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

Linear Models

What are the parameters of a linear model?

• A linear model is of the form 

• The value of represents the rate of change of the function

  • This is the amount the function increases/decreases when increases by 1

    • e.g. the function increasing by 2 every time increases by 1

  • If the function is increasing then is positive

  • If the function is decreasing then is negative

  • When the model is represented as a graph this is the gradient of the line

• The value of represents the value of the function when

  • This is the value of the function when the independent variable is not present

  • This is usually referred to as the initial value

  • When the model is represented as a graph this is the y-intercept of the line

• For example, suppose the vertical height of a bird above the ground in metres after seconds

  • The bird starts 0.75 metres above the ground

  • The height is increasing by 3 metres per second

What can be modelled as a linear model?

• If the graph of the data resembles a straight line

• Anything with a constant rate of change

• Examples include:

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

  • B(m) is the monthly mobile phone bill when m minutes have been used

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

  • d(t) is the distance travelled by a car moving at a constant speed for t seconds

What are possible limitations of a linear model?

• Linear models assume that the variable continuously increases (or decreases) at the same rate

  • In real-life this might not be the case

    • The function might reach a maximum (or minimum)

    • The rate of increase might vary

• If the value of is negative then for some inputs the model will predict negative values

  • In some real-life situations negative values will not make sense

    • To overcome this you can decide on an appropriate domain so that the outputs are never negative

Linear Piecewise Models

What are the parameters of a piecewise linear model?

• 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,

  • The value if the independent variable was not present,

What can be modelled as a piecewise linear model?

• 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

• Examples include:

  • 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

What are possible limitations of a piecewise linear model?

• Piecewise linear models assume that the variable has 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 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