Statistical Modelling Functions part-3

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)

sigmoid-two-parameter

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

sigmoid

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

sigmoid-four-parameter

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

y=20+ 100/1+e^(−0.8*3−x)

negative-four-parameter

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")

negative-gompertzFor 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")

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:

biexpo

biexpo1

 

 

 

 

 

 

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.

 

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