In [1]:
%matplotlib inline
In [2]:
import scipy.linalg as la
import numpy as np
import scipy.optimize as opt
import matplotlib.pyplot as plt
import pandas as pd
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np.set_printoptions(precision=3, suppress=True)
In [4]:
def f(x):
return x**3-3*x+1
In [5]:
x = np.linspace(-3,3,100)
plt.axhline(0, c='red')
plt.plot(x, f(x));
In [6]:
from scipy.optimize import brentq, newton
In [7]:
brentq(f, -3, 0), brentq(f, 0, 1), brentq(f, 1,3)
Out[7]:
In [8]:
newton(f, -3), newton(f, 0), newton(f, 3)
Out[8]:
In [9]:
fprime = lambda x: 3*x**2 - 3
newton(f, -3, fprime), newton(f, 0, fprime), newton(f, 3, fprime)
Out[9]:
In [10]:
from sympy import symbols, N, real_roots
In [11]:
x = symbols('x', real=True)
sol = real_roots(x**3 - 3*x + 1)
list(map(N, sol))
Out[11]:
In [12]:
from scipy.optimize import fixed_point
In [13]:
x = np.linspace(-3,3,100)
plt.plot(x, f(x), color='red')
plt.plot(x, x)
pass
In [14]:
fixed_point(f, 0), fixed_point(f, -3), fixed_point(f, 3)
Out[14]:
In [15]:
newton(lambda x: f(x) - x, 0), newton(lambda x: f(x) - x, -3), newton(lambda x: f(x) - x, 3)
Out[15]:
In [16]:
def f(x, r):
"""Discrete logistic equation."""
return r*x*(1-x)
In [17]:
n = 100
fps = np.zeros(n)
for i, r in enumerate(np.linspace(0, 4, n)):
fps[i] = fixed_point(f, 0.5, args=(r, ))
plt.plot(np.linspace(0, 4, n), fps)
plt.axis([0,4,-0.1, 1.1])
pass
In [18]:
xs = []
for i, r in enumerate(np.linspace(0, 4, 400)):
x = 0.5
for j in range(10000):
x = f(x, r)
for j in range(50):
x = f(x, r)
xs.append((r, x))
xs = np.array(xs)
plt.scatter(xs[:,0], xs[:,1], s=1)
plt.axis([0,4,-0.1, 1.1])
pass
In [19]:
from scipy.optimize import root, fsolve
Suppose we want to solve a sysetm of $m$ equations with $n$ unknowns
\begin{align} f(x_0, x_1) &= x_1 - 3x_0(x_0+1)(x_0-1) \\ g(x_0, x_1) &= 0.25 x_0^2 + x_1^2 - 1 \end{align}Note that the equations are non-linear and there can be multiple solutions. These can be interpreted as fixed points of a system of differential equations.
In [20]:
def f(x):
return [x[1] - 3*x[0]*(x[0]+1)*(x[0]-1),
.25*x[0]**2 + x[1]**2 - 1]
In [21]:
sol = root(f, (0.5, 0.5))
sol.x
Out[21]:
In [22]:
fsolve(f, (0.5, 0.5))
Out[22]:
In [23]:
r0 = opt.root(f,[1,1])
r1 = opt.root(f,[0,1])
r2 = opt.root(f,[-1,1.1])
r3 = opt.root(f,[-1,-1])
r4 = opt.root(f,[2,-0.5])
roots = np.c_[r0.x, r1.x, r2.x, r3.x, r4.x]
In [24]:
Y, X = np.mgrid[-3:3:100j, -3:3:100j]
U = Y - 3*X*(X + 1)*(X-1)
V = .25*X**2 + Y**2 - 1
plt.streamplot(X, Y, U, V, color=U, linewidth=2, cmap=plt.cm.autumn)
plt.scatter(roots[0], roots[1], s=50, c='none', edgecolors='k', linewidth=2)
pass
In [25]:
def jac(x):
return [[-6*x[0], 1], [0.5*x[0], 2*x[1]]]
In [26]:
sol = root(f, (0.5, 0.5), jac=jac)
sol.x, sol.fun
Out[26]:
In [27]:
np.allclose(f(sol.x), 0)
Out[27]:
In [28]:
sol = root(f, (12,12))
sol.x
Out[28]:
In [29]:
np.allclose(f(sol.x), 0)
Out[29]:
Optimization Primer
We will assume that our optimization problem is to minimize some univariate or multivariate function $f(x)$. This is without loss of generality, since to find the maximum, we can simply minimize $-f(x)$. We will also assume that we are dealing with multivariate or real-valued smooth functions - non-smooth or discrete functions (e.g. integer-valued) are outside the scope of this course.
To find the minimum of a function, we first need to be able to express the function as a mathematical expresssion. For example, in least squares regression, the function that we are optimizing is of the form $y_i - f(x_i, \theta)$ for some parameter(s) $\theta$. To choose an appropirate optimization algorithm, we should at least answer these two questions if possible:
- Is the function convex?
- Are there any constraints that the solution must meet?
Finally, we need to realize that optimization methods are nearly always designed to find local optima. For convex problems, there is only one minimum and so this is not a problem. However, if there are multiple local minima, often heuristics such as multiple random starts must be adopted to find a "good" enough solution.
Is the function convex?
Convex functions are very nice because they have a single global minimum, and there are very efficient algorithms for solving large convex systems.
Intuitively, a function is convex if every chord joining two points on the function lies above the function. More formally, a function is convex if $$ f(ta + (1-t)b) \lt tf(a) + (1-t)f(b) $$ for some $t$ between 0 and 1 - this is shown in the figure below.
In [30]:
def f(x):
return (x-4)**2 + x + 1
with plt.xkcd():
x = np.linspace(0, 10, 100)
plt.plot(x, f(x))
ymin, ymax = plt.ylim()
plt.axvline(2, ymin, f(2)/ymax, c='red')
plt.axvline(8, ymin, f(8)/ymax, c='red')
plt.scatter([4, 4], [f(4), f(2) + ((4-2)/(8-2.))*(f(8)-f(2))],
edgecolor=['blue', 'red'], facecolor='none', s=100, linewidth=2)
plt.plot([2,8], [f(2), f(8)])
plt.xticks([2,4,8], ('a', 'ta + (1-t)b', 'b'), fontsize=20)
plt.text(0.2, 40, 'f(ta + (1-t)b) < tf(a) + (1-t)f(b)', fontsize=20)
plt.xlim([0,10])
plt.yticks([])
plt.suptitle('Convex function', fontsize=20)
Checking if a function is convex using the Hessian
The formal definition is only useful for checking if a function is convex if you can find a counter-example. More practically, a twice differentiable function is convex if its Hessian is positive semi-definite, and strictly convex if the Hessian is positive definite.
For example, suppose we want to minimize the scalar-valued function
$$ f(x_1, x_2, x_3) = x_1^2 + 2x_2^2 + 3x_3^2 + 2x_1x_2 + 2x_1x_3 $$
In [31]:
from sympy import symbols, hessian, Function, init_printing, expand, Matrix, diff
x, y, z = symbols('x y z')
f = symbols('f', cls=Function)
init_printing()
In [32]:
f = x**2 + 2*y**2 + 3*z**2 + 2*x*y + 2*x*z
f
Out[32]:
In [33]:
H = hessian(f, (x, y, z))
H
Out[33]:
In [34]:
np.real_if_close(la.eigvals(np.array(H).astype('float')))
Out[34]:
Since the matrix is symmetric and all eigenvalues are positive, the Hessian is positive defintie and the function is convex.
Combining convex functions
The following rules may be useful to determine if more complex functions are convex:
- The intersection of convex functions is convex
- If the functions $f$ and $g$ are convex and $a \ge 0$ and $b \ge 0$ then the function $af + bg$ is convex.
- If the function $U$ is convex and the function $g$ is nondecreasing and convex then the function $f$ defined by $f (x) = g(U(x))$ is convex.
Many more technical details about convexity and convex optimization can be found in this book.
Are there any constraints that the solution must meet?
In general, optimization without constraints is easier to perform than optimization in the presence of constraints. The solutions may be very different in the presence or absence of constraints, so it is important to know if there are any constraints.
We will see some examples of two general strategies:
- convert a problem with constraints into one without constraints or
- use an algorithm that can optimize with constraints.
In [35]:
from scipy import optimize as opt
In [36]:
def f(x):
return x**4 + 3*(x-2)**3 - 15*(x)**2 + 1
In [37]:
x = np.linspace(-8, 5, 100)
plt.plot(x, f(x));
The minimize_scalar function will find the minimum, and can also be told to search within given bounds. By default, it uses the Brent algorithm, which combines a bracketing strategy with a parabolic approximation.
In [38]:
opt.minimize_scalar(f, method='Brent')
Out[38]:
In [39]:
opt.minimize_scalar(f, method='bounded', bounds=[0, 6])
Out[39]:
In [40]:
def f(x, offset):
return -np.sinc(x-offset)
In [41]:
x = np.linspace(-20, 20, 100)
plt.plot(x, f(x, 5));
In [42]:
# note how additional function arguments are passed in
sol = opt.minimize_scalar(f, args=(5,))
sol
Out[42]:
In [43]:
plt.plot(x, f(x, 5))
plt.axvline(sol.x, c='red')
pass
In [44]:
lower = np.random.uniform(-20, 20, 100)
upper = lower + 1
sols = [opt.minimize_scalar(f, args=(5,), bracket=(l, u)) for (l, u) in zip(lower, upper)]
In [45]:
idx = np.argmin([sol.fun for sol in sols])
sol = sols[idx]
In [46]:
plt.plot(x, f(x, 5))
plt.axvline(sol.x, c='red');
Using a stochastic algorithm
See documentation for the basinhopping algorithm, which also works with multivariate scalar optimization. Note that this is heuristic and not guaranteed to find a global minimum.
In [47]:
from scipy.optimize import basinhopping
x0 = 0
sol = basinhopping(f, x0, stepsize=1, minimizer_kwargs={'args': (5,)})
sol
Out[47]:
In [48]:
plt.plot(x, f(x, 5))
plt.axvline(sol.x, c='red');
Minimizing a multivariate function $f: \mathbb{R}^n \rightarrow \mathbb{R}$
We will next move on to optimization of multivariate scalar functions, where the scalar may (say) be the norm of a vector. Minimizing a multivariable set of equations $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is not well-defined, but we will later see how to solve the closely related problem of finding roots or fixed points of such a set of equations.
We will use the Rosenbrock "banana" function to illustrate unconstrained multivariate optimization. In 2D, this is $$ f(x, y) = b(y - x^2)^2 + (a - x)^2 $$
The function has a global minimum at (1,1) and the standard expression takes $a = 1$ and $b = 100$.
Conditioning of optimization problem
With these values for $a$ and $b$, the problem is ill-conditioned. As we shall see, one of the factors affecting the ease of optimization is the condition number of the curvature (Hessian). When the condition number is high, the gradient may not point in the direction of the minimum, and simple gradient descent methods may be inefficient since they may be forced to take many sharp turns.
In [49]:
from sympy import symbols, hessian, Function, N
x, y = symbols('x y')
f = symbols('f', cls=Function)
f = 100*(y - x**2)**2 + (1 - x)**2
H = hessian(f, [x, y]).subs([(x,1), (y,1)])
H, N(H.condition_number())
Out[49]:
In [50]:
import scipy.linalg as la
mu = la.eigvals(np.array([802, -400, -400, 200]).reshape((2,2)))
np.real_if_close(mu[0]/mu[1])
Out[50]:
In [51]:
def rosen(x):
"""Generalized n-dimensional version of the Rosenbrock function"""
return sum(100*(x[1:]-x[:-1]**2.0)**2.0 +(1-x[:-1])**2.0)
In [52]:
x = np.linspace(-5, 5, 100)
y = np.linspace(-5, 5, 100)
X, Y = np.meshgrid(x, y)
Z = rosen(np.vstack([X.ravel(), Y.ravel()])).reshape((100,100))
In [53]:
# Note: the global minimum is at (1,1) in a tiny contour island
plt.contour(X, Y, Z, np.arange(10)**5, cmap='jet')
plt.text(1, 1, 'x', va='center', ha='center', color='red', fontsize=20);
In [54]:
x = np.linspace(0, 2, 100)
y = np.linspace(0, 2, 100)
X, Y = np.meshgrid(x, y)
Z = rosen(np.vstack([X.ravel(), Y.ravel()])).reshape((100,100))
In [55]:
plt.contour(X, Y, Z, [rosen(np.array([k, k])) for k in np.linspace(1, 1.5, 10)], cmap='jet')
plt.text(1, 1, 'x', va='center', ha='center', color='red', fontsize=20);
Gradient descent
The gradient (or Jacobian) at a point indicates the direction of steepest ascent. Since we are looking for a minimum, one obvious possibility is to take a step in the opposite direction to the gradient. We weight the size of the step by a factor $\alpha$ known in the machine learning literature as the learning rate. If $\alpha$ is small, the algorithm will eventually converge towards a local minimum, but it may take long time. If $\alpha$ is large, the algorithm may converge faster, but it may also overshoot and never find the minimum. Gradient descent is also known as a first order method because it requires calculation of the first derivative at each iteration.
Some algorithms also determine the appropriate value of $\alpha$ at each stage by using a line search, i.e., $$ \alpha^* = \arg\min_\alpha f(x_k - \alpha \nabla{f(x_k)}) $$ which is a 1D optimization problem.
As suggested above, the problem is that the gradient may not point towards the global minimum especially when the condition number is large, and we are forced to use a small $\alpha$ for convergence.
In [56]:
def f(x):
return x[0]**2 + x[1]**2
def grad(x):
return np.array([2*x[0], 2*x[1]])
a = 0.1 # learning rate
x0 = np.array([1.0,1.0])
print('Start', x0)
for i in range(41):
x0 -= a * grad(x0)
if i%5 == 0:
print(i, x0)
Gradient descent for least squares minimization
Usually, when we optimize, we are not just finding the minimum, but also want to know the parameters that give us the minimum. As a simple example, suppose we want to find parameters that minimize the least squares difference between a linear model and some data. Suppose we have some data $(0,1), (1,2), (2,3), (3,3.5), (4,6), (5,9), (6,8)$ and want to find a line $y = \beta_0 +\beta_1 x$ that is the best least squares fit. One way to do this is to solve $X^TX\hat{\beta} = X^Ty$, but we want to show how this can be formulated as a gradient descent problem.
We want to find $\beta = (\beta_0, \beta_1)$ that minimize the squared differences
$$ r = \sum(\beta_0 + \beta_1 x - y)^2 $$We calculate the gradient with respect to $\beta$ as
$$\nabla r = \pmatrix{ \frac{\delta r}{\delta \beta_0} \\ \frac{\delta r}{\delta \beta_0}}$$and apply gradient descent.
In [57]:
def f(x, y, b):
"""Helper function."""
return (b[0] + b[1]*x - y)
def grad(x, y, b):
"""Gradient of objective function with respect to parameters b."""
n = len(x)
return np.array([
sum(f(x, y, b)),
sum(x*f(x, y, b))
])
In [58]:
x, y = map(np.array, zip((0,1), (1,2), (2,3), (3,3.5), (4,6), (5,9), (6,8)))
In [59]:
a = 0.001 # learning rate
b = np.zeros(2)
for i in range(10000):
b -= a * grad(x, y, b)
b
Out[59]:
In [60]:
plt.scatter(x, y, s=30)
plt.plot(x, b[0] + b[1]*x, color='red')
pass
Gradient descent to minimize the Rosen function using scipy.optimize
Because gradient descent is unreliable in practice, it is not part of the scipy optimize suite of functions, but we will write a custom function below to illustrate how to use gradient descent while maintaining the scipy.optimize interface.
In [61]:
def rosen_der(x):
"""Derivative of generalized Rosen function."""
xm = x[1:-1]
xm_m1 = x[:-2]
xm_p1 = x[2:]
der = np.zeros_like(x)
der[1:-1] = 200*(xm-xm_m1**2) - 400*(xm_p1 - xm**2)*xm - 2*(1-xm)
der[0] = -400*x[0]*(x[1]-x[0]**2) - 2*(1-x[0])
der[-1] = 200*(x[-1]-x[-2]**2)
return der
Warning One of the most common causes of failure of optimization is because the gradient or Hessian function is specified incorrectly. You can check for this using check_grad which compares the analytical gradient with one calculated using finite differences.
In [62]:
from scipy.optimize import check_grad
for x in np.random.uniform(-2,2,(10,2)):
print(x, check_grad(rosen, rosen_der, x))
In [63]:
def custmin(fun, x0, args=(), maxfev=None, alpha=0.0002,
maxiter=100000, tol=1e-10, callback=None, **options):
"""Implements simple gradient descent for the Rosen function."""
bestx = x0
bestf = fun(x0)
funcalls = 1
niter = 0
improved = True
stop = False
while improved and not stop and niter < maxiter:
niter += 1
# the next 2 lines are gradient descent
step = alpha * rosen_der(bestx)
bestx = bestx - step
bestf = fun(bestx)
funcalls += 1
if la.norm(step) < tol:
improved = False
if callback is not None:
callback(bestx)
if maxfev is not None and funcalls >= maxfev:
stop = True
break
return opt.OptimizeResult(fun=bestf, x=bestx, nit=niter,
nfev=funcalls, success=(niter > 1))
In [64]:
def reporter(p):
"""Reporter function to capture intermediate states of optimization."""
global ps
ps.append(p)
In [65]:
# Initial starting position
x0 = np.array([4,-4.1])
In [66]:
ps = [x0]
opt.minimize(rosen, x0, method=custmin, callback=reporter)
Out[66]:
In [67]:
x = np.linspace(-5, 5, 100)
y = np.linspace(-5, 5, 100)
X, Y = np.meshgrid(x, y)
Z = rosen(np.vstack([X.ravel(), Y.ravel()])).reshape((100,100))
In [68]:
ps = np.array(ps)
plt.figure(figsize=(12,4))
plt.subplot(121)
plt.contour(X, Y, Z, np.arange(10)**5, cmap='jet')
plt.plot(ps[:, 0], ps[:, 1], '-ro')
plt.subplot(122)
plt.semilogy(range(len(ps)), rosen(ps.T));
Newton's method and variants
Recall Newton's method for finding roots of a univariate function
$$ x_{K+1} = x_k - \frac{f(x_k)}{f'(x_k)} $$When we are looking for a minimum, we are looking for the roots of the derivative $f'(x)$, so
$$ x_{K+1} = x_k - \frac{f'(x_k}{f''(x_k)} $$Newton's method can also be seen as a Taylor series approximation
$$ f(x+h) = f(x) + h f'(x) + \frac{h^2}{2}f''(x) $$At the function minimum, the derivative is 0, so \begin{align} \frac{f(x+h) - f(x)}{h} &= f'(x) + \frac{h}{2}f''(x) \\ 0 &= f'(x) + \frac{h}{2}f''(x) \end{align}
and letting $\Delta x = \frac{h}{2}$, we get that the Newton step is
$$ \Delta x = - \frac{f'(x)}{f''(x)} $$The multivariate analog replaces $f'$ with the Jacobian and $f''$ with the Hessian, so the Newton step is
$$ \Delta x = -H^{-1}(x) \nabla f(x) $$Second order methods
Second order methods solve for $H^{-1}$ and so require calculation of the Hessian (either provided or approximated using finite differences). For efficiency reasons, the Hessian is not directly inverted, but solved for using a variety of methods such as conjugate gradient. An example of a second order method in the optimize package is Newton-GC.
In [69]:
from scipy.optimize import rosen, rosen_der, rosen_hess
In [70]:
ps = [x0]
opt.minimize(rosen, x0, method='Newton-CG', jac=rosen_der, hess=rosen_hess, callback=reporter)
Out[70]:
In [71]:
ps = np.array(ps)
plt.figure(figsize=(12,4))
plt.subplot(121)
plt.contour(X, Y, Z, np.arange(10)**5, cmap='jet')
plt.plot(ps[:, 0], ps[:, 1], '-ro')
plt.subplot(122)
plt.semilogy(range(len(ps)), rosen(ps.T));
First order methods
As calculating the Hessian is computationally expensive, sometimes first order methods that only use the first derivatives are preferred. Quasi-Newton methods use functions of the first derivatives to approximate the inverse Hessian. A well know example of the Quasi-Newoton class of algorithjms is BFGS, named after the initials of the creators. As usual, the first derivatives can either be provided via the jac= argument or approximated by finite difference methods.
In [72]:
ps = [x0]
opt.minimize(rosen, x0, method='BFGS', callback=reporter)
Out[72]:
In [73]:
ps = np.array(ps)
plt.figure(figsize=(12,4))
plt.subplot(121)
plt.contour(X, Y, Z, np.arange(10)**5, cmap='jet')
plt.plot(ps[:, 0], ps[:, 1], '-ro')
plt.subplot(122)
plt.semilogy(range(len(ps)), rosen(ps.T));
In [74]:
ps = [x0]
opt.minimize(rosen, x0, method='nelder-mead', callback=reporter)
Out[74]:
In [75]:
ps = np.array(ps)
plt.figure(figsize=(12,4))
plt.subplot(121)
plt.contour(X, Y, Z, np.arange(10)**5, cmap='jet')
plt.plot(ps[:, 0], ps[:, 1], '-ro')
plt.subplot(122)
plt.semilogy(range(len(ps)), rosen(ps.T));
Lagrange multipliers and constrained optimization
Recall why Lagrange multipliers are useful for constrained optimization - a stationary point must be where the constraint surface $g$ touches a level set of the function $f$ (since the value of $f$ does not change on a level set). At that point, $f$ and $g$ are parallel, and hence their gradients are also parallel (since the gradient is normal to the level set). So we want to solve
$$\nabla f = -\lambda \nabla g$$or equivalently,
$$\nabla f + \lambda \nabla g = 0$$Maximize $f (x, y, z) = xy + yz$ subject to the constraints $x + 2y = 6$ and $x − 3z = 0$.
We set up the equations
$$ F (x, y, z, λ, μ) = xy + yz − λ(x + 2y − 6) − μ(x − 3z) $$Now set partial derivatives to zero and solve the following set of equations
\begin{align} y - \lambda - \mu &= 0 \\ x + z - 2\lambda &= 0 \\ y +3\mu &= 0 \\ x + 2y - 6 &= 0 \\ x - 3z &= 0 \end{align}which is a linear equation in $x, y, z, \lambda, \mu$
$$ \begin{pmatrix} 0 & 1 & 0 & -1 & -1 \\ 1 & 0 & 1 & -2 & 0 \\ 0 & 1 & 0 & 0 & 3 \\ 1 & 2 & 0 & 0 & 0 \\ 1 & 0 & -3 & 0 & 0 \\ \end{pmatrix}\pmatrix{x \\ y \\ z \\ \lambda \\ \mu} = \pmatrix{0 \\ 0 \\ 0 \\ 6 \\ 0} $$
In [76]:
A = np.array([
[0, 1, 0, -1, -1],
[1, 0, 1, -2, 0],
[0, 1, 0, 0, 3],
[1, 2, 0, 0, 0],
[1, 0,-3, 0, 0]])
b = np.array([0,0,0,6,0])
sol = la.solve(A, b)
In [77]:
sol
Out[77]:
In [78]:
def f(x, y, z):
return x*y + y*z
In [79]:
f(*sol[:3])
Out[79]:
In [80]:
# Convert to minimization problem by negating function
def f(x):
return -(x[0]*x[1] + x[1]*x[2])
In [81]:
cons = ({'type': 'eq',
'fun' : lambda x: np.array([x[0] + 2*x[1] - 6, x[0] - 3*x[2]])})
In [82]:
x0 = np.array([2,2,0.67])
cx = opt.minimize(f, x0, constraints=cons)
cx
Out[82]:
Another example of constrained optimization
Many real-world optimization problems have constraints - for example, a set of parameters may have to sum to 1.0 (equality constraint), or some parameters may have to be non-negative (inequality constraint). Sometimes, the constraints can be incorporated into the function to be minimized, for example, the non-negativity constraint $p \gt 0$ can be removed by substituting $p = e^q$ and optimizing for $q$. Using such workarounds, it may be possible to convert a constrained optimization problem into an unconstrained one, and use the methods discussed above to solve the problem.
Alternatively, we can use optimization methods that allow the specification of constraints directly in the problem statement as shown in this section. Internally, constraint violation penalties, barriers and Lagrange multipliers are some of the methods used used to handle these constraints. We use the example provided in the Scipy tutorial to illustrate how to set constraints.
We will optimize:
$$ f(x) = -(2xy + 2x - x^2 -2y^2) $$subject to the constraint $$ x^3 - y = 0 \\ y - (x-1)^4 - 2 \ge 0 $$ and the bounds $$ 0.5 \le x \le 1.5 \\ 1.5 \le y \le 2.5 $$
In [83]:
def f(x):
return -(2*x[0]*x[1] + 2*x[0] - x[0]**2 - 2*x[1]**2)
In [84]:
x = np.linspace(0, 3, 100)
y = np.linspace(0, 3, 100)
X, Y = np.meshgrid(x, y)
Z = f(np.vstack([X.ravel(), Y.ravel()])).reshape((100,100))
plt.contour(X, Y, Z, np.arange(-1.99,10, 1), cmap='jet');
plt.plot(x, x**3, 'k:', linewidth=1)
plt.plot(x, (x-1)**4+2, 'k:', linewidth=1)
plt.fill([0.5,0.5,1.5,1.5], [2.5,1.5,1.5,2.5], alpha=0.3)
plt.axis([0,3,0,3])
Out[84]:
To set constraints, we pass in a dictionary with keys type, fun and jac. Note that the inequality constraint assumes a $C_j x \ge 0$ form. As usual, the jac is optional and will be numerically estimated if not provided.
In [85]:
cons = ({'type': 'eq',
'fun' : lambda x: np.array([x[0]**3 - x[1]]),
'jac' : lambda x: np.array([3.0*(x[0]**2.0), -1.0])},
{'type': 'ineq',
'fun' : lambda x: np.array([x[1] - (x[0]-1)**4 - 2])})
bnds = ((0.5, 1.5), (1.5, 2.5))
In [86]:
x0 = [0, 2.5]
Unconstrained optimization
In [87]:
ux = opt.minimize(f, x0, constraints=None)
ux
Out[87]:
Constrained optimization
In [88]:
cx = opt.minimize(f, x0, bounds=bnds, constraints=cons)
cx
Out[88]:
In [89]:
x = np.linspace(0, 3, 100)
y = np.linspace(0, 3, 100)
X, Y = np.meshgrid(x, y)
Z = f(np.vstack([X.ravel(), Y.ravel()])).reshape((100,100))
plt.contour(X, Y, Z, np.arange(-1.99,10, 1), cmap='jet');
plt.plot(x, x**3, 'k:', linewidth=1)
plt.plot(x, (x-1)**4+2, 'k:', linewidth=1)
plt.text(ux['x'][0], ux['x'][1], 'x', va='center', ha='center', size=20, color='blue')
plt.text(cx['x'][0], cx['x'][1], 'x', va='center', ha='center', size=20, color='red')
plt.fill([0.5,0.5,1.5,1.5], [2.5,1.5,1.5,2.5], alpha=0.3)
plt.axis([0,3,0,3]);
Curve fitting
Sometimes, we simply want to use non-linear least squares to fit a function to data, perhaps to estimate parameters for a mechanistic or phenomenological model. The curve_fit function uses the quasi-Newton Levenberg-Marquadt algorithm to perform such fits. Behind the scenes, curve_fit is just a wrapper around the leastsq function that does nonlinear least squares fitting.
In [90]:
from scipy.optimize import curve_fit
In [91]:
def logistic4(x, a, b, c, d):
"""The four paramter logistic function is often used to fit dose-response relationships."""
return ((a-d)/(1.0+((x/c)**b))) + d
In [92]:
nobs = 24
xdata = np.linspace(0.5, 3.5, nobs)
ptrue = [10, 3, 1.5, 12]
ydata = logistic4(xdata, *ptrue) + 0.5*np.random.random(nobs)
In [93]:
popt, pcov = curve_fit(logistic4, xdata, ydata)
In [94]:
perr = yerr=np.sqrt(np.diag(pcov))
print('Param\tTrue\tEstim (+/- 1 SD)')
for p, pt, po, pe in zip('abcd', ptrue, popt, perr):
print('%s\t%5.2f\t%5.2f (+/-%5.2f)' % (p, pt, po, pe))
In [95]:
x = np.linspace(0, 4, 100)
y = logistic4(x, *popt)
plt.plot(xdata, ydata, 'o')
plt.plot(x, y);
Finding paraemeters for ODE models
This is a specialized application of curve_fit, in which the curve to be fitted is defined implicitly by an ordinary differential equation
$$
\frac{dx}{dt} = -kx
$$
and we want to use observed data to estimate the parameters $k$ and the initial value $x_0$. Of course this can be explicitly solved but the same approach can be used to find multiple parameters for $n$-dimensional systems of ODEs.
A more elaborate example for fitting a system of ODEs to model the zombie apocalypse
In [96]:
from scipy.integrate import odeint
def f(x, t, k):
"""Simple exponential decay."""
return -k*x
def x(t, k, x0):
"""
Solution to the ODE x'(t) = f(t,x,k) with initial condition x(0) = x0
"""
x = odeint(f, x0, t, args=(k,))
return x.ravel()
In [97]:
# True parameter values
x0_ = 10
k_ = 0.1*np.pi
# Some random data genererated from closed form solution plus Gaussian noise
ts = np.sort(np.random.uniform(0, 10, 200))
xs = x0_*np.exp(-k_*ts) + np.random.normal(0,0.1,200)
popt, cov = curve_fit(x, ts, xs)
k_opt, x0_opt = popt
print("k = %g" % k_opt)
print("x0 = %g" % x0_opt)
In [98]:
import matplotlib.pyplot as plt
t = np.linspace(0, 10, 100)
plt.plot(ts, xs, 'r.', t, x(t, k_opt, x0_opt), '-');
Another example of fitting a system of ODEs using the lmfit package
You may have to install the lmfit package using pip and restart your kernel. The lmfit algorithm is another wrapper around scipy.optimize.leastsq but allows for richer model specification and more diagnostics.
In [99]:
! pip install lmfit
In [100]:
from lmfit import minimize, Parameters, Parameter, report_fit
import warnings
In [101]:
def f(xs, t, ps):
"""Lotka-Volterra predator-prey model."""
try:
a = ps['a'].value
b = ps['b'].value
c = ps['c'].value
d = ps['d'].value
except:
a, b, c, d = ps
x, y = xs
return [a*x - b*x*y, c*x*y - d*y]
def g(t, x0, ps):
"""
Solution to the ODE x'(t) = f(t,x,k) with initial condition x(0) = x0
"""
x = odeint(f, x0, t, args=(ps,))
return x
def residual(ps, ts, data):
x0 = ps['x0'].value, ps['y0'].value
model = g(ts, x0, ps)
return (model - data).ravel()
t = np.linspace(0, 10, 100)
x0 = np.array([1,1])
a, b, c, d = 3,1,1,1
true_params = np.array((a, b, c, d))
np.random.seed(123)
data = g(t, x0, true_params)
data += np.random.normal(size=data.shape)
# set parameters incluing bounds
params = Parameters()
params.add('x0', value= float(data[0, 0]), min=0, max=10)
params.add('y0', value=float(data[0, 1]), min=0, max=10)
params.add('a', value=2.0, min=0, max=10)
params.add('b', value=2.0, min=0, max=10)
params.add('c', value=2.0, min=0, max=10)
params.add('d', value=2.0, min=0, max=10)
# fit model and find predicted values
result = minimize(residual, params, args=(t, data), method='leastsq')
final = data + result.residual.reshape(data.shape)
# plot data and fitted curves
plt.plot(t, data, 'o')
plt.plot(t, final, '-', linewidth=2);
# display fitted statistics
report_fit(result)
Optimization of graph node placement
To show the many different applications of optimization, here is an example using optimization to change the layout of nodes of a graph. We use a physical analogy - nodes are connected by springs, and the springs resist deformation from their natural length $l_{ij}$. Some nodes are pinned to their initial locations while others are free to move. Because the initial configuration of nodes does not have springs at their natural length, there is tension resulting in a high potential energy $U$, given by the physics formula shown below. Optimization finds the configuration of lowest potential energy given that some nodes are fixed (set up as boundary constraints on the positions of the nodes).
$$ U = \frac{1}{2}\sum_{i,j=1}^n ka_{ij}\left(||p_i - p_j||-l_{ij}\right)^2 $$Note that the ordination algorithm Multi-Dimensional Scaling (MDS) works on a very similar idea - take a high dimensional data set in $\mathbb{R}^n$, and project down to a lower dimension ($\mathbb{R}^k$) such that the sum of distances $d_n(x_i, x_j) - d_k(x_i, x_j)$, where $d_n$ and $d_k$ are some measure of distance between two points $x_i$ and $x_j$ in $n$ and $d$ dimension respectively, is minimized. MDS is often used in exploratory analysis of high-dimensional data to get some intuitive understanding of its "structure".
In [102]:
from scipy.spatial.distance import pdist, squareform
- P0 is the initial location of nodes
- P is the minimal energy location of nodes given constraints
- A is a connectivity matrix - there is a spring between $i$ and $j$ if $A_{ij} = 1$
- $L_{ij}$ is the resting length of the spring connecting $i$ and $j$
- In addition, there are a number of
fixednodes whose positions are pinned.
In [103]:
n = 20
k = 1 # spring stiffness
P0 = np.random.uniform(0, 5, (n,2))
A = np.ones((n, n))
A[np.tril_indices_from(A)] = 0
L = A.copy()
In [104]:
L.astype('int')
Out[104]:
In [105]:
def energy(P):
P = P.reshape((-1, 2))
D = squareform(pdist(P))
return 0.5*(k * A * (D - L)**2).sum()
In [106]:
D0 = squareform(pdist(P0))
E0 = 0.5* k * A * (D0 - L)**2
In [107]:
D0[:5, :5]
Out[107]:
In [108]:
E0[:5, :5]
Out[108]:
In [109]:
energy(P0.ravel())
Out[109]:
In [110]:
# fix the position of the first few nodes just to show constraints
fixed = 4
bounds = (np.repeat(P0[:fixed,:].ravel(), 2).reshape((-1,2)).tolist() +
[[None, None]] * (2*(n-fixed)))
bounds[:fixed*2+4]
Out[110]:
In [111]:
sol = opt.minimize(energy, P0.ravel(), bounds=bounds)
In [112]:
plt.scatter(P0[:, 0], P0[:, 1], s=25)
P = sol.x.reshape((-1,2))
plt.scatter(P[:, 0], P[:, 1], edgecolors='red', facecolors='none', s=30, linewidth=2);
Optimization of standard statistical models
When we solve standard statistical problems, an optimization procedure similar to the ones discussed here is performed. For example, consider multivariate logistic regression - typically, a Newton-like algorithm known as iteratively reweighted least squares (IRLS) is used to find the maximum likelihood estimate for the generalized linear model family. However, using one of the multivariate scalar minimization methods shown above will also work, for example, the BFGS minimization algorithm.
The take home message is that there is nothing magic going on when Python or R fits a statistical model using a formula - all that is happening is that the objective function is set to be the negative of the log likelihood, and the minimum found using some first or second order optimization algorithm.
In [113]:
import statsmodels.api as sm
Logistic regression as optimization
Suppose we have a binary outcome measure $Y \in {0,1}$ that is conditinal on some input variable (vector) $x \in (-\infty, +\infty)$. Let the conditioanl probability be $p(x) = P(Y=y | X=x)$. Given some data, one simple probability model is $p(x) = \beta_0 + x\cdot\beta$ - i.e. linear regression. This doesn't really work for the obvious reason that $p(x)$ must be between 0 and 1 as $x$ ranges across the real line. One simple way to fix this is to use the transformation $g(x) = \frac{p(x)}{1 - p(x)} = \beta_0 + x.\beta$. Solving for $p$, we get $$ p(x) = \frac{1}{1 + e^{-(\beta_0 + x\cdot\beta)}} $$ As you all know very well, this is logistic regression.
Suppose we have $n$ data points $(x_i, y_i)$ where $x_i$ is a vector of features and $y_i$ is an observed class (0 or 1). For each event, we either have "success" ($y = 1$) or "failure" ($Y = 0$), so the likelihood looks like the product of Bernoulli random variables. According to the logistic model, the probability of success is $p(x_i)$ if $y_i = 1$ and $1-p(x_i)$ if $y_i = 0$. So the likelihood is $$ L(\beta_0, \beta) = \prod_{i=1}^n p(x_i)^y(1-p(x_i))^{1-y} $$ and the log-likelihood is \begin{align} l(\beta_0, \beta) &= \sum_{i=1}^{n} y_i \log{p(x_i)} + (1-y_i)\log{1-p(x_i)} \\ &= \sum_{i=1}^{n} \log{1-p(x_i)} + \sum_{i=1}^{n} y_i \log{\frac{p(x_i)}{1-p(x_i)}} \\ &= \sum_{i=1}^{n} -\log 1 + e^{\beta_0 + x_i\cdot\beta} + \sum_{i=1}^{n} y_i(\beta_0 + x_i\cdot\beta) \end{align}
Using the standard 'trick', if we augment the matrix $X$ with a column of 1s, we can write $\beta_0 + x_i\cdot\beta$ as just $X\beta$.
In [114]:
df_ = pd.read_csv("http://www.ats.ucla.edu/stat/data/binary.csv")
df_.head()
Out[114]:
In [115]:
# We will ignore the rank categorical value
cols_to_keep = ['admit', 'gre', 'gpa']
df = df_[cols_to_keep]
df.insert(1, 'dummy', 1)
df.head()
Out[115]:
Solving as a GLM with IRLS
This is very similar to what you would do in R, only using Python's statsmodels package. The GLM solver uses a special variant of Newton's method known as iteratively reweighted least squares (IRLS), which will be further desribed in the lecture on multivarite and constrained optimizaiton.
In [116]:
model = sm.GLM.from_formula('admit ~ gre + gpa',
data=df, family=sm.families.Binomial())
fit = model.fit()
fit.summary()
Out[116]:
In [117]:
%load_ext rpy2.ipython
In [118]:
%%R -i df
m <- glm(admit ~ gre + gpa, data=df, family="binomial")
summary(m)
Home-brew logistic regression using a generic minimization function
This is to show that there is no magic going on - you can write the function to minimize directly from the log-likelihood equation and run a minimizer. It will be more accurate if you also provide the derivative (+/- the Hessian for second order methods), but using just the function and numerical approximations to the derivative will also work. As usual, this is for illustration so you understand what is going on - when there is a library function available, you should probably use that instead.
In [119]:
def f(beta, y, x):
"""Minus log likelihood function for logistic regression."""
return -((-np.log(1 + np.exp(np.dot(x, beta)))).sum() + (y*(np.dot(x, beta))).sum())
In [120]:
beta0 = np.zeros(3)
opt.minimize(f, beta0, args=(df['admit'], df.ix[:, 'dummy':]), method='BFGS', options={'gtol':1e-2})
Out[120]: