Visualizing the tangent

The tangent to a curve is the straight line that touches the curve at a given point. One reason that tangents are so important is that they give the slopes of straight lines. So, if your curve represents a time series you can tell the ratio of change of your values just looking at the tangent.
Suppose that a curve is given as the graph of a function, y = f(x). We have that the slope in the point (a, f(a)) is equal to its derivative in a

and the equation of the tangent line can be stated as follows

With this in mind we can write a snippet of code which visualize the tangent of a curve:

from numpy import sin,linspace,power
from pylab import plot,show

def f(x): # sample function
 return x*sin(power(x,2))

# evaluation of the function
x = linspace(-2,4,150)
y = f(x)

a = 1.4
h = 0.1
fprime = (f(a+h)-f(a))/h # derivative
tan = f(a)+fprime*(x-a)  # tangent

# plot of the function and the tangent
plot(x,y,'b',a,f(a),'om',x,tan,'--r')
show()

The result is as follows:

The f is plotted with a blue curve and the tangent is the dashed line. Looking at the graph it is actually easy to observe that the tangent gives us a way to visualize the slope of a curve in a point. Let's see how this can help us in a practical example. Consider the fresh potatoes consumer price index between the years 1949 and 2006:

from numpy import arange
# Fresh potatoes: Annual Consumer price index, 1949-2006
# obatined at https://explore.data.gov/Agriculture/U-S-Potato-Statistics/cgk7-6ccj
price_index = [21.0,17.6,19.3,28.9,21.1,20.5,22.1,26.4,22.3,24.4,
24.6,28.0,24.7,24.9,25.7,31.6,39.1,31.3,31.3,32.1,34.4,38.0,36.7,
39.6,58.8,71.8,57.7,62.6,63.8,66.3,63.6,81.0,109.5,92.7,91.3,116.0,
101.5,96.1,116.0,119.1,153.5,162.6,144.6,141.5,154.6,174.3,174.7,
180.6,174.1,185.2,193.1,196.3,202.3,238.6,228.1,231.1,247.7,273.1]
t = np.arange(1949,2007)

From the calculus we have that the derivative is positive when f is increasing, it is negative when f is decreasing and zero when f has a saddle point. So, if we look at the tangent of the curve of the consumer price index in a certain year we have that it has a positive slope when the price index is increasing, a negative slope when the price are decreasing and it is constant when the trend is going to change. Using an interpolation of the data we loaded above we can plot the tangent in each year we want:

from scipy import interpolate

def draw_tangent(x,y,a):
 # interpolate the data with a spline
 spl = interpolate.splrep(x,y)
 small_t = arange(a-5,a+5)
 fa = interpolate.splev(a,spl,der=0)     # f(a)
 fprime = interpolate.splev(a,spl,der=1) # f'(a)
 tan = fa+fprime*(small_t-a) # tangent
 plot(a,fa,'om',small_t,tan,'--r')

draw_tangent(t,price_index,1991)
draw_tangent(t,price_index,1998)

plot(t,price_index,alpha=0.5)
show()

The graph shows the data contained in the array price_index and shows the tangent of the curve for the years 1991 and 1998. Using the tangent, this graph gives an emphasis about the fact that the price index is decreasing during the years around 1991 and increasing around 1998.

Find out more about derivative approximation in the post Finite differences with Toeplitz matrix.

Finite differences with Toeplitz matrix

A Toeplitz matrix is a band matrix in which each descending diagonal from left to right is constant. In this post we will see how to approximate the derivative of a function f(x) as matrix-vector products between a Toeplitz matrix and a vector of equally spaced values of f. Let's see how to generate the matrices we need using the function toeplitz(...) provided by numpy:

from numpy import *
from scipy.linalg import toeplitz
import pylab

def forward(size):
 """ returns a toeplitz matrix
   for forward differences
 """
 r = zeros(size)
 c = zeros(size)
 r[0] = -1
 r[size-1] = 1
 c[1] = 1
 return toeplitz(r,c)

def backward(size):
 """ returns a toeplitz matrix
   for backward differences
 """
 r = zeros(size)
 c = zeros(size)
 r[0] = 1
 r[size-1] = -1
 c[1] = -1
 return toeplitz(r,c).T

def central(size):
 """ returns a toeplitz matrix
   for central differences
 """
 r = zeros(size)
 c = zeros(size)
 r[1] = .5
 r[size-1] = -.5
 c[1] = -.5
 c[size-1] = .5
 return toeplitz(r,c).T

# testing the functions printing some 4-by-4 matrices
print 'Forward matrix'
print forward(4)
print 'Backward matrix'
print backward(4)
print 'Central matrix'
print central(4)

The result of the test above is as follows:

Forward matrix
[[-1.  1.  0.  0.]
 [ 0. -1.  1.  0.]
 [ 0.  0. -1.  1.]
 [ 1.  0.  0. -1.]]
Backward matrix
[[ 1.  0.  0. -1.]
 [-1.  1.  0.  0.]
 [ 0. -1.  1.  0.]
 [ 0.  0. -1.  1.]]
Central matrix
[[ 0.   0.5  0.  -0.5]
 [-0.5  0.   0.5  0. ]
 [ 0.  -0.5  0.   0.5]
 [ 0.5  0.  -0.5  0. ]]

We can observe that the matrix-vector product between those matrices and the vector of equally spaced values of f(x) implements, respectively, the following equations:

Forward difference,

Backward difference,

And central difference,

where h is the step size between the samples. Those equations are called Finite Differences and they give us an approximate derivative of f. So, let's approximate some derivatives!

x = linspace(0,10,15)
y = cos(x) # recall, the derivative of cos(x) is sin(x)
# we need the step h to compute f'(x) 
# because the product gives h*f'(x)
h = x[1]-x[2]
# generating the matrices
Tf = forward(15)/h 
Tb = backward(15)/h
Tc = central(15)/h

pylab.subplot(211)
# approximation and plotting
pylab.plot(x,dot(Tf,y),'g',x,dot(Tb,y),'r',x,dot(Tc,y),'m')
pylab.plot(x,sin(x),'b--',linewidth=3)
pylab.axis([0,10,-1,1])

# the same experiment with more samples (h is smaller)
x = linspace(0,10,50)
y = cos(x)
h = x[1]-x[2]
Tf = forward(50)/h
Tb = backward(50)/h
Tc = central(50)/h

pylab.subplot(212)
pylab.plot(x,dot(Tf,y),'g',x,dot(Tb,y),'r',x,dot(Tc,y),'m')
pylab.plot(x,sin(x),'b--',linewidth=3)
pylab.axis([0,10,-1,1])
pylab.legend(['Forward', 'Backward', 'Central', 'True f prime'],loc=4)
pylab.show()

The resulting plot would appear as follows:

As the theory suggests, the approximation is better when h is smaller and the central differences are more accurate (note that, they have a higher order of accuracy respect to the backward and forward ones).