pynkowski.theory.chi2
This submodule contains the classes for the theoretical $\chi^2$ fields.
1'''This submodule contains the classes for the theoretical $\chi^2$ fields.''' 2import numpy as np 3from scipy.special import comb 4from .utils_th import LKC_Chi2, get_σ, get_μ, lkc_ambient_dict 5from .base_th import TheoryField 6 7class Chi2(TheoryField): 8 """General class for Isotropic Chi² fields, to be used directly or as the base for specific Chi² fields. 9 10 Parameters 11 ---------- 12 dim : int 13 Dimension of the space where the field is defined. 14 15 dof : int, optional 16 Number of degrees of freedom $dof$ of the field: $\Chi^2 = \sum_{i=0}{dof} N_i^2$. 17 Default: 2 18 19 sigma : float, optional 20 The standard deviation of the constituent Gaussian fields $N_i$, such that $\Chi^2 = \sum_{i=0}{dof} N_i^2$. 21 Default: 1. 22 23 mu : float, optional 24 The derivative of the covariance function at the origin for the constituent Gaussian fields. 25 Default: 1. 26 27 nu : float, optional 28 The second derivative of the covariance function at the origin for the constituent Gaussian fields. 29 Default: 1. 30 31 lkc_ambient : np.array or None, optional 32 The values for the Lipschitz–Killing Curvatures of the ambient space. If `None`, it is assumed that the volume is 1 and the rest is 0. This is exact for many spaces like Euclidean spaces or the sphere, and exact to leading order in μ for the rest. 33 Default: None 34 35 Attributes 36 ---------- 37 dim : int 38 Dimension of the space where the field is defined. 39 40 name : str 41 Name of the field, `"Isotropic Chi²"` by default. 42 43 dof : int, optional 44 Number of degrees of freedom $dof$ of the field: $\Chi^2 = \sum_{i=0}{dof} N_i^2$. 45 46 sigma : float 47 The standard deviation of the field. 48 49 mu : float 50 The derivative of the covariance function at the origin. 51 52 nu : float, optional 53 The second derivative of the covariance function at the origin for the constituent Gaussian fields. 54 55 lkc_ambient : np.array or None 56 The values for the Lipschitz–Killing Curvatures of the ambient space. 57 58 """ 59 def __init__(self, dim, dof=2, sigma=1., mu=1., nu=1., lkc_ambient=None): 60 super().__init__(dim, name='Isotropic Chi²', sigma=sigma, mu=mu, nu=nu, lkc_ambient=lkc_ambient) 61 self.dof = dof 62 63 def LKC(self, j, us): 64 """Compute the expected values of the Lipschitz–Killing Curvatures of the excursion sets at thresholds `us`, $\mathbb{L}_j(A_u(f))$. 65 66 Parameters 67 ---------- 68 j : int 69 Index of the LKC to compute, `0 < j < dim`. 70 71 us : np.array 72 The thresholds considered for the computation. 73 74 Returns 75 ------- 76 lkc : np.array 77 Expected value of LKC evaluated at the thresholds. 78 79 """ 80 us /= self.sigma 81 return LKC_Chi2(j, us, self.mu, dof=self.dof, dim=self.dim, lkc_ambient=self.lkc_ambient)/self.lkc_ambient[-1] 82 83 def MF(self, j, us): 84 """Compute the expected values of the Minkowski Functionals of the excursion sets at thresholds `us`, $V_j(A_u(f))$. 85 86 Parameters 87 ---------- 88 j : int 89 Index of the MF to compute, `0 < j < dim`. 90 91 us : np.array 92 The thresholds considered for the computation. 93 94 Returns 95 ------- 96 mf : np.array 97 Expected value of MFs evaluated at the thresholds. 98 99 """ 100 us /= self.sigma 101 return 1./comb(self.dim,j)*LKC_Chi2(self.dim-j, us, self.mu, dof=self.dof, dim=self.dim, lkc_ambient=self.lkc_ambient)/self.lkc_ambient[-1] 102 103 def V0(self, us): 104 """Compute the expected values of the first Minkowski Functionals of the excursion sets at thresholds `us`, $V_0(A_u(f))$. 105 106 Parameters 107 ---------- 108 us : np.array 109 The thresholds considered for the computation. 110 111 Returns 112 ------- 113 v0 : np.array 114 Expected value of $V_0$ evaluated at the thresholds. 115 116 """ 117 us /= self.sigma 118 return self.MF(0, us) 119 120 def V1(self, us): 121 """Compute the expected values of the first Minkowski Functionals of the excursion sets at thresholds `us`, $V_1(A_u(f))$. 122 123 Parameters 124 ---------- 125 us : np.array 126 The thresholds considered for the computation. 127 128 Returns 129 ------- 130 v1 : np.array 131 Expected value of $V_1$ evaluated at the thresholds. 132 133 """ 134 us /= self.sigma 135 return self.MF(1, us) 136 137 def V2(self, us): 138 """Compute the expected values of the first Minkowski Functionals of the excursion sets at thresholds `us`, $V_2(A_u(f))$. Only for fields with `dim ≥ 2`. 139 140 Parameters 141 ---------- 142 us : np.array 143 The thresholds considered for the computation. 144 145 Returns 146 ------- 147 v2 : np.array 148 Expected value of $V_2$ evaluated at the thresholds. 149 150 """ 151 assert self.dim>=2, 'V2 is defined only for fields with dim≥2' 152 us /= self.sigma 153 return self.MF(2, us) 154 155 def V3(self, us): 156 """Compute the expected values of the first Minkowski Functionals of the excursion sets at thresholds `us`, $V_3(A_u(f))$. Only for fields with `dim ≥ 3`. 157 158 Parameters 159 ---------- 160 us : np.array 161 The thresholds considered for the computation. 162 163 Returns 164 ------- 165 v3 : np.array 166 Expected value of $V_3$ evaluated at the thresholds. 167 168 """ 169 assert self.dim>=3, 'V3 is defined only for fields with dim≥3' 170 us /= self.sigma 171 return self.MF(3, us) 172 173 # def V4(self, us): 174 # assert self.dim>=4, 'V4 is defined only for fields with dim≥4' 175 # us /= self.sigma 176 # return self.MF(4, us) 177 178 179class SphericalChi2(Chi2): 180 """Class for Spherical Isotropic Gaussian fields, to be used directly or as the base for specific Gaussian fields. 181 182 Parameters 183 ---------- 184 cls : np.array 185 Angular power spectrum of the polarization field (EE, BB). The shape is `(2,lmax+1)` 186 187 normalise : bool, optional 188 If `True`, normalise the Q and U fields to unit variance 189 Default : True 190 191 Attributes 192 ---------- 193 cls : np.array 194 Angular Power Spectrum of the polarization field. 195 196 dim : int 197 Dimension of the space where the field is defined, in this case this is 2. 198 199 name : str 200 Name of the field, `"Spherical Isotropic Chi²"` by default. 201 202 dof : int, optional 203 Number of degrees of freedom $dof$ of the field, $2$. 204 205 sigma : float 206 The standard deviation of the field. 207 208 mu : float 209 The derivative of the covariance function at the origin. 210 211 nu : float, optional 212 The second derivative of the covariance function at the origin for the constituent Gaussian fields. 213 214 lkc_ambient : np.array or None 215 The values for the Lipschitz–Killing Curvatures of the ambient space. 216 217 """ 218 def __init__(self, cls, normalise=True): 219 if normalise: 220 cls /= get_σ((cls[0]+cls[1])/2.) 221 self.sigma = 1. 222 else: 223 self.sigma = np.sqrt(get_σ(cls)) 224 self.cls = cls 225 self.mu = get_μ(cls[0]+cls[1]) 226 super().__init__(dim=2, dof=2, sigma=self.sigma, mu=self.mu, lkc_ambient=lkc_ambient_dict["sphere"]) 227 self.name = 'Spherical Isotropic Chi²' 228 229 230 231__all__ = ["SphericalChi2", "Chi2"] 232 233__docformat__ = "numpy" 234 235
180class SphericalChi2(Chi2): 181 """Class for Spherical Isotropic Gaussian fields, to be used directly or as the base for specific Gaussian fields. 182 183 Parameters 184 ---------- 185 cls : np.array 186 Angular power spectrum of the polarization field (EE, BB). The shape is `(2,lmax+1)` 187 188 normalise : bool, optional 189 If `True`, normalise the Q and U fields to unit variance 190 Default : True 191 192 Attributes 193 ---------- 194 cls : np.array 195 Angular Power Spectrum of the polarization field. 196 197 dim : int 198 Dimension of the space where the field is defined, in this case this is 2. 199 200 name : str 201 Name of the field, `"Spherical Isotropic Chi²"` by default. 202 203 dof : int, optional 204 Number of degrees of freedom $dof$ of the field, $2$. 205 206 sigma : float 207 The standard deviation of the field. 208 209 mu : float 210 The derivative of the covariance function at the origin. 211 212 nu : float, optional 213 The second derivative of the covariance function at the origin for the constituent Gaussian fields. 214 215 lkc_ambient : np.array or None 216 The values for the Lipschitz–Killing Curvatures of the ambient space. 217 218 """ 219 def __init__(self, cls, normalise=True): 220 if normalise: 221 cls /= get_σ((cls[0]+cls[1])/2.) 222 self.sigma = 1. 223 else: 224 self.sigma = np.sqrt(get_σ(cls)) 225 self.cls = cls 226 self.mu = get_μ(cls[0]+cls[1]) 227 super().__init__(dim=2, dof=2, sigma=self.sigma, mu=self.mu, lkc_ambient=lkc_ambient_dict["sphere"]) 228 self.name = 'Spherical Isotropic Chi²'
Class for Spherical Isotropic Gaussian fields, to be used directly or as the base for specific Gaussian fields.
Parameters
- cls (np.array):
Angular power spectrum of the polarization field (EE, BB). The shape is
(2,lmax+1) - normalise (bool, optional):
If
True, normalise the Q and U fields to unit variance Default : True
Attributes
- cls (np.array): Angular Power Spectrum of the polarization field.
- dim (int): Dimension of the space where the field is defined, in this case this is 2.
- name (str):
Name of the field,
"Spherical Isotropic Chi²"by default. - dof (int, optional): Number of degrees of freedom $dof$ of the field, $2$.
- sigma (float): The standard deviation of the field.
- mu (float): The derivative of the covariance function at the origin.
- nu (float, optional): The second derivative of the covariance function at the origin for the constituent Gaussian fields.
- lkc_ambient (np.array or None): The values for the Lipschitz–Killing Curvatures of the ambient space.
SphericalChi2(cls, normalise=True)
219 def __init__(self, cls, normalise=True): 220 if normalise: 221 cls /= get_σ((cls[0]+cls[1])/2.) 222 self.sigma = 1. 223 else: 224 self.sigma = np.sqrt(get_σ(cls)) 225 self.cls = cls 226 self.mu = get_μ(cls[0]+cls[1]) 227 super().__init__(dim=2, dof=2, sigma=self.sigma, mu=self.mu, lkc_ambient=lkc_ambient_dict["sphere"]) 228 self.name = 'Spherical Isotropic Chi²'
8class Chi2(TheoryField): 9 """General class for Isotropic Chi² fields, to be used directly or as the base for specific Chi² fields. 10 11 Parameters 12 ---------- 13 dim : int 14 Dimension of the space where the field is defined. 15 16 dof : int, optional 17 Number of degrees of freedom $dof$ of the field: $\Chi^2 = \sum_{i=0}{dof} N_i^2$. 18 Default: 2 19 20 sigma : float, optional 21 The standard deviation of the constituent Gaussian fields $N_i$, such that $\Chi^2 = \sum_{i=0}{dof} N_i^2$. 22 Default: 1. 23 24 mu : float, optional 25 The derivative of the covariance function at the origin for the constituent Gaussian fields. 26 Default: 1. 27 28 nu : float, optional 29 The second derivative of the covariance function at the origin for the constituent Gaussian fields. 30 Default: 1. 31 32 lkc_ambient : np.array or None, optional 33 The values for the Lipschitz–Killing Curvatures of the ambient space. If `None`, it is assumed that the volume is 1 and the rest is 0. This is exact for many spaces like Euclidean spaces or the sphere, and exact to leading order in μ for the rest. 34 Default: None 35 36 Attributes 37 ---------- 38 dim : int 39 Dimension of the space where the field is defined. 40 41 name : str 42 Name of the field, `"Isotropic Chi²"` by default. 43 44 dof : int, optional 45 Number of degrees of freedom $dof$ of the field: $\Chi^2 = \sum_{i=0}{dof} N_i^2$. 46 47 sigma : float 48 The standard deviation of the field. 49 50 mu : float 51 The derivative of the covariance function at the origin. 52 53 nu : float, optional 54 The second derivative of the covariance function at the origin for the constituent Gaussian fields. 55 56 lkc_ambient : np.array or None 57 The values for the Lipschitz–Killing Curvatures of the ambient space. 58 59 """ 60 def __init__(self, dim, dof=2, sigma=1., mu=1., nu=1., lkc_ambient=None): 61 super().__init__(dim, name='Isotropic Chi²', sigma=sigma, mu=mu, nu=nu, lkc_ambient=lkc_ambient) 62 self.dof = dof 63 64 def LKC(self, j, us): 65 """Compute the expected values of the Lipschitz–Killing Curvatures of the excursion sets at thresholds `us`, $\mathbb{L}_j(A_u(f))$. 66 67 Parameters 68 ---------- 69 j : int 70 Index of the LKC to compute, `0 < j < dim`. 71 72 us : np.array 73 The thresholds considered for the computation. 74 75 Returns 76 ------- 77 lkc : np.array 78 Expected value of LKC evaluated at the thresholds. 79 80 """ 81 us /= self.sigma 82 return LKC_Chi2(j, us, self.mu, dof=self.dof, dim=self.dim, lkc_ambient=self.lkc_ambient)/self.lkc_ambient[-1] 83 84 def MF(self, j, us): 85 """Compute the expected values of the Minkowski Functionals of the excursion sets at thresholds `us`, $V_j(A_u(f))$. 86 87 Parameters 88 ---------- 89 j : int 90 Index of the MF to compute, `0 < j < dim`. 91 92 us : np.array 93 The thresholds considered for the computation. 94 95 Returns 96 ------- 97 mf : np.array 98 Expected value of MFs evaluated at the thresholds. 99 100 """ 101 us /= self.sigma 102 return 1./comb(self.dim,j)*LKC_Chi2(self.dim-j, us, self.mu, dof=self.dof, dim=self.dim, lkc_ambient=self.lkc_ambient)/self.lkc_ambient[-1] 103 104 def V0(self, us): 105 """Compute the expected values of the first Minkowski Functionals of the excursion sets at thresholds `us`, $V_0(A_u(f))$. 106 107 Parameters 108 ---------- 109 us : np.array 110 The thresholds considered for the computation. 111 112 Returns 113 ------- 114 v0 : np.array 115 Expected value of $V_0$ evaluated at the thresholds. 116 117 """ 118 us /= self.sigma 119 return self.MF(0, us) 120 121 def V1(self, us): 122 """Compute the expected values of the first Minkowski Functionals of the excursion sets at thresholds `us`, $V_1(A_u(f))$. 123 124 Parameters 125 ---------- 126 us : np.array 127 The thresholds considered for the computation. 128 129 Returns 130 ------- 131 v1 : np.array 132 Expected value of $V_1$ evaluated at the thresholds. 133 134 """ 135 us /= self.sigma 136 return self.MF(1, us) 137 138 def V2(self, us): 139 """Compute the expected values of the first Minkowski Functionals of the excursion sets at thresholds `us`, $V_2(A_u(f))$. Only for fields with `dim ≥ 2`. 140 141 Parameters 142 ---------- 143 us : np.array 144 The thresholds considered for the computation. 145 146 Returns 147 ------- 148 v2 : np.array 149 Expected value of $V_2$ evaluated at the thresholds. 150 151 """ 152 assert self.dim>=2, 'V2 is defined only for fields with dim≥2' 153 us /= self.sigma 154 return self.MF(2, us) 155 156 def V3(self, us): 157 """Compute the expected values of the first Minkowski Functionals of the excursion sets at thresholds `us`, $V_3(A_u(f))$. Only for fields with `dim ≥ 3`. 158 159 Parameters 160 ---------- 161 us : np.array 162 The thresholds considered for the computation. 163 164 Returns 165 ------- 166 v3 : np.array 167 Expected value of $V_3$ evaluated at the thresholds. 168 169 """ 170 assert self.dim>=3, 'V3 is defined only for fields with dim≥3' 171 us /= self.sigma 172 return self.MF(3, us) 173 174 # def V4(self, us): 175 # assert self.dim>=4, 'V4 is defined only for fields with dim≥4' 176 # us /= self.sigma 177 # return self.MF(4, us)
General class for Isotropic Chi² fields, to be used directly or as the base for specific Chi² fields.
Parameters
- dim (int): Dimension of the space where the field is defined.
- dof (int, optional): Number of degrees of freedom $dof$ of the field: $\Chi^2 = \sum_{i=0}{dof} N_i^2$. Default: 2
- sigma (float, optional): The standard deviation of the constituent Gaussian fields $N_i$, such that $\Chi^2 = \sum_{i=0}{dof} N_i^2$. Default: 1.
- mu (float, optional): The derivative of the covariance function at the origin for the constituent Gaussian fields. Default: 1.
- nu (float, optional): The second derivative of the covariance function at the origin for the constituent Gaussian fields. Default: 1.
- lkc_ambient (np.array or None, optional):
The values for the Lipschitz–Killing Curvatures of the ambient space. If
None, it is assumed that the volume is 1 and the rest is 0. This is exact for many spaces like Euclidean spaces or the sphere, and exact to leading order in μ for the rest. Default: None
Attributes
- dim (int): Dimension of the space where the field is defined.
- name (str):
Name of the field,
"Isotropic Chi²"by default. - dof (int, optional): Number of degrees of freedom $dof$ of the field: $\Chi^2 = \sum_{i=0}{dof} N_i^2$.
- sigma (float): The standard deviation of the field.
- mu (float): The derivative of the covariance function at the origin.
- nu (float, optional): The second derivative of the covariance function at the origin for the constituent Gaussian fields.
- lkc_ambient (np.array or None): The values for the Lipschitz–Killing Curvatures of the ambient space.
def
LKC(self, j, us):
64 def LKC(self, j, us): 65 """Compute the expected values of the Lipschitz–Killing Curvatures of the excursion sets at thresholds `us`, $\mathbb{L}_j(A_u(f))$. 66 67 Parameters 68 ---------- 69 j : int 70 Index of the LKC to compute, `0 < j < dim`. 71 72 us : np.array 73 The thresholds considered for the computation. 74 75 Returns 76 ------- 77 lkc : np.array 78 Expected value of LKC evaluated at the thresholds. 79 80 """ 81 us /= self.sigma 82 return LKC_Chi2(j, us, self.mu, dof=self.dof, dim=self.dim, lkc_ambient=self.lkc_ambient)/self.lkc_ambient[-1]
Compute the expected values of the Lipschitz–Killing Curvatures of the excursion sets at thresholds us, $\mathbb{L}_j(A_u(f))$.
Parameters
- j (int):
Index of the LKC to compute,
0 < j < dim. - us (np.array): The thresholds considered for the computation.
Returns
- lkc (np.array): Expected value of LKC evaluated at the thresholds.
def
MF(self, j, us):
84 def MF(self, j, us): 85 """Compute the expected values of the Minkowski Functionals of the excursion sets at thresholds `us`, $V_j(A_u(f))$. 86 87 Parameters 88 ---------- 89 j : int 90 Index of the MF to compute, `0 < j < dim`. 91 92 us : np.array 93 The thresholds considered for the computation. 94 95 Returns 96 ------- 97 mf : np.array 98 Expected value of MFs evaluated at the thresholds. 99 100 """ 101 us /= self.sigma 102 return 1./comb(self.dim,j)*LKC_Chi2(self.dim-j, us, self.mu, dof=self.dof, dim=self.dim, lkc_ambient=self.lkc_ambient)/self.lkc_ambient[-1]
Compute the expected values of the Minkowski Functionals of the excursion sets at thresholds us, $V_j(A_u(f))$.
Parameters
- j (int):
Index of the MF to compute,
0 < j < dim. - us (np.array): The thresholds considered for the computation.
Returns
- mf (np.array): Expected value of MFs evaluated at the thresholds.
def
V0(self, us):
104 def V0(self, us): 105 """Compute the expected values of the first Minkowski Functionals of the excursion sets at thresholds `us`, $V_0(A_u(f))$. 106 107 Parameters 108 ---------- 109 us : np.array 110 The thresholds considered for the computation. 111 112 Returns 113 ------- 114 v0 : np.array 115 Expected value of $V_0$ evaluated at the thresholds. 116 117 """ 118 us /= self.sigma 119 return self.MF(0, us)
Compute the expected values of the first Minkowski Functionals of the excursion sets at thresholds us, $V_0(A_u(f))$.
Parameters
- us (np.array): The thresholds considered for the computation.
Returns
- v0 (np.array): Expected value of $V_0$ evaluated at the thresholds.
def
V1(self, us):
121 def V1(self, us): 122 """Compute the expected values of the first Minkowski Functionals of the excursion sets at thresholds `us`, $V_1(A_u(f))$. 123 124 Parameters 125 ---------- 126 us : np.array 127 The thresholds considered for the computation. 128 129 Returns 130 ------- 131 v1 : np.array 132 Expected value of $V_1$ evaluated at the thresholds. 133 134 """ 135 us /= self.sigma 136 return self.MF(1, us)
Compute the expected values of the first Minkowski Functionals of the excursion sets at thresholds us, $V_1(A_u(f))$.
Parameters
- us (np.array): The thresholds considered for the computation.
Returns
- v1 (np.array): Expected value of $V_1$ evaluated at the thresholds.
def
V2(self, us):
138 def V2(self, us): 139 """Compute the expected values of the first Minkowski Functionals of the excursion sets at thresholds `us`, $V_2(A_u(f))$. Only for fields with `dim ≥ 2`. 140 141 Parameters 142 ---------- 143 us : np.array 144 The thresholds considered for the computation. 145 146 Returns 147 ------- 148 v2 : np.array 149 Expected value of $V_2$ evaluated at the thresholds. 150 151 """ 152 assert self.dim>=2, 'V2 is defined only for fields with dim≥2' 153 us /= self.sigma 154 return self.MF(2, us)
Compute the expected values of the first Minkowski Functionals of the excursion sets at thresholds us, $V_2(A_u(f))$. Only for fields with dim ≥ 2.
Parameters
- us (np.array): The thresholds considered for the computation.
Returns
- v2 (np.array): Expected value of $V_2$ evaluated at the thresholds.
def
V3(self, us):
156 def V3(self, us): 157 """Compute the expected values of the first Minkowski Functionals of the excursion sets at thresholds `us`, $V_3(A_u(f))$. Only for fields with `dim ≥ 3`. 158 159 Parameters 160 ---------- 161 us : np.array 162 The thresholds considered for the computation. 163 164 Returns 165 ------- 166 v3 : np.array 167 Expected value of $V_3$ evaluated at the thresholds. 168 169 """ 170 assert self.dim>=3, 'V3 is defined only for fields with dim≥3' 171 us /= self.sigma 172 return self.MF(3, us)
Compute the expected values of the first Minkowski Functionals of the excursion sets at thresholds us, $V_3(A_u(f))$. Only for fields with dim ≥ 3.
Parameters
- us (np.array): The thresholds considered for the computation.
Returns
- v3 (np.array): Expected value of $V_3$ evaluated at the thresholds.