diff --git a/simpleTest/exampleAutoapproximate_with_cv.py b/simpleTest/exampleAutoapproximate_with_cv.py
index 23c2753..af1ded7 100644
--- a/simpleTest/exampleAutoapproximate_with_cv.py
+++ b/simpleTest/exampleAutoapproximate_with_cv.py
@@ -3,15 +3,13 @@
 # Example for approximating an periodic function
 
 import math
+import os
+import sys
 
 import matplotlib.pyplot as plt
 import numpy as np
 from TestFunctionPeriodic import *
 
-
-import os
-import sys
-
 src_aa = os.path.abspath(os.path.join(os.getcwd(), "src"))
 sys.path.insert(0, src_aa)
 
diff --git a/simpleTest/exampleMixed.py b/simpleTest/exampleMixed.py
index b201a08..32aab52 100644
--- a/simpleTest/exampleMixed.py
+++ b/simpleTest/exampleMixed.py
@@ -38,9 +38,9 @@ def TestFunction(x):
 d = 4  # dimension
 basis_vect = ["exp", "alg", "cos", "alg"]  # basis per dimension
 
-M = 10000       # number of training points
+M = 10000  # number of training points
 M_test = 100000  # number of test points
-max_iter = 50   # maximum number of LSQR iterations
+max_iter = 50  # maximum number of LSQR iterations
 
 # there are 3 possibilities with varying degree of freedom to define the number of used frequencies
 ########### Variant 1:
@@ -98,10 +98,10 @@ def cheb_nodes(rng, size):
     return (1.0 - np.cos(np.pi * rng.random(size))) / 2.0
 
 
-X = rng.random((M, d))          # initially all columns uniform in [0, 1]
-X[:, 0] -= 0.5                  # x₁ (exp):  shift to [-0.5, 0.5)
-X[:, 1] = cheb_nodes(rng, M)    # x₂ (alg):  Chebyshev-weight on [0, 1]
-X[:, 3] = cheb_nodes(rng, M)    # x₄ (alg):  Chebyshev-weight on [0, 1]
+X = rng.random((M, d))  # initially all columns uniform in [0, 1]
+X[:, 0] -= 0.5  # x₁ (exp):  shift to [-0.5, 0.5)
+X[:, 1] = cheb_nodes(rng, M)  # x₂ (alg):  Chebyshev-weight on [0, 1]
+X[:, 3] = cheb_nodes(rng, M)  # x₄ (alg):  Chebyshev-weight on [0, 1]
 y = np.array(
     [TestFunction(X[i, :]) for i in range(M)], dtype=complex
 )  # function values (complex because the mixed basis with exp uses complex coefficients)
@@ -110,9 +110,7 @@ def cheb_nodes(rng, size):
 X_test[:, 0] -= 0.5
 X_test[:, 1] = cheb_nodes(rng, M_test)
 X_test[:, 3] = cheb_nodes(rng, M_test)
-y_test = np.array(
-    [TestFunction(X_test[i, :]) for i in range(M_test)], dtype=complex
-)
+y_test = np.array([TestFunction(X_test[i, :]) for i in range(M_test)], dtype=complex)
 
 ##########################
 ## Do the approximation ##
diff --git a/src/pyANOVAapprox/__init__.py b/src/pyANOVAapprox/__init__.py
index c41618d..f286d38 100644
--- a/src/pyANOVAapprox/__init__.py
+++ b/src/pyANOVAapprox/__init__.py
@@ -1,4 +1,6 @@
+import csv
 import math
+import os
 import threading
 from math import acos, isnan
 
@@ -8,8 +10,6 @@
 from scipy.optimize import bisect
 from scipy.sparse.linalg import lsqr
 from scipy.special import erf
-import csv
-import os
 
 # from sklearn.metrics import roc_auc_score
 
diff --git a/src/pyANOVAapprox/analysis.py b/src/pyANOVAapprox/analysis.py
index c72d17a..aedd105 100644
--- a/src/pyANOVAapprox/analysis.py
+++ b/src/pyANOVAapprox/analysis.py
@@ -37,10 +37,7 @@ def get_variances(self, settingnr=None, lam=None, Dict=False):
     elif (
         lam == None
     ):  # get_variances( a::approx; dict::Bool = false )::Dict{Float64,Union{Vector{Float64},Dict{Vector{Int},Float64}}}
-        return {
-            l: self._variances(settingnr, l, Dict)
-            for l in self.lam.keys()
-        }
+        return {l: self._variances(settingnr, l, Dict) for l in self.lam.keys()}
 
 
 def _GSI(self, settingnr, lam, Dict):  # helpfunction for get_GSI
@@ -78,9 +75,7 @@ def get_GSI(self, settingnr=None, lam=None, Dict=False):
     ):  # get_GSI(a::approx, λ::Float64; dict::Bool = false,)::Union{Vector{Float64},Dict{Vector{Int},Float64}}
         return self._GSI(settingnr, lam, Dict)
     else:  # get_GSI( a::approx; dict::Bool = false )::Dict{Float64,Union{Vector{Float64},Dict{Vector{Int},Float64}}}
-        return {
-            l: self._GSI(settingnr, l, Dict) for l in self.lam.keys()
-        }
+        return {l: self._GSI(settingnr, l, Dict) for l in self.lam.keys()}
 
 
 def _AttributeRanking(self, settingnr, lam):  # helpfunction for get_AttributeRanking
@@ -120,10 +115,7 @@ def get_AttributeRanking(self, settingnr=None, lam=None):
     if (
         lam is None
     ):  # get_AttributeRanking( a::approx, λ::Float64 )::Dict{Float64,Vector{Float64}}
-        return {
-            l: self._AttributeRanking(settingnr, l)
-            for l in self.lam.keys()
-        }
+        return {l: self._AttributeRanking(settingnr, l) for l in self.lam.keys()}
     else:  # get_AttributeRanking( a::approx, λ::Float64 )::Vector{Float64}
         return self._AttributeRanking(settingnr, lam)
 
@@ -162,10 +154,7 @@ def get_ActiveSet(self, eps, settingnr=None, lam=None):
     if (
         lam is None
     ):  # get_ActiveSet(a::approx, eps::Vector{Float64})::Dict{Float64,Vector{Vector{Int}}}
-        return {
-            l: self._ActiveSet(eps, settingnr, l)
-            for l in self.lam.keys()
-        }
+        return {l: self._ActiveSet(eps, settingnr, l) for l in self.lam.keys()}
     else:  # get_ActiveSet(a::approx, eps::Vector{Float64}, λ::Float64)::Vector{Vector{Int}}
         return self._ActiveSet(eps, settingnr, lam)
 
@@ -190,9 +179,7 @@ def get_ShapleyValues(self, settingnr=None, lam=None):
     This function returns the Shapley values of the approximation for all reg. parameters ``\lambda``, if lam == None, as a dictionary of vectors of length `a.d`. Otherwise for the provided lam as a vector of length `a.d`.
     """
     if lam is None:  # get_ShapleyValues(a::approx)::Dict{Float64,Vector{Float64}}
-        return {
-            l: self._ShapleyValues(settingnr, l) for l in self.lam.keys()
-        }
+        return {l: self._ShapleyValues(settingnr, l) for l in self.lam.keys()}
     else:  # get_ShapleyValues(a::approx, λ::Float64)::Vector{Float64}
         return self._ShapleyValues(settingnr, lam)
 
diff --git a/src/pyANOVAapprox/approx.py b/src/pyANOVAapprox/approx.py
index 616bb06..29b2895 100644
--- a/src/pyANOVAapprox/approx.py
+++ b/src/pyANOVAapprox/approx.py
@@ -1,8 +1,8 @@
 import copy
-import warnings
-import os
 import csv
 import math
+import os
+import warnings
 from bisect import bisect
 
 import pyANOVAapprox.analysis as ANOVAanalysis
@@ -176,7 +176,9 @@ def __init__(
 
         if basis == "mixed":
             if len(basis_vect) == 0:
-                raise ValueError("please call approx with basis_vect for a NFMT transform.")
+                raise ValueError(
+                    "please call approx with basis_vect for a NFMT transform."
+                )
             if len(basis_vect) < self.parent.X.shape[1]:
                 raise ValueError("basis_vect must have an entry for every dimension.")
 
@@ -225,15 +227,14 @@ def __init__(
         ds=None,
         lam={0.0},
     ):
-    
+
         if type(lam) is np.ndarray:
             lam = {float(i) for i in lam}
         if type(lam) is list:
             lam = set(lam)
 
-
         self.X = X
-        
+
         if X.shape[0] != y.shape[0]:
             raise ValueError("X and y have different lengths.")
 
@@ -271,7 +272,7 @@ def addSetting(self, setting):
         self.trafo.append(None)
         self.fc.append({})
 
-    def getSettingNr(self, settingnr=None, lam = None, warn = True):
+    def getSettingNr(self, settingnr=None, lam=None, warn=True):
         if lam is None:
             return self.aktsetting if settingnr is None else settingnr
         else:
@@ -281,13 +282,13 @@ def getSettingNr(self, settingnr=None, lam = None, warn = True):
                 warnings.warn("Not yet approximated for this lambda")
                 return self.aktsetting if settingnr is None else settingnr
 
-    def getSetting(self, settingnr=None, lam = None):
+    def getSetting(self, settingnr=None, lam=None):
         return self.setting[self.getSettingNr(settingnr, lam)]
 
-    def getTrafo(self, settingnr=None, lam = None):
+    def getTrafo(self, settingnr=None, lam=None):
         return self.trafo[self.getSettingNr(settingnr, lam)]
 
-    def getFc(self, settingnr=None, lam = None):
+    def getFc(self, settingnr=None, lam=None):
         return self.fc[self.getSettingNr(settingnr, lam)]
 
     def addTrafo(self, settingnr=None):
@@ -401,11 +402,11 @@ def approximate(
         If lam is a np.ndarray of dtype float, this function computes the approximation for the regularization parameters contained in lam.
         If lam is a float, this function computes the approximation for the regularization parameter lam.
         """
-        
+
         setting = self.getSetting(settingnr)
-        
+
         if lam is None:
-            lam = setting.lam      
+            lam = setting.lam
         else:
             if isinstance(lam, np.ndarray):
                 lam = {float(l) for l in lam}
@@ -461,7 +462,6 @@ def _cv_for_budget(B_curr):
             settingnr_temp = self.aktsetting
             self.addTrafo(settingnr_temp)
 
-            
             self._approximate(
                 lam,
                 settingnr=settingnr_temp,
@@ -473,8 +473,16 @@ def _cv_for_budget(B_curr):
             )
 
             M = self.X.shape[0]
-            nfreqs = np.sum([np.prod(np.array(bw[u])-1) for u in temp_setting.U])+1
-            return 1/M * np.linalg.norm(self.evaluate(settingnr=settingnr_temp, lam=lam) - self.y)**2/(1-nfreqs/M)**2
+            nfreqs = np.sum([np.prod(np.array(bw[u]) - 1) for u in temp_setting.U]) + 1
+            return (
+                1
+                / M
+                * np.linalg.norm(
+                    self.evaluate(settingnr=settingnr_temp, lam=lam) - self.y
+                )
+                ** 2
+                / (1 - nfreqs / M) ** 2
+            )
 
         B_min = float(max(sum(5 ** len(u) for u in setting.U), 1))
         B_max = float(max(B_min + 1.0, len(self.y) / 3.0))
@@ -533,21 +541,23 @@ def _autoapproximate(
 
         D = dict([(u, tuple([1.0] * len(u))) for u in setting.U])
         t = dict([(u, tuple([1.0] * len(u))) for u in setting.U])
-        
-        if verbosity>3:
+
+        if verbosity > 3:
             if not os.path.exists("log"):
                 os.mkdir("log")
-            with open('log/log.csv', 'w', newline='') as csvfile:
-                csv.writer(csvfile, delimiter=',').writerow(["it"] + setting.U)
+            with open("log/log.csv", "w", newline="") as csvfile:
+                csv.writer(csvfile, delimiter=",").writerow(["it"] + setting.U)
 
         for idx in range(maxiter):
             if verbosity > 0:
                 print("===== Iteration ", str(idx + 1), " =====")
-            
+
             # Optional: Use cross-validation to find optimal B
             if cross_validation:
                 if verbosity > 0:
-                    print("Running binary search for optimal bandwidth via cross-validation...")
+                    print(
+                        "Running binary search for optimal bandwidth via cross-validation..."
+                    )
 
                 B, cv_val = self._compute_cv_error(
                     D=D,
@@ -565,7 +575,7 @@ def _autoapproximate(
                 )
                 if verbosity > 0:
                     print(f"Optimal B from CV: {B:.1f} (cv={cv_val:.6e})")
-            
+
             bw = compute_bandwidth(B, D, t)
             if setting.N is not None:
                 self.addSetting(setting)
@@ -579,11 +589,13 @@ def _autoapproximate(
             if verbosity > 0:
                 for i in setting.U:
                     print("bw in", str(i), ":", bw[i])
-                #print("bw in iteration", str(idx + 1), "are", str(bw))
+                # print("bw in iteration", str(idx + 1), "are", str(bw))
                 print()
             if verbosity > 3:
-                with open('log/log.csv', 'a', newline='') as csvfile:
-                    csv.writer(csvfile, delimiter=',').writerow(["bw in it"+str(idx+1)] + [str(bw[i]) for i in setting.U])
+                with open("log/log.csv", "a", newline="") as csvfile:
+                    csv.writer(csvfile, delimiter=",").writerow(
+                        ["bw in it" + str(idx + 1)] + [str(bw[i]) for i in setting.U]
+                    )
             self.approximate(
                 lam=lam,
                 solver=solver,
@@ -596,22 +608,33 @@ def _autoapproximate(
             D, t = self.estimate_rates(lam=lam, verbosity=verbosity)
             if verbosity > 1:
                 for i in setting.U:
-                    print("estimated rates for", str(i), ": D = ", str(D[i]), "and t = ", str(t[i]) )
+                    print(
+                        "estimated rates for",
+                        str(i),
+                        ": D = ",
+                        str(D[i]),
+                        "and t = ",
+                        str(t[i]),
+                    )
                 print()
-                
-                #print(
+
+                # print(
                 #    "estimated rates in iteration",
                 #    str(idx + 1),
                 #    "are D =",
                 #    str(D),
                 #    "and t =",
                 #    str(t),
-                #)
+                # )
             if verbosity > 3:
-                with open('log/log.csv', 'a', newline='') as csvfile:
-                    wr = csv.writer(csvfile, delimiter=',')
-                    wr.writerow(["D in it"+str(idx+1)] + [str(D[i]) for i in setting.U])
-                    wr.writerow(["t in it"+str(idx+1)] + [str(t[i]) for i in setting.U])
+                with open("log/log.csv", "a", newline="") as csvfile:
+                    wr = csv.writer(csvfile, delimiter=",")
+                    wr.writerow(
+                        ["D in it" + str(idx + 1)] + [str(D[i]) for i in setting.U]
+                    )
+                    wr.writerow(
+                        ["t in it" + str(idx + 1)] + [str(t[i]) for i in setting.U]
+                    )
         return settingnrs
 
     def autoapproximate(
@@ -632,7 +655,7 @@ def autoapproximate(
     ):
         """
         Automatic approximation with optional cross-validation for bandwidth selection.
-        
+
         Parameters:
         -----------
         use_cross_validation : bool, optional
@@ -673,7 +696,6 @@ def autoapproximate(
             )
             self.lam[l] = self.lam[l] + settingnrs
 
-
     def evaluate(self, settingnr=None, lam=None, X=None):
         """
         This function evaluates the approximation with optional node matrix X and regularization lam.
@@ -783,7 +805,9 @@ def evaluateSHAPterms(self, settingnr=None, X=None, lam=None):
             for l in self.lam.keys():
                 terms = self.evaluateANOVAterms(settingnr=settingnr, X=X, lam=l)
 
-                Dtype = np.complex128 if setting.basis in {"per", "mixed"} else np.float64
+                Dtype = (
+                    np.complex128 if setting.basis in {"per", "mixed"} else np.float64
+                )
                 values = np.zeros((M, d), dtype=Dtype)
 
                 for i in range(d):
diff --git a/src/pyANOVAapprox/bandwidth.py b/src/pyANOVAapprox/bandwidth.py
index e9fe215..ff0b6f4 100644
--- a/src/pyANOVAapprox/bandwidth.py
+++ b/src/pyANOVAapprox/bandwidth.py
@@ -5,11 +5,11 @@ def getfcu(ghat, u):
     idx = [s.u for s in ghat.settings].index(u)
     bws = ghat.settings[idx].bandwidths
 
-    #fcu = ghat[u].reshape(bws[::-1] - 1)
-    #fcu = np.permute_dims(fcu, range(len(bws))[::-1])
+    # fcu = ghat[u].reshape(bws[::-1] - 1)
+    # fcu = np.permute_dims(fcu, range(len(bws))[::-1])
     fcu = ghat[u].reshape(bws - 1)
     fcu = np.permute_dims(fcu, range(len(bws)))
-    
+
     return fcu
 
 
@@ -37,7 +37,7 @@ def compute_bandwidth(B, D, t):
     us = set(D.keys()) - {()}
     bw = {u: [6] * len(u) for u in us}
     bw[()] = []
-    
+
     minfreqs = sum(math.prod((bw[u][j] - 1) for j in range(len(u))) for u in us)
     if B < minfreqs:
         raise ValueError(f"Budget too small: {B} < {minfreqs}")
@@ -123,19 +123,19 @@ def estimate_rates(self, lam, settingnr=None, verbosity=0):
 
     mcl = np.exp(most_common_value(np.log(abs(self.getFc(settingnr)[lam].data))))
     threshold = 100 * mcl**2
-    
-    if verbosity>5:
-        if not os.path.exists(os.path.join("log","figures")):
-            os.mkdir(os.path.join("log","figures"))
+
+    if verbosity > 5:
+        if not os.path.exists(os.path.join("log", "figures")):
+            os.mkdir(os.path.join("log", "figures"))
         num = 0
-        
+
     system = self.getTrafo(settingnr).system
 
     for u in us:
         if len(u) == 0:
             continue
 
-        if verbosity>5:
+        if verbosity > 5:
             fig, ax = plt.subplots()
             ax.set_xscale("log")
             ax.set_yscale("log")
@@ -154,7 +154,7 @@ def estimate_rates(self, lam, settingnr=None, verbosity=0):
                 None,
             )
 
-            if verbosity>5:
+            if verbosity > 5:
 
                 y = np.cumsum(axissum[::-1])[::-1]
                 ax.plot(
@@ -174,7 +174,7 @@ def estimate_rates(self, lam, settingnr=None, verbosity=0):
                 D[u][j] = Duj
                 t[u][j] = -tuj / 2
 
-                if verbosity>5:
+                if verbosity > 5:
                     x = np.arange(1, idx + 1)
                     ax.plot(
                         x,
@@ -182,12 +182,18 @@ def estimate_rates(self, lam, settingnr=None, verbosity=0):
                         linewidth=2,
                         #                        color=j
                     )
-        if verbosity>5:
-            #plt.figure(figsize=(12, 9))
-            #for ax in ps:
+        if verbosity > 5:
+            # plt.figure(figsize=(12, 9))
+            # for ax in ps:
             #    plt.sca(ax)
-            fig.savefig(os.path.join("log","figures", str(num).strip()+ "_rates_" + str(u).strip() + ".png"))
+            fig.savefig(
+                os.path.join(
+                    "log",
+                    "figures",
+                    str(num).strip() + "_rates_" + str(u).strip() + ".png",
+                )
+            )
             num = num + 1
-            plt.close(fig) 
+            plt.close(fig)
 
     return D, t
diff --git a/src/pyANOVAapprox/errors.py b/src/pyANOVAapprox/errors.py
index e484203..57eedbf 100644
--- a/src/pyANOVAapprox/errors.py
+++ b/src/pyANOVAapprox/errors.py
@@ -37,10 +37,7 @@ def get_l2_error(self, settingnr=None, X=None, y=None, lam=None):
     if lam is not None:
         return self._l2_error(settingnr, lam, X, y)
     else:
-        return {
-            l: self._l2_error(settingnr, l, X, y)
-            for l in self.lam.keys()
-        }
+        return {l: self._l2_error(settingnr, l, X, y) for l in self.lam.keys()}
 
 
 def _mse(self, settingnr, lam, X, y):  # helpfunction for get_mse
@@ -77,9 +74,7 @@ def get_mse(self, settingnr=None, X=None, y=None, lam=None):
     if lam is not None:
         return self._mse(settingnr, lam, X, y)
     else:
-        return {
-            l: self._mse(settingnr, l, X, y) for l in self.lam.keys()
-        }
+        return {l: self._mse(settingnr, l, X, y) for l in self.lam.keys()}
 
 
 def _mad(self, settingnr, lam, X, y):  # helpfunction for get_mad
@@ -116,16 +111,16 @@ def get_mad(self, settingnr=None, X=None, y=None, lam=None):
     if lam is not None:
         return self._mad(settingnr, lam, X, y)
     else:
-        return {
-            l: self._mad(settingnr, l, X, y) for l in self.lam.keys()
-        }
+        return {l: self._mad(settingnr, l, X, y) for l in self.lam.keys()}
 
 
 def _L2_error(self, norm, bc_fun, settingnr, lam):
     # print(settingnr)
     if self.getSetting(settingnr, lam).basis in {"per", "cos", "cheb", "std", "mixed"}:
         error = norm**2
-        index_set = gt.get_IndexSet(self.getTrafo(settingnr, lam).settings, self.X.shape[1])
+        index_set = gt.get_IndexSet(
+            self.getTrafo(settingnr, lam).settings, self.X.shape[1]
+        )
 
         for i in range(index_set.shape[1]):
             k = index_set[:, i]
@@ -153,10 +148,7 @@ def get_L2_error(self, norm, bc_fun, settingnr=None, lam=None):
     if lam is not None:
         return self._L2_error(norm, bc_fun, settingnr, lam)
     else:
-        return {
-            l: self._L2_error(norm, bc_fun, settingnr, l)
-            for l in self.lam.keys()
-        }
+        return {l: self._L2_error(norm, bc_fun, settingnr, l) for l in self.lam.keys()}
 
 
 def _acc(self, settingnr, lam, X, y):  # helpfunction for get_acc
@@ -176,9 +168,7 @@ def get_acc(self, settingnr=None, X=None, y=None, lam=None):
     if lam is not None:
         return self._acc(settingnr, lam, X, y)
     else:
-        return {
-            l: self._acc(settingnr, l, X, y) for l in self.lam.keys()
-        }
+        return {l: self._acc(settingnr, l, X, y) for l in self.lam.keys()}
 
 
 def auc_score(y_true, y_pred_proba):
@@ -240,9 +230,7 @@ def get_auc(self, settingnr=None, X=None, y=None, lam=None):
     if lam is not None:
         return self._auc(settingnr, lam, X, y)
     else:
-        return {
-            l: self._auc(settingnr, l, X, y) for l in self.lam.keys()
-        }
+        return {l: self._auc(settingnr, l, X, y) for l in self.lam.keys()}
 
 
 __all__ = [