9.2.9  Logistic regression: logistic_regression logistic_regression_plot

Differential equations of the form y′ = y(a*y + b) come up often, particularly when studying bounded population growth. With the initial condition y(x0) = y0, the solution is the logistic equation

However, you often don’t know the values of a and b. You can approximate these values given (x₀,y₀) and [y′(x0),y′(x0+1),…,y′(x0+n−1)] by taking the initial value y(x0) = y0 and the approximation y(t+1) ≈ y(t) + y′(t) to get the approximations

Since y′/y = a + by, you can take the approximate values of y′(x0+j)/y(x0+j) and use linear interpolation to get the best fit values of a and b, and then solve the differential equation.

The logistic_regression command uses this approach to find the best fit logistic equation for given data.

• logistic_regression takes three arguments:
  • L, a list representing [y₁₀,y₁₁,…,y_1(n−1)], where y_1j represents the value of y′(x₀ + j).
  • x₀, the initial x value.
  • y₀, the initial y value.
• logistic_regression(L,x₀,y₀) returns a list [y,y′,C,y_max,x_max,R,Y] where y is the logistic function, y′ is the derivative, C=−b/a, y_max is the maximum value of y′, x_max is where y′ has its maximum, R is linear correlation coefficient R of Y=y′/y as a function of y with Y=a*y + b.

Example.
Input:

Output:

The logistic_regression_plot command draws the best fit logistic equation.

• logistic_regression_plot takes three arguments:
  • L, a list representing [y₁₀,y₁₁,…,y_1(n−1)], where y_1j represents the value of y′(x₀ + j).
  • x₀, the initial x value.
  • y₀, the initial y value.
• logistic_regression_plot(L,x₀,y₀) draws the best fit logistic equation.

Example.
Input:

Output: