Numpy Lab: Linear Functions

Numpy Lab: Linear Functions#

Topics: elementwise operations

Linear functions
We will be using the term linear equation to mean a weighted sum of inputs plus an offset. If there is just one input $x$ , then this is a straight line:

Equation-1

(1)#

y = β + ω x,

where $β$ is the y-intercept of the linear and $ω$ is the slope of the line. When there are two inputs $x_{1}$ and $x_{2}$ , then this becomes:

Equation-2

(2)#

y = β + ω_{1} x_{1} + ω_{2} x_{2} .

Any other functions are by definition non-linear.

Type your code below#

# Define a linear function with just one input, x (Eq 1)
def linear_function_1D(x, beta, omega):

  return y

# Compute y using the function you filled in above

x = np.arange(0.0,10.0, 0.01)
beta = 0.0; omega = 1.0

y = linear_function_1D(x,beta,omega)

# Plot this function
fig, ax = plt.subplots()
ax.plot(x,y,'r-')
ax.set_ylim([0,10]);ax.set_xlim([0,10])
ax.set_xlabel('x'); ax.set_ylabel('y')
plt.show

# TODO -- experiment with changing the values of beta and omega
# to understand what they do.  Try to make a line
# that crosses the y-axis at y=10 and the x-axis at x=5

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[2], line 3
      1 # Compute y using the function you filled in above
----> 3 x = np.arange(0.0,10.0, 0.01)
      4 beta = 0.0; omega = 1.0
      6 y = linear_function_1D(x,beta,omega)

NameError: name 'np' is not defined

Now let’s investigate a 2D linear function#

# We will use this function to draw 2D contour plots. Nothing to do here
def draw_2D_function(x1_mesh, x2_mesh, y):

    fig, ax = plt.subplots()

    pos = ax.contourf(x1_mesh, x2_mesh, y, levels=256 ,cmap = 'hot', vmin=-10,vmax=10.0)
    fig.colorbar(pos, ax=ax)
    
    ax.contour(x1_mesh, 
               x2_mesh, 
               y, 
               levels= np.arange(-10,10,1.0), 
               cmap='winter')
    
    ax.set_xlabel('x1')
    ax.set_ylabel('x2')
    
    plt.show()

# Define a linear function with two inputs, x1 and x2
def linear_function_2D(x1,x2,beta,omega1,omega2):


  return y

Using numpy to write general linear functions in compact form#

Often we will want to compute many linear functions at the same time. For example, we might have three inputs, $x_{1}$ , $x_{2}$ , and $x_{3}$ and want to compute two linear functions giving $y_{1}$ and $y_{2}$ . Of course, we could do this by just running each equation separately,

(3)#

\begin{aligned} y_{1} & = & β_{1} + ω_{11} x_{1} + ω_{12} x_{2} + ω_{13} x_{3} \\ y_{2} & = & β_{2} + ω_{21} x_{1} + ω_{22} x_{2} + ω_{23} x_{3} . \end{aligned}

However, we can write it more compactly with vectors and matrices:

(4)#

[\begin{matrix} y_{1} \\ y_{2} \end{matrix}] = [\begin{matrix} β_{1} \\ β_{2} \end{matrix}] + [\begin{matrix} ω_{11} & ω_{12} & ω_{13} \\ ω_{21} & ω_{22} & ω_{23} \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}],

(5)#

y = β + Ω x .

Here, lowercase bold symbols are used for vectors. Upper case bold symbols are used for matrices.

# Define a linear function with three inputs, x1, x2, and x_3
def linear_function_3D(x1,x2,x3,beta,omega1,omega2,omega3):
  # TODO -- replace the code below with formula for a single 3D linear equation
  y = x1

  return y

# Define a linear function using vector input x=[x_1, x_2, x_3], and omega = [omega_1, omega_2, omega_3]
def linear_function_3D_la(x, beta, omega):
  # TODO -- replace the code below with formula for a single 3D linear equation
  y = x1

  return y

Numpy Lab: Linear Functions

Contents

Numpy Lab: Linear Functions#

Type your code below#

Now let’s investigate a 2D linear function#

Using numpy to write general linear functions in compact form#

Check that both functions agree#