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FM Modelling

1.7 Linearisation Using Exponentials

Note: if you cannot remember the appearance of graphs of the form y=kx^{n}, revise notes for 1.6 Exponential Graphs.

Linearisation using x^{-1}

• If a scatterplot appears to show a relationship of the form y=kx^{-1}, it can be linearised by taking the inverse of the data points.

Example

A company running a streaming service records the proportion of first-time users which return to the service over a period of 7 months, the results are plotted above. They require the data to be linearised in order to perform further analysis. From the dot plot, we can see that the relationship appears to be of the form: y=kx^{-1} and so we will take the inverse of the x data in order to linearise it:

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2.3 Extended Matrix Recurrence Relations

Note: if you cannot remember how to use simple matrix recurrence relations, revise notes for 2.1 Simple Matrix Recurrence Relations.

The Extended Matrix Recurrence Formula

• The simple matrix recurrence formula we have analysed previously (in 2.1 Simple Matrix Recurrence Relations and 2.2 Applications of Matrix Recurrence Relations) only allows the values for the next state to be calculated based on proportional values of the current state. In some systems, this is not enough to produce an accurate model. The extended matrix recurrence formula allows us to model systems which have fixed values added or removed between states.
• The general form for the extended recurrence formula is:

S_{n+1}=T S_{n}+B

Where S_n is the state matrix for the nth state, T is the transition matrix and B is a matrix containing constant values.

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2.2 Applications of Matrix Recurrence Relations

Note: if you cannot remember how to use a matrix recurrence relation, revise notes for 2.1 Simple Matrix Recurrence Relations.

Transition Diagrams

• Transition diagrams provide a visual method for modelling how a system changes between states.
• They consist of a series of nodes (dots) representing the variables in the system, with arrows drawn between them showing the movement of individuals/objects in the system between variables (e.g. the percentage of people who go from playing soccer one month to playing baseball the next).
• By convention numerical values should be shown as percentages in transition diagrams.
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5.3 Modelling Annuity Investments using Technology

Note: if you cannot remember how to model reducing balance relations with regular repayments, revise notes for 5.1 Modelling Annuity Investments.

Guide to Analysing Annuity Investments using Technology (Casio Graphics Calculator)

Note: if you cannot remember how to use the Casio Financial Calculator, revise notes for 3.3 Modelling Reducing Balance Systems with Regular Repayments using Technology.

• The annual interest rate should be entered/calculated as a positive value.
• The initial value (PV) should be entered/calculated as a negative value (remember we justify this by saying we must lose this amount to create the investment).
• The payment (PMT) (i.e. the amount withdrawn) should be entered/calculated as negative (we can justify this by saying we lose this amount per compounding period to the investment)
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5.2 Modelling Annuity Investments using Tables

Note: if you cannot remember how to model reducing balance systems with regular repayments, revise notes for 5.1 Modelling Annuity Investments.

Guidelines to use a Table for Annuity Investments

• Tables provide a convenient method for analysing annuity investments, especially when there are regular systemic changes (e.g. the interest rate or repayments change regularly).
• The columns of the table should list, the compounding period, payment, interest earned, principal addition (i.e. the amount the system has increased during each compounding period) and the balance at the end of the compounding period.
• The principal addition can be calculated by as follows:
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5.1 Modelling Annuity Investments

Modelling using Recursion Relations

• An annuity investment describes a system which has a positive interest rate, and positive deposits made at the end of each compounding period.
• We can use the simple recursion formula to model this system:
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4.1 Modelling Compound Interest Systems with Regular Withdrawals

Note: if you cannot remember the recurrence relation formula for a compound interest system, revise notes for 2.2 Analysis of Compound Interest.

Modelling using a Recurrence Relation

• A compound interest system with regular withdrawals describes a system which has a positive interest rate, and withdrawals (negative) made at the end of each compounding period.
• We can use the simple recursion formula to model this system:

A_{n+1}=d+(1+I) A_{n}

Where d<0 and I>0.

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3.1 Modelling Reducing Balance Systems with Regular Repayments

Note: if you cannot remember what a reducing balance system is, revise notes for 1.2 Forms of Depreciation and 1.3 Predicting Future Values for Depreciation Systems.

Modelling using Recursion Relations

• A reducing balance system with regular repayments describes a system which has a negative interest rate, and positive deposits made at the end of each compounding period.
• We can use the simple recursion formula to model this system:

A_{n+1}=d+(1+I) A_{n}

Where d>0 and I<0.

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2.2 Analysis of Compound Interest

Compound Interest with Regular Additions or Withdrawals

Note: if you cannot remember the basic recursive formula for a compound interest system, revise notes for 2.1 Simple and Compound Interest.

• A compound interest system with regular additions or withdrawals has both a percentage increase/decrease and a fixed amount increase/decrease at each compounding period.
• This type of system can be modelled using the following recursive formula:
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3.2 Modelling Linear Associations

Identifying Explanatory and Response Variables

• It is important to correctly select the explanatory and response variables when using regression, or the relationship will be incorrect.
• The explanatory variable is the variable which is used to explain or predict the response variable.
• In a conventional x-y dataset, the x variable is the explanatory variable and y is the response variable.

Fitting Least Squares Models

• Start by identifying the explanatory and response variables.
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