def linear_model(data, precision = 4):
"""
Generates a linear regression model from a given data set
Parameters
----------
data : list of lists of int or float
List of lists of numbers representing a collection of coordinate pairs; it must include at least 10 pairs
precision : int, default=4
Maximum number of digits that can appear after the decimal place of the results
Raises
------
TypeError
First argument must be a 2-dimensional list
TypeError
Elements nested within first argument must be integers or floats
ValueError
First argument must contain at least 10 elements
ValueError
Last argument must be a positive integer
Returns
-------
model['constants'] : list of float
Coefficients of the resultant linear model; the first element is the coefficient of the linear term, and the second element is the coefficient of the constant term
model['evaluations']['equation'] : func
Function that evaluates the equation of the linear model at a given numeric input (e.g., model['evaluations']['equation'](10) would evaluate the equation of the linear model when the independent variable is 10)
model['evaluations']['derivative'] : func
Function that evaluates the first derivative of the linear model at a given numeric input (e.g., model['evaluations']['derivative'](10) would evaluate the first derivative of the linear model when the independent variable is 10)
model['evaluations']['integral'] : func
Function that evaluates the integral of the linear model at a given numeric input (e.g., model['evaluations']['integral'](10) would evaluate the integral of the linear model when the independent variable is 10)
model['points']['roots'] : list of lists of float
List of lists of numbers representing the coordinate pairs of all the x-intercepts of the linear model (will contain exactly one point)
model['points']['maxima'] : list of lists of float
List of lists of numbers representing the coordinate pairs of all the maxima of the linear model (will always be `None`)
model['points']['minima'] : list of lists of float
List of lists of numbers representing the coordinate pairs of all the minima of the linear model (will always be `None`)
model['points']['inflections'] : list of lists of float
List of lists of numbers representing the coordinate pairs of all the inflection points of the linear model (will always be `None`)
model['accumulations']['range'] : float
Total area under the curve represented by the linear model between the smallest independent coordinate originally provided and the largest independent coordinate originally provided (i.e., over the range)
model['accumulations']['iqr'] : float
Total area under the curve represented by the linear model between the first and third quartiles of all the independent coordinates originally provided (i.e., over the interquartile range)
model['averages']['range']['average_value_derivative'] : float
Average rate of change of the curve represented by the linear model between the smallest independent coordinate originally provided and the largest independent coordinate originally provided
model['averages']['range']['mean_values_derivative'] : list of float
All points between the smallest independent coordinate originally provided and the largest independent coordinate originally provided where their instantaneous rate of change equals the function's average rate of change over that interval
model['averages']['range']['average_value_integral'] : float
Average value of the curve represented by the linear model between the smallest independent coordinate originally provided and the largest independent coordinate originally provided
model['averages']['range']['mean_values_integral'] : list of float
All points between the smallest independent coordinate originally provided and the largest independent coordinate originally provided where their value equals the function's average value over that interval
model['averages']['iqr']['average_value_derivative'] : float
Average rate of change of the curve represented by the linear model between the first and third quartiles of all the independent coordinates originally provided
model['averages']['iqr']['mean_values_derivative'] : list of float
All points between the first and third quartiles of all the independent coordinates originally provided where their instantaneous rate of change equals the function's average rate of change over that interval
model['averages']['iqr']['average_value_integral'] : float
Average value of the curve represented by the linear model between the first and third quartiles of all the independent coordinates originally provided
model['averages']['iqr']['mean_values_integral'] : list of float
All points between the first and third quartiles of all the independent coordinates originally provided where their value equals the function's average value over that interval
model['correlation'] : float
Correlation coefficient indicating how well the model fits the original data set (values range between 0.0, implying no fit, and 1.0, implying a perfect fit)
See Also
--------
:func:`~regressions.analyses.equations.linear.linear_equation`, :func:`~regressions.analyses.derivatives.linear.linear_derivatives`, :func:`~regressions.analyses.integrals.linear.linear_integral`, :func:`~regressions.analyses.roots.linear.linear_roots`, :func:`~regressions.statistics.correlation.correlation_coefficient`, :func:`~regressions.execute.run_all`
Notes
-----
- Provided ordered pairs for the data set: :math:`p_i = \\{ (p_{1,x}, p_{1,y}), (p_{2,x}, p_{2,y}), \\cdots, (p_{n,x}, p_{n,y}) \\}`
- Provided values for the independent variable: :math:`X_i = \\{ p_{1,x}, p_{2,x}, \\cdots, p_{n,x} \\}`
- Provided values for the dependent variable: :math:`Y_i = \\{ p_{1,y}, p_{2,y}, \\cdots, p_{n,y} \\}`
- Minimum value of the provided values for the independent variable: :math:`X_{min} \\leq p_{j,x}, \\forall p_{j,x} \\in X_i`
- Maximum value of the provided values for the independent variable: :math:`X_{max} \\geq p_{j,x}, \\forall p_{j,x} \\in X_i`
- First quartile of the provided values for the independent variable: :math:`X_{Q1}`
- Third quartile of the provided values for the independent variable: :math:`X_{Q3}`
- Mean of all provided values for the dependent variable: :math:`\\bar{y} = \\frac{1}{n}\\cdot{\\sum\\limits_{i=1}^n Y_i}`
- Resultant values for the coefficients of the linear model: :math:`C_i = \\{ a, b \\}`
- Standard form for the equation of the linear model: :math:`f(x) = a\\cdot{x} + b`
- First derivative of the linear model: :math:`f'(x) = a`
- Second derivative of the linear model: :math:`f''(x) = 0`
- Integral of the linear model: :math:`F(x) = \\frac{a}{2}\\cdot{x^2} + b\\cdot{x}`
- Potential x-values of the roots of the linear model: :math:`x_{intercepts} = \\{ -\\frac{b}{a} \\}`
- Potential x-values of the maxima of the linear model: :math:`x_{maxima} = \\{ \\varnothing \\}`
- Potential x-values of the minima of the linear model: :math:`x_{minima} = \\{ \\varnothing \\}`
- Potential x-values of the inflection points of the linear model: :math:`x_{inflections} = \\{ \\varnothing \\}`
- Accumulatation of the linear model over its range: :math:`A_{range} = \\int_{X_{min}}^{X_{max}} f(x) \\,dx`
- Accumulatation of the linear model over its interquartile range: :math:`A_{iqr} = \\int_{X_{Q1}}^{X_{Q3}} f(x) \\,dx`
- Average rate of change of the linear model over its range: :math:`m_{range} = \\frac{f(X_{max}) - f(X_{min})}{X_{max} - X_{min}}`
- Potential x-values at which the linear model's instantaneous rate of change equals its average rate of change over its range: :math:`x_{m,range} = \\{ [X_{min}, X_{max}] \\}`
- Average value of the linear model over its range: :math:`v_{range} = \\frac{1}{X_{max} - X_{min}}\\cdot{A_{range}}`
- Potential x-values at which the linear model's value equals its average value over its range: :math:`x_{v,range} = \\{ -\\frac{b - v_{range}}{a} \\}`
- Average rate of change of the linear model over its interquartile range: :math:`m_{iqr} = \\frac{f(X_{Q3}) - f(X_{Q1})}{X_{Q3} - X_{Q1}}`
- Potential x-values at which the linear model's instantaneous rate of change equals its average rate of change over its interquartile range: :math:`x_{m,iqr} = \\{ [X_{Q1}, X_{Q3}] \\}`
- Average value of the linear model over its interquartile range: :math:`v_{iqr} = \\frac{1}{X_{Q3} - X_{Q1}}\\cdot{A_{iqr}}`
- Potential x-values at which the linear model's value equals its average value over its interquartile range: :math:`x_{v,iqr} = \\{ -\\frac{b - v_{iqr}}{a} \\}`
- Predicted values based on the linear model: :math:`\\hat{y}_i = \\{ \\hat{y}_1, \\hat{y}_2, \\cdots, \\hat{y}_n \\}`
- Residuals of the dependent variable: :math:`e_i = \\{ p_{1,y} - \\hat{y}_1, p_{2,y} - \\hat{y}_2, \\cdots, p_{n,y} - \\hat{y}_n \\}`
- Deviations of the dependent variable: :math:`d_i = \\{ p_{1,y} - \\bar{y}, p_{2,y} - \\bar{y}, \\cdots, p_{n,y} - \\bar{y} \\}`
- Sum of squares of residuals: :math:`SS_{res} = \\sum\\limits_{i=1}^n e_i^2`
- Sum of squares of deviations: :math:`SS_{dev} = \\sum\\limits_{i=1}^n d_i^2`
- Correlation coefficient for the linear model: :math:`r = \\sqrt{1 - \\frac{SS_{res}}{SS_{dev}}}`
- |regression_analysis|
Examples
--------
Import `linear_model` function from `regressions` library
>>> from regressions.models.linear import linear_model
Generate a linear regression model for the data set [[1, 30], [2, 27], [3, 24], [4, 21], [5, 18], [6, 15], [7, 12], [8, 9], [9, 6], [10, 3]], then print its coefficients, roots, total accumulation over its interquartile range, and correlation
>>> model_perfect = linear_model([[1, 30], [2, 27], [3, 24], [4, 21], [5, 18], [6, 15], [7, 12], [8, 9], [9, 6], [10, 3]])
>>> print(model_perfect['constants'])
[-3.0, 33.0]
>>> print(model_perfect['points']['roots'])
[[11.0, 0.0]]
>>> print(model_perfect['accumulations']['iqr'])
82.5
>>> print(model_perfect['correlation'])
1.0
Generate a linear regression model for the data set [[1, 32], [2, 25], [3, 14], [4, 23], [5, 39], [6, 45], [7, 42], [8, 49], [9, 36], [10, 33]], then print its coefficients, inflections, total accumulation over its range, and correlation
>>> model_agnostic = linear_model([[1, 32], [2, 25], [3, 14], [4, 23], [5, 39], [6, 45], [7, 42], [8, 49], [9, 36], [10, 33]])
>>> print(model_agnostic['constants'])
[1.9636, 23.0]
>>> print(model_agnostic['points']['inflections'])
[None]
>>> print(model_agnostic['accumulations']['range'])
304.1982
>>> print(model_agnostic['correlation'])
0.5516
"""
# Handle input errors
matrix_of_scalars(data, 'first')
long_vector(data)
positive_integer(precision)
# Store independent and dependent variable values separately
independent_variable = single_dimension(data, 1)
dependent_variable = single_dimension(data, 2)
# Create matrices for independent and dependent variables
independent_matrix = []
dependent_matrix = column_conversion(dependent_variable)
# Iterate over inputted data
for element in independent_variable:
# Store linear and constant evaluations of original independent elements together as lists within independent matrix
independent_matrix.append([element, 1])
# Solve system of equations
solution = system_solution(independent_matrix, dependent_matrix, precision)
# Eliminate zeroes from solution
coefficients = no_zeroes(solution, precision)
# Generate evaluations for function, derivatives, and integral
equation = linear_equation(*coefficients, precision)
derivative = linear_derivatives(*coefficients, precision)['first']['evaluation']
integral = linear_integral(*coefficients, precision)['evaluation']
# Determine key points of graph
points = key_coordinates('linear', coefficients, precision)
# Generate values for lower and upper bounds
five_numbers = five_number_summary(independent_variable, precision)
min_value = five_numbers['minimum']
max_value = five_numbers['maximum']
q1 = five_numbers['q1']
q3 = five_numbers['q3']
# Calculate accumulations
accumulated_range = accumulated_area('linear', coefficients, min_value, max_value, precision)
accumulated_iqr = accumulated_area('linear', coefficients, q1, q3, precision)
# Determine average values and their points
averages_range = average_values('linear', coefficients, min_value, max_value, precision)
averages_iqr = average_values('linear', coefficients, q1, q3, precision)
# Create list of predicted outputs
predicted = []
for element in independent_variable:
predicted.append(equation(element))
# Calculate correlation coefficient for model
accuracy = correlation_coefficient(dependent_variable, predicted, precision)
# Package preceding results in multiple dictionaries
evaluations = {
'equation': equation,
'derivative': derivative,
'integral': integral
}
points = {
'roots': points['roots'],
'maxima': points['maxima'],
'minima': points['minima'],
'inflections': points['inflections']
}
accumulations = {
'range': accumulated_range,
'iqr': accumulated_iqr
}
averages = {
'range': averages_range,
'iqr': averages_iqr
}
# Package all dictionaries in single dictionary to return
result = {
'constants': coefficients,
'evaluations': evaluations,
'points': points,
'accumulations': accumulations,
'averages': averages,
'correlation': accuracy
}
return result
def logistic_model(data, precision=4):
"""
Generates a logistic regression model from a given data set
Parameters
----------
data : list of lists of int or float
List of lists of numbers representing a collection of coordinate pairs; it must include at least 10 pairs
precision : int, default=4
Maximum number of digits that can appear after the decimal place of the results
Raises
------
TypeError
First argument must be a 2-dimensional list
TypeError
Elements nested within first argument must be integers or floats
ValueError
First argument must contain at least 10 elements
ValueError
Last argument must be a positive integer
Returns
-------
model['constants'] : list of float
Coefficients of the resultant logistic model; the first element is the carrying capacity, the second element is the growth rate, and the third element is the sigmoid's midpoint
model['evaluations']['equation'] : func
Function that evaluates the equation of the logistic model at a given numeric input (e.g., model['evaluations']['equation'](10) would evaluate the equation of the logistic model when the independent variable is 10)
model['evaluations']['derivative'] : func
Function that evaluates the first derivative of the logistic model at a given numeric input (e.g., model['evaluations']['derivative'](10) would evaluate the first derivative of the logistic model when the independent variable is 10)
model['evaluations']['integral'] : func
Function that evaluates the integral of the logistic model at a given numeric input (e.g., model['evaluations']['integral'](10) would evaluate the integral of the logistic model when the independent variable is 10)
model['points']['roots'] : list of lists of float
List of lists of numbers representing the coordinate pairs of all the x-intercepts of the logistic model (will always be `None`)
model['points']['maxima'] : list of lists of float
List of lists of numbers representing the coordinate pairs of all the maxima of the logistic model (will always be `None`)
model['points']['minima'] : list of lists of float
List of lists of numbers representing the coordinate pairs of all the minima of the logistic model (will always be `None`)
model['points']['inflections'] : list of lists of float
List of lists of numbers representing the coordinate pairs of all the inflection points of the logistic model (will contain exactly one point)
model['accumulations']['range'] : float
Total area under the curve represented by the logistic model between the smallest independent coordinate originally provided and the largest independent coordinate originally provided (i.e., over the range)
model['accumulations']['iqr'] : float
Total area under the curve represented by the logistic model between the first and third quartiles of all the independent coordinates originally provided (i.e., over the interquartile range)
model['averages']['range']['average_value_derivative'] : float
Average rate of change of the curve represented by the logistic model between the smallest independent coordinate originally provided and the largest independent coordinate originally provided
model['averages']['range']['mean_values_derivative'] : list of float
All points between the smallest independent coordinate originally provided and the largest independent coordinate originally provided where their instantaneous rate of change equals the function's average rate of change over that interval
model['averages']['range']['average_value_integral'] : float
Average value of the curve represented by the logistic model between the smallest independent coordinate originally provided and the largest independent coordinate originally provided
model['averages']['range']['mean_values_integral'] : list of float
All points between the smallest independent coordinate originally provided and the largest independent coordinate originally provided where their value equals the function's average value over that interval
model['averages']['iqr']['average_value_derivative'] : float
Average rate of change of the curve represented by the logistic model between the first and third quartiles of all the independent coordinates originally provided
model['averages']['iqr']['mean_values_derivative'] : list of float
All points between the first and third quartiles of all the independent coordinates originally provided where their instantaneous rate of change equals the function's average rate of change over that interval
model['averages']['iqr']['average_value_integral'] : float
Average value of the curve represented by the logistic model between the first and third quartiles of all the independent coordinates originally provided
model['averages']['iqr']['mean_values_integral'] : list of float
All points between the first and third quartiles of all the independent coordinates originally provided where their value equals the function's average value over that interval
model['correlation'] : float
Correlation coefficient indicating how well the model fits the original data set (values range between 0.0, implying no fit, and 1.0, implying a perfect fit)
See Also
--------
:func:`~regressions.analyses.equations.logistic.logistic_equation`, :func:`~regressions.analyses.derivatives.logistic.logistic_derivatives`, :func:`~regressions.analyses.integrals.logistic.logistic_integral`, :func:`~regressions.analyses.roots.logistic.logistic_roots`, :func:`~regressions.statistics.correlation.correlation_coefficient`, :func:`~regressions.execute.run_all`
Notes
-----
- Provided ordered pairs for the data set: :math:`p_i = \\{ (p_{1,x}, p_{1,y}), (p_{2,x}, p_{2,y}), \\cdots, (p_{n,x}, p_{n,y}) \\}`
- Provided values for the independent variable: :math:`X_i = \\{ p_{1,x}, p_{2,x}, \\cdots, p_{n,x} \\}`
- Provided values for the dependent variable: :math:`Y_i = \\{ p_{1,y}, p_{2,y}, \\cdots, p_{n,y} \\}`
- Minimum value of the provided values for the independent variable: :math:`X_{min} \\leq p_{j,x}, \\forall p_{j,x} \\in X_i`
- Maximum value of the provided values for the independent variable: :math:`X_{max} \\geq p_{j,x}, \\forall p_{j,x} \\in X_i`
- First quartile of the provided values for the independent variable: :math:`X_{Q1}`
- Third quartile of the provided values for the independent variable: :math:`X_{Q3}`
- Mean of all provided values for the dependent variable: :math:`\\bar{y} = \\frac{1}{n}\\cdot{\\sum\\limits_{i=1}^n Y_i}`
- Resultant values for the coefficients of the logistic model: :math:`C_i = \\{ a, b, c \\}`
- Standard form for the equation of the logistic model: :math:`f(x) = \\frac{a}{1 + \\text{e}^{-b\\cdot(x - c)}}`
- First derivative of the logistic model: :math:`f'(x) = \\frac{ab\\cdot{\\text{e}^{-b\\cdot(x - c)}}}{(1 + \\text{e}^{-b\\cdot(x - c)})^2}`
- Second derivative of the logistic model: :math:`f''(x) = \\frac{2ab^2\\cdot{\\text{e}^{-2b\\cdot(x - c)}}}{(1 + \\text{e}^{-b\\cdot(x - c)})^3} - \\frac{ab^2\\cdot{\\text{e}^{-b\\cdot(x - c)}}}{(1 + \\text{e}^{-b\\cdot(x - c)})^2}`
- Integral of the logistic model: :math:`F(x) = \\frac{a}{b}\\cdot{\\ln|\\text{e}^{b\\cdot(x - c)} + 1|}`
- Potential x-values of the roots of the logistic model: :math:`x_{intercepts} = \\{ \\varnothing \\}`
- Potential x-values of the maxima of the logistic model: :math:`x_{maxima} = \\{ \\varnothing \\}`
- Potential x-values of the minima of the logistic model: :math:`x_{minima} = \\{ \\varnothing \\}`
- Potential x-values of the inflection points of the logistic model: :math:`x_{inflections} = \\{ c \\}`
- Accumulatation of the logistic model over its range: :math:`A_{range} = \\int_{X_{min}}^{X_{max}} f(x) \\,dx`
- Accumulatation of the logistic model over its interquartile range: :math:`A_{iqr} = \\int_{X_{Q1}}^{X_{Q3}} f(x) \\,dx`
- Average rate of change of the logistic model over its range: :math:`m_{range} = \\frac{f(X_{max}) - f(X_{min})}{X_{max} - X_{min}}`
- Potential x-values at which the logistic model's instantaneous rate of change equals its average rate of change over its range: :math:`x_{m,range} = \\{ c + \\frac{1}{b}\\cdot{\\ln(2m_{range})} - \\frac{1}{b}\\cdot{\\ln\\left(ab - 2m_{range} - \\sqrt{(2m_{range} - ab)^2 - 4m_{range}^2}\\right)}, \\\\ c + \\frac{1}{b}\\cdot{\\ln(2m_{range})} - \\frac{1}{b}\\cdot{\\ln\\left(ab - 2m_{range} + \\sqrt{(2m_{range} - ab)^2 - 4m_{range}^2}\\right)} \\}`
- Average value of the logistic model over its range: :math:`v_{range} = \\frac{1}{X_{max} - X_{min}}\\cdot{A_{range}}`
- Potential x-values at which the logistic model's value equals its average value over its range: :math:`x_{v,range} = \\{ c - \\frac{1}{b}\\cdot{\\ln(\\frac{a}{v_{range}} - 1)} \\}`
- Average rate of change of the logistic model over its interquartile range: :math:`m_{iqr} = \\frac{f(X_{Q3}) - f(X_{Q1})}{X_{Q3} - X_{Q1}}`
- Potential x-values at which the logistic model's instantaneous rate of change equals its average rate of change over its interquartile range: :math:`x_{m,iqr} = \\{ c + \\frac{1}{b}\\cdot{\\ln(2m_{iqr})} - \\frac{1}{b}\\cdot{\\ln\\left(ab - 2m_{iqr} - \\sqrt{(2m_{iqr} - ab)^2 - 4m_{iqr}^2}\\right)}, \\\\ c + \\frac{1}{b}\\cdot{\\ln(2m_{iqr})} - \\frac{1}{b}\\cdot{\\ln\\left(ab - 2m_{iqr} + \\sqrt{(2m_{iqr} - ab)^2 - 4m_{iqr}^2}\\right)} \\}`
- Average value of the logistic model over its interquartile range: :math:`v_{iqr} = \\frac{1}{X_{Q3} - X_{Q1}}\\cdot{A_{iqr}}`
- Potential x-values at which the logistic model's value equals its average value over its interquartile range: :math:`x_{v,iqr} = \\{ c - \\frac{1}{b}\\cdot{\\ln(\\frac{a}{v_{iqr}} - 1)} \\}`
- Predicted values based on the logistic model: :math:`\\hat{y}_i = \\{ \\hat{y}_1, \\hat{y}_2, \\cdots, \\hat{y}_n \\}`
- Residuals of the dependent variable: :math:`e_i = \\{ p_{1,y} - \\hat{y}_1, p_{2,y} - \\hat{y}_2, \\cdots, p_{n,y} - \\hat{y}_n \\}`
- Deviations of the dependent variable: :math:`d_i = \\{ p_{1,y} - \\bar{y}, p_{2,y} - \\bar{y}, \\cdots, p_{n,y} - \\bar{y} \\}`
- Sum of squares of residuals: :math:`SS_{res} = \\sum\\limits_{i=1}^n e_i^2`
- Sum of squares of deviations: :math:`SS_{dev} = \\sum\\limits_{i=1}^n d_i^2`
- Correlation coefficient for the logistic model: :math:`r = \\sqrt{1 - \\frac{SS_{res}}{SS_{dev}}}`
- |regression_analysis|
Examples
--------
Import `logistic_model` function from `regressions` library
>>> from regressions.models.logistic import logistic_model
Generate a logistic regression model for the data set [[1, 0.0000122], [2, 0.000247], [3, 0.004945], [4, 0.094852], [5, 1.0], [6, 1.905148], [7, 1.995055], [8, 1.999753], [9, 1.999988], [10, 1.999999]], then print its coefficients, roots, total accumulation over its interquartile range, and correlation
>>> model_perfect = logistic_model([[1, 0.0000122], [2, 0.000247], [3, 0.004945], [4, 0.094852], [5, 1.0], [6, 1.905148], [7, 1.995055], [8, 1.999753], [9, 1.999988], [10, 1.999999]])
>>> print(model_perfect['constants'])
[2.0, 3.0, 5.0]
>>> print(model_perfect['points']['roots'])
[None]
>>> print(model_perfect['accumulations']['iqr'])
5.9987
>>> print(model_perfect['correlation'])
1.0
Generate a logistic regression model for the data set [[1, 32], [2, 25], [3, 14], [4, 23], [5, 39], [6, 45], [7, 42], [8, 49], [9, 36], [10, 33]], then print its coefficients, inflections, total accumulation over its range, and correlation
>>> model_agnostic = logistic_model([[1, 32], [2, 25], [3, 14], [4, 23], [5, 39], [6, 45], [7, 42], [8, 49], [9, 36], [10, 33]])
>>> print(model_agnostic['constants'])
[43.9838, 0.3076, 0.9747]
>>> print(model_agnostic['points']['inflections'])
[[0.9747, 21.9919]]
>>> print(model_agnostic['accumulations']['range'])
305.9347
>>> print(model_agnostic['correlation'])
0.5875
"""
# Handle input errors
matrix_of_scalars(data, 'first')
long_vector(data)
positive_integer(precision)
# Store independent and dependent variable values separately
independent_variable = single_dimension(data, 1)
dependent_variable = single_dimension(data, 2)
# Determine key values for bounds
halved_data = half_dimension(data, 1)
dependent_lower = single_dimension(halved_data['lower'], 2)
dependent_upper = single_dimension(halved_data['upper'], 2)
mean_lower = mean_value(dependent_lower)
mean_upper = mean_value(dependent_upper)
dependent_max = max(dependent_variable)
dependent_min = min(dependent_variable)
dependent_range = dependent_max - dependent_min
independent_max = max(independent_variable)
independent_min = min(independent_variable)
independent_range = independent_max - independent_min
independent_avg = (independent_max + independent_min) / 2
# Circumvent error with bounds
if dependent_range == 0:
dependent_range = 1
if independent_range == 0:
independent_range = 1
# Create function to guide model generation
def logistic_fit(variable, first_constant, second_constant,
third_constant):
evaluation = first_constant / (1 + exp(-1 * second_constant *
(variable - third_constant)))
return evaluation
# Create list to store coefficients of generated equation
solution = []
# Handle normal case where values appear to increase in the set
if mean_upper >= mean_lower:
# Generate model
parameters, covariance = curve_fit(
logistic_fit,
independent_variable,
dependent_variable,
bounds=[(dependent_max - dependent_range, 0,
independent_avg - independent_range),
(dependent_max + dependent_range, inf,
independent_avg + independent_range)])
solution = list(parameters)
# Handle case where values do not appear to increase in the set
else:
# Generate model with inverted negative infinity and zero values
parameters, covariance = curve_fit(
logistic_fit,
independent_variable,
dependent_variable,
bounds=[(dependent_max - dependent_range, -inf,
independent_avg - independent_range),
(dependent_max + dependent_range, 0,
independent_avg + independent_range)])
solution = list(parameters)
# Eliminate zeroes from solution
coefficients = no_zeroes(solution, precision)
# Generate evaluations for function, derivative, and integral
equation = logistic_equation(*coefficients, precision)
derivative = logistic_derivatives(*coefficients,
precision)['first']['evaluation']
integral = logistic_integral(*coefficients, precision)['evaluation']
# Determine key points of graph
points = key_coordinates('logistic', coefficients, precision)
# Generate values for lower and upper bounds
five_numbers = five_number_summary(independent_variable, precision)
min_value = five_numbers['minimum']
max_value = five_numbers['maximum']
q1 = five_numbers['q1']
q3 = five_numbers['q3']
# Calculate accumulations
accumulated_range = accumulated_area('logistic', coefficients, min_value,
max_value, precision)
accumulated_iqr = accumulated_area('logistic', coefficients, q1, q3,
precision)
# Determine average values and their points
averages_range = average_values('logistic', coefficients, min_value,
max_value, precision)
averages_iqr = average_values('logistic', coefficients, q1, q3, precision)
# Create list of predicted outputs
predicted = []
for element in independent_variable:
predicted.append(equation(element))
# Calculate correlation coefficient for model
accuracy = correlation_coefficient(dependent_variable, predicted,
precision)
# Package preceding results in multiple dictionaries
evaluations = {
'equation': equation,
'derivative': derivative,
'integral': integral
}
points = {
'roots': points['roots'],
'maxima': points['maxima'],
'minima': points['minima'],
'inflections': points['inflections']
}
accumulations = {'range': accumulated_range, 'iqr': accumulated_iqr}
averages = {'range': averages_range, 'iqr': averages_iqr}
# Package all dictionaries in single dictionary to return
result = {
'constants': coefficients,
'evaluations': evaluations,
'points': points,
'accumulations': accumulations,
'averages': averages,
'correlation': accuracy
}
return result
def sinusoidal_model(data, precision=4):
"""
Generates a sinusoidal regression model from a given data set
Parameters
----------
data : list of lists of int or float
List of lists of numbers representing a collection of coordinate pairs; it must include at least 10 pairs
precision : int, default=4
Maximum number of digits that can appear after the decimal place of the results
Raises
------
TypeError
First argument must be a 2-dimensional list
TypeError
Elements nested within first argument must be integers or floats
ValueError
First argument must contain at least 10 elements
ValueError
Last argument must be a positive integer
Returns
-------
model['constants'] : list of float
Coefficients of the resultant sinusoidal model; the first element is the vertical stretch factor, the second element is the horizontal stretch factor, the third element is the horizontal shift, and the fourth element is the vertical shift
model['evaluations']['equation'] : func
Function that evaluates the equation of the sinusoidal model at a given numeric input (e.g., model['evaluations']['equation'](10) would evaluate the equation of the sinusoidal model when the independent variable is 10)
model['evaluations']['derivative'] : func
Function that evaluates the first derivative of the sinusoidal model at a given numeric input (e.g., model['evaluations']['derivative'](10) would evaluate the first derivative of the sinusoidal model when the independent variable is 10)
model['evaluations']['integral'] : func
Function that evaluates the integral of the sinusoidal model at a given numeric input (e.g., model['evaluations']['integral'](10) would evaluate the integral of the sinusoidal model when the independent variable is 10)
model['points']['roots'] : list of lists of float or str
List of lists of numbers representing the coordinate pairs of all the x-intercepts of the sinusoidal model (will contain either `None` or an initial set of points within two periods along with general terms for finding the other points)
model['points']['maxima'] : list of lists of float or str
List of lists of numbers representing the coordinate pairs of all the maxima of the sinusoidal model (will contain an initial set of points within two periods along with a general term for finding the other points)
model['points']['minima'] : list of lists of float or str
List of lists of numbers representing the coordinate pairs of all the minima of the sinusoidal model (will contain an initial set of points within two periods along with a general term for finding the other points)
model['points']['inflections'] : list of lists of float or str
List of lists of numbers representing the coordinate pairs of all the inflection points of the sinusoidal model (will contain an initial set of points within two periods along with a general term for finding the other points)
model['accumulations']['range'] : float
Total area under the curve represented by the sinusoidal model between the smallest independent coordinate originally provided and the largest independent coordinate originally provided (i.e., over the range)
model['accumulations']['iqr'] : float
Total area under the curve represented by the sinusoidal model between the first and third quartiles of all the independent coordinates originally provided (i.e., over the interquartile range)
model['averages']['range']['average_value_derivative'] : float
Average rate of change of the curve represented by the sinusoidal model between the smallest independent coordinate originally provided and the largest independent coordinate originally provided
model['averages']['range']['mean_values_derivative'] : list of float or str
All points between the smallest independent coordinate originally provided and the largest independent coordinate originally provided where their instantaneous rate of change equals the function's average rate of change over that interval
model['averages']['range']['average_value_integral'] : float
Average value of the curve represented by the sinusoidal model between the smallest independent coordinate originally provided and the largest independent coordinate originally provided
model['averages']['range']['mean_values_integral'] : list of float or str
All points between the smallest independent coordinate originally provided and the largest independent coordinate originally provided where their value equals the function's average value over that interval
model['averages']['iqr']['average_value_derivative'] : float
Average rate of change of the curve represented by the sinusoidal model between the first and third quartiles of all the independent coordinates originally provided
model['averages']['iqr']['mean_values_derivative'] : list of float or str
All points between the first and third quartiles of all the independent coordinates originally provided where their instantaneous rate of change equals the function's average rate of change over that interval
model['averages']['iqr']['average_value_integral'] : float
Average value of the curve represented by the sinusoidal model between the first and third quartiles of all the independent coordinates originally provided
model['averages']['iqr']['mean_values_integral'] : list of float or str
All points between the first and third quartiles of all the independent coordinates originally provided where their value equals the function's average value over that interval
model['correlation'] : float
Correlation coefficient indicating how well the model fits the original data set (values range between 0.0, implying no fit, and 1.0, implying a perfect fit)
See Also
--------
:func:`~regressions.analyses.equations.sinusoidal.sinusoidal_equation`, :func:`~regressions.analyses.derivatives.sinusoidal.sinusoidal_derivatives`, :func:`~regressions.analyses.integrals.sinusoidal.sinusoidal_integral`, :func:`~regressions.analyses.roots.sinusoidal.sinusoidal_roots`, :func:`~regressions.statistics.correlation.correlation_coefficient`, :func:`~regressions.execute.run_all`
Notes
-----
- Provided ordered pairs for the data set: :math:`p_i = \\{ (p_{1,x}, p_{1,y}), (p_{2,x}, p_{2,y}), \\cdots, (p_{n,x}, p_{n,y}) \\}`
- Provided values for the independent variable: :math:`X_i = \\{ p_{1,x}, p_{2,x}, \\cdots, p_{n,x} \\}`
- Provided values for the dependent variable: :math:`Y_i = \\{ p_{1,y}, p_{2,y}, \\cdots, p_{n,y} \\}`
- Minimum value of the provided values for the independent variable: :math:`X_{min} \\leq p_{j,x}, \\forall p_{j,x} \\in X_i`
- Maximum value of the provided values for the independent variable: :math:`X_{max} \\geq p_{j,x}, \\forall p_{j,x} \\in X_i`
- First quartile of the provided values for the independent variable: :math:`X_{Q1}`
- Third quartile of the provided values for the independent variable: :math:`X_{Q3}`
- Mean of all provided values for the dependent variable: :math:`\\bar{y} = \\frac{1}{n}\\cdot{\\sum\\limits_{i=1}^n Y_i}`
- Resultant values for the coefficients of the sinusoidal model: :math:`C_i = \\{ a, b, c, d \\}`
- Standard form for the equation of the sinusoidal model: :math:`f(x) = a\\cdot{\\sin(b\\cdot(x - c))} + d`
- First derivative of the sinusoidal model: :math:`f'(x) = ab\\cdot{\\cos(b\\cdot(x - c))}`
- Second derivative of the sinusoidal model: :math:`f''(x) = -ab^2\\cdot{\\sin(b\\cdot(x - c))}`
- Integral of the sinusoidal model: :math:`F(x) = -\\frac{a}{b}\\cdot{\\cos(b\\cdot(x - c))} + d\\cdot{x}`
- Potential x-values of the roots of the sinusoidal model: :math:`x_{intercepts} = \\{ c + \\frac{1}{b}\\cdot{\\left(\\sin^{-1}(-\\frac{d}{a}) + 2\\pi\\cdot{k} \\right)}, c + \\frac{1}{b}\\cdot{\\left(-\\sin^{-1}(-\\frac{d}{a}) + \\pi\\cdot(2k - 1) \\right)}, \\\\ c - \\frac{\\pi}{b}\\cdot(2k - 1) \\}`
- :math:`k \\in \\mathbb{Z}`
- Potential x-values of the maxima of the sinusoidal model: :math:`x_{maxima} = \\{ c + \\frac{\\pi}{b}\\cdot(\\frac{1}{2} + k) \\}`
- :math:`k \\in \\mathbb{Z}`
- Potential x-values of the minima of the sinusoidal model: :math:`x_{maxima} = \\{ c + \\frac{\\pi}{b}\\cdot(\\frac{1}{2} + k) \\}`
- :math:`k \\in \\mathbb{Z}`
- Potential x-values of the inflection points of the sinusoidal model: :math:`x_{inflections} = \\{ c + \\frac{\\pi}{b}\\cdot{k} \\}`
- :math:`k \\in \\mathbb{Z}`
- Accumulatation of the sinusoidal model over its range: :math:`A_{range} = \\int_{X_{min}}^{X_{max}} f(x) \\,dx`
- Accumulatation of the sinusoidal model over its interquartile range: :math:`A_{iqr} = \\int_{X_{Q1}}^{X_{Q3}} f(x) \\,dx`
- Average rate of change of the sinusoidal model over its range: :math:`m_{range} = \\frac{f(X_{max}) - f(X_{min})}{X_{max} - X_{min}}`
- Potential x-values at which the sinusoidal model's instantaneous rate of change equals its average rate of change over its range: :math:`x_{m,range} = \\{ c + \\frac{1}{b}\\cdot{\\left(\\cos^{-1}(\\frac{m_{range}}{ab}) + \\pi\\cdot{k} \\right)}, c + \\frac{1}{b}\\cdot{\\left(-\\cos^{-1}(\\frac{m_{range}}{ab}) + 2\\pi\\cdot{k} \\right)} \\}`
- :math:`k \\in \\mathbb{Z}`
- Average value of the sinusoidal model over its range: :math:`v_{range} = \\frac{1}{X_{max} - X_{min}}\\cdot{A_{range}}`
- Potential x-values at which the sinusoidal model's value equals its average value over its range: :math:`x_{v,range} = \\{ c + \\frac{1}{b}\\cdot{\\left(\\sin^{-1}(-\\frac{d - v_{range}}{a}) + 2\\pi\\cdot{k} \\right)}, c + \\frac{1}{b}\\cdot{\\left(-\\sin^{-1}(-\\frac{d - v_{range}}{a}) + \\pi\\cdot(2k - 1) \\right)}, \\\\ c + \\frac{\\pi}{b}\\cdot(2k - 1) \\}`
- :math:`k \\in \\mathbb{Z}`
- Average rate of change of the sinusoidal model over its interquartile range: :math:`m_{iqr} = \\frac{f(X_{Q3}) - f(X_{Q1})}{X_{Q3} - X_{Q1}}`
- Potential x-values at which the sinusoidal model's instantaneous rate of change equals its average rate of change over its interquartile range: :math:`x_{m,iqr} = \\{ c + \\frac{1}{b}\\cdot{\\left(\\cos^{-1}(\\frac{m_{iqr}}{ab}) + \\pi\\cdot{k} \\right)}, c + \\frac{1}{b}\\cdot{\\left(-\\cos^{-1}(\\frac{m_{iqr}}{ab}) + 2\\pi\\cdot{k} \\right)} \\}`
- :math:`k \\in \\mathbb{Z}`
- Average value of the sinusoidal model over its interquartile range: :math:`v_{iqr} = \\frac{1}{X_{Q3} - X_{Q1}}\\cdot{A_{iqr}}`
- Potential x-values at which the sinusoidal model's value equals its average value over its interquartile range: :math:`x_{v,iqr} = \\{ c + \\frac{1}{b}\\cdot{\\left(\\sin^{-1}(-\\frac{d - v_{iqr}}{a}) + 2\\pi\\cdot{k} \\right)}, c + \\frac{1}{b}\\cdot{\\left(-\\sin^{-1}(-\\frac{d - v_{iqr}}{a}) + \\pi\\cdot(2k - 1) \\right)}, \\\\ c + \\frac{\\pi}{b}\\cdot(2k - 1) \\}`
- :math:`k \\in \\mathbb{Z}`
- Predicted values based on the sinusoidal model: :math:`\\hat{y}_i = \\{ \\hat{y}_1, \\hat{y}_2, \\cdots, \\hat{y}_n \\}`
- Residuals of the dependent variable: :math:`e_i = \\{ p_{1,y} - \\hat{y}_1, p_{2,y} - \\hat{y}_2, \\cdots, p_{n,y} - \\hat{y}_n \\}`
- Deviations of the dependent variable: :math:`d_i = \\{ p_{1,y} - \\bar{y}, p_{2,y} - \\bar{y}, \\cdots, p_{n,y} - \\bar{y} \\}`
- Sum of squares of residuals: :math:`SS_{res} = \\sum\\limits_{i=1}^n e_i^2`
- Sum of squares of deviations: :math:`SS_{dev} = \\sum\\limits_{i=1}^n d_i^2`
- Correlation coefficient for the sinusoidal model: :math:`r = \\sqrt{1 - \\frac{SS_{res}}{SS_{dev}}}`
- |regression_analysis|
Examples
--------
Import `sinusoidal_model` function from `regressions` library
>>> from regressions.models.sinusoidal import sinusoidal_model
Generate a sinusoidal regression model for the data set [[1, 3], [2, 8], [3, 3], [4, -2], [5, 3], [6, 8], [7, 3], [8, -2], [9, 3], [10, 8]], then print its coefficients, roots, total accumulation over its interquartile range, and correlation
>>> model_perfect = sinusoidal_model([[1, 3], [2, 8], [3, 3], [4, -2], [5, 3], [6, 8], [7, 3], [8, -2], [9, 3], [10, 8]])
>>> print(model_perfect['constants'])
[-5.0, 1.5708, 3.0, 3.0]
>>> print(model_perfect['points']['roots'])
[[3.4097, 0.0], [4.5903, 0.0], [7.4097, 0.0], [8.5903, 0.0], ['3.4097 + 4.0k', 0.0], ['4.5903 + 4.0k', 0.0]]
>>> print(model_perfect['accumulations']['iqr'])
11.8168
>>> print(model_perfect['correlation'])
1.0
Generate a sinusoidal regression model for the data set [[1, 32], [2, 25], [3, 14], [4, 23], [5, 39], [6, 45], [7, 42], [8, 49], [9, 36], [10, 33]], then print its coefficients, inflections, total accumulation over its range, and correlation
>>> model_agnostic = sinusoidal_model([[1, 32], [2, 25], [3, 14], [4, 23], [5, 39], [6, 45], [7, 42], [8, 49], [9, 36], [10, 33]])
>>> print(model_agnostic['constants'])
[14.0875, 0.7119, -3.7531, 34.2915]
>>> print(model_agnostic['points']['inflections'])
[[5.0729, 34.2915], [9.4859, 34.2915], [13.8985, 34.2915], [18.3114, 34.2915], ['5.0729 + 4.413k', 34.2915]]
>>> print(model_agnostic['accumulations']['range'])
307.8897
>>> print(model_agnostic['correlation'])
0.9264
"""
# Handle input errors
matrix_of_scalars(data, 'first')
long_vector(data)
positive_integer(precision)
# Store independent and dependent variable values separately
independent_variable = single_dimension(data, 1)
dependent_variable = single_dimension(data, 2)
# Determine key values for bounds
independent_max = max(independent_variable)
independent_min = min(independent_variable)
independent_range = independent_max - independent_min
dependent_max = max(dependent_variable)
dependent_min = min(dependent_variable)
dependent_range = dependent_max - dependent_min
# Circumvent errors with bounds
if independent_range == 0:
independent_range = 1
if dependent_range == 0:
dependent_range = 1
dependent_max += 1
# Create function to guide model generation
def sinusoidal_fit(variable, first_constant, second_constant,
third_constant, fourth_constant):
evaluation = first_constant * sin(
second_constant * (variable - third_constant)) + fourth_constant
return evaluation
# Create list to store coefficients of generated equation
solution = []
# Handle normal case
try:
# Generate model
parameters, covariance = curve_fit(
sinusoidal_fit,
independent_variable,
dependent_variable,
bounds=[(-dependent_range, -inf, -independent_range,
dependent_min),
(dependent_range, inf, independent_range, dependent_max)])
solution = list(parameters)
# Narrow bounds in event of runtime error
except RuntimeError:
# Regenerate model within tighter parameters
parameters, covariance = curve_fit(
sinusoidal_fit,
independent_variable,
dependent_variable,
bounds=[(dependent_range - 1, -independent_range,
-independent_range, dependent_min),
(dependent_range + 1, independent_range, independent_range,
dependent_max)])
solution = list(parameters)
# Eliminate zeroes from solution
coefficients = no_zeroes(solution, precision)
# Generate evaluations for function, derivative, and integral
equation = sinusoidal_equation(*coefficients, precision)
derivative = sinusoidal_derivatives(*coefficients,
precision)['first']['evaluation']
integral = sinusoidal_integral(*coefficients, precision)['evaluation']
# Determine key points of graph
points = key_coordinates('sinusoidal', coefficients, precision)
final_roots = shifted_coordinates_within_range(points['roots'],
independent_min,
independent_max, precision)
final_maxima = shifted_coordinates_within_range(points['maxima'],
independent_min,
independent_max, precision)
final_minima = shifted_coordinates_within_range(points['minima'],
independent_min,
independent_max, precision)
final_inflections = shifted_coordinates_within_range(
points['inflections'], independent_min, independent_max, precision)
# Generate values for lower and upper bounds
five_numbers = five_number_summary(independent_variable, precision)
min_value = five_numbers['minimum']
max_value = five_numbers['maximum']
q1 = five_numbers['q1']
q3 = five_numbers['q3']
# Calculate accumulations
accumulated_range = accumulated_area('sinusoidal', coefficients, min_value,
max_value, precision)
accumulated_iqr = accumulated_area('sinusoidal', coefficients, q1, q3,
precision)
# Determine average values and their points
averages_range = average_values('sinusoidal', coefficients, min_value,
max_value, precision)
averages_iqr = average_values('sinusoidal', coefficients, q1, q3,
precision)
# Create list of predicted outputs
predicted = []
for element in independent_variable:
predicted.append(equation(element))
# Calculate correlation coefficient for model
accuracy = correlation_coefficient(dependent_variable, predicted,
precision)
# Package preceding results in multiple dictionaries
evaluations = {
'equation': equation,
'derivative': derivative,
'integral': integral
}
points = {
'roots': final_roots,
'maxima': final_maxima,
'minima': final_minima,
'inflections': final_inflections
}
accumulations = {'range': accumulated_range, 'iqr': accumulated_iqr}
averages = {'range': averages_range, 'iqr': averages_iqr}
# Package all dictionaries in single dictionary to return
result = {
'constants': coefficients,
'evaluations': evaluations,
'points': points,
'accumulations': accumulations,
'averages': averages,
'correlation': accuracy
}
return result