diff --git a/wafo/kdetools/kernels.py b/wafo/kdetools/kernels.py index 94e013c..928d067 100644 --- a/wafo/kdetools/kernels.py +++ b/wafo/kdetools/kernels.py @@ -22,6 +22,12 @@ def _assert(cond, msg): if not cond: raise ValueError(msg) + +def _assert_warn(cond, msg): + if not cond: + warnings.warn(msg) + + # stats = (mu2, R, Rdd) where # mu2 : 2'nd order moment, i.e.,int(x^2*kernel(x)) # R : integral of squared kernel, i.e., int(kernel(x)^2) @@ -36,31 +42,39 @@ _stats_logi = (pi ** 2 / 3, 1. / 6, 1 / 42) _stats_gaus = (1, 1. / (2 * sqrt(pi)), 3. / (8 * sqrt(pi))) -def qlevels(pdf, p=(10, 30, 50, 70, 90, 95, 99, 99.9), x1=None, x2=None): - """QLEVELS Calculates quantile levels which encloses P% of PDF. +def qlevels(pdf, p=(10, 30, 50, 70, 90, 95, 99, 99.9), xi=(), indexing='xy'): + """QLEVELS Calculates quantile levels which encloses P% of pdf. - CALL: [ql PL] = qlevels(pdf,PL,x1,x2); + Parameters + ---------- + pdf: array-like + joint point density function given as array or vector + p : float in range of [0,100] (or sequence of floats) + Percentage to compute which must be between 0 and 100 inclusive. + xi : tuple + input arguments to the pdf, i.e., (x0, x1,...., xn) + indexing : {'xy', 'ij'}, optional + Cartesian ('xy', default) or matrix ('ij') indexing of pdf. + See numpy.meshgrid for more details. - ql = the discrete quantile levels. - pdf = joint point density function matrix or vector - PL = percent level (default [10:20:90 95 99 99.9]) - x1,x2 = vectors of the spacing of the variables - (Default unit spacing) + Returns + ------ + levels: array-like + discrete levels which encloses P% of pdf QLEVELS numerically integrates PDF by decreasing height and find the - quantile levels which encloses P% of the distribution. If X1 and - (or) X2 is unspecified it is assumed that dX1 and dX2 is constant. - NB! QLEVELS normalizes the integral of PDF to N/(N+0.001) before - calculating QL in order to reflect the sampling of PDF is finite. - Currently only able to handle 1D and 2D PDF's if dXi is not constant - (i=1,2). + quantile levels which encloses P% of the distribution. + + If Xi is unspecified it is assumed that dX0, dX1,..., and dXn is constant. + NB! QLEVELS normalizes the integral of PDF to n/(n+0.001) before + calculating 'levels' in order to reflect the sampling of PDF is finite. Example ------- >>> import wafo.stats as ws >>> x = np.linspace(-8,8,2001); >>> PL = np.r_[10:90:20, 90, 95, 99, 99.9] - >>> qlevels(ws.norm.pdf(x),p=PL, x1=x); + >>> qlevels(ws.norm.pdf(x),p=PL, xi=(x,)); array([ 0.39591707, 0.37058719, 0.31830968, 0.23402133, 0.10362052, 0.05862129, 0.01449505, 0.00178806]) @@ -74,73 +88,58 @@ def qlevels(pdf, p=(10, 30, 50, 70, 90, 95, 99, 99.9), x1=None, x2=None): qlevels2, tranproc """ + def _dx(x): + dx = np.diff(x.ravel()) * 0.5 + return np.r_[0, dx] + np.r_[dx, 0] + + def _init(pdf, xi, indexing): + if not xi: + return pdf.ravel() + if not isinstance(xi, tuple): + xi = (xi,) + dx = np.meshgrid(*[_dx(x) for x in xi], sparse=True, indexing=indexing) + dxij = np.ones((1)) + for dxi in dx: + dxij = dxij * dxi + _assert(dxij.shape == pdf.shape, + 'Shape of pdf does not match the arguments') + return (pdf * dxij).ravel() + + def _check_levels(levels, pdf): + _assert_warn(not np.any(levels >= max(pdf.ravel())), + 'The lowest percent level is too close to 0%') + _assert_warn(not np.any(levels <= min(pdf.ravel())), + 'The given pdf is too sparsely sampled or the highest ' + 'percent level is too close to 100%') + + pdf, p = np.atleast_1d(pdf, p) + _assert(not any(pdf.ravel() < 0), + 'This is not a pdf since one or more values of pdf is negative') + _assert(not np.any((p < 0) | (100 < p)), 'PL must satisfy 0 <= PL <= 100') - norm = 1 # normalize cdf to unity - pdf = np.atleast_1d(pdf) - _assert(not any(pdf.ravel() < 0), 'This is not a pdf since one or more ' - 'values of pdf is negative') - - fsiz = pdf.shape - fsizmin = min(fsiz) - if fsizmin == 0: + if min(pdf.shape) == 0: return [] - N = np.prod(fsiz) - d = len(fsiz) - if x1 is None or ((x2 is None) and d > 2): - fdfi = pdf.ravel() - else: - if d == 1: # pdf in one dimension - dx22 = np.ones(1) - else: # % pdf in two dimensions - dx2 = np.diff(x2.ravel()) * 0.5 - dx22 = np.r_[0, dx2] + np.r_[dx2, 0] - - dx1 = np.diff(x1.ravel()) * 0.5 - dx11 = np.r_[0, dx1] + np.r_[dx1, 0] - dx1x2 = dx22[:, None] * dx11 - fdfi = (pdf * dx1x2).ravel() - - p = np.atleast_1d(p) - _assert(not np.any((p < 0) | (100 < p)), 'PL must satisfy 0 <= PL <= 100') - - p2 = p / 100.0 ind = np.argsort(pdf.ravel()) # sort by height of pdf ind = ind[::-1] - fi = pdf.flat[ind] + sorted_pdf = pdf.flat[ind] + pdf_dx = _init(pdf, xi, indexing=indexing) # integration in the order of decreasing height of pdf - Fi = np.cumsum(fdfi[ind]) - - if norm: # normalize Fi to make sure int pdf dx1 dx2 approx 1 - Fi = Fi / Fi[-1] * N / (N + 1.5e-8) - - maxFi = np.max(Fi) - if maxFi > 1: - warnings.warn('this is not a pdf since cdf>1! normalizing') - - Fi = Fi / Fi[-1] * N / (N + 1.5e-8) + cdf = np.cumsum(pdf_dx[ind]) + n = pdf_dx.size + # normalize cdf to make sure int pdf dx1 dx2 approx 1 + cdf = cdf / cdf[-1] * n / (n + 1.5e-8) - elif maxFi < .95: - msg = '''The given pdf is too sparsely sampled since cdf<.95. - Thus QL is questionable''' - warnings.warn(msg) - - # make sure Fi is strictly increasing by not considering duplicate values - ind, = np.where(np.diff(np.r_[Fi, 1]) > 0) - # calculating the inverse of Fi to find the index - ui = tranproc(Fi[ind], fi[ind], p2) + # make sure cdf is strictly increasing by not considering duplicate values + ind, = np.where(np.diff(np.r_[cdf, 1]) > 0) - if np.any(ui >= max(pdf.ravel())): - warnings.warn('The lowest percent level is too close to 0%') + # calculating the inverse of cdf to find the levels + levels = tranproc(cdf[ind], sorted_pdf[ind], p / 100.0) - if np.any(ui <= min(pdf.ravel())): - msg = '''The given pdf is too sparsely sampled or - the highest percent level is too close to 100%''' - warnings.warn(msg) - ui[ui < 0] = 0.0 - - return ui + _check_levels(levels, pdf) + levels[levels < 0] = 0.0 + return levels def qlevels2(data, p=(10, 30, 50, 70, 90, 95, 99, 99.9), method=1): @@ -262,9 +261,16 @@ def sphere_volume(d, r=1.0): class _Kernel(object): __metaclass__ = ABCMeta - def __init__(self, r=1.0, stats=None): - self.r = r # radius of kernel + def __init__(self, r=1.0, stats=None, name=''): + self.r = r # radius of effective support of kernel self.stats = stats + if not name: + name = self.__class__.__name__.replace('_Kernel', '') + self._name = name + + @property + def name(self): + return self._name def norm_factor(self, d=1, n=None): _assert(0 < d, "D") @@ -305,12 +311,12 @@ class _KernelMulti(_Kernel): p=0; Sphere = rect for 1D p=1; Multivariate Epanechnikov kernel. p=2; Multivariate Bi-weight Kernel - p=3; Multi variate Tri-weight Kernel - p=4; Multi variate Four-weight Kernel + p=3; Multivariate Tri-weight Kernel + p=4; Multivariate Four-weight Kernel """ - def __init__(self, r=1.0, p=1, stats=None): + def __init__(self, r=1.0, p=1, stats=None, name=''): self.p = p - super(_KernelMulti, self).__init__(r, stats) + super(_KernelMulti, self).__init__(r, stats, name) def norm_factor(self, d=1, n=None): r = self.r @@ -325,9 +331,10 @@ class _KernelMulti(_Kernel): x2 = x ** 2 return ((1.0 - x2.sum(axis=0) / r ** 2).clip(min=0.0)) ** p -mkernel_epanechnikov = _KernelMulti(p=1, stats=_stats_epan) -mkernel_biweight = _KernelMulti(p=2, stats=_stats_biwe) -mkernel_triweight = _KernelMulti(p=3, stats=_stats_triw) +mkernel_epanechnikov = _KernelMulti(p=1, stats=_stats_epan, + name='epanechnikov') +mkernel_biweight = _KernelMulti(p=2, stats=_stats_biwe, name='biweight') +mkernel_triweight = _KernelMulti(p=3, stats=_stats_triw, name='triweight') class _KernelProduct(_KernelMulti): @@ -350,9 +357,11 @@ class _KernelProduct(_KernelMulti): pdf = (1 - (x / r) ** 2).clip(min=0.0) ** self.p return pdf.prod(axis=0) -mkernel_p1epanechnikov = _KernelProduct(p=1, stats=_stats_epan) -mkernel_p1biweight = _KernelProduct(p=2, stats=_stats_biwe) -mkernel_p1triweight = _KernelProduct(p=3, stats=_stats_triw) +mkernel_p1epanechnikov = _KernelProduct(p=1, stats=_stats_epan, + name='p1epanechnikov') +mkernel_p1biweight = _KernelProduct(p=2, stats=_stats_biwe, name='p1biweight') +mkernel_p1triweight = _KernelProduct(p=3, stats=_stats_triw, + name='p1triweight') class _KernelRectangular(_Kernel): @@ -404,11 +413,6 @@ class _KernelGaussian(_Kernel): mkernel_gaussian = _KernelGaussian(r=4.0, stats=_stats_gaus) -# def mkernel_gaussian(X): -# x2 = X ** 2 -# d = X.shape[0] -# return (2 * pi) ** (-d / 2) * exp(-0.5 * x2.sum(axis=0)) - class _KernelLaplace(_Kernel): @@ -439,8 +443,8 @@ _MKERNEL_DICT = dict( tria=mkernel_triangular, lapl=mkernel_laplace, logi=mkernel_logistic, - gaus=mkernel_gaussian -) + gaus=mkernel_gaussian) + _KERNEL_EXPONENT_DICT = dict( re=0, sp=0, ep=1, bi=2, tr=3, fo=4, fi=5, si=6, se=7) @@ -530,7 +534,7 @@ class Kernel(object): @property def name(self): - return self.kernel.__class__.__name__.replace('_Kernel', '').title() + return self.kernel.name def stats(self): """Return some 1D statistics of the kernel. @@ -586,8 +590,13 @@ class Kernel(object): visual check by eye. Example: - data = rndnorm(0, 1,20,1) - h = hns(data,'epan') + ------- + >>> import numpy as np + >>> import wafo.kdetools as wk + >>> import wafo.stats as ws + >>> kernel = wk.Kernel('epan') + >>> data = ws.norm.rvs(0, 1, size=(1,20)) + >>> h = kernel.hns(data) See also: --------- @@ -601,7 +610,6 @@ class Kernel(object): Wand,M.P. and Jones, M.C. (1995) 'Kernel smoothing' Chapman and Hall, pp 60--63 - """ a = np.atleast_2d(data) @@ -611,13 +619,13 @@ class Kernel(object): mu2, R, _Rdd = self.stats() amise_constant = (8 * sqrt(pi) * R / (3 * mu2 ** 2 * n)) ** (1. / 5) iqr = iqrange(a, axis=1) # interquartile range - stdA = np.std(a, axis=1, ddof=1) + std_a = np.std(a, axis=1, ddof=1) # use of interquartile range guards against outliers. # the use of interquartile range is better if # the distribution is skew or have heavy tails # This lessen the chance of oversmoothing. return np.where(iqr > 0, - np.minimum(stdA, iqr / 1.349), stdA) * amise_constant + np.minimum(std_a, iqr / 1.349), std_a) * amise_constant def hos(self, data): """Returns Oversmoothing Parameter. @@ -680,7 +688,8 @@ class Kernel(object): elif name == 'gaus': # Gaussian kernel a = (4.0 / (d + 2.0)) ** (1. / (d + 4.0)) else: - raise ValueError('Unknown kernel.') + raise NotImplementedError('Hmns bandwidth not implemented for ' + 'kernel {}.'.format(name)) return a def hmns(self, data): @@ -696,7 +705,6 @@ class Kernel(object): 'triweight' - Tri-weight kernel. 'gaussian' - Gaussian kernel - Note that only the first 4 letters of the kernel name is needed. HMNS only gives a optimal value with respect to mean integrated square error, when the true underlying distribution is Multivariate @@ -725,7 +733,6 @@ class Kernel(object): Chapman and Hall, pp 60--63, 86--88 """ - # TODO: implement more kernels a = np.atleast_2d(data) d, n = a.shape @@ -1081,35 +1088,53 @@ class Kernel(object): warnings.warn('The obtained value did not converge.') h[dim] = h1 * s - # end % for dim loop + # end # for dim loop return h def hscv(self, data, hvec=None, inc=128, maxit=100, fulloutput=False): ''' HSCV Smoothed cross-validation estimate of smoothing parameter. - CALL: [hs,hvec,score] = hscv(data,kernel,hvec) - hs = smoothing parameter - hvec = vector defining possible values of hs + Parameters + ---------- + data = data vector + hvec = vector defining possible values of hs (default linspace(0.25*h0,h0,100), h0=0.62) - score = score vector - data = data vector - kernel = 'gaussian' - Gaussian kernel the only supported - - Note that only the first 4 letters of the kernel name is needed. + inc = length of estimated kerneldensity estimate + maxit = maximum number of iterations + fulloutput = True if fulloutput is wanted - Example: - data = rndnorm(0,1,20,1) - [hs hvec score] = hscv(data,'epan'); - plot(hvec,score) - See also hste, hbcv, hboot, hos, hldpi, hlscv, hstt, kde, kdefun + Returns + ------- + hs = smoothing parameter + hvec = vector defining possible values of hs + score = score vector + + Example + ------ + >>> import wafo.kdetools as wk + >>> import wafo.stats as ws + >>> data = ws.norm.rvs(0,1, size=(1,20)) + >>> kernel = wk.Kernel('epan') + >>> hs0 = kernel.hscv(data, fulloutput=False) + >>> hs, hvec, score = kernel.hscv(data, fulloutput=True) + >>> np.allclose(hs, hs0) + True + + import matplotlib.pyplot as plt + plt.plot(hvec,score) + + See also: + hste, hbcv, hboot, hos, hldpi, hlscv, hstt, kde, kdefun - Wand,M.P. and Jones, M.C. (1986) - 'Kernel smoothing' - Chapman and Hall, pp 75--79 + Reference + --------- + Wand,M.P. and Jones, M.C. (1986) + 'Kernel smoothing' + Chapman and Hall, pp 75--79 ''' - # TODO: Add support for other kernels than Gaussian + A = np.atleast_2d(data) d, n = A.shape @@ -1209,18 +1234,17 @@ class Kernel(object): idx = score.argmin() # Kernel other than Gaussian scale bandwidth h[dim] = hvec[idx] * (ste_constant / ste_constant2) ** (1 / 5) - if idx == 0: - warnings.warn("Optimum is probably lower than " - "hs={0:g} for dim={1:d}".format(h[dim] * s, dim)) - elif idx == maxit - 1: - msg = "Optimum is probably higher than hs={0:g] for dim={1:d}" - warnings.warn(msg.format(h[dim] * s, dim)) + _assert_warn(0 < idx, + "Optimum is probably lower than " + "hs={0:g} for dim={1:d}".format(h[dim] * s, dim)) + _assert_warn(idx < maxit - 1, + "Optimum is probably higher than " + "hs={0:g} for dim={1:d}".format(h[dim] * s, dim)) hvec = hvec * (ste_constant / ste_constant2) ** (1 / 5) if fulloutput: - return h * sigmaA, score, hvec, sigmaA - else: - return h * sigmaA + return h * sigmaA, score, hvec + return h * sigmaA def hldpi(self, data, L=2, inc=128): '''HLDPI L-stage Direct Plug-In estimate of smoothing parameter. @@ -1361,7 +1385,7 @@ def mkernel(X, kernel): See also -------- - kde, kdefun, kdebin + KDE References ---------- diff --git a/wafo/spectrum/core.py b/wafo/spectrum/core.py index fabaefb..d4909cf 100644 --- a/wafo/spectrum/core.py +++ b/wafo/spectrum/core.py @@ -1131,7 +1131,7 @@ class SpecData1D(PlotData): title='Joint density of maximum and minimum') try: pl = [10, 30, 50, 70, 90, 95, 99, 99.9] - mmpdf.cl = qlevels(uvdens, pl, h, h) + mmpdf.cl = qlevels(uvdens, pl, xi=(h, h)) mmpdf.pl = pl except: pass diff --git a/wafo/tests/test_kdetools.py b/wafo/tests/test_kdetools.py index bcbc0a7..9e936b9 100644 --- a/wafo/tests/test_kdetools.py +++ b/wafo/tests/test_kdetools.py @@ -449,6 +449,27 @@ class TestSmoothing(unittest.TestCase): assert_allclose(hs, [[3.25196193e-01, -2.68892467e-02, 3.18932448e-04], [-2.68892467e-02, 3.91283306e-01, 2.38654678e-02], [3.18932448e-04, 2.38654678e-02, 4.05123874e-01]]) + hs = self.gauss.hmns(self.data[0]) + assert_allclose(hs, self.gauss.hns(self.data[0])) + + hs = wk.Kernel('epan').hmns(self.data) + assert_allclose(hs, + [[8.363847e-01, -6.915749e-02, 8.202747e-04], + [-6.915749e-02, 1.006357e+00, 6.138052e-02], + [8.202747e-04, 6.138052e-02, 1.041954e+00]], + rtol=1e-5) + hs = wk.Kernel('biwe').hmns(self.data[:2]) + assert_allclose(hs, [[0.868428, -0.071705], + [-0.071705, 1.04685]], rtol=1e-5) + hs = wk.Kernel('triwe').hmns(self.data[:2]) + assert_allclose(hs, [[0.975375, -0.080535], + [-0.080535, 1.17577]], rtol=1e-5) + self.assertRaises(NotImplementedError, + wk.Kernel('biwe').hmns, self.data) + self.assertRaises(NotImplementedError, + wk.Kernel('triwe').hmns, self.data) + self.assertRaises(NotImplementedError, + wk.Kernel('triangular').hmns, self.data) def test_hscv(self): hs = self.gauss.hscv(self.data)