def test_custom_function(self):
""" Define a custom sin function, which is the unique solution to
f'' = -f, f(0) = 0, f'(0) = 1
"""
f = Function(func_type="sin", func=lambda j: math.sin(j))
f_prime = f.derivative()
f_prime_prime = f.derivative().derivative()
self.assertEqual(str(f), "Sin function")
self.assertEqual(f(0), 0)
self.assertAlmostEqual(f_prime(0), 1, places=4)
for x in self.test_points:
self.assertAlmostEqual(f(x), -f_prime_prime(x), places=4)
def test_custom_function(self):
""" Define a custom sin function, which is the unique solution to
f'' = -f, f(0) = 0, f'(0) = 1
"""
f = Function(func_type="sin", func=lambda j: math.sin(j))
f_prime = f.derivative()
f_prime_prime = f.derivative().derivative()
self.assertEqual(str(f), "Sin function")
self.assertEqual(f(0), 0)
self.assertAlmostEqual(f_prime(0), 1, places=4)
for x in self.test_points:
self.assertAlmostEqual(f(x), -f_prime_prime(x), places=4)
def setUp(self):
self.bias = Function("bias")
self.identity = Function("identity")
self.degree = 2 # polynomial degree
self.poly = Function("polynomial", degree=self.degree)
self.poly_derivative = self.poly.derivative()
self.mu = 3
self.std_dev = 4
self.standard_gaussian = Function("gaussian")
self.gaussian = Function("gaussian", mu=self.mu, sigma=self.std_dev)
self.gaussian_derivative = self.gaussian.derivative()
self.logistic_sigmoid = Function("sigmoid", mu=self.mu, sigma=self.std_dev, sig_func="logistic")
self.logistic_sigmoid_derivative = self.logistic_sigmoid.derivative()
self.tanh_sigmoid = Function("sigmoid", mu=self.mu, sigma=self.std_dev, sig_func="tanh")
self.tanh_sigmoid_derivative = self.tanh_sigmoid.derivative()
self.test_points = range(10)
def setUp(self):
self.bias = Function('bias')
self.identity = Function('identity')
self.degree = 2 # polynomial degree
self.poly = Function('polynomial', degree=self.degree)
self.poly_derivative = self.poly.derivative()
self.mu = 3
self.std_dev = 4
self.standard_gaussian = Function('gaussian')
self.gaussian = Function('gaussian', mu=self.mu,
sigma=self.std_dev)
self.gaussian_derivative = self.gaussian.derivative()
self.logistic_sigmoid = Function('sigmoid', mu=self.mu,
sigma=self.std_dev,
sig_func='logistic')
self.logistic_sigmoid_derivative = self.logistic_sigmoid.derivative()
self.tanh_sigmoid = Function('sigmoid', mu=self.mu,
sigma=self.std_dev,
sig_func='tanh')
self.tanh_sigmoid_derivative = self.tanh_sigmoid.derivative()
self.test_points = range(10)
class TestFunction(TestCase):
def setUp(self):
self.bias = Function('bias')
self.identity = Function('identity')
self.degree = 2 # polynomial degree
self.poly = Function('polynomial', degree=self.degree)
self.poly_derivative = self.poly.derivative()
self.mu = 3
self.std_dev = 4
self.standard_gaussian = Function('gaussian')
self.gaussian = Function('gaussian', mu=self.mu,
sigma=self.std_dev)
self.gaussian_derivative = self.gaussian.derivative()
self.logistic_sigmoid = Function('sigmoid', mu=self.mu,
sigma=self.std_dev,
sig_func='logistic')
self.logistic_sigmoid_derivative = self.logistic_sigmoid.derivative()
self.tanh_sigmoid = Function('sigmoid', mu=self.mu,
sigma=self.std_dev,
sig_func='tanh')
self.tanh_sigmoid_derivative = self.tanh_sigmoid.derivative()
self.test_points = range(10)
def test_bias(self):
for x in self.test_points:
self.assertEqual(1, self.bias(x), "Bias function should always equal 1")
def test_identity(self):
for x in self.test_points:
self.assertEqual(x, self.identity(x), "Identity function returns argument")
def test_poly(self):
for x in self.test_points:
self.assertEqual(x ** self.degree, self.poly(x),
"Polynomial function should raise number to {:d} degree".format(self.degree))
def test_poly_derivative(self):
""" For any polynomial, (x^n)' = n x^(n - 1), so x (x^n)' = n (x^n)
"""
for x in self.test_points:
self.assertAlmostEqual(x * self.poly_derivative(x), self.degree * self.poly(x), places=4,
msg="{0:d} * x^{0:d} = x * (x^{0:d})'".format(self.degree))
def test_gaussian(self):
""" Gaussians are symmetric about their mean, and can be transformed to a standard
gaussian by multiplying by the std deviation and adding mu
"""
for x in self.test_points:
self.assertAlmostEqual(self.gaussian(self.mu + x), self.gaussian(self.mu - x), places=4,
msg="The gaussian is symmetric")
for x in self.test_points:
self.assertAlmostEqual(self.gaussian(self.mu + x * self.std_dev), self.standard_gaussian(x), places=4,
msg="The gaussian is symmetric")
def test_gaussian_derivative(self):
""" f' = ((mu - x) / sigma^2) * f
"""
for x in self.test_points:
self.assertAlmostEqual(self.gaussian_derivative(x),
float(self.mu - x) / float(self.std_dev ** 2) * self.gaussian(x),
msg="""Gaussian derivative must satisfy
f'({0:.0f})=((mu - {0:.0f})/sigma^2) f({0:.0f})""".format(x))
def test_logistic_sigmoid_derivative(self):
""" f' = f (1 - f)/sigma
"""
f = self.logistic_sigmoid
f_prime = self.logistic_sigmoid_derivative
for x in self.test_points:
self.assertAlmostEqual(f_prime(x),
(f(x) * (1 - f(x))) / self.std_dev,
msg="Logistic sigmoid satisfies f' = f(1-f)/sigma")
def test_tanh_sigmoid_derivative(self):
""" f' = (1 - f^2)/sigma and f(mu) = 0.5 defines the tanh sigmoid
"""
f = self.tanh_sigmoid
self.assertEqual(f(self.mu), 0., msg="Initial condition for tanh sigmoid is f(mu) = 0")
f_prime = self.tanh_sigmoid_derivative
for x in self.test_points:
self.assertAlmostEqual(f_prime(x),
(1 - f(x) * f(x)) / (2 * self.std_dev),
msg="Tanh sigmoid must satisfy f'({0:.0f}) = (1-f({0:.0f})^2)/(2 * sigma)".format(x))
def test_undefined_function(self):
self.assertRaises(NotImplementedError, Function, "foo")
def test_undefined_sigmoid(self):
self.assertRaises(NotImplementedError,
lambda x: Function("sigmoid", sig_func=x),
"foo")
def test_prints(self):
""" Tests the __repr__ functions
"""
self.assertEqual(str(self.logistic_sigmoid), "Sigmoid function")
self.assertEqual(str(Function(func=lambda j: j)), "Custom function")
def test_custom_function(self):
""" Define a custom sin function, which is the unique solution to
f'' = -f, f(0) = 0, f'(0) = 1
"""
f = Function(func_type="sin", func=lambda j: math.sin(j))
f_prime = f.derivative()
f_prime_prime = f.derivative().derivative()
self.assertEqual(str(f), "Sin function")
self.assertEqual(f(0), 0)
self.assertAlmostEqual(f_prime(0), 1, places=4)
for x in self.test_points:
self.assertAlmostEqual(f(x), -f_prime_prime(x), places=4)
def test_prints(self):
""" Tests the __repr__ functions
"""
self.assertEqual(str(self.logistic_sigmoid), "Sigmoid function")
self.assertEqual(str(Function(func=lambda j: j)), "Custom function")
def test_undefined_sigmoid(self):
self.assertRaises(NotImplementedError,
lambda x: Function("sigmoid", sig_func=x),
"foo")
class TestFunction(TestCase):
def setUp(self):
self.bias = Function("bias")
self.identity = Function("identity")
self.degree = 2 # polynomial degree
self.poly = Function("polynomial", degree=self.degree)
self.poly_derivative = self.poly.derivative()
self.mu = 3
self.std_dev = 4
self.standard_gaussian = Function("gaussian")
self.gaussian = Function("gaussian", mu=self.mu, sigma=self.std_dev)
self.gaussian_derivative = self.gaussian.derivative()
self.logistic_sigmoid = Function("sigmoid", mu=self.mu, sigma=self.std_dev, sig_func="logistic")
self.logistic_sigmoid_derivative = self.logistic_sigmoid.derivative()
self.tanh_sigmoid = Function("sigmoid", mu=self.mu, sigma=self.std_dev, sig_func="tanh")
self.tanh_sigmoid_derivative = self.tanh_sigmoid.derivative()
self.test_points = range(10)
def test_bias(self):
for x in self.test_points:
self.assertEqual(1, self.bias(x), "Bias function should always equal 1")
def test_identity(self):
for x in self.test_points:
self.assertEqual(x, self.identity(x), "Identity function returns argument")
def test_poly(self):
for x in self.test_points:
self.assertEqual(
x ** self.degree,
self.poly(x),
"Polynomial function should raise number to {:d} degree".format(self.degree),
)
def test_poly_derivative(self):
""" For any polynomial, (x^n)' = n x^(n - 1), so x (x^n)' = n (x^n)
"""
for x in self.test_points:
self.assertAlmostEqual(
x * self.poly_derivative(x),
self.degree * self.poly(x),
places=4,
msg="{0:d} * x^{0:d} = x * (x^{0:d})'".format(self.degree),
)
def test_gaussian(self):
""" Gaussians are symmetric about their mean, and can be transformed to a standard
gaussian by multiplying by the std deviation and adding mu
"""
for x in self.test_points:
self.assertAlmostEqual(
self.gaussian(self.mu + x), self.gaussian(self.mu - x), places=4, msg="The gaussian is symmetric"
)
for x in self.test_points:
self.assertAlmostEqual(
self.gaussian(self.mu + x * self.std_dev),
self.standard_gaussian(x),
places=4,
msg="The gaussian is symmetric",
)
def test_gaussian_derivative(self):
""" f' = ((mu - x) / sigma^2) * f
"""
for x in self.test_points:
self.assertAlmostEqual(
self.gaussian_derivative(x),
float(self.mu - x) / float(self.std_dev ** 2) * self.gaussian(x),
msg="""Gaussian derivative must satisfy
f'({0:.0f})=((mu - {0:.0f})/sigma^2) f({0:.0f})""".format(
x
),
)
def test_logistic_sigmoid_derivative(self):
""" f' = f (1 - f)/sigma
"""
f = self.logistic_sigmoid
f_prime = self.logistic_sigmoid_derivative
for x in self.test_points:
self.assertAlmostEqual(
f_prime(x), (f(x) * (1 - f(x))) / self.std_dev, msg="Logistic sigmoid satisfies f' = f(1-f)/sigma"
)
def test_tanh_sigmoid_derivative(self):
""" f' = (1 - f^2)/sigma and f(mu) = 0.5 defines the tanh sigmoid
"""
f = self.tanh_sigmoid
self.assertEqual(f(self.mu), 0.0, msg="Initial condition for tanh sigmoid is f(mu) = 0")
f_prime = self.tanh_sigmoid_derivative
for x in self.test_points:
self.assertAlmostEqual(
f_prime(x),
(1 - f(x) * f(x)) / (2 * self.std_dev),
msg="Tanh sigmoid must satisfy f'({0:.0f}) = (1-f({0:.0f})^2)/(2 * sigma)".format(x),
)
def test_undefined_function(self):
self.assertRaises(NotImplementedError, Function, "foo")
def test_undefined_sigmoid(self):
self.assertRaises(NotImplementedError, lambda x: Function("sigmoid", sig_func=x), "foo")
def test_prints(self):
""" Tests the __repr__ functions
"""
self.assertEqual(str(self.logistic_sigmoid), "Sigmoid function")
self.assertEqual(str(Function(func=lambda j: j)), "Custom function")
def test_custom_function(self):
""" Define a custom sin function, which is the unique solution to
f'' = -f, f(0) = 0, f'(0) = 1
"""
f = Function(func_type="sin", func=lambda j: math.sin(j))
f_prime = f.derivative()
f_prime_prime = f.derivative().derivative()
self.assertEqual(str(f), "Sin function")
self.assertEqual(f(0), 0)
self.assertAlmostEqual(f_prime(0), 1, places=4)
for x in self.test_points:
self.assertAlmostEqual(f(x), -f_prime_prime(x), places=4)
