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p1_pandas.py
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p1_pandas.py
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import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
def marginalize(dist_table: pd.DataFrame, var_to_keep, leave_values=False):
"""
Marginalize variables out of probability distribution table.
dist_table: The probability distribution table is a DataFrame of which
the first columns are the random variables values and the
last column is the probability of having this combination
of values (joint probability). For example : [X, Y, P(X ^ Y)].
Each column are labelled with capital letter
denoting the name of the variable (for the columns containing
variables).
var_to_keep: the marginalization target. So If one has P(X^Y) and
want to get P(X) out of it, then X is such a target.
"""
p_column = dist_table.columns[-1]
r = dist_table.groupby(var_to_keep).agg(
summed=(p_column, 'sum')).reset_index()
# Normalize probabilities so they sum to one.
r['summed'] /= r['summed'].sum()
if not leave_values:
# Only return probabilities, not a whole
# contingency table
return r['summed']
else:
# Return a fulle contingency table, with
# nice colums headers.
if type(var_to_keep) == str:
# Only one variable was given
p_name = f"P({var_to_keep})"
else:
# Several variables were given
p_name = f"P({'^'.join(var_to_keep)})"
r.rename(columns={'summed': p_name}, inplace=True)
return r
def entropy(probabilities: np.array):
"""
Computes H(X)
X is given as a numpy array, all elements of the array
are assumed to represent the distribution (to the shape
of the array is not meaningful)
"""
# Avoid situations where log can't be computed
non_zero = probabilities[probabilities != 0]
return - np.sum(non_zero * np.log2(non_zero))
def joint_entropy(x_and_y: pd.DataFrame):
"""
Computes the joint entropy H(X,Y)
Expects a dataframe with three columns :
- values of X
- values of Y
- P(X=x, Y=y) : probability distribution; must sum to one.
"""
return entropy(x_and_y["P(X^Y)"])
def cond_entropy(x_given_y: pd.DataFrame, y: pd.DataFrame):
"""
Compute the conditional entropy
Expects a dataframe with three columns :
- x_given_y: values of X|Y as table of rows (x,y,X=x|Y=y)
- y: values of P(Y=y) as one column table
"""
# First, relate P(X_i|Y_j) to P(Y_j)
r = pd.merge(x_given_y, y)
return - np.sum(r["P(X|Y)"] * r["P(Y)"] * np.log2(r["P(X|Y)"]))
def mutual_information(x_and_y: pd.DataFrame,
var_x: str = "X",
var_y: str = "Y"):
""" Computes :
I(X;Y) = H(X) + H(Y) - H(X,Y)
Expects parameters :
- x_and_y : a table (DataFrame) giving P(one row of the table)
- var_x : name of the variable X, must be in the columns of x_and_y
- var_y : name of the variable Y, must be in the columns of x_and_y
"""
# The code here is a bit more dynamic so we can use this function
# in part 2 of the problem statement.
# Compute probabilities for all values of random variable X.
x = marginalize(x_and_y, var_x)
y = marginalize(x_and_y, var_y)
# last column is the probabilities
p_x_and_y = x_and_y[x_and_y.columns[-1]]
# FIXME What happen sif the table has these
# columns : X,Y,a,b,c,P(row)
return entropy(x) + entropy(y) - entropy(p_x_and_y)
def joint_entropy3(x_and_y_and_z):
# Compute joint entropy on three variables
return entropy(x_and_y_and_z["P(X^Y^Z)"])
def cond_mutual_information(x_and_y_and_z: pd.DataFrame):
# I(X;Y|Z) = H(X,Z)+ H(Y,Z) - H(Z) - H(X,Y,Z)
vx,vy,vz,joint_p = list(x_and_y_and_z.columns)
x_and_z = marginalize(x_and_y_and_z, [vx, vz]) # X and Z
y_and_z = marginalize(x_and_y_and_z, [vy, vz])
z = marginalize(x_and_y_and_z, vz)
return entropy(x_and_z) - entropy(z) - entropy(x_and_y_and_z[joint_p]) + entropy(y_and_z)
def cond_joint_entropy(x_and_y_and_z):
# H(X,Y|Z) = H(X,Y,Z) - H(Z) (see above)
z = marginalize(x_and_y_and_z, "Z")
return entropy(x_and_y_and_z['P(X^Y^Z)']) - entropy(z)
def implementation():
# Some test data sets
x_given_y = pd.DataFrame(
[[True, True, 0.25],
[True, False, 0.25],
[False, True, 0.2],
[False, False, 0.3]],
columns=["X", "Y", "P(X|Y)"])
x_and_y = pd.DataFrame(
[[True, True, 0.25],
[True, False, 0.25],
[False, True, 0.2],
[False, False, 0.3]],
columns=["X", "Y", "P(X^Y)"])
y = pd.DataFrame([[True, 0.3],
[False, 0.7]],
columns=["Y", "P(Y)"])
x_and_y_and_z = pd.DataFrame(
[[True, True, True, 0.25],
[True, True, False, 0.25],
[True, False, True, 0.2],
[True, False, False, 0.3],
[False, True, True, 0.25],
[False, True, False, 0.25],
[False, False, True, 0.2],
[False, False, False, 0.3]],
columns=["X", "Y", "Z", "P(X^Y^Z)"])
# Q1
entropy(x_and_y)
# Q2
joint_entropy(x_and_y)
# Q3
cond_entropy(x_given_y, y)
# Q4
mutual_information(x_and_y)
# Q5
joint_entropy3(x_and_y_and_z)
cond_joint_entropy(x_and_y_and_z)
cond_mutual_information(x_and_y_and_z)
def medical_diagnosis():
df = pd.read_csv('P1_medicalDB.csv')
print(df)
# Compute joint proabilities and add them to the table
"""
a,i,w
a,i,w
a,i,z
=>
a,i,w 2
a,i,z 1
"""
jpd = df.groupby(list(df.columns)).size().reset_index()
"""
a,i,w 2 / sum(1+2) => P(a,i,w)
a,i,z 1 / sum(1+2) => P(a,i,z)
"""
jpd[0] /= jpd[0].sum() # r[0] is the new counts columns; it's the last column
# Question 6
entropies = []
cardinalities = []
names = []
"""
age, obseity, BIL, ...
----------------------------
>40, y, 10
<40, n, 11
...
"""
for var_name in jpd.columns[0:-1]: # every column but the last one
card = len(jpd[var_name].unique())
# => >40, <40. => cardinality
# H(age) = P(>40)*log(P(>40)) + P(<40)*log(P(<40))
e = entropy(marginalize(jpd, var_name))
names.append(var_name)
cardinalities.append(card)
entropies.append(e)
with open("question6.inc","w") as fout:
for vname, ent in sorted( zip(names, entropies), key=lambda t:t[1]):
fout.write(f"{vname} & {ent:.3f} \\\\\n")
plt.figure()
plt.scatter(cardinalities, entropies)
for i, name in enumerate(names):
plt.annotate(name, (cardinalities[i], entropies[i]))
plt.xticks([2,3,4])
plt.xlabel("Cardinalities")
plt.ylabel("Entropies")
plt.savefig("entropiescardinalities.pdf")
plt.show()
# Question 7
# All variables names excluding the disease (DIS)
# Make a list(...) to keep order
vnames = list(set(list(df.columns)) - set(['DIS']))
entropies = []
for vname in vnames:
""" Compute the conditional entropy H(DIS|variable) based on joint
probability table.
age, obseity, BIL, ...
----------------------------
>40, y, 10
<40, n, 11
... for missing combinations of variables we have :
?, ?, ? => P(?,?,?) = 0
X Y Z P(X,y,Z)
-------------- --> FULL P(x,y,z) table.
n n n 0.1
n y n
n n y
n y y
y n n
y y n
y n y
y y y
We can't reuse our funtion cond_entropy() because it expects
P(X|Y) but the most direct thing we have is P(X^Y).
Slide 3, course 2 : H(X|Y) = − Σ Σ P(Xi ∩ Yj) log P(Xi | Yj)
Applying P(a|b) = P(a ^ b) / P(b), we get :
H(X|Y) = − Σ Σ P(Xi ∩ Yj) log P(Xi ∩ Yj) / P(Yj)
Renaming X -> D, T -> S(ymptom)
H(D|S) = − Σ Σ P(Di ∩ Sj) log P(Di ∩ Sj) / P(Sj)
"""
dis_symptom = marginalize(jpd, ["DIS", vname], True) # P(D ∩ S)
symptom = marginalize(dis_symptom, vname, True) # P(S)
m = pd.merge(dis_symptom, symptom) # relate (d_i ^ s_j)'s to s_j's
p_di_sj = m[m.columns[-2]] # P(d_i^s_j)
p_sj = m[m.columns[-1]] # P(s_j)
e = - np.sum(p_di_sj * np.log2(p_di_sj / p_sj)) # compute H
entropies.append( (vname,e) )
for vname, e in sorted(entropies, key=lambda p:p[1]):
e = f"{e:.3f}"
if vname in ('JAU', 'BIL'):
vname = f"\\textbf{{{vname}}}"
e = f"\\textbf{{{e}}}"
print(f"{vname} & {e} \\\\")
# Question 8
obesity_age = marginalize(jpd, ["obesity", "age"], True)
print(mutual_information(obesity_age, "obesity", "age"))
print(mutual_information(obesity_age, "age", "obesity")) # swap variabkes to checking for bugs
# Try with different variables to see if the results are the same than in question 9
obesity_dic = marginalize(jpd, ["obesity", "DIS"], True)
print(mutual_information(obesity_dic, "obesity", "DIS"))
print(mutual_information(obesity_dic, "DIS", "obesity"))
# Question 9
print("-"*80)
mutual_info = []
for vname in vnames:
dis_symptom = marginalize(jpd, ["DIS", vname], True)
mi = mutual_information(dis_symptom, "DIS", vname)
mutual_info.append((vname,mi))
for vname, i in sorted(mutual_info, key=lambda p:p[1]):
i = f"{i:.3f}"
print(f"{vname} & {i} \\\\")
# Question 10
# Recompute joint proabilities excluding steatosis/healthy
jpd = df[df.DIS.isin(['steatosis', 'healthy'])].groupby(
list(df.columns)).size().reset_index()
jpd[0] /= jpd[0].sum() # r[0] is the counts columns
mutual_info = []
for vname in vnames:
dis_symptom = marginalize(jpd, ["DIS", vname], True)
symptom = marginalize(dis_symptom, vname, True)
m = pd.merge(dis_symptom, symptom) # relate (x_i^y_j)'s to y_j's
p_di_sj = m[m.columns[-2]] # P(x_i^y_j)
p_sj = m[m.columns[-1]] # P(y_j)
e = - np.sum(p_di_sj * np.log2(p_di_sj / p_sj))
mi = mutual_information(dis_symptom, "DIS", vname)
mutual_info.append((vname,mi))
# print(f"DIS|{vname:10s}:\tH={e:.3f} I={mi:.3f}")
with open("question10.inc","w") as fout:
for vname, mi in sorted(mutual_info, key=lambda p:p[1]):
fout.write(f"{vname} & {mi:.3f} \\\\\n")
# Question 11
jpd = df[df.age.isin(['morethan40', 'healthy'])].groupby(
list(df.columns)).size().reset_index()
jpd[0] /= jpd[0].sum() # r[0] is the counts columns
mutual_info = []
vnames = list(set(list(df.columns)) - set(['DIS','age']))
for vname in vnames:
dis_symptom_age = marginalize(jpd, ["DIS", vname, 'age'], True)
# I(DIS;symptom|age)
cmi = cond_mutual_information(dis_symptom_age)
mutual_info.append((vname, cmi))
with open("question11.inc","w") as fout:
for vname, cmi in sorted(mutual_info, key=lambda p:p[1]):
fout.write(f"{vname} & {cmi:.3f} \\\\\n")
print(f"I(DIS;{vname:10s}|age) :\tI={cmi:.3f}")
if __name__ == "__main__":
implementation()
medical_diagnosis()