### 20.2.9 Logistic regression

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

y=

−b y0
a y0−(a y0+b)eb(x0−x)

.

However, you often do not 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

y(x0+1) | ≈y0+y′(x0) |  |  |  |  |  |  |  |  |
y(x0+2) | ≈y0+y′(x0)+y′(x0+1) |  |  |  |  |  |  |  |  |
⋮ |  |  |  |  |  |  |  |  |  |
y(x0+n) | ≈y0+y′(x0) +⋯+y′(x0+n−1) |  |  |  |  |  |  |  |  |

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,…,y1(n−1)], where y1j 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,ymax,xmax,R,Y] where y is the logistic function, y′ is the derivative, C=−b/a, ymax is the maximum value of y′, xmax 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

logistic_regression([0.0,1.0,2.5],0,1)

Pinstant=0.132478632479*Pcumul+0.0206552706553
Correlation 0.780548607383, Estimated total P=-0.155913978495
Returning estimated Pcumul, Pinstant, Ptotal, Pinstantmax, tmax, R

| ⎡ ⎢ ⎢ ⎣

−

0.155913978495

1+e

−0.0554152581707x+ | ⎛ ⎝ | 0.140088513344+3.14159265359i | ⎞ ⎠

,

−

0.00161022271237

1+cos | ⎛ ⎝ | −i | ⎛ ⎝ | −0.0554152581707x+ | ⎛ ⎝ | 0.140088513344+3.14159265359i | ⎞ ⎠ | ⎞ ⎠ | ⎞ ⎠

,

| −0.155913978495,−0.000805111356186,2.52797727501+56.6918346552i, |  |  |  |  |  |  |  |  |

0.780548607383 | ⎤ ⎥ ⎥ ⎦

The logistic_regression_plot command draws the best fit logistic equation.

- logistic_regression_plot takes three arguments:
  - L, a list representing [y10,y11,…,y1(n−1)], where y1j 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

logistic_regression_plot([1,2,4,6,8,7,5],0,2.0)