be selected. Note that very small values of the u-resolution are possible but increase the cost for the setup step. The interpolating polynomials have to be computed in a setup step. However, it only works for distributions with bounded domain; for distributions with unbounded domain the tails are cut off such that the probability for the tail regions is small compared to the given u-resolution. The construction of the interpolation polynomial only works when the PDF is unimodal or when the PDF does not vanish between two modes. There are some restrictions for the given distribution: * The support of the distribution (i.e., the region where the PDF is strictly positive) must be connected. In practice this means, that the region where PDF is "not too small" must be connected. Unimodal densities satisfy this condition. If this condition is violated then the domain of the distribution might be truncated. * When the PDF is integrated numerically, then the given PDF must be continuous and should be smooth. * The PDF must be bounded. * The algorithm has problems when the distribution has heavy tails (as then the inverse CDF becomes very steep at 0 or 1) and the requested u-resolution is very small. E.g., the Cauchy distribution is likely to show this problem when the requested u-resolution is less then 1.e-12. Parameters ---------- dist : object An instance of a class with a ``pdf`` or ``logpdf`` method, optionally a ``cdf`` method. * ``pdf``: PDF of the distribution. The signature of the PDF is expected to be: ``def pdf(self, x: float) -> float``, i.e., the PDF should accept a Python float and return a Python float. It doesn't need to integrate to 1, i.e., the PDF doesn't need to be normalized. This method is optional, but either ``pdf`` or ``logpdf`` need to be specified. If both are given, ``logpdf`` is used. * ``logpdf``: The log of the PDF of the distribution. The signature is the same as for ``pdf``. Similarly, log of the normalization constant of the PDF can be ignored. This method is optional, but either ``pdf`` or ``logpdf`` need to be specified. If both are given, ``logpdf`` is used. * ``cdf``: CDF of the distribution. This method is optional. If provided, it enables the calculation of "u-error". See `u_error`. Must have the same signature as the PDF. mode : float, optional (Exact) Mode of the distribution. Default is ``None``. center : float, optional Approximate location of the mode or the mean of the distribution. This location provides some information about the main part of the PDF and is used to avoid numerical problems. Default is ``None``. domain : list or tuple of length 2, optional The support of the distribution. Default is ``None``. When ``None``: * If a ``support`` method is provided by the distribution object `dist`, it is used to set the domain of the distribution. * Otherwise the support is assumed to be :math:`(-\infty, \infty)`. order : int, optional Order of the interpolating polynomial. Valid orders are between 3 and 17. Higher orders result in fewer intervals for the approximations. Default is 5. u_resolution : float, optional Set maximal tolerated u-error. Values of u_resolution must at least 1.e-15 and 1.e-5 at most. Notice that the resolution of most uniform random number sources is 2-32= 2.3e-10. Thus a value of 1.e-10 leads to an inversion algorithm that could be called exact. For most simulations slightly bigger values for the maximal error are enough as well. Default is 1e-10. random_state : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional A NumPy random number generator or seed for the underlying NumPy random number generator used to generate the stream of uniform random numbers. If `random_state` is None (or `np.random`), the `numpy.random.RandomState` singleton is used. If `random_state` is an int, a new ``RandomState`` instance is used, seeded with `random_state`. If `random_state` is already a ``Generator`` or ``RandomState`` instance then that instance is used. References ---------- .. [1] Derflinger, Gerhard, Wolfgang Hörmann, and Josef Leydold. "Random variate generation by numerical inversion when only the density is known." ACM Transactions on Modeling and Computer Simulation (TOMACS) 20.4 (2010): 1-25. .. [2] UNU.RAN reference manual, Section 5.3.12, "PINV - Polynomial interpolation based INVersion of CDF", https://statmath.wu.ac.at/software/unuran/doc/unuran.html#PINV Examples -------- >>> from scipy.stats.sampling import NumericalInversePolynomial >>> from scipy.stats import norm >>> import numpy as np To create a generator to sample from the standard normal distribution, do: >>> class StandardNormal: ... def pdf(self, x): ... return np.exp(-0.5 * x*x) ... >>> dist = StandardNormal() >>> urng = np.random.default_rng() >>> rng = NumericalInversePolynomial(dist, random_state=urng) Once a generator is created, samples can be drawn from the distribution by calling the `rvs` method: >>> rng.rvs() -1.5244996276336318 To check that the random variates closely follow the given distribution, we can look at it's histogram: >>> import matplotlib.pyplot as plt >>> rvs = rng.rvs(10000) >>> x = np.linspace(rvs.min()-0.1, rvs.max()+0.1, 1000) >>> fx = norm.pdf(x) >>> plt.plot(x, fx, 'r-', lw=2, label='true distribution') >>> plt.hist(rvs, bins=20, density=True, alpha=0.8, label='random variates') >>> plt.xlabel('x') >>> plt.ylabel('PDF(x)') >>> plt.title('Numerical Inverse Polynomial Samples') >>> plt.legend() >>> plt.show() It is possible to estimate the u-error of the approximated PPF if the exact CDF is available during setup. To do so, pass a `dist` object with exact CDF of the distribution during initialization: >>> from scipy.special import ndtr >>> class StandardNormal: ... def pdf(self, x): ... return np.exp(-0.5 * x*x) ... def cdf(self, x): ... return ndtr(x) ... >>> dist = StandardNormal() >>> urng = np.random.default_rng() >>> rng = NumericalInversePolynomial(dist, random_state=urng) Now, the u-error can be estimated by calling the `u_error` method. It runs a Monte-Carlo simulation to estimate the u-error. By default, 100000 samples are used. To change this, you can pass the number of samples as an argument: >>> rng.u_error(sample_size=1000000) # uses one million samples UError(max_error=8.785994154436594e-11, mean_absolute_error=2.930890027826552e-11) This returns a namedtuple which contains the maximum u-error and the mean absolute u-error. The u-error can be reduced by decreasing the u-resolution (maximum allowed u-error): >>> urng = np.random.default_rng() >>> rng = NumericalInversePolynomial(dist, u_resolution=1.e-12, random_state=urng) >>> rng.u_error(sample_size=1000000) UError(max_error=9.07496300328603e-13, mean_absolute_error=3.5255644517257716e-13) Note that this comes at the cost of increased setup time. The approximated PPF can be evaluated by calling the `ppf` method: >>> rng.ppf(0.975) 1.9599639857012559 >>> norm.ppf(0.975) 1.959963984540054 Since the PPF of the normal distribution is available as a special function, we can also check the x-error, i.e. the difference between the approximated PPF and exact PPF:: >>> import matplotlib.pyplot as plt >>> u = np.linspace(0.01, 0.99, 1000) >>> approxppf = rng.ppf(u) >>> exactppf = norm.ppf(u) >>> error = np.abs(exactppf - approxppf) >>> plt.plot(u, error) >>> plt.xlabel('u') >>> plt.ylabel('error') >>> plt.title('Error between exact and approximated PPF (x-error)') >>> plt.show() Get the number of intervals used in the computation.scipy.stats._unuran.unuran_wrapper.SimpleRatioUniforms SimpleRatioUniforms(dist, *, mode=None, pdf_area=1, domain=None, cdf_at_mode=None, random_state=None) Simple Ratio-of-Uniforms (SROU) Method. SROU is based on the ratio-of-uniforms method that uses universal inequalities for constructing a (universal) bounding rectangle. It works for T-concave distributions with ``T(x) = -1/sqrt(x)``. The main advantage of the method is a fast setup. This can be beneficial if one repeatedly needs to generate small to moderate samples of a distribution with different shape parameters. In such a situation, the setup step of `NumericalInverseHermite` or `NumericalInversePolynomial` will lead to poor performance. Parameters ---------- dist : object An instance of a class with ``pdf`` method. * ``pdf``: PDF of the distribution. The signature of the PDF is expected to be: ``def pdf(self, x: float) -> float``. i.e. the PDF should accept a Python float and return a Python float. It doesn't need to integrate to 1 i.e. the PDF doesn't need to be normalized. If not normalized, `pdf_area` should be set to the area under the PDF. mode : float, optional (Exact) Mode of the distribution. When the mode is ``None``, a slow numerical routine is used to approximate it. Default is ``None``. pdf_area : float, optional Area under the PDF. Optionally, an upper bound to the area under the PDF can be passed at the cost of increased rejection constant. Default is 1. domain : list or tuple of length 2, optional The support of the distribution. Default is ``None``. When ``None``: * If a ``support`` method is provided by the distribution object `dist`, it is used to set the domain of the distribution. * Otherwise the support is assumed to be :math:`(-\infty, \infty)`. cdf_at_mode : float, optional CDF at the mode. It can be given to increase the performance of the algorithm. The rejection constant is halfed when CDF at mode is given. Default is ``None``. random_state : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional A NumPy random number generator or seed for the underlying NumPy random number generator used to generate the stream of uniform random numbers. If `random_state` is None (or `np.random`), the `numpy.random.RandomState` singleton is used. If `random_state` is an int, a new ``RandomState`` instance is used, seeded with `random_state`. If `random_state` is already a ``Generator`` or ``RandomState`` instance then that instance is used. References ---------- .. [1] UNU.RAN reference manual, Section 5.3.16, "SROU - Simple Ratio-of-Uniforms method", http://statmath.wu.ac.at/software/unuran/doc/unuran.html#SROU .. [2] Leydold, Josef. "A simple universal generator for continuous and discrete univariate T-concave distributions." ACM Transactions on Mathematical Software (TOMS) 27.1 (2001): 66-82 .. [3] Leydold, Josef. "Short universal generators via generalized ratio-of-uniforms method." Mathematics of Computation 72.243 (2003): 1453-1471 Examples -------- >>> from scipy.stats.sampling import SimpleRatioUniforms >>> import numpy as np Suppose we have the normal distribution: >>> class StdNorm: ... def pdf(self, x): ... return np.exp(-0.5 * x**2) Notice that the PDF doesn't integrate to 1. We can either pass the exact area under the PDF during initialization of the generator or an upper bound to the exact area under the PDF. Also, it is recommended to pass the mode of the distribution to speed up the setup: >>> urng = np.random.default_rng() >>> dist = StdNorm() >>> rng = SimpleRatioUniforms(dist, mode=0, ... pdf_area=np.sqrt(2*np.pi), ... random_state=urng) Now, we can use the `rvs` method to generate samples from the distribution: >>> rvs = rng.rvs(10) If the CDF at mode is available, it can be set to improve the performance of `rvs`: >>> from scipy.stats import norm >>> rng = SimpleRatioUniforms(dist, mode=0, ... pdf_area=np.sqrt(2*np.pi), ... cdf_at_mode=norm.cdf(0), ... random_state=urng) >>> rvs = rng.rvs(1000) We can check that the samples are from the given distribution by visualizing its histogram: >>> import matplotlib.pyplot as plt >>> x = np.linspace(rvs.min()-0.1, rvs.max()+0.1, 1000) >>> fx = 1/np.sqrt(2*np.pi) * dist.pdf(x) >>> fig, ax = plt.subplots() >>> ax.plot(x, fx, 'r-', lw=2, label='true distribution') >>> ax.hist(rvs, bins=10, density=True, alpha=0.8, label='random variates') >>> ax.set_xlabel('x') >>> ax.set_ylabel('PDF(x)') >>> ax.set_title('Simple Ratio-of-Uniforms Samples') >>> ax.legend() >>> plt.show() scipy.stats._unuran.unuran_wrapper.TransformedDensityRejection TransformedDensityRejection(dist, *, mode=None, center=None, domain=None, c=-0.5, construction_points=30, use_dars=True, max_squeeze_hat_ratio=0.99, random_state=None) Transformed Density Rejection (TDR) Method. TDR is an acceptance/rejection method that uses the concavity of a transformed density to construct hat function and squeezes automatically. Most universal algorithms are very slow compared to algorithms that are specialized to that distribution. Algorithms that are fast have a slow setup and require large tables. The aim of this universal method is to provide an algorithm that is not too slow and needs only a short setup. This method can be applied to univariate and unimodal continuous distributions with T-concave density function. See [1]_ and [2]_ for more details. Parameters ---------- dist : object An instance of a class with ``pdf`` and ``dpdf`` methods. * ``pdf``: PDF of the distribution. The signature of the PDF is expected to be: ``def pdf(self, x: float) -> float``. i.e. the PDF should accept a Python float and return a Python float. It doesn't need to integrate to 1 i.e. the PDF doesn't need to be normalized. * ``dpdf``: Derivative of the PDF w.r.t x (i.e. the variate). Must have the same signature as the PDF. mode : float, optional (Exact) Mode of the distribution. Default is ``None``. center : float, optional Approximate location of the mode or the mean of the distribution. This location provides some information about the main part of the PDF and is used to avoid numerical problems. Default is ``None``. domain : list or tuple of length 2, optional The support of the distribution. Default is ``None``. When ``None``: * If a ``support`` method is provided by the distribution object `dist`, it is used to set the domain of the distribution. * Otherwise the support is assumed to be :math:`(-\infty, \infty)`. c : {-0.5, 0.}, optional Set parameter ``c`` for the transformation function ``T``. The default is -0.5. The transformation of the PDF must be concave in order to construct the hat function. Such a PDF is called T-concave. Currently the following transformations are supported: .. math:: c = 0.: T(x) &= \log(x)\\ c = -0.5: T(x) &= \frac{1}{\sqrt{x}} \text{ (Default)} construction_points : int or array_like, optional If an integer, it defines the number of construction points. If it is array-like, the elements of the array are used as construction points. Default is 30. use_dars : bool, optional If True, "derandomized adaptive rejection sampling" (DARS) is used in setup. See [1]_ for the details of the DARS algorithm. Default is True. max_squeeze_hat_ratio : float, optional Set upper bound for the ratio (area below squeeze) / (area below hat). It must be a number between 0 and 1. Default is 0.99. random_state : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional A NumPy random number generator or seed for the underlying NumPy random number generator used to generate the stream of uniform random numbers. If `random_state` is None (or `np.random`), the `numpy.random.RandomState` singleton is used. If `random_state` is an int, a new ``RandomState`` instance is used, seeded with `random_state`. If `random_state` is already a ``Generator`` or ``RandomState`` instance then that instance is used. References ---------- .. [1] UNU.RAN reference manual, Section 5.3.16, "TDR - Transformed Density Rejection", http://statmath.wu.ac.at/software/unuran/doc/unuran.html#TDR .. [2] Hörmann, Wolfgang. "A rejection technique for sampling from T-concave distributions." ACM Transactions on Mathematical Software (TOMS) 21.2 (1995): 182-193 .. [3] W.R. Gilks and P. Wild (1992). Adaptive rejection sampling for Gibbs sampling, Applied Statistics 41, pp. 337-348. Examples -------- >>> from scipy.stats.sampling import TransformedDensityRejection >>> import numpy as np Suppose we have a density: .. math:: f(x) = \begin{cases} 1 - x^2, & -1 \leq x \leq 1 \\ 0, & \text{otherwise} \end{cases} The derivative of this density function is: .. math:: \frac{df(x)}{dx} = \begin{cases} -2x, & -1 \leq x \leq 1 \\ 0, & \text{otherwise} \end{cases} Notice that the PDF doesn't integrate to 1. As this is a rejection based method, we need not have a normalized PDF. To initialize the generator, we can use: >>> urng = np.random.default_rng() >>> class MyDist: ... def pdf(self, x): ... return 1-x*x ... def dpdf(self, x): ... return -2*x ... >>> dist = MyDist() >>> rng = TransformedDensityRejection(dist, domain=(-1, 1), ... random_state=urng) Domain can be very useful to truncate the distribution but to avoid passing it every time to the constructor, a default domain can be set by providing a `support` method in the distribution object (`dist`): >>> class MyDist: ... def pdf(self, x): ... return 1-x*x ... def dpdf(self, x): ... return -2*x ... def support(self): ... return (-1, 1) ... >>> dist = MyDist() >>> rng = TransformedDensityRejection(dist, random_state=urng) Now, we can use the `rvs` method to generate samples from the distribution: >>> rvs = rng.rvs(1000) We can check that the samples are from the given distribution by visualizing its histogram: >>> import matplotlib.pyplot as plt >>> x = np.linspace(-1, 1, 1000) >>> fx = 3/4 * dist.pdf(x) # 3/4 is the normalizing constant >>> plt.plot(x, fx, 'r-', lw=2, label='true distribution') >>> plt.hist(rvs, bins=20, density=True, alpha=0.8, label='random variates') >>> plt.xlabel('x') >>> plt.ylabel('PDF(x)') >>> plt.title('Transformed Density Rejection Samples') >>> plt.legend() >>> plt.show() Get the current ratio (area below squeeze) / (area below hat) for the generator. Get the area below the hat for the generator.Get the area below the squeeze for the generator.scipy.stats._unuran.unuran_wrapper.Method A base class for all the wrapped generators. There are 6 basic functions of this base class: * It provides a `_set_rng` method to initialize and set a `unur_gen` object. It should be called during the setup stage in the `__cinit__` method. As it uses MessageStream, the call must be protected under the module-level lock. * `_check_errorcode` must be called after calling a UNU.RAN function that returns a error code. It raises an error if an error has occurred in UNU.RAN. * It implements the `rvs` public method for sampling. No child class should override this method. * Provides a `set_random_state` method to change the seed. * Implements the __dealloc__ method. The child class must not override this method. * Implements __reduce__ method to allow pickling. scipy.stats._unuran.unuran_wrapper._URNG Build a UNU.RAN's uniform random number generator from a NumPy random number generator. Parameters ---------- numpy_rng : object An instance of NumPy's Generator or RandomState class. i.e. a NumPy random number generator. `00`00000 00000000000000000000``0 p 00000`0`<311111111111111111111111111111111111111111111111111111111111111T311$3111112311 3113v2211211111111111112`32122H3^211j2111M1031M1"0&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/&/v0&/&/.0&/&/&/&/&/R0^0&/&/j0&/&////&/&/ 0&/&/&/&/&/&/&/&/&/&/&/&/&/00/&/////&/&/:0&/&/&/t/F0&/t/2|1|12|1|1|1|1|1|3l3|1|1l3|1|1\3\3\3|1|1|1|1|1|1|1|1|1|1|1|1|1|1|1|1223|133|3l3|1|1l3|1|1|12\3|123//3/////33//3//333////////////////2K4 3/ 3 322//2///22/20//0/////2|2//l2//l2l2l2////////////////002/222|2//l2///0l2/04p3p3p3p3p3p3p3p3p3V4p3p3V4p3p3p3p3p3p3p3p3p3p3p3p3p3p3p3p3p3p3V4k4p3p3p3p3p3p33p3p3p3p3p3p3p3p3p3p3p3p3p3p3p3p3p35p35_4k4 3_4p3 3p3p3p3p3p3 3 3p3p3 3p3p3 3p3 3p3p35p3p3p3p3p3>5p3p3p3_4p3p3p3 3 3 3p3 3 3 3 3p3p3 3p3p3p3 3 3p36p3p3p3p35p3p3p3p34ygVB(U$HA@}B?h>e$@20B/.B A@Bb yB gB eB=B,BH#BPb EB(b/bKBPQB04B1__pyx_fatalerror ppf(u) PPF of the given distribution. Parameters ---------- u : array_like Quantiles. Returns ------- ppf : array_like Percentiles corresponding to given quantiles `u`. qrvs(size=None, d=None, qmc_engine=None) Quasi-random variates of the given RV. The `qmc_engine` is used to draw uniform quasi-random variates, and these are converted to quasi-random variates of the given RV using inverse transform sampling. Parameters ---------- size : int, tuple of ints, or None; optional Defines shape of random variates array. Default is ``None``. d : int or None, optional Defines dimension of uniform quasi-random variates to be transformed. Default is ``None``. qmc_engine : scipy.stats.qmc.QMCEngine(d=1), optional Defines the object to use for drawing quasi-random variates. Default is ``None``, which uses `scipy.stats.qmc.Halton(1)`. Returns ------- rvs : ndarray or scalar Quasi-random variates. See Notes for shape information. Notes ----- The shape of the output array depends on `size`, `d`, and `qmc_engine`. The intent is for the interface to be natural, but the detailed rules to achieve this are complicated. - If `qmc_engine` is ``None``, a `scipy.stats.qmc.Halton` instance is created with dimension `d`. If `d` is not provided, ``d=1``. - If `qmc_engine` is not ``None`` and `d` is ``None``, `d` is determined from the dimension of the `qmc_engine`. - If `qmc_engine` is not ``None`` and `d` is not ``None`` but the dimensions are inconsistent, a ``ValueError`` is raised. - After `d` is determined according to the rules above, the output shape is ``tuple_shape + d_shape``, where: - ``tuple_shape = tuple()`` if `size` is ``None``, - ``tuple_shape = (size,)`` if `size` is an ``int``, - ``tuple_shape = size`` if `size` is a sequence, - ``d_shape = tuple()`` if `d` is ``None`` or `d` is 1, and - ``d_shape = (d,)`` if `d` is greater than 1. The elements of the returned array are part of a low-discrepancy sequence. If `d` is 1, this means that none of the samples are truly independent. If `d` > 1, each slice ``rvs[..., i]`` will be of a quasi-independent sequence; see `scipy.stats.qmc.QMCEngine` for details. Note that when `d` > 1, the samples returned are still those of the provided univariate distribution, not a multivariate generalization of that distribution. u_error(sample_size=100000) Estimate the u-error of the approximation using Monte Carlo simulation. This is only available if the generator was initialized with a `dist` object containing the implementation of the exact CDF under `cdf` method. Parameters ---------- sample_size : int, optional Number of samples to use for the estimation. It must be greater than or equal to 1000. Returns ------- max_error : float Maximum u-error. mean_absolute_error : float Mean absolute u-error. ppf(u) Approximated PPF of the given distribution. Parameters ---------- u : array_like Quantiles. Returns ------- ppf : array_like Percentiles corresponding to given quantiles `u`. qrvs(size=None, d=None, qmc_engine=None) Quasi-random variates of the given RV. The `qmc_engine` is used to draw uniform quasi-random variates, and these are converted to quasi-random variates of the given RV using inverse transform sampling. Parameters ---------- size : int, tuple of ints, or None; optional Defines shape of random variates array. Default is ``None``. d : int or None, optional Defines dimension of uniform quasi-random variates to be transformed. Default is ``None``. qmc_engine : scipy.stats.qmc.QMCEngine(d=1), optional Defines the object to use for drawing quasi-random variates. Default is ``None``, which uses `scipy.stats.qmc.Halton(1)`. Returns ------- rvs : ndarray or scalar Quasi-random variates. See Notes for shape information. Notes ----- The shape of the output array depends on `size`, `d`, and `qmc_engine`. The intent is for the interface to be natural, but the detailed rules to achieve this are complicated. - If `qmc_engine` is ``None``, a `scipy.stats.qmc.Halton` instance is created with dimension `d`. If `d` is not provided, ``d=1``. - If `qmc_engine` is not ``None`` and `d` is ``None``, `d` is determined from the dimension of the `qmc_engine`. - If `qmc_engine` is not ``None`` and `d` is not ``None`` but the dimensions are inconsistent, a ``ValueError`` is raised. - After `d` is determined according to the rules above, the output shape is ``tuple_shape + d_shape``, where: - ``tuple_shape = tuple()`` if `size` is ``None``, - ``tuple_shape = (size,)`` if `size` is an ``int``, - ``tuple_shape = size`` if `size` is a sequence, - ``d_shape = tuple()`` if `d` is ``None`` or `d` is 1, and - ``d_shape = (d,)`` if `d` is greater than 1. The elements of the returned array are part of a low-discrepancy sequence. If `d` is 1, this means that none of the samples are truly independent. If `d` > 1, each slice ``rvs[..., i]`` will be of a quasi-independent sequence; see `scipy.stats.qmc.QMCEngine` for details. Note that when `d` > 1, the samples returned are still those of the provided univariate distribution, not a multivariate generalization of that distribution. u_error(sample_size=100000) Estimate the u-error of the approximation using Monte Carlo simulation. This is only available if the generator was initialized with a `dist` object containing the implementation of the exact CDF under `cdf` method. Parameters ---------- sample_size : int, optional Number of samples to use for the estimation. It must be greater than or equal to 1000. Returns ------- max_error : float Maximum u-error. mean_absolute_error : float Mean absolute u-error. ppf(u) Approximated PPF of the given distribution. Parameters ---------- u : array_like Quantiles. Returns ------- ppf : array_like Percentiles corresponding to given quantiles `u`. cdf(x) Approximated cumulative distribution function of the given distribution. Parameters ---------- x : array_like Quantiles, with the last axis of `x` denoting the components. Returns ------- cdf : array_like Approximated cumulative distribution function evaluated at `x`. ppf_hat(u) Evaluate the inverse of the CDF of the hat distribution at `u`. Parameters ---------- u : array_like An array of percentiles Returns ------- ppf_hat : array_like Array of quantiles corresponding to the given percentiles. Examples -------- >>> from scipy.stats.sampling import TransformedDensityRejection >>> from scipy.stats import norm >>> import numpy as np >>> from math import exp >>> >>> class MyDist: ... def pdf(self, x): ... return exp(-0.5 * x**2) ... def dpdf(self, x): ... return -x * exp(-0.5 * x**2) ... >>> dist = MyDist() >>> rng = TransformedDensityRejection(dist) >>> >>> rng.ppf_hat(0.5) -0.00018050266342393984 >>> norm.ppf(0.5) 0.0 >>> u = np.linspace(0, 1, num=1000) >>> ppf_hat = rng.ppf_hat(u) set_random_state(random_state=None) Set the underlying uniform random number generator. Parameters ---------- random_state : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional A NumPy random number generator or seed for the underlying NumPy random number generator used to generate the stream of uniform random numbers. If `random_state` is None (or `np.random`), the `numpy.random.RandomState` singleton is used. If `random_state` is an int, a new ``RandomState`` instance is used, seeded with `random_state`. If `random_state` is already a ``Generator`` or ``RandomState`` instance then that instance is used. rvs(size=None, random_state=None) Sample from the distribution. Parameters ---------- size : int or tuple, optional The shape of samples. Default is ``None`` in which case a scalar sample is returned. random_state : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional A NumPy random number generator or seed for the underlying NumPy random number generator used to generate the stream of uniform random numbers. If `random_state` is None (or `np.random`), `random_state` provided during initialization is used. If `random_state` is an int, a new ``RandomState`` instance is used, seeded with `random_state`. If `random_state` is already a ``Generator`` or ``RandomState`` instance then that instance is used. Returns ------- rvs : array_like A NumPy array of random variates. scipy/stats/_unuran/unuran_wrapper.pyxA Bhas&! AV1 #Q#Qb !7"AQb 1A""$6aq#2[ 5A QavQd!7!4s!1 &gQ &awb 4xwa fAU"Cs$gQc1 fAU"Ct1r$hawaqprobability vector must contain at least one non-zero value.probability vector must contain only finite / non-nan values.TransformedDensityRejection.ppf_hat (line 932)scipy.stats._unuran.unuran_wrapperscipy.stats._distn_infrastructureonly univariate continuous and discrete distributions supportednumpy._core.umath failed to importnumpy._core.multiarray failed to importguide_factor sizes larger than 3 are not recommended.Raised when an error occurs in the UNU.RAN library.NumericalInversePolynomial.u_errorNumericalInverseHermite._validate_argsInvalid pointer to anon_func_state.Incompatible checksums (0x%x vs (0x74c7631, 0xc5c3b56, 0x0453d1e) = (i, numpy_rng, qrvs_array))If the relative size (guide_factor) is set to 0, sequential search is used. However, this is not recommended, except in exceptional cases, since the discrete sequential search method has almost no setup and is thus faster.Failed to initialize the default URNG.Exact CDF required but not found. Reinitialize the generator with a `dist` object that contains a `cdf` method to enable the estimation of u-error.Either of the methods `pdf` or `logpdf` must be specified for the distribution object `dist`.A Bhas&! AV1 #Q#Qb !7"AQb 1A""$6aq#2[ 5A QavQd!7!4s!1 &gQ &awbr$hawaqz,e4q 2WEZs#Q a *AQ JfBc1 Cq E3jU$gQa1j .wrap_dist.pmf_unpack_dist..wrap_dist.pdf_unpack_dist..wrap_dist.logpdf_unpack_dist..wrap_dist.cdf`u_resolution` must be between 1e-15 and 1e-5.`sample_size` must be greater than or equal to 1000.probability vector must contain at least one element.`order` must be an integer in the range [3, 17].`domain` must contain only non-nan values.`domain` must be provided when the probability vector is not available.`domain` must be a length 2 tuple.`d` must be consistent with dimension of `qmc_engine`.`construction_points` must either be a scalar or a non-empty array.`construction_points` must be a positive integer.`construction_points` must be a non-empty array.TransformedDensityRejection.ppf_hatTTT G1F,avWA!qt;gQq&d!7+WA&d!7+QNumericalInversePolynomial.qrvsNumericalInverseHermite.u_errorA Bhas&! AV1""$6aqrAQ 1F#Q 2XQd(!1 E!6$as!4s!1r$hawaqA!(! 6Cq *AQuAQ )+=Qa>DBa 3at/s! *AQqAN Bhas&! AV1 #Q#Qb !7"AQb 1A""$6aq#2[ 5A  !6$awat3aq &gQ &awbr$hawaq 1r D+1L uCqaU!1 ! Ec2U!1 AQaBfASb XQ $nJfN! $fA  !1 !$a 2XQjCq  51 4xr! rAt81At81Ks!_unpack_dist..wrap_dist.__init__"!, >> from scipy.stats.sampling import TransformedDensityRejection >>> from scipy.stats import norm >>> import numpy as np >>> from math import exp >>> >>> class MyDist: ... def pdf(self, x): ... return exp(-0.5 * x**2) ... def dpdf(self, x): ... return -x * exp(-0.5 * x**2) ... >>> dist = MyDist() >>> rng = TransformedDensityRejection(dist) >>> >>> rng.ppf_hat(0.5) -0.00018050266342393984 >>> norm.ppf(0.5) 0.0 >>> u = np.linspace(0, 1, num=1000) >>> ppf_hat = rng.ppf_hat(u) __cinit__.._callback_wrapperNumericalInversePolynomial.ppfNumericalInversePolynomial.cdf*!< Ec2U!1D =q !! QnAT rq61  !1 }G1%Qa uCqxq 2XQixq qA &F"A  1A }G1%Qa uCqy 2XQj %Qa,A( Maq XQ  & 1D ! !4q !at1 N)1 t6Ak$jA $fD ("AQd%qAt4q2Qt5XQd$gRrQdRS1Ba`order` must be either 1, 3, or 5._unpack_dist..wrap_distNumericalInverseHermite.qrvs("AQd%qA$e5Qd$at2QqRqq1"!, 0cline_in_tracebackcheck_random_stateasyncio.coroutinesDiscreteGuideTableprev_random_stateascontiguousarrayset_random_statescipy._lib._util_callback_wrapperDiscreteAliasUrn_validate_domain__setstate_cython____pyx_PickleErrorbit_generatorMethod.__reduce__RuntimeWarning_validate_argsrv_continuous__reduce_cython____class_getitem___bit_generatorA$erandom_state__pyx_checksum_is_coroutine_initializingguide_factorRuntimeErrorA4uL$a;?q 4qunuran_urngscipy.statssample_sizerv_discrete__mro_entries__default_rngcollectionscdf_at_modeUNURANErrorRandomStatePickleErrorImportErrorurn_factortuple_sizeqmc_engine__pyx_vtable____pyx_result_parse_argsnamedtupleempty_likeargsreduceValueErrorMethod.rvswrap_distvectorizethreadingrv_frozen__reduce_ex__qrvs_view__pyx_stateout_discrnumpy_rng__metaclass__max_errorisenabledfunctoolsTypeErrorQMCEnginewarningsuse_dars__setstate____set_name____qualname____pyx_typepdf_areaout_contlogpdfisscalarisfinite__getstate__callbacku_errorsupportsqueezereshaperelease__prepare__ppf_hatpartialmessagegoodoutfloat64disablecapsuleasarrayacquirex_viewupdateu_view__reduce__randompickleoshape__name____module___logpdfenabledomain_dictcenterastypearangeUErrorMethodHaltonA $nDqstatsstateshapescipyscaleravelrangepmfplacepdfordernumpyklassisnanint32gooduemptydtypecond2cond1cond0clear__class__cdf__all__RLockwarn__test____spec__sizeselfqrvsprodnamemode__main__kwds__init____func__dpdfdist__dict__contargs_URNG 4z%qrvsresqmcppfpop_pmf_pdfout__new__nanmaeloglocinfget__doc___cdfanyallnpgc_?.#1F!xudcN`Gz?h㈵>|=-q=V瞯<@@C?@@________________warning(empirical)sample sizehistogram sizehistogram, min >= maxhistogram, unbounded domainbins not strictly increasing../scipy/_lib/unuran/unuran/src/distr/cemp.cprobabilities of histogram not setk < 0 or k >= dim../scipy/_lib/unuran/unuran/src/distr/condi.cconditionalOverwriting of HR not alloweddlogPDFlogCDFHRinvalid parameter positiondomain, left >= rightmode not in domainpdf area <= 0upd area <= 0../scipy/_lib/unuran/unuran/src/distr/cont.cPDF required for finding mode numericallyOverwriting of PDF not allowedOverwriting of dPDF not allowedOverwriting of logPDF not allowedOverwriting of dlogPDF not allowedOverwriting of CDF not allowedOverwriting of inverse CDF not allowedOverwriting of logCDF not allowedSyntax error in function string=?n < 2 or k < 1 or k > n../scipy/_lib/unuran/unuran/src/distr/corder.cNo order statistics of order statistics allowedorder statistics@?dimension < 1invalid coordinatemeanvariance <= 0diagonals <= 0covariance matrixcovariance matrix not knowndiagonals != 1rank-correlation matrixrank correlation matrixmarginals == NULLn not in 1 .. dimmarginalsmarginals not equalPDF volume <= 0upd volume <= 0volume../scipy/_lib/unuran/unuran/src/distr/cvec.cOverwriting of pdPDF not allowedOverwriting of pdlogPDF not allowedcovariance matrix not symmetriccovariance matrix not positive definiteinverse of covariance matrix not symmetriccannot compute inverse of covariancerank-correlation matrix not symmetricrankcorriance matrix not positive definite?= b../scipy/_lib/unuran/unuran/src/distributions/c_beta.cbeta?_unur_stdgen_sample_beta_bb_unur_stdgen_sample_beta_bc_unur_stdgen_sample_beta_b00_unur_stdgen_sample_beta_binv_unur_stdgen_sample_beta_b01_unur_stdgen_sample_beta_b1prs?T@B.?7 @QW2XUU?C q?:?@k <= 0 || c <= 0unkown typetype < 1 || type > 12../scipy/_lib/unuran/unuran/src/distributions/c_burr.cq^ZZ0[[\\\ ]P]]^hZnf_b^\`$c|cc`abtb,dburr-DT!-DT!?-DT! @-DT!-DT!?m0_?-DT!@m0_?lambda <= 0../scipy/_lib/unuran/unuran/src/distributions/c_cauchy.ccauchy../scipy/_lib/unuran/unuran/src/distributions/c_chi.cchi9B.?_unur_stdgen_sample_chi_chru../scipy/_lib/unuran/unuran/src/distributions/c_chi_gen.c3?r?@?nvd?ffffff?mhf?mh?../scipy/_lib/unuran/unuran/src/distributions/c_chisquare.cchisquare../scipy/_lib/unuran/unuran/src/distributions/c_exponential.cexponential_unur_stdgen_sample_exponential_inv../scipy/_lib/unuran/unuran/src/distributions/c_exponential_gen.ctheta <= 0../scipy/_lib/unuran/unuran/src/distributions/c_extremeI.cextremeIk <= 0../scipy/_lib/unuran/unuran/src/distributions/c_extremeII.cextremeIIalpha <= 0.beta <= 0.../scipy/_lib/unuran/unuran/src/distributions/c_gamma.cgamma_unur_stdgen_sample_gamma_gs_unur_stdgen_sample_gamma_gd_unur_stdgen_sample_gamma_gll../scipy/_lib/unuran/unuran/src/distributions/c_gamma_gen.cy$Y@@NP*? =ݼ?vA?j,}»׿?I?dU?{?tz?TUU?@,bӊ70??[.V?#vu?! U?9cVU??%y?(@Jvf@ȍqj&?3d>?H,_C?v%5?u]5?vݦY? +_nZ?WU?Y SUU?rh| @%C *@RQ??1Zd?ŏ1w-!?X9v?zG??X9v?~jt?oʡ?bX9?(\?(\?MbX9?Mb?alpha <= |beta|../scipy/_lib/unuran/unuran/src/distributions/c_ghyp.cghypomega <= 0../scipy/_lib/unuran/unuran/src/distributions/c_gig.cgigpsi <= 0chi <= 0../scipy/_lib/unuran/unuran/src/distributions/c_gig2.cgig2_unur_stdgen_sample_gig_gigru../scipy/_lib/unuran/unuran/src/distributions/c_gig_gen.cп@UUUUUU?c/?es-8R@../scipy/_lib/unuran/unuran/src/distributions/c_hyperbolic.chyperbolicmu <= 0../scipy/_lib/unuran/unuran/src/distributions/c_ig.cig"@' @phi <= 0../scipy/_lib/unuran/unuran/src/distributions/c_laplace.claplacebeta <= 0../scipy/_lib/unuran/unuran/src/distributions/c_logistic.clogistic../scipy/_lib/unuran/unuran/src/distributions/c_lognormal.clognormalC <= 0../scipy/_lib/unuran/unuran/src/distributions/c_lomax.clomaxalpha or delta <= 0beta not in (-PI,PI)../scipy/_lib/unuran/unuran/src/distributions/c_meixner.cmeixner../scipy/_lib/unuran/unuran/src/distributions/c_normal.cnormal' @_unur_stdgen_sample_normal_bm_unur_stdgen_sample_normal_kr_unur_stdgen_sample_normal_pol_unur_stdgen_sample_normal_acr_unur_stdgen_sample_normal_nquo_unur_stdgen_sample_normal_quo_unur_stdgen_sample_normal_leva_unur_stdgen_sample_normal_sum p  8 h H P  H ( z8?r?HPs??s?J +?hUM?U+~?QOI? @ ?ϡKER@?2R7u?'NJ?B[|?Aq@~]%?@E?Ё?K׭?mw/?\J-??O?3E?=  ?xky)?J&?Wڳ?BC?͐129?DJd?T'?@ZMT?|8?Ub@t @:6?V2@(?C {?Ϥ,?lݽ u @/Vq6_?_ @.@"%?+$n ?`u? ?Cf D?FL{B?k T?]SҺ?.*d?-??ď?../scipy/_lib/unuran/unuran/src/distributions/c_pareto.cparetotau <= 0../scipy/_lib/unuran/unuran/src/distributions/c_powerexponential.cpowerexponential../scipy/_lib/unuran/unuran/src/distributions/c_powerexponential_gen.c_unur_stdgen_sample_powerexponential_epdsigma <= 0.../scipy/_lib/unuran/unuran/src/distributions/c_rayleigh.crayleigh../scipy/_lib/unuran/unuran/src/distributions/c_slash.cslashP63E?_unur_stdgen_sample_slash_slash../scipy/_lib/unuran/unuran/src/distributions/c_slash_gen.cnu <= 0.../scipy/_lib/unuran/unuran/src/distributions/c_student.cstudent_unur_stdgen_sample_student_tpol../scipy/_lib/unuran/unuran/src/distributions/c_student_gen.c_unur_stdgen_sample_student_trouo0@H < 0 || H > 1../scipy/_lib/unuran/unuran/src/distributions/c_triangular.ctriangular../scipy/_lib/unuran/unuran/src/distributions/c_uniform.cuniform../scipy/_lib/unuran/unuran/src/distributions/c_vg.cvgHP?alpha <= 0../scipy/_lib/unuran/unuran/src/distributions/c_weibull.cweibullp <= 0 || p >= 1 || n <= 0../scipy/_lib/unuran/unuran/src/distributions/d_binomial.cn was rounded to the closest integer valuebinomialMbP?_unur_stdgen_sample_binomial_bruec$@@p <= 0 || p >= 1../scipy/_lib/unuran/unuran/src/distributions/d_geometric.cgeometric0CA../scipy/_lib/unuran/unuran/src/distributions/d_hypergeometric.cM, N, n must be > 0 and n= 1../scipy/_lib/unuran/unuran/src/distributions/d_logarithmic.clogarithmic_unur_stdgen_sample_logarithmic_lsk ףp= ?p <= 0 || p >= 1 || r <= 0../scipy/_lib/unuran/unuran/src/distributions/d_negativebinomial.cnegativebinomiald?../scipy/_lib/unuran/unuran/src/distributions/d_poisson.cpoisson_unur_stdgen_sample_poisson_pdtabl_unur_stdgen_sample_poisson_pdac../scipy/_lib/unuran/unuran/src/distributions/d_poisson_gen.c_unur_stdgen_sample_poisson_pprscxZd;O?TUUU? @333333@6S?hy@?~[B4?ۦx\T˿?;J? _U?ɨ?l ?t _VUU?|??M OV_?lHE?hUUUU?333333?-@6$I?.@F@ Ac]?rho <= 0tau < 0../scipy/_lib/unuran/unuran/src/distributions/d_zipf.czipf_unur_stdgen_sample_zipf_zetcorrelationcopula../scipy/_lib/unuran/unuran/src/distributions/vc_multicauchy.cmulticauchyHP?sigma is too low../scipy/_lib/unuran/unuran/src/distributions/vc_multiexponential.cmultiexponentiale͍>??../scipy/_lib/unuran/unuran/src/distributions/vc_multinormal.cmultinormaldg?truncated domain not allowed../scipy/_lib/unuran/unuran/src/distributions/vc_multinormal_gen.c_unur_stdgen_sample_multinormal_cholesky../scipy/_lib/unuran/unuran/src/distributions/vc_multistudent.cmultistudentgenerator ID: %s distribution: functions = PDF dPDF domain = (%g, %g) center = %g [= mode] [ Hint: %s ] performance characteristics: area(hat) = %g rejection constant <= %g # segments = %d parameters: max_sqhratio = %g %s max_segments = %d %s verify = on pedantic = on PDF(x) < 0.PDF(x) < 0.!PDF not T-concave.AROUstarting point out of domainPDF not unimodalPDF < 0PDF not T-concavehat/squeeze ratio too small.bad construction pointsderivative of PDFDARS factor < 0number of starting points < 0guide table size < 0You may provide a point near the mode as "center".method: AROU (Automatic Ratio-Of-Uniforms) area ratio squeeze/hat = %g You can set "max_sqhratio" closer to 1 to decrease rejection constant.You should increase "max_segments" to obtain the desired rejection constant.../scipy/_lib/unuran/unuran/src/methods/arou.cCannot chop segment at given pointCannot split segment at given point.starting points not increasing -> skipcannot create bounded envelope!maximal number of segments too small. increase.DARS aborted: no intervals could be splitted.DARS aborted: maximum number of intervals exceeded.starting points not strictly monotonically increasingratio A(squeeze)/A(hat) not in [0,1]maximum number of segments < 1:0yE>.A= %.3f [approx.] # intervals = %d cpoints = %d %s logPDF(x) overflowcannot create bounded hat!PDF not T-concave!PDF not unimodal!PDF not log-concaveempty generator objectgenerated point out of domainPDF > hat. Not log-concave!derivative of logPDFpercentiles out of rangebad construction points.maximum number of iterationsargument u not in [0,1] functions = logPDF dlogPDF method: ARS (Adaptive Rejection Sampling) T_c(x) = log(x) ... c = 0 area(hat) = %g [ log = %g ] ../scipy/_lib/unuran/unuran/src/methods/ars.cdTfx0 < dTfx1 (x0 100. using 100percentiles not strictly monotonically increasingnumber of construction points < 10bad construction points for reinit@ư>.Q@ysKp]ysKpP>? {Gz?'??AUTOlog < 0../scipy/_lib/unuran/unuran/src/methods/auto.cparameters: none CEXTsampling routine missingmethod: CEXT (wrapper for Continuous EXTernal generators) E [#urn] = %.2f [approx.] init for external generator failed../scipy/_lib/unuran/unuran/src/methods/cext.c@[implements inversion method] variant = %d %s [%d] = %g CSTD_unur_cstd_sample_invparametersdomain changed, CDF requiredvariant for special generatortruncated domain too largeCDF values very closeinversion CDF requiredU not in [0,1]method: CSTD (special generator for Continuous STandarD distribution) table of precomputed constants: ../scipy/_lib/unuran/unuran/src/methods/cstd.cdomain changed for non inversion methodinit() for special generators or inverse CDFtruncated domain for non inversion methodtruncated domain, CDF requiredCDF values at boundary points too close[unknown] functions = PMF domain = (%d, %d) mode = %d %s sum(PMF) = %g %s You may provide the "mode".You may provide the "pmfsum". no table use table of size %d use squeeze sum(hat) = %g = %.2f [approx.] tablesize = %d %s squeeze = on cpfactor = %g DARIPMF(i) > hat(i) for tailpartk %d h %.20e H(k-0.5) %.20e PMF(mode)=0sum over PMF; use defaultcp-factor <= 0invalid table sizemethod: DARI (Discrete Automatic Rejection Inversion) PMF(i) > hat(i) for centerpart../scipy/_lib/unuran/unuran/src/methods/dari.ci %d PMF(x) %.20e hat(x) %.20efor tailpart hat too low, ie hp[k] < H(k-0.5)k %d hp %.20e H(k-0.5) %.20e Area below hat too large or zero!! possible reasons: PDF, mode or area below PMF wrong; or PMF not T-concavemode: try finding it (numerically)domain contains negative numberscp-factor > 2 not recommended. skip=Y@sh|??d@, created from PMFmethod: DAU (Alias-Urn) E [#look-ups] = %g urnfactor = %g %s probability < 0squared histogramDAUPV. Try to compute it.relative urn size < 1. functions = PV [length=%d%s] ../scipy/_lib/unuran/unuran/src/methods/dau.c?DEXTmethod: DEXT (wrapper for Discrete EXTernal generators) ../scipy/_lib/unuran/unuran/src/methods/dext.cmethod: DGT (Guide Table) guidefactor = %g %s variant = %d DGTrelative table size < 0../scipy/_lib/unuran/unuran/src/methods/dgt.c sum(PMF) = %g F(mode) = %g F(mode) = [unknown] You may provide the "mode" area(hat) = %g + %g = %g rejection constant = %g cdfatmode = %g cdfatmode = [not set] DSROUPMF(x) > hat(x)PMF(mode) <= 0.sum over PMFarea and/or CDF at modeCDF(mode)method: DSROU (Discrete Simple Ratio-Of-Uniforms) enveloping rectangle = (%g,%g) x (%g,%g) [left] (%g,%g) x (%g,%g) [right] You can set "cdfatmode" to reduce the rejection constant.../scipy/_lib/unuran/unuran/src/methods/dsrou.c@ functions = CDF DSSPV+sum, PMF+sum, or CDF../scipy/_lib/unuran/unuran/src/methods/dss.c functions = PV [length=%d] method: DSS (Simple Sequential Search) performance characteristics: slow [%d] = %d DSTD_unur_dstd_sample_invmethod: DSTD (special generator for Discrete STandarD distribution) table of precomputed double constants: table of precomputed integer constants: ../scipy/_lib/unuran/unuran/src/methods/dstd.ctruncated domain for non-inversion method[kernel generator set] [standard kernel] [default kernel] smoothing factor = %g no variance correction smoothing = %g %s beta = %g varcor = on positive = on number of observed sampleCannot overwrite kernelvariance correction disabledsmoothing factor < 0 functions = DATA [length=%d] method: EMPK (EMPirical distribution with Kernel smoothing) kernel type = %s (alpha=%g) window width = %g (opt = %g) positive random variable only; use mirroring variance correction factor = %g ../scipy/_lib/unuran/unuran/src/methods/empk.cUnknown kernel. make it manuallyCould not initialize kernel generator_|,?s?@@?@??_|,?V-?ah?-?SbQ @q= ףp?EMPLmethod: EMPL (EMPirical distribution with Linear interpolation) ../scipy/_lib/unuran/unuran/src/methods/empl.ccoordinate sampling [default]random direction sampling dimension = %d center = [= mode] [default] variant = %s T_c(x) = log(x) ... c = 0 -1/sqrt(x) ... c = -1/2 -x^(%g) ... c = %g thinning = %d variant_random_direction c = %g %s thinning = %d %s burnin = %d %s GIBBSc > 0c < -0.5 not implemented yetthinning < 1burnin < 0reset chainmethod: GIBBS (GIBBS sampler [MCMC]) rejection constant = %.2f [approx.] variant_coordinate [default] ../scipy/_lib/unuran/unuran/src/methods/gibbs.cCannot create generator for conditional distributionsCannot create aux Gaussian generator-0.5 < c < 0 not recommended. using c = -0.5 instead.CDF(x) < 0.CDF(x) > 1.HINVCDF not increasing functions = CDF dPDF domain = (%g, %g) [truncated from (%g, %g)] mode = %g order of polynomial = %d Prob(Xdomain) = %g # intervals = %d too many intervalscannot find r.h.s. of domaincannot find l.h.s. of domainu-resolutionnumber of starting points < 1../scipy/_lib/unuran/unuran/src/methods/hinv.cYou may set the "mode" of the distribution in case of a high peakmethod: HINV (Hermite approximation of INVerse CDF) truncated domain = (%g,%g) You can set "order=5" to decrease #intervals(it is bounded by the machine epsilon, however.)You can decrease the u-error by decreasing "u_resolution". u-error <= %g (mean = %g) one or more intervals very short; possibly due to numerical problems with a pole or very flat tailNaN occured; possibly due to numerical problems with a pole or very flat tailu-resolution so small that problems may occurdomain (+/- UNUR_INFINITY not allowed)maximum number of intervals < 1000domain, increase left boundarydomain, decrease right boundaryP hat(x)PDF(x) < squeeze(x)cannot compute bxinverse pdf not T_cp concavepdf not T_ct concavemode (pole)xi out of domaincp > -0.1 or <= -1ct > -0.1 or <= -1../scipy/_lib/unuran/unuran/src/methods/itdr.cmethod: ITDR (Inverse Transformed Density Rejection -- 2 point method) area(hat) = %g [ = %g + %g + %g ] cannot compute hat for pole: cpcannot compute hat for pole: xpcannot compute hat for tail: ctpole not on boundary of domain??rZ| ?0.++@@+}Ô%ITTKEl@ _B eigenvalues = parameters: MCORReigenvalue <= 0 dimension = %d x %d (= %d) method: MCORR (Random CORRelation matrix) generate correlation matrix with given eigenvalues ../scipy/_lib/unuran/unuran/src/methods/mcorr.call eigenvalues are ~1 -> identity matrixscaling sum(eigenvalues) -> dim ?? # components = %d probabilities = (%g components = [%d] %s - continuousdiscrete useinversion = off [default] TRUE inversion method = %s FALSEMIXTinvalid probabilitiescomponent is NULLcomponent not univariate(mixture)performance characteristics: depends on components method: MIXT (MIXTure of distributions -- meta method) select component = method DGT ../scipy/_lib/unuran/unuran/src/methods/mixt.ccomponent does not implement inversiondomains of components overlap or are unsorted?MVSTDtruncated domainstandard distributioninit() for special generatorsmethod: MVSTD (special generator for MultiVariate continuous STandarD distribution) E [#urn] = %.2f x %d = %.2f [approx.] ../scipy/_lib/unuran/unuran/src/methods/mvstd.c volume(hat) = %g # cones = %d # vertices = %d triangulation levels = %d stepsmin = %d %s maxcones = %d %s boundsplitting = %g %s no domain givenMVTDRcenter out of support of PDFcannot create hatdim < 2d/(log)PDFstepsmin < 0../scipy/_lib/unuran/unuran/src/methods/mvtdr_init.hYou can set the mode to improve the rejection constant.method: MVTDR (Multi-Variate Transformed Density Rejection) triangulation levels = %d-%d You can increase "stepsmin" to improve the rejection constant.You can increase "maxcones" to improve the rejection constant.You can change "boundsplitting" to change the creating of the hat function.../scipy/_lib/unuran/unuran/src/methods/mvtdr_sample.hCannot create aux Gamma generator../scipy/_lib/unuran/unuran/src/methods/mvtdr_newset.h @`pgu  YftLXes@KWdr5?JVcq#+4>IUbP  "*3=H\d > !)2An46i P80 P+Z V (B2b @€Jz5ffffff?'[disabled] Newton method Bisection method Regula falsi [ u-resolution = %g ] starting points = %g (CDF = %g) %s %g, %g (CDF = %g, %g) %s usenewton usebisect useregula [default] u_resolution = %g %s %s x_resolution = %g %s %s max_iter = %d %s NINVmaximal iterationsx-resolution too smallu-resolution too smallmethod: NINV (Numerical INVersion) average number of iterations = %.2f [approx.] u-error <= %g (mean = %g) [rough estimate] u-error NA [requires CDF] starting points = table of size %d You can increase accuracy by decreasing "x_resolution".You can increase "max_iter" if you encounter problems with accuracy.Regula Falsi cannot find interval with sign change../scipy/_lib/unuran/unuran/src/methods/ninv_regula.hflat region: accuracy goal in x cannot be reachedsharp peak or pole: accuracy goal in u cannot be reachedmax number of iterations exceeded: accuracy goal might not be reached../scipy/_lib/unuran/unuran/src/methods/ninv_init.hNewton's method cannot leave flat region../scipy/_lib/unuran/unuran/src/methods/ninv_newton.hboth x-resolution and u-resolution negativ. using defaults.cannot compute normalization constant../scipy/_lib/unuran/unuran/src/methods/ninv_newset.h../scipy/_lib/unuran/unuran/src/methods/ninv_sample.h?$ʉvR$@6v#4@?:0yE> marginals =NORTA functions = MARGINAL distributions method: NORTA (NORmal To Anything) ../scipy/_lib/unuran/unuran/src/methods/norta.cdata for (numerical) inversion of marginal missingcannot handle non-rectangular domaincannot compute eigenvalues for given sigma_ysigma_y not positive definite -> corrected matrix(corrected) sigma_y not positive definitinit of marginal generators failedes-8R? [ Hint: %s %s ] r = %g = %.2f [approx.] center = %g %s v = %g %s u = (%g, %g) %s vmax not finiteumin or umax not finiter<=0the "center" (a point near the mode).You may provide the "mode" or at leastmethod: NROU (Naive Ratio-Of-Uniforms) bounding rectangle = (%g,%g) x (%g,%g) You can set "v" to avoid numerical estimate.You can set "u" to avoid slow (and inexact) numerical estimates.../scipy/_lib/unuran/unuran/src/methods/nrou.cCannot compute bounding rectangleMbPMbP-C6?PINVPDF(center) too smallPDF(x) < 0 [guess] functions = %s smoothness = %d [continuous] [differentiable] [twice differentiable] use CDF %s # CDF table size = %d order = %d [corrected] use_upoints = %s %s smoothness must be 0, 1, or 2PDF(center) <= 0.cannot compute area below PDFtruncated domain too narrowPDF or CDForder <3 or >17PDF missingCDF missing'keepcdf' not set../scipy/_lib/unuran/unuran/src/methods/pinv_prep.hCDF too small/large on given domain../scipy/_lib/unuran/unuran/src/methods/pinv_init.hmethod: PINV (Polynomial interpolation based INVerse CDF) use PDF + Lobatto integration %s Chebyshev points in u scale area below PDF = %18.17g search for boundary: left=%s, right=%s %s maximum number of interval = %d %s keep table of CDF values = %s %s You can increase "order" to decrease #intervalsmaximum number of intervals exceeded../scipy/_lib/unuran/unuran/src/methods/pinv_newton.hPDF too close to 0 on relevant part of domain --> abortnumerical problems with cut-off point, PDF too steepPDF increasing towards boundarytail probability gives NaN --> assume 0.numerical problems with cut-off point, out of domainorder must be >= 5 when smoothness = 2order must be 2 mod 3 when smoothness = 2PDF and dPDF required for smoothness = 2 --> try smoothness=1 insteadorder must be odd when smoothness = 1PDF required --> use smoothness=0 insteadcenter moved into domain of distributionCannot get left boundary of relevant domain.Cannot get right boundary of relevant domain.cannot approximate area below PDFcannot find left boundary for computational domaincannot find right boundary for computational domainintegration of pdf: numerical problems with cut-off points../scipy/_lib/unuran/unuran/src/methods/pinv_newset.hu-resolution too large --> use 1.e-5 insteadu-resolution too small --> use 1.e-15 insteadmaximum number of intervals < 100 or > 1000000../scipy/_lib/unuran/unuran/src/methods/pinv_sample.h vIh%<=??<X@P@H@Q?333333?? =u<7~>r<@|=}Ô%I[generalized version] mode = %g %s area(PDF) = %g use CDF at mode use mirror principle r = %g %s usesqueeze usemirror PDF(mode) <= 0.PDF(mode) overflowr < 1PDF(mode)method: SROU (Simple Ratio-Of-Uniforms) enveloping rectangle = (%g,%g) x (%g,%g) ../scipy/_lib/unuran/unuran/src/methods/srou.c;f?oʡ@ʡE?{Gz?"~@ )\(??SSR../scipy/_lib/unuran/unuran/src/methods/ssr.cmethod: SSR (Simple Setup Rejection) rejection constant <= %g [approx. = %.2f]  area(PDF) = [not set: use 1.0] variant = variant_ia = off max_intervals = %d %s PDF not monotone in slopehat unboundedsum of areas not validcannot create guide tableTABLinvalid variantarea factor <= 0invalid slopesCannot make hat functionCannot compute slopesCannot split intervalsnumber of slopes <= 0slopes must be boundedmethod: TABL (Ahrens' TABLe Method) immediate acceptance [ia = on] acceptance/rejection [ia = off] variant_ia = on [default] You should increase "max_intervals" to obtain the desired rejection constant.../scipy/_lib/unuran/unuran/src/methods/tabl_init.h../scipy/_lib/unuran/unuran/src/methods/tabl_sample.hPDF > hat. PDF not monotone in intervalPDF < squeeze. PDF not monotone in interval../scipy/_lib/unuran/unuran/src/methods/tabl_newset.hnumber of starting points <= 0area below PDF, use default insteadPDF discontinuous or slope not monotonesplit A stopped, maximal number of intervals reached.slopes (overlapping or not in ascending order)adaptive rejection sampling disabled for truncated distributioncannot use IA for truncated distribution, switch to RHtruncated domain not subset of domain !?Gz?GW (original Gilks & Wild) PS (proportional squeeze) IA (immediate acceptance) T_c(x) = variant_gw = on variant_ia = on hat(x) might be < PDF(x)hat(x) < PDF(x)error below hat (almost) 0PDF > hat. Not T-concave!PDF < squeeze. Not T-concave!mode -> ignoreinvalid rule for DARS%s:Intervals: %d %s: No intervals ! %s:Areas in intervals: %s: A(total) = %-12.6g %s: generated point: x = %g %s: h(x) = %.20g %s: f(x) = %.20g %s: s(x) = %.20g %s: hat: x - x0 = %g <-- error %s: h(x) - f(x) = %g%s: squeeze: x - x0 = %g%s: f(x) - s(x) = %g%s: old interval: %s: inserted point: %s: new intervals: %s: left interval: %s: interval chopped. %s: right interval: %s: total areas: %s: squeeze ratio = %g %s: old intervals: %s: A(squeeze) = %s: %-12.6g (%6.3f%%) %s: A(hat\squeeze) = %s: A(hat) = method: TDR (Transformed Density Rejection) variant_ps = on [default] You may use "variant_ia" for faster generation times.../scipy/_lib/unuran/unuran/src/methods/tdr_init.h../scipy/_lib/unuran/unuran/src/methods/tdr_ps_sample.h../scipy/_lib/unuran/unuran/src/methods/tdr_gw_sample.h../scipy/_lib/unuran/unuran/src/methods/tdr_newset.hdTfx0 < dTfx1 (x0 A(hat). PDF not T-concave!c != 0. and c != -0.5 not implemented!cannot use IA for truncated distribution, switch to PS../scipy/_lib/unuran/unuran/src/methods/tdr_sample.h%s: Nr. tp ip f(tp) T(f(tp)) d(T(f(tp))) squeeze %s:[%3d]: %#12.6g %#12.6g %#12.6g %#12.6g %#12.6g %#12.6g %s:[...]: %#12.6g %#12.6g %#12.6g %#12.6g %s: Nr. below squeeze below hat (left and right) cumulated %s:[%3d]: %-12.6g(%6.3f%%) | %-12.6g+ %-12.6g(%6.3f%%) | %-12.6g(%6.3f%%) %s: ---------- --------- | ------------------------ --------- + %s: Sum : %-12.6g(%6.3f%%) %-12.6g (%6.3f%%) %s: A(squeeze) = %-12.6g (%6.3f%%) %s: A(hat\squeeze) = %-12.6g (%6.3f%%) %s: point generated in left part: %s: point generated in right part: %s: construction point: x0 = %g %s: transformed hat Th(x) = %g + %g * (x - %g) %s: transformed squeeze Ts(x) = %g + %g * (x - %g) %s: split interval at x = %g f(x) = %g %s: left construction point = %-12.6g f(x) = %-12.6g %s: right construction point = %-12.6g f(x) = %-12.6g %s: A(squeeze) = %-12.6g (%6.3f%%) %s: A(hat\squeeze) = %-12.6g (%6.3f%%) %s: A(hat) = %-12.6g + %-12.6g(%6.3f%%) %s: x = %g, f(x) = %g, Tf(x)=%g, dTf(x) = %g, squeeze = %g: %s: left construction point = %g %s: middle construction point = %g %s: right construction point = %g %s: A(total) = %-12.6g %s: Nr. left ip tp f(tp) T(f(tp)) d(T(f(tp))) f(ip) squ. ratio %s:[%3d]:%#12.6g %#12.6g %#12.6g %#12.6g %#12.6g %#12.6g %#12.6g %s:[...]:%#12.6g %#12.6g %s: transformed hat Th(x) = %g + %g * (x - %g) %s: left boundary point = %-12.6g f(x) = %-12.6g %s: left construction point = %-12.6g f(x) = %-12.6g %s: middle boundary point = %-12.6g f(x) = %-12.6g %s: right boundary point = %-12.6g f(x) = %-12.6g %s: middle construction point= %-12.6g f(x) = %-12.6g %s: A(squeeze) = %-12.6g (%6.3f%%) %s: A(hat\squeeze) = %-12.6g (%6.3f%%) 9B.UUUUUU?UUUUUU? _"2  Gz??࿮Gz?distribution: uniform (0,1) UNIFmethod: UNIF (wrapper for UNIForm random number generator) [Remark: allows using uniform random number generator in UNU.RAN framework] ../scipy/_lib/unuran/unuran/src/methods/unif.c deltafactor = %g %s pdfatmode = %g UTDRdelta must be smallmethod: UTDR (Universal Transformed Density Rejection -- 3 point method) ../scipy/_lib/unuran/unuran/src/methods/utdr.cx %e PDF(x) %e hat(x) %e squeeze(x) %eDelta larger than c/100!!, perhaps you can use a mode closer to 0 to remove this problem?; Area below hat too large or zero!! possible reasons: PDF, mode or area below PDF wrong; density not T-concave sh|??h㈵>333333? kernel type = multinormal bandwith = %g VEMPKsize of observed samplemethod: VEMPK (EMPirical distribution with Kernel smoothing) ../scipy/_lib/unuran/unuran/src/methods/vempk.c [numeric.] bounding rectangle = u = -- VNROUmethod: VNROU (Naive Ratio-Of-Uniforms) ../scipy/_lib/unuran/unuran/src/methods/vnrou.creinitgen_list_freegen_list_setgen_list_clone../scipy/_lib/unuran/unuran/src/methods/x_gen.cuser identifier (variable name) expectedincomplete. not all tokens parsedinvalid number of parameters for functionunknown symbol in function string../scipy/_lib/unuran/unuran/src/parser/functparser_parser.hcannot derivate subtree at '%s'../scipy/_lib/unuran/unuran/src/parser/functparser_deriv.h../scipy/_lib/unuran/unuran/src/parser/functparser_init.h../scipy/_lib/unuran/unuran/src/parser/functparser_scanner.h../scipy/_lib/unuran/unuran/src/parser/functparser_stringgen.hexpected symbol: '='expected symbol: '('expected symbol: ')'function (name) expected%s: %s -->%s<-- <-- FSTRING%.16g*/cossgn+tansin-sec^log_ROS_NAS_ANS_ENDempty stringunknown symbol '%s': %c --> %s\\ true-inffalse) not followed by commatoo many arguments,)arouautocstddaridsroudstdemplgibbshinvhisthitroitdrmcorrmvstdmvtdrninvnortapinvtablunifutdrvempkvnrouunknown %s: '%s'invalid data for %s '%s'betacauchychisquareextremeiextremeiigammagig2hyperbolichypergeometriclaplacelogarithmiclogisticlognormallomaxnegativebinomialparetopoissonpowerexponentialrayleighslashstudenttriangularuniformweibullcempcontdiscrhist_binshist_domainhist_proborderstatisticshrstrlogcdflogcdfstrlogpdfstrpdfareapdfparamspmfparamspmfstrpmfsumset callsetting method faileddarsfactorguidefactormax_segmentsmax_sqhratiopedanticusecenterusedarsverifymax_intervalsmax_iterreinit_ncpointsreinit_percentileslogsscpfactortablesizeurnfactorcdfatmodepositivesmoothingvarcorburninthinningvariant_coordinatevariant_random_directionu_resolutionadaptive_multiplieruse_adaptivelineuse_adaptiverectangleuse_boundingrectangleupperboundboundsplittingmaxconesstepsminstartusebisectusenewtonuseregulax_resolutionkeepcdfsearchboundarysmoothnessuse_upointsusecdfusepdfpdfatmodeusemirrorusesqueezeareafractionnstpuseearvariant_iavariant_splitmodeusemodevariant_gwvariant_psdeltafactordebugparameter for given method&method=urng=category../scipy/_lib/unuran/unuran/src/parser/stringparser.cclosing '"' not followed by commainvalid argument string for '%s'../scipy/_lib/unuran/unuran/src/parser/stringparser_lists.hkey for distribution does not start with 'd'parameter for given distributionkey for method does not start with 'm'setting URNG requires PRNG library enabledn\n$o\oo\n$p\pp\n\n\nqq\nox?A*@%m]LWdg?חAJ?llf?UUUUUU?9B.<9B.@Ҽz+#?<;f?WU?́R?UUUUUU?llfJ?88C$+K?<ٰj_AAz?SˆB8?5gGdg?too few classes! GOODNESS-OF-FIT TESTS: i = %d length = %d Fdelta <= 0.F(x) > Fmax (out of domain).F(x) < 0 (out of domain). intervals = %d distribution dimension < 1 ? Summary: ../scipy/_lib/unuran/unuran/src/tests/chi2test.cClass #%d: observed %d expected %.2f Cannot run chi^2-Test: too few classes CDF for CHI^2 distribution required Result of chi^2-Test: samplesize = %d classes = %d (minimum per class = %d) chi2-value = %g p-value = %g Chi^2-Test for discrete distribution with given probability vector:CDF required for continuous random variates! Chi^2-Test for continuous distribution: Chi^2-Test for continuous empirical distribution: (Assumes standard normal distribution!)correlated and domain truncated --> test might failCDF of continuous standardized marginalcannot compute inverse of Cholesky factor Chi^2-Test for marginal distribution [%d] Not implemented for such distributions! Minimal p-value * number_of_tests = %g: marginal distributionsCDF of continuous marginal rank correlation =../scipy/_lib/unuran/unuran/src/tests/correlation.cdont know how to compute correlation coefficient for distribution Correlation coefficient: %g Rank correlations of random vector: rank correlation coefficients cannot be computedcannot run test for method! COUNT: Running Generator: total: %7d (%g) PDF: %7d (%g) dPDF: %7d (%g) logPDF: %7d (%g) dlogPDF: %7d (%g) CDF: %7d (%g) HR: %7d (%g) pdPDF: %7d (%g) pdlogPDF:%7d (%g) PMF: %7d (%g) function calls total: %7d PDF: %7d dPDF: %7d logPDF: %7d dlogPDF: %7d CDF: %7d HR: %7d pdPDF: %7d pdlogPDF:%7d PMF: %7d ../scipy/_lib/unuran/unuran/src/tests/countpdf.c COUNT-PDF: cannot count PDF for distribution type) function calls (per generated number) COUNT: Initializing Generator: ../scipy/_lib/unuran/unuran/src/tests/counturn.c COUNT: %g urng per generated number (total = %ld) inversion method required../scipy/_lib/unuran/unuran/src/tests/inverror.csamplesize too small --> increased to 1000 max u-error exceeded at %g: %g (>%g) max u-error exceeded at U=%g: %g (>%g) j@number of moments < 1 or > 4 Central MOMENTS: [%d] = %g ../scipy/_lib/unuran/unuran/src/tests/moments.cdont know how to compute moments for distribution Central MOMENTS for dimension #%d: SAMPLE: %04d %8.5f ( %8.5f, %8.5f ) ../scipy/_lib/unuran/unuran/src/tests/printsample.c Quartiles: min = %6.5g 25%% = %6.5g 50%% = %6.5g 75%% = %6.5g max = %6.5g ../scipy/_lib/unuran/unuran/src/tests/quantiles.cdont know how to compute quartiles for distributionTeststype of method unknown!METHOD: special (DSTD) METHOD: special (CSTD) METHOD: special (MVSTD) ../scipy/_lib/unuran/unuran/src/tests/tests.c TYPE: discrete univariate distribution TYPE: continuous univariate distribution TYPE: continuous univariate empirical distribution TYPE: continuous multivariate distribution METHOD: automatic selection (AUTO) METHOD: alias and alias-urn method (DAU) METHOD: indexed search -- guide table (DGT) METHOD: discrete simple universal ratio-of-uniforms search (DSROU) METHOD: sequential search (DSS) METHOD: external generator (DEXT) METHOD: automatic ratio-of-uniforms method (NINV) METHOD: numerical inversion of CDF by Hermite Interpolation (HINV) METHOD: inverse transformed density rejection (ITDR) METHOD: numerical inversion of CDF (NINV) METHOD: naive universal ratio-of-uniforms method (NROU) METHOD: polynomial interpolation based inversion of CDF (PINV) METHOD: simple universal ratio-of-uniforms method (SROU) METHOD: simple transformed density rejection with universal bounds (SSR) METHOD: rejection from piecewise constant hat (TABL) METHOD: transformed density rejection (TDR) METHOD: transformed density rejection, 3-point method (UTDR) METHOD: external generator (CEXT) METHOD: empirical distribution with kernel smoothing (EMPK) METHOD: Markov Chain - GIBBS sampler (GIBBS) METHOD: hit&run ratio-of-uniforms (HITRO) METHOD: multivariate transformed density rejection (MVTDR) METHOD: normal to anything (NORTA) METHOD: vector matrix transformation (VMT) METHOD: vector naive ratio-of-uniforms (VNROU) METHOD: wrapper for uniform (UNIF) uniform exponential 10^%ld: %#g %#g %#g ../scipy/_lib/unuran/unuran/src/tests/timing.c TIMING: usec relative to relative to setup time: %#g %#g %#g generation time: %#g %#g %#g average generation time for samplesize: YUk@ ףp= ?reset../scipy/_lib/unuran/unuran/src/urng/urng.cDefault URNG not set. EXIT !!!../scipy/_lib/unuran/unuran/src/urng/urng_default.cDefault auxilliary URNG not set. EXIT !!!URNGaux../scipy/_lib/unuran/unuran/src/urng/urng_set.csyncseeding functionantithetic flagnext substreamreset substream../scipy/_lib/unuran/unuran/src/urng/urng_unuran.cdebugging not enabledDEBUG%s.%03d../scipy/_lib/unuran/unuran/src/utils/debug.cp9222UNURAN(no error)(generator) sampling error(generator)(URNG)(URNG) missing functionality(parser) invalid string(parser) unknown keyword(parser) syntax error(parser) invalid parameterargument out of domain(serious) round-off errorvirtual memory exhaustedinvalid NULL pointerinvalid cookie(silent error)invalid infinity occuredNaN occured%s: [%s] %s:%d - %s: %s: ..> %s error should not happen, report this!(distribution) parameter out of domain(distribution) invalid variant for special generator(distribution) invalid distribution object(distribution) incomplete distribution object, entry missing(distribution) unknown distribution, cannot handle(distribution) set failed (invalid parameter)(distribution) get failed (parameter not set)(distribution) data are missing (cannot execute)(distribution) desired property does not exist(parameter) set failed, invalid parameter -> using default(parameter) invalid variant -> using default(parameter) invalid parameter object(generator) (possible) invalid data(generator) condition for method violated(generator) invalid generator object(generator) reinit not implemented(generator) quantile not implemented(function parser) syntax error(function parser) cannot derivate functionnot available, recompile library(distribution) invalid number of parametersHLLLLLLLLLLLLLLLL<0T$ LLLLLLܚКĚLLLLLLLLLLLLL|pdLLLLLLLLLXLLLLLLLLLLLLLLL@4(LLLLLLLLLLԙ˙LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL™p@@@@@@@@@@@@@@@@h`xXPH@80@@@@@@( @@@@@@@@@@@@@ؙ@@@@@@@@@Йș@@@@@@@@@@@@@@@@@@@@@@@@xph`XP@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@HCMAX../scipy/_lib/unuran/unuran/src/utils/fmax.cHz>8boundaries of integration domain not finite../scipy/_lib/unuran/unuran/src/utils/lobatto.cnumeric integration did not reach full accuracyadaptive numeric integration aborted (too many function calls)%s: subintervals for Lobatto integration: %d %s: [%3d] x = %.16g, u = %.16g area below PDF 0.size<2}'?Z vy?H@P@f@v@ҰD?J-O max datasizeread_datacannot open filedata file not valid../scipy/_lib/unuran/unuran/src/utils/stream.cUNURAN../scipy/_lib/unuran/unuran/src/utils/umalloc.c@;$4f@4Ps$7s7ht8(v :v4:xwl:x:-y<;z<z=}=><>b?Ќ?BtNxSSْS*VߟPu,w|wL^`h4|4404P4p4445 5P05h5|5 5555@6,6pH6@l666066D77`8@8 8@88@9h9P9 99@:@x;;;D<`<@<<=`|=>p>@>>L??0@Pt@p@@TAhAAApAAPAB(B`@BXBpBPBB@B,CPLClCC0CD(D@`D D Dp4EpE0E EF dF@#FP%,G&PGP'pG(G)GP*G +H0.@H`.\H0H 1H3H5LI8I0:I`J@KB@KChKHK KLL4LMdLOL0SMVtMYM[N^PN@^dNPaOpdhOfOh PmxPPmPmPmPqDQvQPvQpz `?8@BB DĻpDػD0EpE,FXIPIJ@KMX Nl`NNNOн@OO0P$PQPTPU@VؾPWpXY@`Zl^pbdd e(@e@eX0fpgphjj@k@pk\kpPl m npoqXrs ttuv0`vHwt`yyPzz {$P|P|p}~0,ЀXЁ0 PdPxЋЌ@@TPh|Pp0@x0DЛxpHPЧ@4г p0`\`0P0x`P0D d0@Ph| @PTt<PP0|`P@h@@0``tp ` p P P <p T0 \p@T p,P!`$P(D0,,`. 2p56E0@GhGHJpKpLM8MXPNx@OOPQ0R TpVV@Y]X^`d@0j@n0stLul`wy}d`P8`0pHЎp4`TВx@(0T@hPpК` ЛDhp0 @X `(T$`Hp0P0p0lp00h` 00l@ 8 T ` 0\P$dP `8d <t`Pd ` 4 p  4lp0$xp!! @"T"hP$'0,<0.l//P122383L 4h456p678P;@<<D0pFhGG0IIJK<0L\M|MNP0RPS`U V `W\XZp[[@`H`mo@pqTPsxt@vwpxPy8yX0$@\pH0`0L`Ъp4pT t@ @ ` 0     $ PP    P  H  p p   0 `l p p    PP  @     X0Pp0 D P   PXPP  `@``p+--@.x13P45X@9P99:p:0?T@@A CpD$D8DLpEtG IJK4LPQPSPVT@a`ac@hPPm`np @|X Ђp<\0|0 \PPLl  `   P!4!Pp!!!0!г"4"PT"P|"""и"#0#P# h#P##04$`$$$0%@P%%%%p(&pX&|&P&&`& ('D'|'0'''(@(( P((`(@ )`X)@)P))P*`d** (+p+%+','0,(P, *x,*,+,,,P.-/8-/X-p0x- 1-2-3 .P48.p5\.6.;.;. =/?4/`Fp/G/J/J/N$0PL0Q0S0T0U1PV(1`WP1Xp1X1Y1@[1\2P_@2b2c2`f2f2i03PkX3l3m3 o4o(4pP4`qp4Pr4s4x<5`|t5p~5 56PT6`6@6 7\77P7778ЎH8Џp8p8888 909`9P9@9`P:@p::@:@:;@;;;;D<<<,=p=0=>P<>>@>>H??@?@P@`@@APPA A"A`$B&TB*B / C0@C0`C1C2C`3C4C4D5(D6HD6hDp7D8D8DP9D@:E@;(E<\Ep=|E>E@?EPDEFFPIFPMLGPGQGUGY\H[HP_HcIcIc0I0ddI@dxI eIeI@fIhJ`jJpjJmJxJxKP|@K}dKPKpKKL0LXLLPLLp$MHMMMM0N8NXNN Np8O|OPOO OP(P`PP` Q@@Q `QQQQQRR ,RPRhRRRRSвSPSpdSxS0SPSpSSTT8T`TpTTTUHUPUpUU V@ V`4VHV\VpVVоVVVV@VpVWп$W8W0LW@`W W0W`WWX$XlX XYPYtYpYYZ\Z Z`Z8[@|[`[\0`\\`\D] ]]`^^,_P$x_'_6(`6<`6P` ;`;``<`=0a>da?a@@aAbChbDb LbPM$cNpcOcPd@R@dSdUdXHe`mee`4f0pff fp8gpg@g g`gг h`Xhh@hhh0iDi XiiiP8jPjkk0kDkXklkk k0k@kPk`kpk l lll@lPl Hmmn@n`$n ,8o@,Lo.o`4pp@q`EXqEpqEq`FqFr@GrG,rG@rPHXrHtrHrPRrRrSs\\s]xs0]s]s^sP^s^ t^$t@_GAE Dq K K E hQD t H xBBE B(D0D8Dc 8G0A(B BBBF ] 8D0A(B BBBF Q 8I0A(B BBBM X ܱk]zI(t O=IUL sABH ?BEB B(A0A8GPk8D0A(B BBB( Af BADp[ ABD ?MBHE A(D08I@T8A0P(D BBB` DDbx ̱AG |ix 'BBB E(A0A8D@_ 8A0A(B BBBF _ 8A0A(B BBBG _ 8F0A(B BBBB L( BBE A(J0 (A BBBF (A BBBx lkAw H j$ LnBDA cABH LFBBB E(A0A8G`'8A0A(B BBB@ HBAD D0  AABG d  DABE P Զ`d 5BBB B(D0D8G` 8A0A(B BBBF  8C0A(B BBBH d BEF E(A0D8Dr 8C0A(B BBBG  8A0A(B BBBA 0 dD 0e|h\ dh t Do E C E  k]zI k]zI Tk]zI CtNCtNCtN4PCtNLCtNdSDN4|nK^BEE D(A0B(D BBBпDM G 4DBDG M ABC D DBF xD~ F o(xD~ F o(H\ADD d AAA (tADD d AAA @dBED G0^  ABBK D  DBBL D~ F o I 4pBDD  ABG AABH<8BEE E(A0A8DP 8C0A(B BBBF (AGG0D AAK X@CBEB D(D0D2 0A(A BBBA P 0A(A BBBH 84 AAG ' CAD  CAC (LBAD ] DBF xdhQBB B(A0D8DP`HPg 8D0A(B BBBD HBBB B(H0D8GP  8D0A(B BBBA `@$BBE D(A0D (A BEBL * (A BBBG q (D BBBE `p KAD D0  AABK   CABK D  DABE C0 6D Q AA ,8Ab M <LSAG D DH G AH J AM Z HF AV I g I AV I g I PAV I g I A  G H H )Ac48BAD  DBI \ DBF ptA} J { E 8`zAAG ] CAF  CAK T.BGB A(A0D@] 0A(A BBBJ  0A(A BBBJ T(|.BGB A(A0D@] 0A(A BBBJ  0A(A BBBJ TT.BGB A(A0D@] 0A(A BBBJ  0A(A BBBJ T,.BGB A(A0D@] 0A(A BBBJ  0A(A BBBJ `0.BBB B(A0A8FP] 8A0A(B BBBI  8A0A(B BBBJ `.BBB B(A0A8FP] 8A0A(B BBBI  8A0A(B BBBJ CAZ E  A $FD gAFH tDPBBE E(D0D8GC 8C0A(B BBBD XHRAb 8A0A(B BBBA $bBT EE (AO H U S \ A ,BAA G ABB 8@dBBA A(D0] (D ABBH \|BBD A(G@ (A ABBI g (G ABBE h (A ABBJ p(oBKF D(D0F (I DBBF u (I DBBJ  (D ABBE (I ABB<P$ BBB D(A0n (A BBBH H FBBB E(A0D8D@^ 8D0A(B BBBA L BBA A(D@ (D ABBI | (D ABBK ,=d@BBB B(D0D8FP 8A0A(B BBBI  8A0A(B BBBE 0?BAA D` AABdHBBB B(A0D8FP  8A0A(B BBBI  8A0A(B BBBE PD:BBA D@  ABBB g  CBBB f  ABBE L:BDG  ABF M KBF D KBO _ HBG hBIB A(D0G`dhVp_hA`S 0D(A BBBG nhOpGhB`phApFhA`T@D W E _t@D W E _@D W E _hBIB A(D0G`dhVp_hA`S 0D(A BBBG nhOpGhB`phApFhA`h  BIB A(D0G`dhVp_hA`S 0D(A BBBG nhOpGhB`phApFhA`X$@D Z B _hx$BIB A(D0G`dhVp_hA`S 0D(A BBBG nhOpGhB`phApFhA`,(@D Z B _P8L(BBD D@   ABBC f  CBBC V  ABBE L* BBB B(A0A8G 8A0A(B BBBK t5 BPO A(A0GLGBzKFA\V_Ay 0D(A BBBJ $T>bAA aAB$|>LBDA AABH ?QBBB E(A0A8D@28D0A(B BBBH?BEE E(A0A8D`Y 8D0A(B BBBC (< XABED v EBH h ,B>Ae J Id LBBBB E(D0A8Gp 8A0A(B BBBH  8A0A(B BBBH EBBB B(A0A8D` 8A0A(B BBBI M 8C0A(B BBBG ~ 8A0A(B BBBH thRp`hA`F 8A0A(B BBBI d!IUBBE E(D0A8Gp 8A0A(B BBBG  8A0A(B BBBH d!P BBB E(D0A8G 8A0A(B BBBF n 8C0A(B BBBF @d"YBBB B(A0D@? 0D(B BBBH H"V> BDB B(A0A8G 8D0A(B BBB"@_BBB B(A0D8G_FiAaFgDVFgBVFgB 8D0A(B BBBH  FfBgJbA`JeAL#v BBE E(D0D8G 8A0A(B BBBA ( $@BBB E(A0A8GGGGGGGGGGGGGGGGGGGGGGGGGGP 8D0A(B BBBI gGGGGGGGGGGGGGGGGGGGGGGGGGX8%JBB B(A0D@0A(B BBBBH@l 0D(B BBBH T 0M(B BBBO L%c BDB E(G0A8Gj 8A0A(B BBBH &<BBB B(A0A8DP@XV`KhFpFxFFFFFFFFFCCCCCCCCCCCCPP 8D0A(B BBBB XN`gXAPH&jIB B(A0A8D@y 8E0A(B BBBD l 'DBIB B(D0A8DeW_Av 8D0A(B BBBA W_Al'GBIB B(D0A8DeW_Au 8A0A(B BBBA W_Al(U BIB B(D0A8DcW_Av 8D0A(B BBBA W_Alp(tPBIB B(D0A8DeW_Au 8A0A(B BBBA W_Al(TrBIB B(D0A8DeW_Av 8D0A(B BBBA W_AlP)dyBIB B(D0A8DeW_Au 8A0A(B BBBA W_Al)tBIB B(D0A8DeW_Av 8D0A(B BBBA 'W_Al0*BIB B(D0A8DeW_Au 8A0A(B BBBA W_Al*d! BIB B(D0A8DeW_Av 8D0A(B BBBA W_Al+-BLE B(A0A8DW_AW 8A0A(B BBBF W_AX+D2 BIB E(A0A8D|HYAW 8A0A(B BBBI X+?dBLB B(A0A8D|HYAW 8A0A(B BBBI L8,CWBLB B(D0A8G  8A0A(B BBBK l,[BIE B(A0A8GNW_A[ 8D0A(B BBBH W_AL,L`m BDB E(G0A8G 8D0A(B BBBK XH-lkdBLB B(A0A8D|HYAW 8A0A(B BBBI (-onBBB B(A0D8GGGGGGGGGGGGGGGGGGGGGGGGGGPGGGGGGGGGGGGGGGGGGGGGGGGGP3 8D0A(B BBBK l.ċBLE B(A0A8DW_AW 8A0A(B BBBF W_A(@/nBBB B(A0D8GGGGGGGGGGGGGGGGGGGGGGGGGGPGGGGGGGGGGGGGGGGGGGGGGGGGP3 8D0A(B BBBK ll0XBLE B(A0A8DW_AW 8A0A(B BBBF W_A(0BBB B(D0A8GGGGGGGGGGGGGGGGGGGGGGGGGGPGGGGGGGGGGGGGGGGGGGGGGGGGP+ 8D0A(B BBBG l2 BLE B(A0A8DW_AW 8A0A(B BBBF W_Ax2\~!BIB E(A0A8J 8D0A(B BBBG LGGGGGGGGGGGGGGGGGGGGGGGGXL43 7 BBB B(D0A8G 8A0A(B BBBC L37 BBB B(D0A8G 8A0A(B BBBA 3 BBB E(A0A8GFABBBBADAA G G B B G G G G G G G G G G G G LWLGGGGGGG G G G G G G G G G G G G G G G G Pl 8D0A(B BBBG DAABBBBBB G G G G G G G G G G G G G G A A LWLGGGGGGG G G G G G G G G G G G G G G G G PX5(lBLB B(A0A8DL`Au 8A0A(B BBBA l46-4BLE B(A0A8DCW_A^ 8A0A(B BBBC gW_A\62N#BPN H(L0A8J= 8D0A(B BBBG W_AL7U BBB B(A0A8J} 8A0A(B BBBE LT7`UBEB E(A0A8G( 8D0A(B BBBD x7xBBE A(D0DPFA\ 0D(A BBBC LV_A 0D(D BBBF L 84}BEE B(A0A8G 8D0A(B BBBK pp8BBA D(GPFAX (D ABBE LW_A (D DBBG 80@"BBB E(A0D8GAFBObAVV_A 8D0A(B BBBJ h9z"BMT P(L0D8GV_A 8D0A(B BBBJ FgAzAFB89BEA  BBD v EBC (:5&BBE E(A0A8GAFBObAV_AZ 8D0A(B BBBD :$&BGB E(I0A8G+LGGGGGGGGGGGGGGGGGGGGGGGGX 8D0A(B BBBC h;(rBMT M(I0A8JV_A 8D0A(B BBBE SKbAAFB8;EBEA  BBD v EBC ((<F&BGB J(A0A8GL GGGGGGGGGGGGGGGGGGGGGGGGGTfGGGGGGGGGGGGGGGGGGGGGGGGGX 8D0A(B BBBD \T=Pl\BPG B(A0A8T 8D0A(B BBBA VRA(=Pr&BGB J(A0A8GL GGGGGGGGGGGGGGGGGGGGGGGGGTfGGGGGGGGGGGGGGGGGGGGGGGGGX 8D0A(B BBBD \>ė\BPG B(A0A8T 8D0A(B BBBA VRA(@?ĝk&BBB B(D0A8GGGGGGGGGGGGGGGGGGGGGGGGGGXGGGGGGGGGGGGGGGGGGGGGGGGGX9 8D0A(B BBBE ll@BLE B(A0A8DW_AW 8A0A(B BBBF W_AL@XVBBB E(D0D8J 8D0A(B BBBF ,A*/2BBB B(A0A8GbHHK T[BxT\BQ[ADQ^ABT\B[T[A[T\BDT[A*T^ABT\BET[A[T\BZT[AET^AAT[B[T\BZVYBDQ[A8A0A(B BBBBA[(BgAAG N CAE CZ CQFB88CLBBA / BBH s BBI tCD\CHhBBA A(D0Y (A ABBG p (F ABBE k (F ABBJ $CXAq N r F mDD[ A r F \4DPhBBA A(D0Z (A ABBF p (F ABBE k (F ABBJ \D`BAA D0  AABG k  FABD p  FABG k  FABD <DiAD } AE k FG p FJ k FG t4EfBBA A(D0 (D ABBJ Z (A ABBH F (D ABBI q (D ABBF @EBAA G0  EABG |  CABN 0ETBAA G0  AABO @$FBAA G0  EABG |  CABN 0hF|BAA G0  AABO pFVBBA A(D0 (A ABBH I (A ABBA p (F ABBE p (F ABBE tGBBB A(A0S (D BBBK W (A EBBG W (D BBBG p (D BBBF $G<Dy C u K u K (GDf F r F r F GG!H!H!,H!@H4!THP!hHl!8|H!KHD  DBN HNNLH|_BBA A(D0 (D ABBK s (D ABBD IYDO M x$(IAG { AD xAPI4YDO M x(pItAG m AJ p FJ IH@D;$Il6DJ B u K u K $I6DL H u C u K $JFDY C u K u K $0JFD[ A u K u K $XJ6DL H u C u K $J6DE G u K u K $JFD[ A u K u K $JD6DE G u K u K LJ\FAAD  AAF s CAA z FAG p FAA LHK\zAAD  AAF  CAC z FAG p FAA <KAy F N B r F r F m C  C r<KLA@ G r F { E r F m K m A r<LAy F p H r F r F m K m C rXLDu G r F xLDu G r F L\Du G r F L Du G r F LDu G r F LlDu G r F MDu G r F 8MDU G r F XM< DV F r F xM DV F r F M DV F r F M DV F r F M DV F r F Ml DV F r F N DV F r F 8NL DZ B x H XN D_ E x H xN D_ E x H N|D_ E x H N<D_ E x H (ND E x H L D D D OD_ E x H $OD_ E x H ,DOp.DU G ^ B D L  C $tOpDu G r F U C `OYBBB B(A0A8D@s 8A0A(B BBBE P 8F0A(B BBBA (PAt K o A q G h ,PD F ^ J PPTD@ D u C tPDM G M K $PD E u C u K 0PdAs D r F r F m K r0PP.A~ A u K u K p H u(QLDc I u K $HQDd H T D t D pQTDq K u K 0Q3AA F G A r F m K r4Q AG x AG p FB E FE (Q Ag H u K @ H p`(R GBBA A(G@n (F ABBJ v (A ABBD g (A ABBC 4R!ADP AH x AG x AG <R$AD@} AE x AG x AG  AM (S%Az U u K u K p 0S&AG  AE \TS'BBA A(D0i (D ABBD s (D ABBD s (D ABBD $S)Df F r F r F ,Sx*VD` D u C u K u K $ T+DC I u K u K H4T,&BBB B(A0A8D@ 8D0A(B BBBG Tt1Dj J G I xDT 2aKBH D(K0 (D ABBH h TH3;A  B j$Ud4&D~ F u C u K $8Ul5&D@ D u C u K $`Ut6&D@ D u C u K U|7DX D r F U7DY C r F U\8DY C r F ,U8AD~ F D D x H x H $V9aD G G I u K $@V4;D I x H x H $hV<6DI C u K u K $V=6DK A u K u K $V>6DK A u K u K V?DY C r F Wd@DY C r F  W@DY C r F ,@WDAQD B D L x H x H LpWtBBBA A(G@U (D ABBE p (G DBBA LWCdBAA G0\  AABE   AABD _ AAB$XEaD G G I u K $8XLFD I x H x H D`XGnBAA  ABH  ABH m FBA 8XIsKHD y CBK AFBGX0JV$X|JDd H I G r F , YDK1AD J AH h FJ PYTLDf F r F pYMBBB B(A0A8Dp 8A0A(B BBBB  8F0A(B BBBD  8C0A(B BBBE ` 8F0A(B BBBA Z,PBBB B(A0A8Dp 8A0A(B BBBA  8F0A(B BBBD  8C0A(B BBBA p 8F0A(B BBBA ZRDf F r F ZDSDf F r F 4ZSAAD0R DAC ` GAH [\UBBB B(A0A8Dp 8A0A(B BBBF  8F0A(B BBBF  8C0A(B BBBH h 8F0A(B BBBA [XDf F r F [TYDf F r F `[Z.BBB A(A0 (A BBBH U (F BBBG m (F BBBG pT\[nBBB A(A0 (A BBBI  (A BBBG m (F BBBG m (F BBBG d\]BBB B(A0A8D 8A0A(B BBBG P 8F0A(B BBBA $zRx ,@]$l]`cD G E r F r F ]Pal]\aRBBB A(D0m (F BBBD  (C BBBH m (F BBBG m(F BBB$^LbDE G U K u K @^cDj B B F `d^PdaBBB B(A0A8D@v 8A0A(B BBBB P 8F0A(B BBBA (^\eAz E o A v J h,^f1AD J AH p FB 0$_ gAv I r F r F m C r0X_ h@AF I o A [ E j F o,_i1AD J AH p FB 0_(j;A H H H o I E K J_4kDt H u K 0`l3AD K r F G A m C r(D`mAy F u K p H up`mb`0niQ`ni`oc``oiQ`oi`0p1Fe8aTp\BBA  BBH s BBI LaxqAp G jDlarpBAA p ABI m FBK h FBH $a0sAu J r F masD^ F r F Da(tBAGP ABA  ABF q ABD 8DbpxBAGPJ ABI F ABG Pb{pBAA GP6  AABK   AABE ,  AABH 4b~FAAG@ AAB x AAF @ cBBA G0  FBBE   ABBI pPcBBB A(A0G`  0A(A BBBF  0A(A BBBI q 0A(A BBBG 4cBAD 1 DBB s DBG $cHDf F r F r F L$dBAD s ABC H FBH p FBH p FBH $tdDd H x H x H PdHvBBA a BBF e GBB m GBJ X BBD $dtDd H x H x H $e<Dd H x H x H $@eDs I u K u K 8heܐQAD Z HI u AJ p FJ 8e|BBA  BBE s BBI $eDDb R H H x H $fܓDb R H H x H 0ftJQ E OTfD\lfȕBBA A(D0 (A ABBH p (F ABBE p (F ABBE LfHVBBB B(A0A8Dm 8D0A(B BBBH gXD[ A r F $AG  CI ) CD @qTBAC G0  AABJ   DABF qLAG0B AE r2AG0 AH }?AGpa EB H~dE]AAG0i AAD  AAD  AAG 4\~xIAAD  AAC h FAA ,~JBAC   ABF ~LV(~LAAD o AAI NGD  G OSG0iAL D0PSG0Ip h QSG0iADQBAC G0  AABA   DABF 4TfAAD q AAG h FAA  VD o A (VeR M @VOG fAC dPWSG0AP ,XSG0nAGDXBAC G0g  AABH   DABE [AG0 EA @\BAA GPX  AABI   CABI \dD0p A (xeBED0Y JBI eH0` A fH0^ A 4܁fvAAD  AAB h FAA 0gJBJE G@  ABBA @HhBAC G0i  AABF   DABE HkD0 A (kVAAD f AAB Ԃ0ma$m<OO0ANE0nT0u A ,HoaG HE$LoOK0ADE0,tp~AG@ CJ R CA DrBAC G0  AABF   DABE u-u4uOMt G 0vD0M W _PvbD} O Q$pvKz [ i W G I |wTK` M S(wAAG j AAK (xAG [ CJ  AA (yEADF0- AAF <{4P{<Mnl|;Dn @|AK0 Ej |=Dp(ą }AAG k AAJ (~AG o CF A,~eADF0 AAE L3AG iAl(AG0C]K0AX]K rĂ؂-QW(VAAD s AAE D BAC G@  AABF   DABE \DD (tAAD _ AAI TAzQG `Aԇ=X[E@Tg(<AG a CD CD40~BAD F0G  AABH   DABF |h?D z@AAD  AAH h FAI p FAI ؈CD z@$BAC G0  DABA   DABF 43H4\ؐAZxܐVAG LA(7T5(ԉAG  AA RC(AAG k AAJ <,BAD  ABH AB lx;AG 1A AG o ED `AG E ԊPAG0q AF L AG  EA (8BAG@q ABB HTAG0g ED l(AG@E EF @AAG u CAF N AAH  AAG ԋ4-M_Ф;X^X0V4pYX p(LfAAD m AAK (xAG ~ CG rC,BAC  ABJ Ԍ%NV(rAG V EE DE)NV8 ALHR N P P(lAO0P AG A(\AAD P AAH ,č0NAG B CC  AE <PBAD N ABH  ABI (40.BAG@ AB8`4AAG0X AAE  AAI 4DpX OȎaR (AAD P AAH  ȲSTz($AG P CE C8PBAD  ABE  ABI `0n WAUdA(ԏ|AK ] AF JA("BAF  ABG ,6AG dA4LAAG T AAA k AAC 0и AK  AG J FP JEAO0wAؐ,ARn`A(AAD P AAH (4pAK  CH | AA @`DXBAD F0n  AABA   DABE `AG E đ AG0 AA @AAG0) CAJ b AAD d CAH ,8Y@]Tih,4|XqQ94AAG q AAD xFA8%BAD  ABG { ABB 30DEXAlX]pY(6AAG  AAC ēؓl,hBAD  ABB 4vAAD  AAB h FAA DT0BEA G@  DBBG g  DBBA ~AK nCHUAK GAܔU^c(AK@ AF bA0 AK0 AD Y AF KA(TPAAD s AAE (AAG r CC  CE 48QBAD F0  DABI `3(9IS0W EH A@$BAA D0  AABF h  FABG hl b(~AG X CE | CI 4bBAC G0`  DABD (0HBAG@L DBD PT BAA G0X  AABI   CABI   CABI d5(x<AG W AH E4AAD U AAC ` FAI ܗGT nKXn( 8AG o CF C08BAD  ABD lxLAG A dQG ADĘ`9AAD0] CAI u AAI h FAA D X9BAC G0  AABG {  DABF @TPBAG@T DEI - AEA CHPBAA G@X  AABI n  CABD   CABA 8OAy F N h- EQO cA4@AAD U AAC ` FAI 8xBAD  ABF  ABJ 80-AG u FE M AJ D FF gJP$BAA G0X  AABI   CABF b AABDSX,kAG aA x|QO mA4AAD g AAA h FAA Dԛ?BAC G0  AABA k  DABF "0lQO yJ(PDAAD O AAI |CXf(PAG o CF C0BAD  ABB p aAv I  H D BBBA GpE  DBBA   DBBH p`BBA A(G` (D ABBG  (A DBBA U (A DBBA L (A DEBA TԝBAA G0.  AABC   CABG ^  AABF (,HAAD [ AAE X-lIU[0BAD  ABE 0!4BAA G@" CAB8" AAG0X AAE x AAF ,#5Dp4D#BAG d DBD ODB|($AG0A$LAf I `$gBBE A(A0< (A OBBM x (I BBBA O (I BBBA `&BBB B(G0A8G@r 8C0A(B BBBK O8F0A(B BBBPT(BBE A(A0 (A WOBH  (I BBBA Ԡ*DIL*"BEE A(FP (D BBBA g(D BBB<+rPd-d-qAR E H@.BEB E(D0A8DP 8C0A(B BBBD С.9D b J DH.BBE E(A0A8D` 8D0A(B BBBF <H3iAG _A\3LAf I Hx3BBA D(J0 (C ABBA D(F ABBPĢ`4BBE A(A0  (A BBBA  (I BBBA  6DIl06BBG A(G0f (A ABBA x (A ABBB \ (A ABBF (A ABBHd7BEE A(F@ (D BBBC g(D BBBH9]BBE B(A0A8DP2 8C0A(B BBBA 88:AAG r AAC m FAD t:AG@A;LAf I |;BBE E(A0A8D@E 8A0A(B BBBM C 8A0A(B BBBA Y 8A0A(B BBBA `0=BBE B(A0A8GP 8C0A(B BBBD O8F0A(B BBBX`?MBBE A(A0G@E 0A(A BBBA  0A(A BBBA TADIP\ABEE A(F`9 (D BBBJ t (D BBBB L\CBDK A(G0 (A ABBG  (M ABBI DGoBEG@ DBI  ABG ` DBJ (@ID ; A T D  E \ MyED D AABFHE  FABJ m  FABJ 4PJDD E DAL HNM@PBBE A(A0D@ 0A(A BBBG L8RBBD A(KP (E ABBA )(A AFB@LVBAA G@S  EABJ  AAF,TXBHC  ABG H$YbBBB B(A0A8Dp 8D0A(B BBBF 4 HhBAD  DBH s DBG DiDn F u C d`jDq K u K $@kD D ] K U K lDm G u C $̩xmDx D u K u K PnD^ F u C nDY C u K 4PoDZ B u K ToDf F u K tpDg E u K 0qDg E u K $qD} G J F I G ܪxrDg E u K LsBDK A(GP (A ABBE  (M ABBO 4LxuBLA A(D0(D ABB4`vJDD W DAJ HNM,vdAG@p AG D AK D(yBAD G@  AABH    AABK 44p}BIG@ DBA `DB@lH~BAA G0t  AABE }  AABG h BFB A(D0D` 0A(A BBBH | 0F(A BBBG 0F(A BBBX>BFB A(D0Dh 0C(A BBBK N 0F(A BBBE 0xBDA DP  EABG @hBBB A(A0G` 0A(A BBBA 4BAD  DBG s DBG (Df F u K ,H\As D r F  F r F Px6AAD  CAJ u AAI p FAI  FAI p̮BBA A(D0 (C ABBD u (A ABBE e (F ABBH  (F ABBA P@4BAA D01  DABH c  DABF @ DABDg E u K Dg E r F ԯ`Df F u K 0Dg E u K $D E J F I G <xDg E u K $\AC D P H uL1AAD@B AAF u AAI x AAF P AAF L԰p=BBB B(A0A8DP@ 8C0A(B BBBF 4$`BAD e DBF  DBG (\HBAD ` DBC ̩De G u K 4&BFA  ABG RKD(QBHA BABī^OLMu 0iFi I P8TPBAA  ABE  ABJ 0ĮBAG o DBA sDBIJPDZ B u K ЯDd H u K (pBAA w ABB 0԰ DаL\ȰBKK A(G0 (A ABBD  (M ABBI (VBHA GAB س̲LG \A 8iFh J P A J g I 8D|BAD u DBF  DBG 4ĸDBAD z DBI s DBG @ܹfAAD ~ FAE U AAI p FAI 4AD0o AK  FE m FE <4AD  AA E AJ { AL P AG 8t>BAG@L DBD  DBB PBDI K(A0 (A BBBE  (K DBBJ (BHA ABL0D BBA A(G`e (D ABBE > (D ABBI |BEB B(A0A8D 8A0A(B BBBC ] 8F0A(B BBBA s 8F0A(B BBBA FA I P$$sA[ D  I r8Lh BAD  DBG  DBG 4LBAD  DBB s DBG (FAD  AK p FJ DY C u K  XDf F u K ,(Dg E u K $LD} G J F I G tpAR E aHBNH A(G0  (A ABBG @(F HBBd$BBB E(A0A8D`x 8A0A(B BBBE  8A0A(B BBBE (HlBHA vAB tF{ O P(<AAG s AAB DĹHBAD  DBE { DBG K DBG 4 BAD  DBF  DBG DpDq K u K 4dPBFA  ABF RKD(8QBHA BABȺl^OLMu iFi I P8 BAA  ABE  ABJ 0HlBAG o DBA sDB|DZ B u K xDd H u K (8pBAA w ABB | xpdAZ80BEH e FBD u ABA LlBNH A(G0  (A ABBG F (M ABBH (BHA {AB LF{ O P( AAG  AAF D8|hBAD  DBK S DBG K DBG 4BAD  DBF  DBG $Dg E u K u K Dm G u C <AG  AC u FE  AD N AA @@DAD  AG u FE f AA N AA LeBDK A(G0' (A ABBJ  (M ABBM Ծ (BAG0P DBH ( BAG@ DBD `@BAA G0t  FABH Z  CABH e  CABE Z  CABH Q4<BAD  DBH  DBG $ {A J  F r4< DBAD  DBG s DBG $PT Dw E u K u K x,Dj B u K $D} G J F I G (tDx D r F r F P8BAG0M DBC t DBF b DBH Y DBA 8@vBMA  ABD R KDO 4|sAH G j F | D } C  A r0 0,BAD  DBD sDBQ8$lBAA  ABF [ ABJ LLXBKK A(G0 (A ABBC ) (M ABBE (BHA AB ,~LG \A yFz H P4AAG  DAG a JAD 8H\BAD  DBD  DBG 4DBAD z DBI s DBG @fAAD ~ FAE U AAI p FAI P!BAA D0   AABJ   FABJ m  FABJ <T$AD  AF u FE { AD N AI &8&~BMA 2 ABK  KDL `(+Df0x(AAG m AAH \AA0(Q4D0)BAD  DBK s DBG P|*vBAA D@  AABB `  FABG z  FABE D-YBAA D@  DABK 3  DABF ,1Do E u C u K O A H3Dn F u C h|4Dn F u C $<5SD u G r F u C t6Dg E u K 7Dq K r F 7Dg E u K `8(l8VAU4D8BMA  ABF RKD|89#D^0P9BAD  DBB sDB:Q4h:qBAD  DBA s DBG P;VBDB K(A0 (A BBBJ  (K DBBJ (h>BHA AB ?Jc K PP @BBB A(A0 (D BBBH p (D BBBF 4 GBAD  DBI s DBG DID` D u C dID` D u C 0J{AT K r F r F r F rLKD_ E u C KDe G u K LDf F u K lMDY C o A 8MDp D r F $XNA| C o A r@O#BDA D0  FABH d  CABF DPBBE A(A0D@ 0A(A BBBA P QBKA G0  DBBF   DBBK p  DBBH (`$SBHA AB4SAG ! CD ~ AA m FE 4pUJDD W DAJ HNM8U4BED  ABG _ ABA 48VBBA D(D0(A ABBptWZ(ZzAG V AQ DE\[GBDK A(G@HBPQHA@I (A ABBF  (M ABBH `$^ BBA A(GP (F ABBJ  (F ABBH X (A ABBB @kLBAD  DBH s DBG L DBF (mAG M AJ kA4ntBAD  DBD s DBG 0doDW E m C ,TpAD U AE  FB $0r\D D T D u K $hsFDI C u K u K tDn F u C PuDj B u K vDX D o A @4pvAD@` AJ  FH  FE  FE 4x{AD  AF E AJ P AG (T}nDGP RAF D u }A \ E [ 4T~BMA  ABE RKD(8~BHA AB dPF{ O PT:BAA  ABB K ABB b ABA Y ABA @DBAD  DBC s DBG s DBG \$BIA A(D@ (A ABBB _ (A ABBK L (J ABBE LBMD D(D@ (A ABBF Y (A ABBI d0vBDL B(A0A8G@ 8A0A(B BBBE   8F0H(B BBBF (<HBHA AB h J M PL؍BBA A(D0 (D ABBH k (D ABBD HyBEB B(A0A8Dp 8C0A(B BBBA L(ܙ!BGB B(D0A8G 8C0A(B BBBA 4xlBAD  DBA s DBG D` D u C tD` D u C Dn F u C Dn F u C 0Dn F u C PDm G u C ppd2BBA A(D0 (F ABBF u (A ABBE p (F ABBE a (A ABBA 40FAD J AH p FB p FJ HDu G u K <(D_ E u C \Df F u K |xDf F u K HDY C o A @AAD0 AAB m FAD m FAD (#A B o A r8,BMA  ABE ` KDI h| $|xAFG@AA0@>BHF G@  AABG LQ4AAG c CAH }FA8$`dBAD  DBG  DBG 4`DBAD  DBG s DBG $Dw E u K u K Dg E u K (D E J F I G 8 ȸSBMA  ABE X KDI H \AG0A(|ȺOAHG@<AAQ08AEG  CAB mFA8ABAD  DBA [ DBG 4,4BAD w DBD s DBG dDg E u K $D E J F I G 8HBMA  ABE ` KDI  4(AFGP AAA ;AA44aAFGPD AAD AAlQ8DBAD V DBE  DBG 4ZAEG  CAH E FAD 4DBAD  DBG s DBG ,(Du G u K LDg E u K $lD E J F I G 4PAK@ E\ r EQ | EO ,AAG  CD  FC hAHP AK L 4BDK A(G0U (A ABBD  (M ABBM pt psAG@` EC 0BBA GP  FBBB BBB B(A0A8G^ 8A0A(B BBBG m 8F0A(B BBBD  8F0A(B BBBD  8F0A(B BBBA x<AQ F cQ4BAD 2 DBA s DBG L|BAD  DBE s DBG s DBG s DBG $4D E u C u K $\D| H u C u K 4&AD G AC Y FA p FJ D] G u C DZ B u K DZ B u K DZ B u K <Dg E u K $\(D} G J F I G 8vBCD  ABD Y DKH ($BHA ABHBBE B(A0C8GP) 8C0A(B BBBA zRx P(aLp BBB B(D0A8G 8A0A(B BBBA 4AAG0 FAA  CAK JM I P4hBAD ` ABF L DBN 8TZBAA  ABJ S ABA 44 BAD  DBE s DBG $ BD D N J u A @4AAD0 AAB ] CAG m FAA @4pdBDB K(A0G@ 0F(A BBBF x" "=Aj E H "tAG K AD #QH`#BBB B(A0A8D` 8D0A(B BBBH 40'TBAD0 DBD s DBG h(Dk A u K ()AD0 AA u AJ H@+LBBD A(M0 (A ABBI U(F HBBD, @,Q$(,A_ H o I o4P,BAD  DBF s DBG 8.BAD  DBD s DBG 00|r,1BAI S ABF Hd26BBB J(A0A8D@ 8D0A(B BBBG L\X3fBDK A(G0  (A ABBH  (M ABBG 4x7BAD z ABD eABH@9BBE B(D0C8D` 8A0A(B BBBE d0; BFB B(A0A8D 8A0A(B BBBA  8A0A(B BBBA |FDQ,FBDA  ABF HG2BEB B(A0A8G`~ 8C0A(B BBBB <,LBBA D(J/ (A ABBA 8lQBED A(D0 (A ABBA <RKDA \ DBE ELPHS) BBB B(A0A8Dpj 8D0A(B BBBA H4_BIB B(D0A8Dp, 8A0A(B BBBA 4tcBAD  DBI { DBG dDf F u K eD` D u C LfD] G u C fDX D o A 8,gD] G u C XgDg E u K (xYBDB K(A0$ (A BBBA 3 (K DBBJ 4BCADG@ AAD m FAD @PFBBE A(A0D@K 0A(A BBBF Hak D z F 0 (A ABBC q (M ABBM l~*^ ]U H ABME E(D0D8D` 8E0A(B BBBE HBME E(D0D8Dp 8E0A(B BBBB ` BED D0   DABG R  DABG   DABD p  DABI 4WADP\ AAF HPH][AD@[ EAG PAAf@L AAJ @ AEG0O AAB w EAC  AAE @L x3BAD G0%  AABI D  FABC L tEBBB B(D0A8G 8D0A(B BBBD 4 tJDD i DAH HNML  BEA A(D0 (A ABBA [ (A ABBA (h l>bHG0 ABE @ BAA G@  EABC   AAFC H ~BEB B(A0A8DK 8A0A(B BBBJ @$ 0gBAA G@  EABG L  EEBH Hh \OBBB B(A0A8G` 8F0A(B BBBC 8 `qBLA C(D02 (D ABBG 8 BAD f ABH A FBG p, (BBB A(D0Jp 0A(A BBBC p 0F(A BBBC  0F(A BBBC 8 BBA A(GP (E ABBA L 8 BBB B(A0A8G 8A0A(B BBBA L, BBB B(A0D8Dy 8D0A(B BBBI 4| 8BAD  DBH s DBG $ D D ] K U K P X6AAD  CAJ u AAI p FAI  FAI p0DBBA A(D0 (C ABBI u (A ABBE m (F ABBH  (F ABBA TnBAA e ABD { ABB y ABD % ABH Dg E u K Dj B r F <xDm G u C $\8D{ A u K u K D^ F u C DY C u K DZ B u K Di C u K PDa K o A $Dg E u K DPDg E u K dD_ E u C `D_ E u C D_ E u C `D} G u C 0Dq K u K 0{AT K r F r F r F r8\Dg E u K $XDg E b F I G Dg E u K 4t AD0 AJ  FE m FE L6BBB A(D@" (A BBBD D (P EBBK D (P BBBF P (A BBBI P (A BBBI d\BBB B(A0A8D@ 8A0A(B BBBF } 8F0A(B BBBA ``BBE B(A0D8D` 8D0H(D BBBF D 8A0A(B BBBB 8(.BED A(GP (D ABBN 0d\BDA G0B DAB8 BEA A(G0  (D ABBJ ` BBE B(A0D8D` 8D0H(D BBBF D 8A0A(B BBBB 08\BDA G@  DABH 8l8zBED A(G@ (D ABBH 8|IBEE D(A0)(D BBB 0 oACM } AAH DMA@ 0TBAD  DBD sDBlQ4BAG e DBK sDB @AG n EE dKBDI I(K0A8G@G 8A0A(B BBBD T 8M0A(B BBBF ` 0tBBA G@ ABB " AGP AI -Q8-YBAD M DBF  DBG 1A M  F 4@2TBAD  DBG s DBG $x3D L H u C u K (4FAD  AK p FJ 5&D| H u C 6Dg E u K $ h7D} G J F I G $4 8D M G r F r F 8\9BMA  ABD b KDG (:pBHA aAB8:BDD  CBA   ABC D=iFi I PL$=BBB B(A0A8D 8A0A(B BBBH 4tDBAD  DBE s DBG 8FDm G u C $F9D c I r F u C HDg E u K HDg E r F d4 IBDL B(D0A8DP 8A0A(B BBBJ  8M0A(B BBBH (MBHA ABH,NEBFE B(A0A8GP 8C0A(B BBB@0OeBIB A(A0GPD0C(A BBB,X\QgBAA 0 EEB R6Dq RF{ O P8,S"BAA W ABB s ABJ 4 U\BAD  DBI s DBG \<HVBBA A(D0 (A ABBC ` (A ABBJ p (F ABBA \WBBA A(D0 (A ABBD s (C ABBE m (F ABBA 8YDn F u C YDn F r F <ZDm G u C \x[Dg E u K $|\D} G J F I G $\Di C u K ]]]m ]AG o FA ,X^BTmXD^BAD b DBA a DBI p DBJ G DBC O DBK ____8_Df F U K T L K E D L q O D,`@a4TaAAD ~ AAJ T CAH xaaa aaF[jaDR J r F $ @b<BAL iAB(< XbBAD ` DBC h bLFO( cBAD }Fr8 cBBA D(D0d (D ABBF 4 xdAAD b DAC p GAH H$!0eBBB B(A0A8D@~ 8D0A(B BBBG 4p!fBAD  ABB NAJ(!gPDD xDB!h! 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