def main(ifname='winequality-red.csv', delimiter=';', has_header=True, folds = 10):
Train_Data, Test_Data = split_data(ifname, delimiter, has_header, [0,1,2,3,4,5,6,7,8,9,10])
Train_Targets, Test_Targets = split_data(ifname, delimiter, has_header, [11])
Train_Second = second_degree_designmtx(Train_Data)
Test_Data_Second = second_degree_designmtx(Test_Data)
Train_Third = third_degree_designmtx(Train_Data)
Test_Data_Third = third_degree_designmtx(Test_Data)
Train_Fourth = fourth_degree_designmtx(Train_Data)
Test_Data_Fourth = fourth_degree_designmtx(Test_Data)
#print("Second degree polynomial regression")
#evaluate_reg_param(Train_Second, Train_Targets, create_cv_folds(len(Train_Second), folds), reg_params = np.logspace(-2,5))
#print("Third degree polynomial regression")
#evaluate_reg_param(Train_Third, Train_Targets, create_cv_folds(len(Train_Third), folds), reg_params = np.logspace(2,7))
#print("Fourth degree polynomial regression")
#evaluate_reg_param(Train_Fourth, Train_Targets, create_cv_folds(len(Train_Third), folds), reg_params = np.logspace(6,10))
#print("Second degree polynomial regression (normalized)")
reg_second = evaluate_reg_param(normalize(Train_Second), Train_Targets, create_cv_folds(len(Train_Second), folds), reg_params = np.logspace(-1,3))
#print("Third degree polynomial regression (normalized)")
reg_third = evaluate_reg_param(normalize(Train_Third), Train_Targets, create_cv_folds(len(Train_Third), folds), reg_params = np.logspace(-1,3))
#print("Fourth degree polynomial regression (normalized)")
reg_fourth =evaluate_reg_param(normalize(Train_Fourth), normalize(Train_Targets), create_cv_folds(len(Train_Third), folds), reg_params = np.logspace(-1,3))
print("Train, Test error (degree 2): ", train_and_test(normalize(Train_Second), Train_Targets, normalize(Test_Data_Second), Test_Targets, reg_param = reg_second))
print("Train, Test error (degree 3): ", train_and_test(normalize(Train_Third), Train_Targets, normalize(Test_Data_Third), Test_Targets, reg_param = reg_third))
print("Train, Test error (degree 4): ", train_and_test(normalize(Train_Fourth), Train_Targets, normalize(Test_Data_Fourth), Test_Targets, reg_param = reg_fourth))
def parameter_search_rbf(inputs, targets, test_fraction):
"""
"""
N = inputs.shape[0]
# run all experiments on the same train-test split of the data
train_part, test_part = train_and_test_split(N,
test_fraction=test_fraction)
# for the centres of the basis functions sample 10% of the data
sample_fraction = 0.15
p = (1 - sample_fraction, sample_fraction)
centres = inputs[np.random.choice([False, True], size=N, p=p), :]
print("centres.shape = %r" % (centres.shape, ))
scales = np.logspace(0, 2, 17) # of the basis functions
reg_params = np.logspace(-15, -4, 11) # choices of regularisation strength
# create empty 2d arrays to store the train and test errors
train_errors = np.empty((scales.size, reg_params.size))
test_errors = np.empty((scales.size, reg_params.size))
# iterate over the scales
for i, scale in enumerate(scales):
# i is the index, scale is the corresponding scale
# we must recreate the feature mapping each time for different scales
feature_mapping = construct_rbf_feature_mapping(centres, scale)
designmtx = feature_mapping(inputs)
# partition the design matrix and targets into train and test
train_designmtx, train_targets, test_designmtx, test_targets = \
train_and_test_partition(
designmtx, targets, train_part, test_part)
# iteratre over the regularisation parameters
for j, reg_param in enumerate(reg_params):
# j is the index, reg_param is the corresponding regularisation
# parameter
# train and test the data
train_error, test_error = train_and_test(train_designmtx,
train_targets,
test_designmtx,
test_targets,
reg_param=reg_param)
# store the train and test errors in our 2d arrays
train_errors[i, j] = train_error
test_errors[i, j] = test_error
# we have a 2d array of train and test errors, we want to know the (i,j)
# index of the best value
best_i = np.argmin(np.argmin(test_errors, axis=1))
best_j = np.argmin(test_errors[i, :])
print("Best joint choice of parameters:")
print("\tscale %.2g and lambda = %.2g" %
(scales[best_i], reg_params[best_j]))
# now we can plot the error for different scales using the best
# regulariation choice
fig, ax = plot_train_test_errors("scale", scales, train_errors[:, best_j],
test_errors[:, best_j])
ax.set_xscale('log')
# ...and the error for different regularisation choices given the best
# scale choice
fig, ax = plot_train_test_errors("$\lambda$", reg_params,
train_errors[best_i, :],
test_errors[best_i, :])
ax.set_xscale('log')
def cv_evaluation_linear_model(inputs, targets, folds, reg_param=None):
"""
Will split inputs and targets into train and test parts, then fit a linear
model to the training part, and test on the both parts.
Inputs can be a data matrix (or design matrix), targets should
be real valued.
parameters
----------
inputs - the input design matrix !!!!!!!!!!!!!!!!!!!!(any feature mapping should already be applied)
targets - the targets as a vector
num_folds - the number of folds
reg_param (optional) - the regularisation strength. If provided, then
regularised least squares fitting is uses with this regularisation
strength. Otherwise, (non-regularised) least squares is used.
returns
-------
train_errors - the training errors for the approximation
test_errors - the test errors for the approximation
"""
# get the number of datapoints
N = inputs.shape[0]
# get th number of folds
num_folds = len(folds)
train_errors = np.empty(num_folds)
test_errors = np.empty(num_folds)
for f,fold in enumerate(folds):
# f is the fold id, fold is the train-test split
train_part, test_part = fold
# break the data into train and test sets
train_inputs, train_targets, test_inputs, test_targets = \
train_and_test_partition(inputs, targets, train_part, test_part)
# now train and evaluate the error on both sets
train_error, test_error,weights = train_and_test(
train_inputs, train_targets, test_inputs, test_targets,
reg_param=reg_param)
#print("train_error = %r" % (train_error,))
#print("test_error = %r" % (test_error,))
train_errors[f] = train_error
test_errors[f] = test_error
return train_errors, test_errors,weights
def parameter_search_rbf(inputs, targets, test_fraction):
"""
"""
n = inputs.shape[0]
# run all experiments on the same train-test split of the data
train_part, test_part = train_and_test_split(n,
test_fraction=test_fraction)
# for the centres of the basis functions sample 10% of the data
sample_fraction = 0.10
p = (1 - sample_fraction, sample_fraction)
centres = inputs[np.random.choice([False, True], size=n, p=p), :]
print("\ncentres.shape = %r" % (centres.shape, ))
scales = np.logspace(0, 6, 20) # of the basis functions
reg_params = np.logspace(-15, 5, 20) # choices of regularisation strength
# create empty 2d arrays to store the train and test errors
train_errors = np.empty((scales.size, reg_params.size))
test_errors = np.empty((scales.size, reg_params.size))
# iterate over the scales
for i, scale in enumerate(scales):
# i is the index, scale is the corresponding scale
# we must recreate the feature mapping each time for different scales
feature_mapping = construct_rbf_feature_mapping(centres, scale)
designmtx = feature_mapping(inputs)
# partition the design matrix and targets into train and test
train_designmtx, train_targets, test_designmtx, test_targets = \
train_and_test_partition(
designmtx, targets, train_part, test_part)
# iterating over the regularisation parameters
for j, reg_param in enumerate(reg_params):
# j is the index, reg_param is the corresponding regularisation
# parameter
# train and test the data
train_error, test_error = train_and_test(train_designmtx,
train_targets,
test_designmtx,
test_targets,
reg_param=reg_param)
# store the train and test errors in our 2d arrays
train_errors[i, j] = train_error
test_errors[i, j] = test_error
# we have a 2d array of train and test errors, we want to know the (i,j)
# index of the best value
best_i = np.argmin(np.argmin(test_errors, axis=1))
best_j = np.argmin(test_errors[i, :])
min_place = np.argmin(test_errors)
best_i_correct = (int)(min_place / test_errors.shape[1])
best_j_correct = min_place % test_errors.shape[1]
print(best_i)
print(best_j)
print(best_i_correct)
print(best_j_correct)
min = test_errors[test_errors != 0].min()
ij_min = np.where(test_errors == min)
ij_min = tuple([i.item() for i in ij_min])
print(ij_min[1])
print("\nBest joint choice of parameters:")
print("\tscale %.2g and lambda = %.2g" %
(scales[best_i_correct], reg_params[best_j_correct]))
# now we can plot the error for different scales using the best
# regularisation choice
fig, ax = plot_train_test_errors("scale", scales,
train_errors[:, best_j_correct],
test_errors[:, best_j_correct])
ax.set_xscale('log')
ax.set_title('Train vs Test Error Across Scales')
fig.savefig("../plots/rbf_searching_scales.pdf", fmt="pdf")
# ...and the error for different regularisation choices given the best
# scale choice
fig, ax = plot_train_test_errors("$\lambda$", reg_params,
train_errors[best_i_correct, :],
test_errors[best_i_correct, :])
ax.set_xscale('log')
ax.set_title('Train vs Test Error Across Reg Params')
fig.savefig("../plots/rbf_searching_reg_params.pdf", fmt="pdf")
# using the best parameters found above,
# we now vary the number of centres and evaluate the performance
reg_param = reg_params[best_j_correct]
scale = scales[best_i_correct]
n_centres_seq = np.arange(1, 20)
train_errors = []
test_errors = []
for n_centres in n_centres_seq:
# constructing the feature mapping anew for each number of centres
centres = np.linspace(0, 1, n_centres)
feature_mapping = construct_rbf_feature_mapping(centres, scale)
design_matrix = feature_mapping(inputs)
train_designmtx, train_targets, test_designmtx, test_targets = \
train_and_test_partition(
design_matrix, targets, train_part, test_part)
# evaluating the test and train error for the given regularisation parameter and scale
train_error, test_error = train_and_test(train_designmtx,
train_targets, test_designmtx,
test_targets, reg_param)
# collecting the errors
train_errors.append(train_error)
test_errors.append(test_error)
# plotting the results
fig, ax = plot_train_test_errors("no. centres", n_centres_seq,
train_errors, test_errors)
ax.set_title('Train vs Test Error Across Centre Number')
fig.savefig("../plots/rbf_searching_number_centres.pdf", fmt="pdf")
def main(ifname,
delimiter=None,
columns=None,
has_header=True,
test_fraction=0.25):
data, field_names = import_data(ifname,
delimiter=delimiter,
has_header=has_header,
columns=columns)
#Exploratory Data Analysis (EDA)
raw_data = pd.read_csv('datafile.csv', sep=";")
# view correlation efficieny result where |r|=1 has the strongest relation and |r|=0 the weakest
df = pd.DataFrame(data=raw_data)
print(df.corr())
# view data if it is normally distributed
plt.hist(raw_data["quality"],
range=(1, 10),
edgecolor='black',
linewidth=1)
plt.xlabel('quality')
plt.ylabel('amount of samples')
plt.title("distribution of red wine quality")
# feature selection
import scipy.stats as stats
from scipy.stats import chi2_contingency
class ChiSquare:
def __init__(self, dataframe):
self.df = dataframe
self.p = None # P-Value
self.chi2 = None # Chi Test Statistic
self.dof = None
self.dfObserved = None
self.dfExpected = None
def _print_chisquare_result(self, colX, alpha):
result = ""
if self.p < alpha:
result = "{0} is IMPORTANT for Prediction".format(colX)
else:
result = "{0} is NOT an important predictor. (Discard {0} from model)".format(
colX)
print(result)
def TestIndependence(self, colX, colY, alpha=0.05):
X = self.df[colX].astype(str)
Y = self.df[colY].astype(str)
self.dfObserved = pd.crosstab(Y, X)
chi2, p, dof, expected = stats.chi2_contingency(
self.dfObserved.values)
self.p = p
self.chi2 = chi2
self.dof = dof
self.dfExpected = pd.DataFrame(expected,
columns=self.dfObserved.columns,
index=self.dfObserved.index)
self._print_chisquare_result(colX, alpha)
print('self:%s' % (self), self.chi2, self.p)
# Initialize ChiSquare Class
cT = ChiSquare(raw_data)
# Feature Selection
testColumns = [
"fixed acidity", "volatile acidity", "citric acid", "residual sugar",
"chlorides", "free sulfur dioxide", "total sulfur dioxide", "density",
"pH", "sulphates", "alcohol"
]
for var in testColumns:
cT.TestIndependence(colX=var, colY="quality")
# split data into inputs and targets
inputs = data[:, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]]
targets = data[:, 11]
# mean normalisation
fixed_acidity = inputs[:, 0]
volatile_acidity = inputs[:, 1]
citric_acid = inputs[:, 2]
residual_sugar = inputs[:, 3]
chlorides = inputs[:, 4]
free_sulfur_dioxide = inputs[:, 5]
total_sulfur_dioxide = inputs[:, 6]
density = inputs[:, 7]
ph = inputs[:, 8]
sulphates = inputs[:, 9]
alcohol = inputs[:, 10]
# draw plot of data set
normalised_data = np.column_stack((inputs, targets))
exploratory_plots(normalised_data, field_names)
# add a colum of x0.ones
inputs[:, 0] = np.ones(len(targets))
# normalize data
inputs[:, 1] = (volatile_acidity -
np.mean(volatile_acidity)) / np.std(volatile_acidity)
inputs[:, 2] = (citric_acid - np.mean(citric_acid)) / np.std(citric_acid)
inputs[:, 7] = (density - np.mean(density)) / np.std(density)
inputs[:, 9] = (sulphates - np.mean(sulphates)) / np.std(sulphates)
inputs[:, 10] = (alcohol - np.mean(alcohol)) / np.std(alcohol)
# run all experiments on the same train-test split of the data
train_part, test_part = train_and_test_split(inputs.shape[0],
test_fraction=test_fraction)
# another evaluation function
def rsquare(test_targets, test_predicts):
y_mean = np.mean(test_targets)
ss_tot = sum((test_targets - y_mean)**2)
ss_res = sum((test_targets - test_predicts)**2)
rsquare = 1 - (ss_res / ss_tot)
return rsquare
print(
'---------------------------Linear Regression-----------------------------------'
)
# linear regression
# add a column of 1 to the data matrix
inputs = inputs[:, [0, 1, 2, 7, 9, 10]]
#train_part, test_part = train_and_test_split(inputs.shape[0], test_fraction=test_fraction)
train_inputs, train_targets, test_inputs, test_targets = train_and_test_partition(
inputs, targets, train_part, test_part)
weights = ml_weights(train_inputs, train_targets)
train_predicts = linear_model_predict(train_inputs, weights)
test_predicts = linear_model_predict(test_inputs, weights)
train_error = root_mean_squared_error(train_targets, train_predicts)
test_error = root_mean_squared_error(test_targets, test_predicts)
print("LR-train_weights", weights)
print("LR-train_error", train_error)
print("LR-test_error", test_error)
print("LR-rsquare score", rsquare(test_targets, test_predicts))
print("LR-prediction:", test_predicts[:20], "LR-original",
test_targets[:20])
print(
'----------------Regularised Linear Regression-----------------------------'
)
#regularised linear regression
reg_params = np.logspace(-15, -4, 11)
train_errors = []
test_errors = []
for reg_param in reg_params:
# print("RLR-Evaluating reg_para " + str(reg_param))
train_inputs, train_targets, test_inputs, test_targets = train_and_test_partition(
inputs, targets, train_part, test_part)
reg_weights = regularised_ml_weights(train_inputs, train_targets,
reg_param)
train_predicts = linear_model_predict(train_inputs, reg_weights)
test_predicts = linear_model_predict(test_inputs, reg_weights)
train_error = root_mean_squared_error(train_targets, train_predicts)
test_error = root_mean_squared_error(test_targets, test_predicts)
train_errors.append(train_error)
test_errors.append(test_error)
#best lambda
test_errors = np.array(test_errors)
best_l = np.argmin(test_errors)
print("RLR-Best joint choice of parameters:")
print("RLR-lambda = %.2g" % (reg_params[best_l]))
# plot train_test_errors in different reg_params
fig, ax = plot_train_test_errors("$\lambda$", reg_params, train_errors,
test_errors)
ax.set_xscale('log')
reg_weights = regularised_ml_weights(train_inputs, train_targets, best_l)
print("RLR-train_weights", reg_weights)
print("RLR-train_error", train_errors[best_l])
print("RLR-test_error", test_errors[best_l])
print("RLR-rsquare score", rsquare(test_targets, test_predicts))
print("RLR-prediction:", test_predicts[:20], "RLR-original",
test_targets[:20])
print(
'-----------------------------kNN Regression------------------------------------'
)
# KNN-regression
# tip out the x0=1 column
inputs = inputs[:, [1, 2, 3, 4, 5]]
train_errors = []
test_errors = []
K = range(2, 9)
for k in K:
train_inputs, train_targets, test_inputs, test_targets = train_and_test_partition(
inputs, targets, train_part, test_part)
knn_approx = construct_knn_approx(train_inputs, train_targets, k)
train_knn_predicts = knn_approx(train_inputs)
train_error = root_mean_squared_error(train_knn_predicts,
train_targets)
test_knn_predicts = knn_approx(test_inputs)
test_error = root_mean_squared_error(test_knn_predicts, test_targets)
train_errors.append(train_error)
test_errors.append(test_error)
# print("knn_predicts: ", np.around(test_knn_predicts), "knn-original", test_targets)
#best k
train_errors = np.array(train_errors)
test_errors = np.array(test_errors)
best_k = np.argmin(test_errors)
print("Best joint choice of parameters:")
print("k = %.2g" % (K[best_k]))
fig, ax = plot_train_test_errors("K", K, train_errors, test_errors)
ax.set_xticks(np.arange(min(K), max(K) + 1, 1.0))
print("kNN-train_error", train_errors[-1])
print("kNN-test_error", test_errors[-1])
knn_approx = construct_knn_approx(train_inputs, train_targets, k=3)
test_predicts = knn_approx(test_inputs)
print("kNN-rsquare score", rsquare(test_targets, test_predicts))
print("kNN-y_predicts", test_predicts[:20], 'y_original',
test_targets[:20])
print(
'----------------------------RBF Function-------------------------------------'
)
# Radinal Basis Functions
# for the centres of the basis functions sample 15% of the data
sample_fraction = 0.15
p = (1 - sample_fraction, sample_fraction)
centres = inputs[np.random.choice([False, True], size=inputs.shape[0],
p=p), :] # !!!
print("centres.shape = %r" % (centres.shape, ))
scales = np.logspace(0, 2, 17) # of the basis functions
reg_params = np.logspace(-15, -4, 11) # choices of regularisation strength
# create empty 2d arrays to store the train and test errors
train_errors = np.empty((scales.size, reg_params.size))
test_errors = np.empty((scales.size, reg_params.size))
# iterate over the scales
for i, scale in enumerate(scales):
# i is the index, scale is the corresponding scale
# we must recreate the feature mapping each time for different scales
feature_mapping = construct_rbf_feature_mapping(centres, scale)
designmtx = feature_mapping(inputs)
# partition the design matrix and targets into train and test
train_designmtx, train_targets, test_designmtx, test_targets = \
train_and_test_partition(designmtx, targets, train_part, test_part)
# iteratre over the regularisation parameters
for j, reg_param in enumerate(reg_params):
# j is the index, reg_param is the corresponding regularisation
# parameter
# train and test the data
train_error, test_error = train_and_test(train_designmtx,
train_targets,
test_designmtx,
test_targets,
reg_param=reg_param)
# store the train and test errors in our 2d arrays
train_errors[i, j] = train_error
test_errors[i, j] = test_error
# we have a 2d array of train and test errors, we want to know the (i,j)
# index of the best value
best_i = np.argmin(np.argmin(test_errors, axis=1))
best_j = np.argmin(test_errors[i, :])
print("Best joint choice of parameters:")
print("\tscale= %.2g and lambda = %.2g" %
(scales[best_i], reg_params[best_j]))
# now we can plot the error for different scales using the best
# regulariation choice
fig, ax = plot_train_test_errors("scale", scales, train_errors[:, best_j],
test_errors[:, best_j])
ax.set_xscale('log')
# ...and the error for different regularisation choices given the best
# scale choice
fig, ax = plot_train_test_errors("$\lambda$", reg_params,
train_errors[best_i, :],
test_errors[best_i, :])
ax.set_xscale('log')
feature_mapping = construct_rbf_feature_mapping(centres, scales[best_i])
reg_weights = regularised_ml_weights(train_designmtx, train_targets,
reg_params[best_j])
# test function
test_predicts = np.matrix(test_designmtx) * np.matrix(reg_weights).reshape(
(len(reg_weights), 1))
test_predicts = np.array(test_predicts).flatten()
print("RBF-train_error", train_errors[best_i, best_j])
print("RBF-test_error", test_errors[best_i, best_j])
print("RBF-rsquare score", rsquare(test_targets, test_predicts))
print('RBF_y_predicts: ', test_predicts[:20], 'rbf_y_originals: ',
test_targets[:20])
print(
'-----------------------------Polynomial---------------------------------------'
)
# Polynomial Basis Function
# set input features as 'alcohol'
degrees = range(1, 10)
train_errors = []
test_errors = []
for degree in degrees:
processed_inputs = 0
for i in range(inputs.shape[1]):
processed_input = expand_to_monomials(inputs[:, i], degree)
processed_inputs += processed_input
processed_inputs = np.array(processed_inputs)
# split data into train and test set
processed_train_inputs, train_targets, processed_test_inputs, test_targets = train_and_test_partition\
(processed_inputs, targets, train_part, test_part)
train_error, test_error = train_and_test(processed_train_inputs,
train_targets,
processed_test_inputs,
test_targets,
reg_param=None)
weights = regularised_least_squares_weights(processed_train_inputs,
train_targets, reg_param)
train_errors.append(train_error)
test_errors.append(test_error)
train_errors = np.array(train_errors)
test_errors = np.array(test_errors)
print("Polynomial-train error: ", train_errors[-1])
print("Polynomial-test error: ", test_errors[-1])
best_d = np.argmin(test_errors)
print("Best joint choice of degree:")
final_degree = degrees[best_d]
print("degree = %.2g" % (final_degree))
fig, ax = plot_train_test_errors("Degree", degrees, train_errors,
test_errors)
ax.set_xticks(np.arange(min(degrees), max(degrees) + 1, 1.0))
# test functionality with the final degree
processed_inputs = 0
for i in range(inputs.shape[1]):
processed_input = expand_to_monomials(inputs[:, i], final_degree)
processed_inputs += processed_input
processed_inputs = np.array(processed_inputs)
processed_train_inputs, train_targets, processed_test_inputs, test_targets = train_and_test_partition \
(processed_inputs, targets, train_part, test_part)
train_error, test_error = train_and_test(processed_train_inputs,
train_targets,
processed_test_inputs,
test_targets,
reg_param=None)
weights = regularised_least_squares_weights(processed_train_inputs,
train_targets, reg_param)
# print("processed_train_inputs.shape", processed_train_inputs.shape)
# print('weights: ', weights, 'weights shape: ', weights.shape)
test_predicts = prediction_function(processed_test_inputs, weights,
final_degree)
print("Polynomial-rsquare score", rsquare(test_targets, test_predicts))
print('Polynomial-y_predicts: ', test_predicts[:20],
'Polynomial-y_original: ', test_targets[:20])
plt.show()
def main():
"""
This function contains example code that demonstrates how to use the
functions defined in poly_fit_base for fitting polynomial curves to data.
"""
# specify the centres of the rbf basis functions
centres = np.linspace(0,1,7)
# the width (analogous to standard deviation) of the basis functions
scale = 0.15
print("centres = %r" % (centres,))
print("scale = %r" % (scale,))
feature_mapping = construct_rbf_feature_mapping(centres,scale)
datamtx = np.linspace(0,1, 51)
designmtx = feature_mapping(datamtx)
fig = plt.figure()
ax = fig.add_subplot(1,1,1)
for colid in range(designmtx.shape[1]):
ax.plot(datamtx, designmtx[:,colid])
ax.set_xlim([0,1])
ax.set_xticks([0,1])
ax.set_yticks([0,1])
# choose number of data-points and sample a pair of vectors: the input
# values and the corresponding target values
N = 20
inputs, targets = sample_data(N, arbitrary_function_1, seed=37)
# define the feature mapping for the data
feature_mapping = construct_rbf_feature_mapping(centres,scale)
# now construct the design matrix
designmtx = feature_mapping(inputs)
#
# find the weights that fit the data in a least squares way
weights = ml_weights(designmtx, targets)
# use weights to create a function that takes inputs and returns predictions
# in python, functions can be passed just like any other object
# those who know MATLAB might call this a function handle
rbf_approx = construct_feature_mapping_approx(feature_mapping, weights)
fig, ax, lines = plot_function_data_and_approximation(
rbf_approx, inputs, targets, arbitrary_function_1)
ax.legend(lines, ['true function', 'data', 'linear approx'])
ax.set_xticks([])
ax.set_yticks([])
fig.tight_layout()
fig.savefig("regression_rbf.pdf", fmt="pdf")
# for a single choice of regularisation strength we can plot the
# approximating function
reg_param = 10**-3
reg_weights = regularised_ml_weights(
designmtx, targets, reg_param)
rbf_reg_approx = construct_feature_mapping_approx(feature_mapping, reg_weights)
fig, ax, lines = plot_function_data_and_approximation(
rbf_reg_approx, inputs, targets, arbitrary_function_1)
ax.set_xticks([])
ax.set_yticks([])
fig.tight_layout()
fig.savefig("regression_rbf_basis_functions_reg.pdf", fmt="pdf")
# to find a good regularisation parameter, we can performa a parameter
# search (a naive way to do this is to simply try a sequence of reasonable
# values within a reasonable range.
# sample some training and testing inputs
train_inputs, train_targets = sample_data(N, arbitrary_function_1, seed=37)
# we need to use a different seed for our test data, otherwise some of our
# sampled points will be the same
test_inputs, test_targets = sample_data(100, arbitrary_function_1, seed=82)
# convert the raw inputs into feature vectors (construct design matrices)
train_designmtx = feature_mapping(train_inputs)
test_designmtx = feature_mapping(test_inputs)
# now we're going to evaluate train and test error for a sequence of
# potential regularisation strengths storing the results
reg_params = np.logspace(-5,1)
train_errors = []
test_errors = []
for reg_param in reg_params:
# evaluate the test and train error for this regularisation parameter
train_error, test_error = train_and_test(
train_designmtx, train_targets, test_designmtx, test_targets,
reg_param=reg_param)
# collect the errors
train_errors.append(train_error)
test_errors.append(test_error)
# plot the results
fig, ax = plot_train_test_errors(
"$\lambda$", reg_params, train_errors, test_errors)
ax.set_xscale('log')
# we may also be interested in choosing the right number of centres, or
# the right width/scale of the rbf functions.
# Here we vary the width and evaluate the performance
reg_param = 10**-3
scales = np.logspace(-2,0)
train_errors = []
test_errors = []
for scale in scales:
# we must construct the feature mapping anew for each scale
feature_mapping = construct_rbf_feature_mapping(centres,scale)
train_designmtx = feature_mapping(train_inputs)
test_designmtx = feature_mapping(test_inputs)
# evaluate the test and train error for this regularisation parameter
train_error, test_error = train_and_test(
train_designmtx, train_targets, test_designmtx, test_targets,
reg_param=reg_param)
# collect the errors
train_errors.append(train_error)
test_errors.append(test_error)
# plot the results
fig, ax = plot_train_test_errors(
"scale", scales, train_errors, test_errors)
ax.set_xscale('log')
# Here we vary the number of centres and evaluate the performance
reg_param = 10**-3
scale = 0.15
n_centres_seq = np.arange(3,20)
train_errors = []
test_errors = []
for n_centres in n_centres_seq:
# we must construct the feature mapping anew for each number of centres
centres = np.linspace(0,1,n_centres)
feature_mapping = construct_rbf_feature_mapping(centres,scale)
train_designmtx = feature_mapping(train_inputs)
test_designmtx = feature_mapping(test_inputs)
# evaluate the test and train error for this regularisation parameter
train_error, test_error = train_and_test(
train_designmtx, train_targets, test_designmtx, test_targets,
reg_param=reg_param)
# collect the errors
train_errors.append(train_error)
test_errors.append(test_error)
# plot the results
fig, ax = plot_train_test_errors(
"Num. Centres", n_centres_seq, train_errors, test_errors)
plt.show()
def parameter_search_rbf_without_cross(inputs, targets, test_fraction,test_error_linear,normalize=True):
"""
"""
if(normalize):
# normalise inputs (meaning radial basis functions are more helpful)
for i in range(inputs.shape[1]):
inputs[:,i]=(inputs[:,i]-np.mean(inputs[:,i]))/np.std(inputs[:,i])
N = inputs.shape[0]
# for the centres of the basis functions sample 10% of the data
sample_fractions = np.array([0.05,0.1,0.15,0.2,0.25])
scales = np.logspace(0,4,20 ) # of the basis functions
reg_params = np.logspace(-16,-1, 20) # choices of regularisation strength.
# create empty 3d arrays to store the train and test errors
train_mean_errors = np.empty((sample_fractions.size,scales.size,reg_params.size))
test_mean_errors = np.empty((sample_fractions.size,scales.size,reg_params.size))
#Randomly generates a train/test split for data of size N. Returns a 2 arrays of boolean true/false.
train_part, test_part = train_and_test_split(N, test_fraction=test_fraction)
best_k=0
best_i=0
best_j=0
test_error_temp=10**100
#loop through the possible centres as a percentage (5%, 10%,15%, 20%, 25%)
for k,sample_fraction in enumerate(sample_fractions):
p = (1-sample_fraction,sample_fraction)
centres = inputs[np.random.choice([False,True], size=N, p=p),:]
# iterate over the scales
for i,scale in enumerate(scales):
# i is the index, scale is the corresponding scale
# we must recreate the feature mapping each time for different scales
feature_mapping = construct_rbf_feature_mapping(centres,scale)
designmtx = feature_mapping(inputs)
# partition the design matrix and targets into train and test. This effectively takes as inputs the boolean arrays train_part, test_part and the whole design matrix and
#creates 2 subsets of the design matrix (train matrix, test matrix). The test data are splitted as well but the values are not affected
train_designmtx, train_targets, test_designmtx, test_targets = train_and_test_partition(designmtx, targets, train_part, test_part)
# iteratre over the regularisation parameters
for j, reg_param in enumerate(reg_params):
# j is the index, reg_param is the corresponding regularisation
# parameter
# train and test the data
train_error, test_error,weights = train_and_test(train_designmtx, train_targets, test_designmtx, test_targets,reg_param=reg_param)
# store the train and test errors in our 2d arrays
train_mean_errors[k,i,j] = train_error
test_mean_errors[k,i,j] = test_error
#When we've found a lowest than stores test error value, we store it's indices
if (np.mean(test_error)<test_error_temp):
test_error_temp=test_error
best_k=k
best_i=i
best_j=j
print ("The value with the lowest error is:",test_mean_errors[best_k][best_i][best_j])
print("Best joint choice of parameters: sample fractions %.2g scale %.2g and lambda = %.2g" % (sample_fractions[best_k],scales[best_i],reg_params[best_j]))
# now we can plot the error for different scales using the best
# regularization choice
# now we can plot the error for different scales using the best regularization choice and centres percentage
fig , ax = plot_train_test_errors("scale", scales, train_mean_errors[best_k,:,best_j], test_mean_errors[best_k,:,best_j])
ax.set_xscale('log')
fig.suptitle('RBF regression for the best reg. parameter & centres', fontsize=10)
xlim = ax.get_xlim()#get the xlim to graph the linear regression
ax.plot(xlim, test_error_linear*np.ones(2), 'g:') #graph the linear regression
# ...and the error for different regularisation choices given the best scale choice and centres percentage
fig , ax = plot_train_test_errors("$\lambda$", reg_params, train_mean_errors[best_k,best_i,:], test_mean_errors[best_k,best_i,:])
ax.set_xscale('log')
fig.suptitle('RBF regression for the best scale parameter & centres', fontsize=10)
xlim = ax.get_xlim()#get the xlim to graph the linear regression
ax.plot(xlim, test_error_linear*np.ones(2), 'g:')
# #ax.set_ylim([0,20])
# ...and the error for different centres given the best reg.parameter and the best scale choice
fig , ax = plot_train_test_errors("sample fractions", sample_fractions, train_mean_errors[:,best_i,best_j], test_mean_errors[:,best_i,best_j])
fig.suptitle('RBF regression for the best scale parameter & reg. parameter', fontsize=10)
ax.set_xlim([0.05, 0.25])
xlim = ax.get_xlim()#get the xlim to graph the linear regression
ax.plot(xlim, test_error_linear*np.ones(2), 'g:')