Logarithmic Model

logarithmic_model(data, precision=4)

Generates a logarithmic 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 logarithmic model; the first element is the coefficient of the logarithmic term, and the second element is the coefficient of the constant term

  • model[‘evaluations’][‘equation’] (func) – Function that evaluates the equation of the logarithmic model at a given numeric input (e.g., model[‘evaluations’][‘equation’](10) would evaluate the equation of the logarithmic model when the independent variable is 10)

  • model[‘evaluations’][‘derivative’] (func) – Function that evaluates the first derivative of the logarithmic model at a given numeric input (e.g., model[‘evaluations’][‘derivative’](10) would evaluate the first derivative of the logarithmic model when the independent variable is 10)

  • model[‘evaluations’][‘integral’] (func) – Function that evaluates the integral of the logarithmic model at a given numeric input (e.g., model[‘evaluations’][‘integral’](10) would evaluate the integral of the logarithmic 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 logarithmic 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 logarithmic 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 logarithmic 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 logarithmic model (will always be None)

  • model[‘accumulations’][‘range’] (float) – Total area under the curve represented by the logarithmic 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 logarithmic 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 logarithmic 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 logarithmic 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 logarithmic 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 logarithmic 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

logarithmic_equation(), logarithmic_derivatives(), logarithmic_integral(), logarithmic_roots(), correlation_coefficient(), run_all()

Notes

  • Provided ordered pairs for the data set: \(p_i = \{ (p_{1,x}, p_{1,y}), (p_{2,x}, p_{2,y}), \cdots, (p_{n,x}, p_{n,y}) \}\)

  • Provided values for the independent variable: \(X_i = \{ p_{1,x}, p_{2,x}, \cdots, p_{n,x} \}\)

  • Provided values for the dependent variable: \(Y_i = \{ p_{1,y}, p_{2,y}, \cdots, p_{n,y} \}\)

  • Minimum value of the provided values for the independent variable: \(X_{min} \leq p_{j,x}, \forall p_{j,x} \in X_i\)

  • Maximum value of the provided values for the independent variable: \(X_{max} \geq p_{j,x}, \forall p_{j,x} \in X_i\)

  • First quartile of the provided values for the independent variable: \(X_{Q1}\)

  • Third quartile of the provided values for the independent variable: \(X_{Q3}\)

  • Mean of all provided values for the dependent variable: \(\bar{y} = \frac{1}{n}\cdot{\sum\limits_{i=1}^n Y_i}\)

  • Resultant values for the coefficients of the logarithmic model: \(C_i = \{ a, b \}\)

  • Standard form for the equation of the logarithmic model: \(f(x) = a\cdot{\ln{x}} + b\)

  • First derivative of the logarithmic model: \(f'(x) = a\cdot{\frac{1}{x}}\)

  • Second derivative of the logarithmic model: \(f''(x) = -a\cdot{\frac{1}{x^2}}\)

  • Integral of the logarithmic model: \(F(x) = a\cdot{x}\cdot(\ln{x} - 1) + b\cdot{x}\)

  • Potential x-values of the roots of the logarithmic model: \(x_{intercepts} = \{ \text{e}^{-\frac{b}{a}} \}\)

  • Potential x-values of the maxima of the logarithmic model: \(x_{maxima} = \{ \varnothing \}\)

  • Potential x-values of the minima of the logarithmic model: \(x_{minima} = \{ \varnothing \}\)

  • Potential x-values of the inflection points of the logarithmic model: \(x_{inflections} = \{ \varnothing \}\)

  • Accumulatation of the logarithmic model over its range: \(A_{range} = \int_{X_{min}}^{X_{max}} f(x) \,dx\)

  • Accumulatation of the logarithmic model over its interquartile range: \(A_{iqr} = \int_{X_{Q1}}^{X_{Q3}} f(x) \,dx\)

  • Average rate of change of the logarithmic model over its range: \(m_{range} = \frac{f(X_{max}) - f(X_{min})}{X_{max} - X_{min}}\)

  • Potential x-values at which the logarithmic model’s instantaneous rate of change equals its average rate of change over its range: \(x_{m,range} = \{ \frac{a}{m_{range}} \}\)

  • Average value of the logarithmic model over its range: \(v_{range} = \frac{1}{X_{max} - X_{min}}\cdot{A_{range}}\)

  • Potential x-values at which the logarithmic model’s value equals its average value over its range: \(x_{v,range} = \{ \text{e}^{-\frac{b - v_{range}}{a}} \}\)

  • Average rate of change of the logarithmic model over its interquartile range: \(m_{iqr} = \frac{f(X_{Q3}) - f(X_{Q1})}{X_{Q3} - X_{Q1}}\)

  • Potential x-values at which the logarithmic model’s instantaneous rate of change equals its average rate of change over its interquartile range: \(x_{m,iqr} = \{ \frac{a}{m_{iqr}} \}\)

  • Average value of the logarithmic model over its interquartile range: \(v_{iqr} = \frac{1}{X_{Q3} - X_{Q1}}\cdot{A_{iqr}}\)

  • Potential x-values at which the logarithmic model’s value equals its average value over its interquartile range: \(x_{v,iqr} = \{ \text{e}^{-\frac{b - v_{iqr}}{a}} \}\)

  • Predicted values based on the logarithmic model: \(\hat{y}_i = \{ \hat{y}_1, \hat{y}_2, \cdots, \hat{y}_n \}\)

  • Residuals of the dependent variable: \(e_i = \{ p_{1,y} - \hat{y}_1, p_{2,y} - \hat{y}_2, \cdots, p_{n,y} - \hat{y}_n \}\)

  • Deviations of the dependent variable: \(d_i = \{ p_{1,y} - \bar{y}, p_{2,y} - \bar{y}, \cdots, p_{n,y} - \bar{y} \}\)

  • Sum of squares of residuals: \(SS_{res} = \sum\limits_{i=1}^n e_i^2\)

  • Sum of squares of deviations: \(SS_{dev} = \sum\limits_{i=1}^n d_i^2\)

  • Correlation coefficient for the logarithmic model: \(r = \sqrt{1 - \frac{SS_{res}}{SS_{dev}}}\)

  • Regression Analysis

Examples

Import logarithmic_model function from regressions library
>>> from regressions.models.logarithmic import logarithmic_model
Generate a logarithmic regression model for the data set [[1, 2], [2, 4.0794], [3, 5.2958], [4, 6.1589], [5, 6.8283], [6, 7.3753], [7, 7.8377], [8, 8.2383], [9, 8.5917], [10, 8.9078]], then print its coefficients, roots, total accumulation over its interquartile range, and correlation
>>> model_perfect = logarithmic_model([[1, 2], [2, 4.0794], [3, 5.2958], [4, 6.1589], [5, 6.8283], [6, 7.3753], [7, 7.8377], [8, 8.2383], [9, 8.5917], [10, 8.9078]])
>>> print(model_perfect['constants'])
[3.0, 2.0]
>>> print(model_perfect['points']['roots'])
[[0.5134, 0.0]]
>>> print(model_perfect['accumulations']['iqr'])
35.0191
>>> print(model_perfect['correlation'])
1.0
Generate a logarithmic 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 = logarithmic_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'])
[7.4791, 22.5032]
>>> print(model_agnostic['points']['inflections'])
[None]
>>> print(model_agnostic['accumulations']['range'])
307.4295
>>> print(model_agnostic['correlation'])
0.5086