def squared_distance(v1, v2):
"""
Compute the squared distance between two vectors.
Args:
v1, v2 (List): The vectors for which a squared distance will be
computed.
Returns:
A scalar of the squared distance between the vectors.
Examples:
>>> x = [1, 2, 3]
>>> y = [3, 2, 1]
>>> squared_distance(x, y)
8
>>> e = [1,2,[1,2]]
>>> squared_distance(x, e)
Traceback (most recent call last):
...
IndexError: One of the vectors passed is not a valid vector
"""
if not (valid.is_vector(v1) and valid.is_vector(v2)):
raise IndexError("One of the vectors passed is not a valid vector")
return sum_of_squares(vector_subtract(v1, v2))
def distance(v1, v2):
"""
Compute the distance between two vectors.
Args:
v1, v2 (List): The vectors for which a distance will be computed.
Returns:
A scalar of the distance between the vectors.
Examples:
>>> x = [1, 2, 3]
>>> y = [3, 2, 1]
>>> distance(x, y)
2.8284271247461903
>>> e = [1, 2, [1, 2]]
>>> distance(x, e)
Traceback (most recent call last):
...
IndexError: One of the vectors passed is not a valid vector
"""
if not (valid.is_vector(v1) and valid.is_vector(v2)):
raise IndexError("One of the vectors passed is not a valid vector")
return magnitude(vector_subtract(v1, v2))
def mode(vector):
"""
Returns a list containing the most common value. Returns a list of
len() > 1 in the case of ties.
Args:
vector (List): A list vector for which a mode is sought.
Returns:
A List containing one or more values which are the most frequent.
Examples:
>>> x = [1, 2, 3]
>>> mode(x)
[1, 2, 3]
>>> y = [1, 2, 3, 2]
>>> mode(y)
[2]
>>> z = [1, 2, [3, 4]]
>>> mode(z)
Traceback (most recent call last):
...
IndexError: The vector passed is not a valid vector
"""
if not (valid.is_vector(vector) and valid.is_vector(vector)):
raise IndexError("The vector passed is not a valid vector")
counts = Counter(vector)
max_count = max(counts.values())
return [x_i for x_i, count in counts.items() if count == max_count]
def mean(vector):
"""
Return the simple arithmatic mean of a vector.
Args:
vector (List): A list vector containing the values to calculate
against.
Returns:
A scalar Float value representing the mean of the vector.
Examples:
>>> x = [2,2,2]
>>> mean(x)
2.0
>>> y = [1,2,3]
>>> mean(y)
2.0
>>> z = [1, 2, 3, [1, 2]]
>>> mean(z)
Traceback (most recent call last):
...
IndexError: The vector passed is not a valid vector
"""
if not (valid.is_vector(vector) and valid.is_vector(vector)):
raise IndexError("The vector passed is not a valid vector")
return sum(vector) / len(vector)
def correlation(v_1, v_2):
"""
Calculates the correlation between two vectors.
Args:
v_1, v_2 (List): List vectors of the same length.
Returns:
A scalar value representing the correlation.
Examples:
>>> x = [1, 2, 3]
>>> y = [3, 2, 1]
>>> correlation(x, y)
-1.0
>>> z = [1, 2]
>>> correlation(x, z)
Traceback (most recent call last):
...
IndexError: The two vectors must be the same length.
>>> zz = ['a', 'b', 'c']
>>> correlation(x, zz) # doctest: +NORMALIZE_WHITESPACE
Traceback (most recent call last):
...
TypeError: The vector passed must contains either Int or Float values.
>>> zzz = [1, 2, [1, 2]]
>>> correlation(x, zzz)
Traceback (most recent call last):
...
IndexError: The vector passed is not a valid vector
"""
if not len(v_1) == len(v_2):
raise IndexError("The two vectors must be the same length.")
if not (valid.is_vector(v_1) and valid.is_vector(v_2)):
raise IndexError("The vector passed is not a valid vector")
chk = v_1.copy()
chk.extend(v_2)
if any(isinstance(i, str) for i in chk):
message = ("The vector passed must contains either " +
"Int or Float values.")
raise TypeError(message)
sd_v_1 = standard_deviation(v_1)
sd_v_2 = standard_deviation(v_2)
if sd_v_1 > 0 and sd_v_2 > 0:
return covariance(v_1, v_2) / sd_v_1 / sd_v_2
else:
return 0
def covariance(v_1, v_2):
"""
Calculates the covariance of two vectors.
Args:
v_1, v_2 (List): List vectors of the same length.
Returns:
A scalar value of the covariance.
Examples:
>>> x = [1, 2, 3]
>>> y = [3, 2, 1]
>>> covariance(x, y)
-1.0
>>> z = [1, 2]
>>> covariance(x, z)
Traceback (most recent call last):
...
IndexError: The two vectors must be the same length.
>>> zz = ['a', 'b', 'c']
>>> covariance(x, zz) # doctest: +NORMALIZE_WHITESPACE
Traceback (most recent call last):
...
TypeError: The vector passed must contains either Int or Float values.
>>> zzz = [1, 2, [1, 2]]
>>> covariance(x, zzz)
Traceback (most recent call last):
...
IndexError: The vector passed is not a valid vector
"""
if not len(v_1) == len(v_2):
raise IndexError("The two vectors must be the same length.")
if not (valid.is_vector(v_1) and valid.is_vector(v_2)):
raise IndexError("The vector passed is not a valid vector")
chk = v_1.copy()
chk.extend(v_2)
if any(isinstance(i, str) for i in chk):
message = ("The vector passed must contains either " +
"Int or Float values.")
raise TypeError(message)
num_elements = len(v_1)
return dot(de_mean(v_1), de_mean(v_2)) / (num_elements - 1)
def variance(vector):
"""
Calculate the variance of a vector with length >= 2.
Args:
vector (List): A vector containing 2 or more Int or Float values.
Returns:
A scalar representing the variance.
Examples:
>>> x = [1, 2, 3]
>>> variance(x)
1.0
>>> y = ['a', 'b', 'c']
>>> variance(y) # doctest: +NORMALIZE_WHITESPACE
Traceback (most recent call last):
...
TypeError: The vector passed must contains either Int or Float values.
>>> z = [1, 2, [1, 2]]
>>> variance(z)
Traceback (most recent call last):
...
IndexError: The vector passed is not a valid vector
>>> zz = [1]
>>> variance(zz)
Traceback (most recent call last):
...
IndexError: The vector must contain at least two values
"""
if not (valid.is_vector(vector) and valid.is_vector(vector)):
raise IndexError("The vector passed is not a valid vector")
if any(isinstance(i, str) for i in vector):
message = ("The vector passed must contains either " +
"Int or Float values.")
raise TypeError(message)
if len(vector) < 2:
raise IndexError("The vector must contain at least two values")
num_elements = len(vector)
deviations = de_mean(vector)
return sum_of_squares(deviations) / (num_elements - 1)
def scalar_multiply(v, sc):
"""
Multiply a vector by a scalar.
Args:
v (List): List representing the vector.
sc (int or float): Scalar to multiply by.
Returns:
A new vector of the same length as all of the vectors, where the
corresponding elements have been summed.
Examples:
>>> scalar_multiply([1, 2, 3], 1)
[1, 2, 3]
>>> scalar_multiply([1, 2, 3], 2)
[2, 4, 6]
>>> scalar_multiply([1, 2, [1, 2]], 1)
Traceback (most recent call last):
...
IndexError: The vector passed is not a valid vector
"""
if not valid.is_vector(v):
raise IndexError("The vector passed is not a valid vector")
return [i * sc for i in v]
def sum_of_squares(v):
"""
Compute the sum of squares of for a vector
Args:
v (List): The vector for which the sum of squares should be calculated.
Returns:
A scalar of the sum of squares of the vector.
Examples:
>>> x = [1, 2, 3]
>>> sum_of_squares(x)
14
>>> e = [1, 2, [1, 2]]
>>> sum_of_squares(e)
Traceback (most recent call last):
...
IndexError: The vector passed is not a valid vector
"""
if not valid.is_vector(v):
raise IndexError("The vector passed is not a valid vector")
return dot(v, v)
def magnitude(v):
"""
Compute the magnitude of a vector
Args:
v (List): The vector for which the magnitude should be calculated.
Returns:
A scalar of the magnitude of the vector.
Examples:
>>> x = [1, 2, 3]
>>> magnitude(x)
3.7416573867739413
>>> e = [1, 2, [1, 2]]
>>> magnitude(e)
Traceback (most recent call last):
...
IndexError: The vector passed is not a valid vector
"""
if not valid.is_vector(v):
raise IndexError("The vector passed is not a valid vector")
return math.sqrt(sum_of_squares(v))
def quantile(vector, percentile):
"""
Returns the desired percentile value of a vector.
Args:
vector (List): A list vector containing the values to calculate
against.
percentile (Float): A value of the sought percentile such that
(0.0 < value < 1.0)
Returns:
A scalar value from the target vector, which represents the value less
than which a certain percentile of the data lies.
Examples:
>>> x = [*range(10)]
>>> quantile(x, 0.30)
3
>>> quantile(x, 0.90)
9
>>> quantile(x, 1.0) # doctest: +NORMALIZE_WHITESPACE
Traceback (most recent call last):
...
ValueError: The percentile must be a Float such that
(0.0 < percentile < 1.0)
>>> z = [1, 2, 3, [1, 2, 3]]
>>> quantile(z, 0.50)
Traceback (most recent call last):
...
IndexError: The vector passed is not a valid vector
"""
if not (percentile > 0.0 and percentile < 1.0):
message = ('The percentile must be a Float such that ' +
'(0.0 < percentile < 1.0)')
raise ValueError(message)
if not (valid.is_vector(vector) and valid.is_vector(vector)):
raise IndexError("The vector passed is not a valid vector")
p_index = int(percentile * len(vector))
return sorted(vector)[p_index]
def standard_deviation(vector):
"""
Computes the standard deviation of a vector.
Args:
vector (List): A vector.
Returns:
A scalar value representing the standard deviation.
Examples:
>>> x = [1, 2, 3]
>>> standard_deviation(x)
1.0
>>> y = ['a', 'b', 'c']
>>> standard_deviation(y) # doctest: +NORMALIZE_WHITESPACE
Traceback (most recent call last):
...
TypeError: The vector passed must contains either Int or Float values.
>>> z = [1, 2, [1, 2]]
>>> standard_deviation(z)
Traceback (most recent call last):
...
IndexError: The vector passed is not a valid vector
>>> zz = [1]
>>> standard_deviation(zz)
Traceback (most recent call last):
...
IndexError: The vector must contain at least two values
"""
if not (valid.is_vector(vector) and valid.is_vector(vector)):
raise IndexError("The vector passed is not a valid vector")
if any(isinstance(i, str) for i in vector):
message = ("The vector passed must contains either " +
"Int or Float values.")
raise TypeError(message)
if len(vector) < 2:
raise IndexError("The vector must contain at least two values")
return math.sqrt(variance(vector))
def median(vector):
"""
Return the median (central value) of a vector.
Args:
vector (List): A list vector containing the values to calculate
against.
Returns:
A scalar Float value representing the median of the vector.
Examples:
>>> w = [2, 2, 2]
>>> median(w)
2.0
>>> x = [1, 2, 5]
>>> median(x)
2.0
>>> y = [1, 2, 3, 4]
>>> median(y)
2.5
>>> z = [1, 2, 3, [1, 2]]
>>> median(z)
Traceback (most recent call last):
...
IndexError: The vector passed is not a valid vector
"""
if not (valid.is_vector(vector) and valid.is_vector(vector)):
raise IndexError("The vector passed is not a valid vector")
elements = len(vector)
sorted_vector = sorted(vector)
midpoint = elements // 2
if elements % 2 == 1:
return float(sorted_vector[midpoint])
else:
low = midpoint - 1
hih = midpoint
return mean([sorted_vector[low], sorted_vector[hih]])
def vector_subtract(v1, v2):
"""
Subtracts Corresponding elements in two vectors (lists) of the same length.
Args:
v1, v2 (List): Lists of the same length representing the vectors.
Returns:
A new vector of the same length as v1 and v2, where the corresponding
elements have been subtracted to each other.
Examples:
>>> w = [3, 2, 1]
>>> x = [1, 2, 3]
>>> vector_subtract(w, x)
[2, 0, -2]
>>> y = [1, 2, 3]
>>> vector_subtract(x, y)
[0, 0, 0]
>>> z = [1, 3]
>>> vector_subtract(x, z)
Traceback (most recent call last):
...
IndexError: Vectors must be the same length
>>> e = [1,2,[1,2]]
>>> vector_subtract(x, e)
Traceback (most recent call last):
...
IndexError: One of the vectors passed is not a valid vector
"""
if not (valid.is_vector(v1) and valid.is_vector(v2)):
raise IndexError("One of the vectors passed is not a valid vector")
if len(v1) != len(v2):
raise IndexError('Vectors must be the same length')
return [v1_i - v2_i
for v1_i, v2_i in zip(v1, v2)]
def data_range(vector):
"""
Returns max(vector) - min(vector). Only works for vectors containing
Int and Float values. Errors on character vectors.
Args:
vector (List): A vector to calculate against.
Returns:
A scalar value of the same type as the vector.
Examples:
>>> w = [1, 2, 3]
>>> data_range(w)
2
>>> x = [1.0, 2.0, 3.0]
>>> data_range(x)
2.0
>>> y = ['a', 'b', 'c']
>>> data_range(y) # doctest: +NORMALIZE_WHITESPACE
Traceback (most recent call last):
...
TypeError: The vector passed must contains either Int or Float values.
>>> z = [1, 2, [1, 2]]
>>> data_range(z)
Traceback (most recent call last):
...
IndexError: The vector passed is not a valid vector
"""
if not (valid.is_vector(vector) and valid.is_vector(vector)):
raise IndexError("The vector passed is not a valid vector")
if any(isinstance(i, str) for i in vector):
message = ("The vector passed must contains either " +
"Int or Float values.")
raise TypeError(message)
return max(vector) - min(vector)
def de_mean(vector):
"""
Translate the vector by subtracting its mean, so the result has mean=0.
Args:
vector (List): a valid vector
Returns:
A vector (List), centered on 0.
Examples:
>>> x = [1, 2, 3]
>>> de_mean(x)
[-1.0, 0.0, 1.0]
>>> y = ['a', 'b', 'c']
>>> de_mean(y) # doctest: +NORMALIZE_WHITESPACE
Traceback (most recent call last):
...
TypeError: The vector passed must contains either Int or Float values.
>>> z = [1, 2, [1, 2]]
>>> de_mean(z)
Traceback (most recent call last):
...
IndexError: The vector passed is not a valid vector
"""
if not (valid.is_vector(vector) and valid.is_vector(vector)):
raise IndexError("The vector passed is not a valid vector")
if any(isinstance(i, str) for i in vector):
message = ("The vector passed must contains either " +
"Int or Float values.")
raise TypeError(message)
x_bar = mean(vector)
return [x_i - x_bar for x_i in vector]
def dot(v1, v2):
"""
Compute the dot product of two vectors.
Args:
v1, v2 (List): The vectors for which a dot product will be computed.
Must be the same length.
Returns:
A scalar of the dot product of the two vectors.
Examples:
>>> w = [1, 2, 3]
>>> x = [1, 2, 3]
>>> dot(w, x)
14
>>> z = [1, 2]
>>> dot(x, z)
Traceback (most recent call last):
...
IndexError: Vectors must be the same length
>>> e = [1, 2, [1, 2]]
>>> dot(x, e)
Traceback (most recent call last):
...
IndexError: One of the vectors passed is not a valid vector
"""
if not (valid.is_vector(v1) and valid.is_vector(v2)):
raise IndexError("One of the vectors passed is not a valid vector")
if not len(v1) == len(v2):
raise IndexError('Vectors must be the same length')
return sum(v1_i * v2_i
for v1_i, v2_i in zip(v1, v2))
def interquartile_range(vector):
"""
Calculates the difference between the 75th and 25th percentile.
Args:
vector (List): A numeric vector.
Returns:
A scalar value representing the interquartile range.
Examples:
>>> x = [*range(10)]
>>> interquartile_range(x)
5
>>> y = ['a', 'b', 'c']
>>> interquartile_range(y) # doctest: +NORMALIZE_WHITESPACE
Traceback (most recent call last):
...
TypeError: The vector passed must contains either Int or Float values.
>>> z = [1, 2, [1, 2]]
>>> interquartile_range(z)
Traceback (most recent call last):
...
IndexError: The vector passed is not a valid vector
"""
if not (valid.is_vector(vector) and valid.is_vector(vector)):
raise IndexError("The vector passed is not a valid vector")
if any(isinstance(i, str) for i in vector):
message = ("The vector passed must contains either " +
"Int or Float values.")
raise TypeError(message)
return quantile(vector, 0.75) - quantile(vector, 0.25)
def vector_sum(vectors):
"""
Sum Corresponding elements in all vectors (lists) of the same length.
Args:
vectors (List): Lists of the same length representing the vectors.
Returns:
A new vector of the same length as all of the vectors, where the
corresponding elements have been summed.
Examples:
>>> w = [1, 2, 3]
>>> x = [1, 2, 3]
>>> y = [1, 2, 3]
>>> myVecs = [w, x, y]
>>> vector_sum(myVecs)
[3, 6, 9]
>>> z = [1, 3]
>>> myVecs = [w, x, y, z]
>>> vector_sum(myVecs)
Traceback (most recent call last):
...
IndexError: Vectors must be the same length
>>> e = [1, 2, [1, 2]]
>>> vector_sum([w, x, y, e])
Traceback (most recent call last):
...
IndexError: One of the vectors passed is not a valid vector
"""
if not all(valid.is_vector(v) for v in vectors):
raise IndexError("One of the vectors passed is not a valid vector")
lengthToTest = len(vectors[0])
if any(len(v) != lengthToTest for v in vectors[1:]):
raise IndexError('Vectors must be the same length')
return reduce(vector_add, vectors)
def vector_mean(vectors):
"""
Compute the vector whose ith element is the mean of the ith elements of
the input vectors
Args:
vectors (List): List containing all of the vectors to be computed
against.
Returns:
A new vector of the same length as all of the vectors, where the
corresponding elements have had a mean calculated on them.
Examples:
>>> myVecs = [ [1, 2, 3], [1, 2, 3], [1, 2, 3] ]
>>> vector_mean(myVecs)
[1.0, 2.0, 3.0]
>>> myVecs = [ [1, 2, 3], [2, 2, 2], [3, 2, 1] ]
>>> vector_mean(myVecs)
[2.0, 2.0, 2.0]
>>> myVecs = [ [1, 2, 3], [2, 2, 2], [2, 1] ]
>>> vector_mean(myVecs)
Traceback (most recent call last):
...
IndexError: Vectors must be the same length
>>> myVecs = [ [1, 2, 3], [2, 2, 2], [2, 1, [1, 2] ] ]
>>> vector_mean(myVecs)
Traceback (most recent call last):
...
IndexError: One of the vectors passed is not a valid vector
"""
if not all(valid.is_vector(v) for v in vectors):
raise IndexError("One of the vectors passed is not a valid vector")
n = len(vectors)
return scalar_multiply(vector_sum(vectors), 1/n)
