Hi guys lets begin the third part of statistical modelling functions

**Sigmoid(s-shaped) function****Biexponential model function**

**Sigmoid(s-shaped) function:** The simplest s-shaped function is **two-parameter logistic**, where **0≤y≤1 **

y=[e^(a+b*x)]/[1+e^(a+b*x)]

here is the cirve with parameter 1,0.1, as you increase the value of a(which is asymptotic value)

**Three-parameter logistic:** This allows **y to vary on any scale**

y=a/[1+b*e(-c*x)]

Here is the curve with parameter 100,90,10

The intercept is =**a/1+b , **a=asymptotic value initial slope is measured by c

**Four- parameter logistic:** The four-parameter logistic function has asymptotes at the left-a and right-hand b ends of the x axis and scales c the response to x about the midpoint d where the curve has its inflexion:

y=a+ b−a/1+e^(c*d−x)

Letting a=20 b=120 c=0.8 and d=3, the function

y=20+ 100/1+e^(0.8*3−x)

looks like this

Negative sigmoid curves have the parameter c<0 as for the function

y=20+ 100/1+e^(−0.8*3−x)

**An asymmetric S-shaped curve much used in demography and life insurance work is the ****Gompertz growth model**,

y=a*e^(b*e^(c*x))

The shape of the function depends on the signs of the parameters b and c.

**For a negative ****sigmoid, b is negative (here −1) and c is positive (here +0.02)**:

x<- -200:100 y<-100*exp(-exp(0.02*x)) plot(x,y,type="l",main="negative Gompertz")

For a positive sigmoid both parameters are negative:

x<- 0:100 y<- 50*exp(-5*exp(-0.08*x)) plot(x,y,type="l",main="positive Gompertz")

**Biexponential model:**This is a useful four-parameter non-linear function, which is the sum of two exponential functions of x:

y=a*e^(b*x) +c*e^(d*x)

Various shapes depend upon the signs of the parameters b, c and d:

When b, c and d are all negative, this function is known as the first-order compartment model in which a drug administered at time 0 passes through the system with its dynamics affected by elimination, absorption and clearance.

**Transformations of the response and explanatory variables**

We have seen the use of transformation to linearize the relationship between the response

and the explanatory variables:

##### • log(y) against x for exponential relationships;

• log(y) against logx for power functions;

• exp(y) against x for logarithmic relationships;

• 1/y against 1/x for asymptotic relationships;

• log(p/1−p) against x for proportion data.

**Other transformations are useful for variance stabilization:**

##### •√y to stabilize the variance for count data;

• arcsin(y) to stabilize the variance of percentage data.

Please shoot your doubt and correction to mail irrfankhann29@gmail.com and in comments section.