Search topics

Search

Set up a transition matrix

Start a set of 8
1 OF 6
Members who attend now and attend next week
\(\displaystyle 85\%\)
Members who skip now and attend next week
\(\displaystyle 30\%\)

A gym records each member as attending (A) or skipping (S) each week. Of members who attend this week, 85% attend next week. Of members who skip this week, 30% attend next week. What is the transition matrix T with rows and columns in the order A, S and current states in columns?

Year 11, VCE Units 1 & 2, Chapter 4, 4 difficulty levels, No calculatorBuild a worksheet

Set up a transition matrix

A transition matrix records how a population moves between named states in one step. Use current states for columns, next states for rows, and make every column add to one. This model assumes that next-state proportions depend only on the current state, not on earlier states

\[T_{ij}=P(\text{next state }i\mid\text{current state }j)\]
  1. Put current states in columns Each column describes the people or objects that are in one state now \(\displaystyle T_{ij}=P(\text{next state }i\mid\text{current state }j)\)
  2. Put next states in rows The row identifies the state reached after one transition \(\displaystyle \text{row }i=\text{next state }i\)
  3. Make each column total one All possible next states together account for 100 percent of the current state \(\displaystyle \sum_{i=1}^{n}T_{ij}=1\)
  4. Use decimal proportions Convert a stated percentage to a decimal before placing it in the matrix \(\displaystyle 74\%=0.74,\quad 1-0.74=0.26\)

Worked example

A two-state model records people who use bus or train from one working day to the next. Columns represent current states and rows represent next states. If 72% of current bus users remain bus users and 18% of current train users change to bus, construct the transition matrix in the order bus, train

\[T=\begin{bmatrix}T_{\text{bus,bus}}&T_{\text{bus,train}}\\T_{\text{train,bus}}&T_{\text{train,train}}\end{bmatrix}\]
  1. \(\displaystyle 72\%=0.72,\quad 18\%=0.18\)Convert the two stated percentages to decimal proportions before placing them in the matrix
  2. \(\displaystyle T_{\text{train,bus}}=1-0.72=0.28\)The bus column must total one, so the proportion changing from bus to train is 0.28
  3. \(\displaystyle T_{\text{train,train}}=1-0.18=0.82\)The train column must total one, so the proportion remaining train is 0.82
  4. \(\displaystyle T=\begin{bmatrix}0.72&0.18\\0.28&0.82\end{bmatrix}\)Rows give next states and columns give current states, and each column totals one

Common mistakes

Reversing current and next states
✕\(\displaystyle T_{ij}=P(\text{current state }i\mid\text{next state }j)\)✓\(\displaystyle T_{ij}=P(\text{next state }i\mid\text{current state }j)\)
The first index names the next-state row and the second index names the current-state column
Entering a percentage as a whole number
✕\(\displaystyle T_{11}=72\)✓\(\displaystyle T_{11}=0.72\)
A transition entry is a proportion, so 72% must be written as 0.72
Forgetting the complement
✕\(\displaystyle T_{21}=0.72\)✓\(\displaystyle T_{21}=1-0.72=0.28\)
The other entry in a two-state column is the complement of the stated retention proportion

Difficulty levels

introComplete a two-state matrix from whole-percent retention proportions
standardBuild a two-state matrix from a full description or transition diagram
challengeBuild a three-state matrix from a fully labelled transition diagram, using decimal proportions
examChoose the transition matrix with the correct orientation, complements and decimal entries

Questions about this topic

Why do the columns of a transition matrix total one?+

A current state must distribute all of its members among the possible next states, so the proportions in that column total 100 percent

What does an entry in row i and column j mean?+

It is the proportion of the members currently in state j who are in state i after one transition

How is an unstated two-state transition found?+

Subtract the stated proportion from one, because the two entries in the current-state column must total one

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, General Mathematics, chapter 4: Matrices

In this chapter

State the order of a matrix and locate its elements4A Matrix types and order: Identify row, column, square, symmetric and zero matricesRead information from a labelled data matrix4B Matrix addition and subtraction: Identify row, column, square, symmetric and zero matricesAdd and subtract matrices4B Matrix addition and subtraction: Find unknown entries in a matrix equationMultiply a matrix by a scalarCombine scalar multiples of matricesDecide whether a matrix product is defined and give its orderMultiply matrices4D Matrix products and powers: Evaluate powers of a square matrixIdentify and construct the identity matrix4E Inverse matrices and simultaneous equations: Find unknown entries in a matrix equationFind the determinant of a 2 × 2 matrixFind the inverse of a 2 × 2 matrixWrite simultaneous equations as a matrix equationSolve simultaneous equations using the inverse matrix4F Matrices for networks: Evaluate powers of a square matrixRepresent a network in a matrixIdentify the network a connection matrix describesInterpret connection sums and symmetryCount routes using powers of a connection matrixSet up a transition matrixUse a transition matrix to predict the next step4H State matrices and recurrence: Evaluate powers of a square matrixGenerate state matrices by recursionDetermine the long-run state of a transition modelUse matrix multiplication to answer an application questionUse row and column matrices of ones to extract totals