Distance Metrics¶

Content:

• Various ways to initialize an equation
• Various ways to compare two equations

Installation¶

The Equation Tree package is available on pyPI:

In [ ]:

!pip install equation-tree

!pip install equation-tree

Initialization of Equations¶

Sampling¶

In the previous tutorial, we learned how to sample an equation:

In [ ]:

# Imports
import numpy as np
from equation_tree import sample

# Set a seed for reproducible results
np.random.seed(42)

# Initialization
sampled_equation = sample()[0]
sampled_equation

# Imports import numpy as np from equation_tree import sample # Set a seed for reproducible results np.random.seed(42) # Initialization sampled_equation = sample()[0] sampled_equation

From Sympy/String¶

We can also use sympy representation to initialize an equation:

In [ ]:

# Import
from equation_tree import EquationTree
from sympy import sympify

sympy_equation = sympify('x_1 * 2 + x_2 ** 3')
equation_from_sympy = EquationTree.from_sympy(sympy_equation)
equation_from_sympy

# Import from equation_tree import EquationTree from sympy import sympify sympy_equation = sympify('x_1 * 2 + x_2 ** 3') equation_from_sympy = EquationTree.from_sympy(sympy_equation) equation_from_sympy

From Prefix¶

Or, we can use prefix notation to initialize an equation:

In [ ]:

equation_from_prefix = EquationTree.from_prefix(['+', '*', 'x_1', '2', '**', 'x_2', '3'])
equation_from_prefix

equation_from_prefix = EquationTree.from_prefix(['+', '*', 'x_1', '2', '**', 'x_2', '3']) equation_from_prefix

!!!Note: This is feature can be used to manipulate equations:

In [ ]:

# Set a random seed for reproducible results
np.random.seed(42)

# Sample an initial equation
equation = sample()[0]
print('original:\t', equation)

# Use sympy to multiply the equation with a scalar and reinitialize it
new_sympy_equation = equation.sympy_expr * 3
equation_scaled = EquationTree.from_sympy(new_sympy_equation)
print('scaled:\t\t', equation_scaled)

# Use prefix notation to replace a function with another
new_prefix_notation = ['sin' if x == 'abs' else x for x in equation_scaled.prefix]
equation_replaced = EquationTree.from_prefix(new_prefix_notation)
print('replaced:\t', equation_replaced)

# Set a random seed for reproducible results np.random.seed(42) # Sample an initial equation equation = sample()[0] print('original:\t', equation) # Use sympy to multiply the equation with a scalar and reinitialize it new_sympy_equation = equation.sympy_expr * 3 equation_scaled = EquationTree.from_sympy(new_sympy_equation) print('scaled:\t\t', equation_scaled) # Use prefix notation to replace a function with another new_prefix_notation = ['sin' if x == 'abs' else x for x in equation_scaled.prefix] equation_replaced = EquationTree.from_prefix(new_prefix_notation) print('replaced:\t', equation_replaced)

Distance Metrics Between Equations¶

For more indepth information about the distances used, please read the respective documentation

Prediction Distance¶

First, we calculate the prediction distance. It is the difference between the values of the equations on a specified input sample.

In [ ]:

# Import
from equation_tree.metrics import prediction_distance

# First, we define our equations
sympy_square = sympify('x_1 ** 2')
equation_square = EquationTree.from_sympy(sympy_square)

sympy_cube = sympify('x_1 ** 3')
equation_cube = EquationTree.from_sympy(sympy_cube)

# For the prediction distance, we need to define a sample of input values on which the metric is evaluated:
sample_1 = {'x_1': [0, 1, 2]}
sample_2 = {'x_1': np.linspace(0,1)}
sample_3 = {'x_1': np.linspace(-10, 10)}

# Now, we can calculate the metrics:
print('sample_1:\t', prediction_distance(equation_square, equation_cube, sample_1))
print('sample_2:\t', prediction_distance(equation_square, equation_cube, sample_2))
print('sample_3:\t', prediction_distance(equation_square, equation_cube, sample_3))

# Import from equation_tree.metrics import prediction_distance # First, we define our equations sympy_square = sympify('x_1 ** 2') equation_square = EquationTree.from_sympy(sympy_square) sympy_cube = sympify('x_1 ** 3') equation_cube = EquationTree.from_sympy(sympy_cube) # For the prediction distance, we need to define a sample of input values on which the metric is evaluated: sample_1 = {'x_1': [0, 1, 2]} sample_2 = {'x_1': np.linspace(0,1)} sample_3 = {'x_1': np.linspace(-10, 10)} # Now, we can calculate the metrics: print('sample_1:\t', prediction_distance(equation_square, equation_cube, sample_1)) print('sample_2:\t', prediction_distance(equation_square, equation_cube, sample_2)) print('sample_3:\t', prediction_distance(equation_square, equation_cube, sample_3))

Symbolic Solution¶

Second, we calculate the symbolic solution metric. This metric checks weather equations differ from each other only through scalars or constants.

In [ ]:

# Import
from equation_tree.metrics import symbolic_solution_quot, symbolic_solution_diff

# Since x ** 2 and x ** 3 differ not only through multiplying with a scalar or adding a constant, the following returns inf
print(f'diff|x_1 ** 2, x_1 ** 3| = {symbolic_solution_diff(equation_square, equation_cube)}')
print(f'quot|x_1 ** 2, x_1 ** 3| = {symbolic_solution_quot(equation_square, equation_cube)}')
print()

equation_identity = EquationTree.from_sympy(sympify('x_1'))
equation_identity_shifted = EquationTree.from_sympy(sympify('x_1 + 3'))
equation_scaled = EquationTree.from_sympy(sympify('2 * x_1'))

# Since x_1 + 3 is shifted x_1 with the value of three, the following returns 3
print(f'diff|x_1 + 3, x_1| = {symbolic_solution_diff(equation_identity_shifted, equation_identity)}')
# This still returns inf:
print(f'quot|x_1 + 3, x_1| = {symbolic_solution_quot(equation_identity_shifted, equation_identity)}')
print()

# and vice vers for the scaled equations:
print(f'diff|x_1 * 2, x_1| = {symbolic_solution_diff(equation_scaled, equation_identity)}')
print(f'quot|x_1 * 2, x_1| = {symbolic_solution_quot(equation_scaled, equation_identity)}')
print()

# Import from equation_tree.metrics import symbolic_solution_quot, symbolic_solution_diff # Since x ** 2 and x ** 3 differ not only through multiplying with a scalar or adding a constant, the following returns inf print(f'diff|x_1 ** 2, x_1 ** 3| = {symbolic_solution_diff(equation_square, equation_cube)}') print(f'quot|x_1 ** 2, x_1 ** 3| = {symbolic_solution_quot(equation_square, equation_cube)}') print() equation_identity = EquationTree.from_sympy(sympify('x_1')) equation_identity_shifted = EquationTree.from_sympy(sympify('x_1 + 3')) equation_scaled = EquationTree.from_sympy(sympify('2 * x_1')) # Since x_1 + 3 is shifted x_1 with the value of three, the following returns 3 print(f'diff|x_1 + 3, x_1| = {symbolic_solution_diff(equation_identity_shifted, equation_identity)}') # This still returns inf: print(f'quot|x_1 + 3, x_1| = {symbolic_solution_quot(equation_identity_shifted, equation_identity)}') print() # and vice vers for the scaled equations: print(f'diff|x_1 * 2, x_1| = {symbolic_solution_diff(equation_scaled, equation_identity)}') print(f'quot|x_1 * 2, x_1| = {symbolic_solution_quot(equation_scaled, equation_identity)}') print()

Normalized Edit Distance¶

For a pair of two trees, edit distance computes the minimum cost to transform one to another with a sequence of operations, each of which either 1) inserts, 2) deletes, or 3) renames a node. It is normalized between 0 and 1 (For more details, see the respective documentation)

In [ ]:

# Import
from src.equation_tree.metrics import normalized_tree_distance

# Here, we use the equations above to give examples:
print(f'ned|x_1 ** 2, x_1 ** 3| = {normalized_tree_distance(equation_square, equation_cube)}')
print(f'ned|x_1 + 3, x_1| = {normalized_tree_distance(equation_identity_shifted, equation_identity)}')
print(f'ned|x_1 * 2, x_1| = {normalized_tree_distance(equation_scaled, equation_identity)}')

# Import from src.equation_tree.metrics import normalized_tree_distance # Here, we use the equations above to give examples: print(f'ned|x_1 ** 2, x_1 ** 3| = {normalized_tree_distance(equation_square, equation_cube)}') print(f'ned|x_1 + 3, x_1| = {normalized_tree_distance(equation_identity_shifted, equation_identity)}') print(f'ned|x_1 * 2, x_1| = {normalized_tree_distance(equation_scaled, equation_identity)}')