def train_and_test(train_inputs,
train_targets,
test_inputs,
test_targets,
reg_param=None):
"""
Will fit a linear model with either least squares, or regularised least
squares to the training data, then evaluate on both test and training data
parameters
----------
train_inputs - the input design matrix for training
train_targets - the training targets as a vector
test_inputs - the input design matrix for testing
test_targets - the test targets as a vector
reg_param (optional) - the regularisation strength. If provided, then
regularised maximum likelihood fitting is uses with this regularisation
strength. Otherwise, (non-regularised) least squares is used.
returns
-------
train_error - the training error for the approximation
test_error - the test error for the approximation
"""
# finding the optimal weights (depends on regularisation)
if reg_param is None:
# using simple least squares approach
weights = ml_weights(train_inputs, train_targets)
else:
# using regularised least squares approach
weights = regularised_ml_weights(train_inputs, train_targets,
reg_param)
# predictions are linear functions of the inputs, we evaluate those here
train_predicts = linear_model_predict(train_inputs, weights)
test_predicts = linear_model_predict(test_inputs, weights)
residuals = (np.array(train_targets).flatten() -
np.array(train_predicts).flatten())
variance_training_error = np.var(residuals)
sum_joint_log_probabilities = 0
for n in range(len(test_predicts)):
if test_predicts[n] <= 0:
continue
else:
sum_joint_log_probabilities += math.log(test_predicts[n])
sum_joint_log_probabilities *= -1
# print("Error as negative joint log probability: %r" % sum_joint_log_probabilities)
# evaluate the error between the predictions and true targets on both sets
train_error = root_mean_squared_error(train_targets, train_predicts)
test_error = root_mean_squared_error(test_targets, test_predicts)
if np.isnan(test_error):
print("test_predicts = %r" % (test_predicts, ))
return train_error, test_error, variance_training_error
def train_and_test(train_inputs,
train_targets,
test_inputs,
test_targets,
reg_param=None):
"""
Will fit a linear model with either least squares, or regularised least
squares to the training data, then evaluate on both test and training data
parameters
----------
train_inputs - the input design matrix for training
train_targets - the training targets as a vector
test_inputs - the input design matrix for testing
test_targets - the test targets as a vector
reg_param (optional) - the regularisation strength. If provided, then
regularised maximum likelihood fitting is uses with this regularisation
strength. Otherwise, (non-regularised) least squares is used.
returns
-------
train_error - the training error for the approximation
test_error - the test error for the approximation
"""
# Find the optimal weights (depends on regularisation)
if reg_param is None:
# use simple least squares approach
weights = ml_weights(train_inputs, train_targets)
else:
# use regularised least squares approach
weights = regularised_ml_weights(train_inputs, train_targets,
reg_param)
# predictions are linear functions of the inputs, we evaluate those here
train_predicts = linear_model_predict(train_inputs, weights)
test_predicts = linear_model_predict(test_inputs, weights)
# evaluate the error between the predictions and true targets on both sets
train_error = root_mean_squared_error(train_targets, train_predicts)
test_error = root_mean_squared_error(test_targets, test_predicts)
if np.isnan(test_error):
print("test_predicts = %r" % (test_predicts, ))
return train_error, test_error
def main(ifname=None,
delimiter=None,
columns=None,
normalise=None,
features=None):
"""
To be called when the script is run. This function fits and plots imported data (if a filename is
provided). Data is 2 dimensional real valued data and is fit
with maximum likelihood 2d gaussian.
parameters
----------
ifname -- filename/path of data file.
delimiter -- delimiter of data values
has_header -- does the data-file have a header line
columns -- a list of integers specifying which columns of the file to import
(counting from 0)
"""
# if no file name is provided then use synthetic data
if ifname is None:
print("You need to ingest the CSV file")
else:
data, field_names = import_data(ifname,
delimiter=delimiter,
has_header=True,
columns=columns)
# DATA PREPARATION-----------------------------------------------
N = data.shape[0]
target = data[:, 11:]
# Ask user to confirm whether to normalise or not
if normalise == None:
normalise_response = input(
"Do you want to normalise the data? (Y/N)")
normalise = normalise_response.upper()
normalise_label = ""
if normalise == "Y":
normalise_label = "_normalised"
# Normalise input data
fixed_acidity = data[:, 0]
volatility_acidity = data[:, 1]
citric_acid = data[:, 2]
residual_sugar = data[:, 3]
chlorides = data[:, 4]
free_sulfur_dioxide = data[:, 5]
total_sulfur_dioxide = data[:, 6]
density = data[:, 7]
pH = data[:, 8]
sulphates = data[:, 9]
alcohol = data[:, 10]
data[:, 0] = (fixed_acidity -
np.mean(fixed_acidity)) / np.std(fixed_acidity)
data[:, 1] = (volatility_acidity - np.mean(volatility_acidity)
) / np.std(volatility_acidity)
data[:, 2] = (citric_acid -
np.mean(citric_acid)) / np.std(citric_acid)
data[:, 3] = (residual_sugar -
np.mean(residual_sugar)) / np.std(residual_sugar)
data[:, 4] = (chlorides - np.mean(chlorides)) / np.std(chlorides)
data[:, 5] = (free_sulfur_dioxide - np.mean(free_sulfur_dioxide)
) / np.std(free_sulfur_dioxide)
data[:, 6] = (total_sulfur_dioxide - np.mean(total_sulfur_dioxide)
) / np.std(total_sulfur_dioxide)
data[:, 7] = (density - np.mean(density)) / np.std(density)
data[:, 8] = (pH - np.mean(pH)) / np.std(pH)
data[:, 9] = (sulphates - np.mean(sulphates)) / np.std(sulphates)
data[:, 10] = (alcohol - np.mean(alcohol)) / np.std(alcohol)
elif normalise != "N":
sys.exit("Please enter valid reponse of Y or N")
if features == None:
feature_response = input(
"Please specify which feature combination you want (e.g.1,2,5,7)"
)
feature_response = feature_response.split(",")
# need to convert list of strings into list of integer
feature_combin = []
for i in range(len(feature_response)):
print(feature_response[i])
feature_combin.append(int(feature_response[i]))
else:
feature_combin = features
inputs = np.array([])
for j in range(len(feature_combin)):
inputs = np.append(inputs, data[:, feature_combin[j]])
inputs = inputs.reshape(len(feature_combin), data.shape[0])
inputs = (np.rot90(inputs, 3))[:, ::-1]
#print("INPUT: ", inputs)
# Plotting RBF Model ----------------------------------------------------------
# specify the centres of the rbf basis functions
centres = np.asarray([
0.35, 0.4, 0.45, 0.459090909, 0.468181818, 0.477272727,
0.486363636, 0.495454545, 0.504545455, 0.513636364, 0.522727273,
0.531818182, 0.540909091, 0.55, 0.56, 0.57, 0.58, 0.59, 0.6, 0.61,
0.62, 0.63, 0.64, 0.65, 0.7, 0.75, 0.8
])
# the width (analogous to standard deviation) of the basis functions
scale = 450
reg_param = 7.906043210907701e-11
print("centres = %r" % (centres, ))
print("scale = %r" % (scale, ))
print("reg param = %r" % (reg_param, ))
# create the feature mapping
feature_mapping = construct_rbf_feature_mapping(centres, scale)
# plot the basis functions themselves for reference
#display_basis_functions(feature_mapping)
# now construct the design matrix for the inputs
designmtx = feature_mapping(inputs)
# the number of features is the widht of this matrix
print("DESIGN MATRIX: ", designmtx)
if reg_param is None:
# use simple least squares approach
weights = ml_weights(designmtx, target)
else:
# use regularised least squares approach
weights = regularised_ml_weights(designmtx, target, reg_param)
# get the cross-validation folds
num_folds = 4
folds = create_cv_folds(N, num_folds)
train_errors, test_errors = cv_evaluation_linear_model(
designmtx, target, folds, reg_param=reg_param)
# we're interested in the average (mean) training and testing errors
train_mean_error = np.mean(train_errors)
test_mean_error = np.mean(test_errors)
train_stdev_error = np.std(train_errors)
test_stdev_error = np.std(test_errors)
print("TRAIN MEAN ERROR: ", train_mean_error)
print("TEST MEAN ERROR: ", test_mean_error)
print("TRAIN STDEV ERROR: ", train_stdev_error)
print("TEST STDEV ERROR: ", test_stdev_error)
print("ML WEIGHTS: ", weights)
apply_validation_set(feature_combin, feature_mapping, weights)
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()