Piecewise Models (DP IB Applications & Interpretation (AI)): Revision Note

A piecewise linear model is made up of multiple linear models 

For each linear model there will be

• The rate of change for that interval m­_i

• The value if the independent variable was not present c_i

Piecewise linear models can be used when the rate of change of a function changes for different intervals

• These commonly apply when there are different tariffs or levels of charges

Anything with a constant rate of change for set intervals

• C(d) is the taxi charge for a journey of d km

  • The charge might double after midnight

• R(d) is the rental fee for a car used for d days

  • The daily fee might triple if the car is rented over bank holidays

• s(t) is the speed of a car travelling for t seconds with constant acceleration

  • The car might reach a maximum speed

Piecewise linear models have a constant rate of change (represented by a straight line) in each interval

• In real-life this might not be the case

• The data in some intervals might have a continuously variable rate of change (represented by a curve) rather than a constant rate

• Or the transition from one constant rate of change to another may be gradual- i.e. a curve rather than a sudden change in gradient

A non-linear piecewise model is made up of multiple functions 

• Each function will be defined for a range of values of x

The individual functions can contain any function

• For example: quadratic, cubic, exponential, etc

When graphed the individual functions should join to make a continuous graph

• This fact can be used to find unknown parameters

  • If then 

Piecewise models can be used when different functions are needed to represent the output for different intervals of the variable

• S(x) is the standardised score on a test with x raw marks

  • For small values of x there might be a quadratic model

  • For large values of x there might be a linear model

• H(t) is the height of water in a bathtub with after t minutes

  • Initially a cubic model might be a appropriate if the bottom of the bathtub is curved

  • Then a linear model might be a appropriate if the sides of top of the bathtub has the shape of a prism

Piecewise models can be used to model real-life accurately

Piecewise models can be difficult to analyse or apply mathematical techniques to