An adjacency matrix for a non-directed graph with `n` vertices is a `n×n` matrix in which the entry in row `i` and column `j` is the number of edges joining the vertices `i` and `j`. In an adjacency matrix a loop is counted as one edge.
For a directed graph, the entry in row `i` and column `j` is the number of directed edges, arcs, joining the vertex `i` and `j` in the direction `i` to `j`.
An algorithm is a precisely defined routine procedure that can be applied and systematically followed through to a conclusion.
angle of depression
When an observer looks at an object that is lower than the eye of the observer, the angle between the line of sight and the horizontal is called the angle of depression.
angle of elevation
When an observer looks at an object that is higher than the eye of the observer, the angle between the line of sight and the horizontal is called the angle of elevation.
An annuity is a compound interest investment from which payments are made on a regular basis for a fixed period of time. At the end of this time the investment has no residual value.
A sequence of numbers such that the difference between any two successive members of the sequence is constant.
For example, the sequence 2, 5, 8, 11, 14, 17, … is an arithmetic sequence with first term 2 and common difference 3, by inspection of the sequence, the rule for the `nth` term `t_n` of this sequence is:
`t_n=2+(n-1)3=3n-1`, `n ≥ 1`, if `t_n` is used to denote the `nth`term in the sequence, then a recursion relation that will generate this sequence is:
A general term used to describe the relationship between two, or more, variables. The term association is often used interchangeably with the term correlation. The latter tends to be used when referring to the strength of a linear relationship between two numerical variables.
average percentage method
In the average percentage method for calculating a seasonal index, the data for each ‘season’ are expressed as percentages of the average for the year. The percentages for the corresponding ‘seasons’ for different years are then averaged using a mean or median to arrive at a seasonal index.
The direction of a fixed point, or the path of an object, from the point of observation.
A bipartite graph is a graph whose set of vertices can be split into two distinct groups in such a way that each edge of the graph joins a vertex in the first group to a vertex in the second group.
The book value is the value of an asset recorded on a balance sheet. The book value is based on the original cost of the asset less depreciation. There are three commonly used methods for calculating yearly depreciation in the value of an asset; namely, reducing balance depreciation, flat rate depreciation or unit cost depreciation.
The break-even point is the point at which revenue begins to exceed the cost of production.
Data associated with a categorical variable is called categorical data.
A categorical variable is a variable whose values are categories.
Examples include blood group (A, B, AB or O) or house construction type, brick, concrete, timber, steel, other.
Categories may have numerical labels; for example, the numbers worn by player in a sporting team, but these labels have no numerical significance, they merely serve as labels.
A relationship between an explanatory and a response variable is said to be causal if the change in the explanatory variable actually causes a change in the response variable. Simply knowing that two variables are associated, no matter how strongly, is not sufficient evidence by itself to conclude that the two variables are causally related.
coefficient of determination
In a linear model between two variables, the coefficient of determination (R2) is the proportion of the total variation that can be explained by the linear relationship existing between the two variables, usually expressed as a percentage. For two variables only, the coefficient of determination is numerically equal to the square of the correlation coefficient (r2).
A response is said to be common when there is no causation, but instead the association is explained by at least one other variable that is associated with both the explanatory and the response variable.
Compass bearings are specified as angles either side of north or south, that describe the direction of a fixed point, or the path of an object. For example, a compass bearing of N50°E is found by facing north and moving through an angle of 50° towards east.
A complete graph is a simple graph in which every vertex is joined to every other vertex by an edge. The complete graph with n vertices is denoted Kn.
The interest earned when each successive interest payment is added to the principal for the purpose of calculating the next interest payment.
For example, if the principal (`P`) earns compound interest (`A`) at the interest rate (`i`) expressed as a percentage per period, then after (`n`) compounding periods the total amount accrued is:
`A = P(1 + i)^n`
When plotted on a graph, the total amount accrued is shown to grow exponentially.
Confounding exists in situations where there may be causation, but the change may also be caused by one or more uncontrolled variables whose effects cannot be disentangled from the effect of the response variable.
A graph is connected if there is a path between each pair of vertices. A bridge is an edge in a connected graph that, if removed, leaves a graph disconnected.
Consumer Price Index
The Consumer price index (CPI) is a measure of changes, over time, in retail prices of a constant basket of goods and services representative of consumption expenditure by resident households in Australian metropolitan areas.
Data associated with a continuous variable is called continuous data.
A continuous variable is a numerical variable that can take any value that lies within an interval. In practice, the values taken are subject to accuracy of the measurement instrument used to obtain these values.
Examples include height, reaction time, temperature and systolic blood pressure.
Correlation is a measure of the strength of the linear relationship between two variables.
The correlation coefficient (`r`) is a measure of the strength of the liner relationship between a pair of variables.
In any right-angled triangle,
`cosθ = (adjacent )/(hypoten use)`
For a triangle of side lengths `a`, `b` and `c` and angles `A`, `B` and `C`, the cosine rule states that:
critical path analysis (CPA)
A project often involves many related activities, some of which cannot be started until one or more earlier tasks have been completed. One way of scheduling such activities that takes this into account is to construct a network diagram.
Critical path analysis is a method for determining the longest path, the critical path, in such a network and hence the minimum time in which the project can be completed. There may be more than one critical path in the network.
cut - in a flow network
In a flow network, a cut is a partition of the vertices of a graph into two separate groups with the source in one group and the sink in the other.
The capacity of the cut is the sum of the capacities of the cut edges directed from source to sink. Cut edges directed from sink to source are ignored.
A cycle is a closed walk that begins and ends at the same vertex and in which has no repeated edges or vertices except the first.
degree of a vertex
In a graph, the degree of a vertex is the number of edges incident with the vertex, with loops counted twice. It is denoted deg v.
directed graph- digraph
A directed graph is a diagram comprising points, called vertices, joined by directed lines called arcs. The directed graphs are commonly called digraphs.
Discrete data is data associated with a discrete variable. Discrete data is sometimes called count data.
A discrete variable is a numerical variable that can take only integer values.
Examples include the number of people in a car, the number of decayed teeth in an 18-year-old male etc.
earliest starting time (EST)
Using a forward scan of a network diagram to determine the earliest time an activity can begin.
In a graph, an edge is a line that connects tow vertices.
effective annual rate of interest
The effective annual rate of interest `i_(effective)` is used to compare the interest paid on loans, or investments, with the same nominal annual interest rate `i` but with different compounding periods, daily, fortnightly, monthly, quarterly, annually, other. If the number of compounding periods per annum is `n`, then:
For a connected planar graph, Euler’s rule states that:
`v + f – e = 2`
where `v` is the number vertices, `e` the number of edges and `f` is the number of faces.
A connected graph is Eularian if it has a closed trail, starts and ends at the same vertex, that is, includes every edge once only; such a trail is called a Eulerian trail. An Eularian trail may include repeated vertices. A connected graph is semi‐Eularian if there is an open trail that includes every edge once only.
When investigating relationships in bivariate data, the explanatory variable, independent variable, is the variable used to explain or predict a difference in the response variable, dependent variable; for example, when investigating the relationship between the temperature of a loaf of bread and the time it has spent in a hot oven, temperature is the response variable and time is the explanatory variable.
In the context of fitting a linear relationship between two variables, extrapolation occurs when the fitted model is used to make predictions using values of the explanatory variable that are outside the range of the original data. Extrapolation is a dangerous process as it can sometimes lead to quite erroneous predictions.
The faces of a planar graph are the regions bounded by the edges including the outer infinitely large region.
first-order linear recurrence relation
A first-order linear recurrence relation is defined by the rule:
`t_0=a,t_(n+1)=bt_n+c`, for `n ≥ 1`
For example, the rule `t_0=10,t_n=5t_(n-1)+1`, for `n ≥ 1` is a first-order recurrence relation.
The sequence generated by this rule starting at `t_0` is: 10, 51, 256, … as shown below.
flat rate depreciation
In flat rate or straight-line depreciation, the value of an asset is depreciated by a fixed amount each year. Usually, this amount is specified as a fixed percentage of the original cost.
Is the amount of time that a task in a project network can be delayed without causing a delay to subsequent tasks. All activities on a critical path have zero floats.
A flow network is a directed graph where each edge has a capacity, e.g. 100 cars per hour, 800 litres per minute etc., and each edge receives a flow. The amount of flow on an edge cannot exceed the capacity of the edge. A flow must satisfy the restriction that the amount of flow into a node equals the amount of flow out of it, except when it is a source, which has more outgoing flow, or a sink, which has more incoming flow. A flow network can be used to model traffic in a road system, fluids in pipes, currents in an electrical circuit, or any situation in which something travels through a network of nodes.
A food web, or food chain, depicts feeding connections, who eats whom, in an ecological community.
geometric growth or decay
A sequence displays geometric growth or decay when each term is some constant multiple, greater or less than one, of the preceding term: a multiple greater than one corresponds to growth, a multiple less than one corresponds to decay, e.g. 1, 2, 4, … displays geometric growth because each term is double the previous term, 100, 10, 0.1, … displays geometric decay because each term is one tenth of the previous term.
A sequence of numbers where each term after the first is found by multiplying the previous term by a fixed non-zero number (excluding ±1) called the common ratio, e.g. 2, 6, 18, ... is a geometric sequence with first term 2 and common ratio 3;
by inspection of the sequence, the rule for the nth term of this sequence is:
`t_n=2×3^((n-1))`, `n>= 1`
if `t_n` is used to denote the nth term in the sequence, then a recursion relation that will generate this sequence is:
`t_1=2`, `t_n+1=3t_n`, `n>=1`
Goods and Services Tax (GST)
The Goods and Services Tax (GST) is a broad sales tax of 10% on most goods and services and other items sold or consumed in Australia.
The gradient of a line describes its steepness, incline, or grade.
Gradient is normally described by the ratio of the "rise" divided by the "run" between two points on a line.
A graph is a diagram that consists of a set of points, called vertices, that are joined by a set of lines called edges. Each edge joins two vertices. A loop is an edge in a graph that joins a vertex in a graph to itself. Two vertices are adjacent if they a joined by an edge. Two or more edges connect the same vertices are called multiple edges.
A Hamiltonian cycle is a cycle that includes each vertex in a graph, except the first, once only.
A Hamiltonian path is path that includes every vertex in a graph once only. A Hamiltonian path that begins and ends at the same vertex is a Hamiltonian cycle.
Heron’s rule is a rule for determining the area of a triangle given the length of its sides. The area `A` of a triangle of side lengths `a`, `b` and `c` is given by:
where `s= (a+b+c)/2`
The Hungarian algorithm is used to solve assignment, allocation, problems.
In the context of fitting a linear relationship between two variables, interpolation occurs when the fitted model is used to make predictions using values of the explanatory variable that lie within the range of the original data.
irregular variation or noise, time series
Irregular variation or noise is erratic and short‐term variation in a time series that is the product of chance occurrences.
latest starting time (LST)
Using a backward scan of a network diagram to determine the latest time an activity can begin.
In fitting a straight-line `y = a + bx` to the relationship between a response variable y and an explanatory variable x, the least-‐squares line is the line for which the sum of the squared residuals is the smallest.
linear growth or decay, sequence
A sequence displays linear growth or decay when the difference between successive terms is constant. A positive constant difference corresponds to linear growth while a negative constant difference corresponds to decay.
The sequence, 1, 4, 7, … displays linear growth because the difference between successive terms is 3.
The sequence, 100, 90, 80, … displays linear decay because the difference between successive terms is –10. By definition, arithmetic sequences display linear growth or decay.
length of a walk
The length of a walk is the number of edges it includes.
The arithmetic mean, `x ̅` of a list of numbers is the sum of the data values divided by the number of values in the list.
In everyday language, the arithmetic mean is commonly called the average.
The median is the value in a set of ordered set of data values that divides the data into two parts of equal size. When there are an odd number of data values, the median is the middle value. When there is an even number of data values, the median is the arithmetic mean of the two central values.
minimum cut-maximum flow theorem
The minimum cut-maximum flow theorem states that in a flow network, the maximum flow from the source to the sink is equal to the capacity of the minimum cut.
The mode is the most frequently occurring value in a data set.
In a time series, a simple moving average is a method used to smooth the time series whereby each observation is replaced by a simple average of the observation and its near neighbours. This process reduces the effect of non-typical data and makes the overall trend easier to see.
The word network is frequently used in everyday life, e.g. television network, rail network, social network etc. Weighted graphs or digraphs can often be used to model such networks.
An outlier in a set of data is an observation that appears to be inconsistent with the remainder of that set of data. An outlier is a surprising observation.
path in a graph
A path in a graph is a walk in which all of the edges and all the vertices are different. A path that starts and finishes at different vertices is said to be open, while a path that starts and finishes at the same vertex is said to be closed. A cycle is a closed path.
A perpetuity is a compound interest investment from which payments are made on a regular basis in perpetuity, forever. This is possible because the payments made at the end of each period exactly equal the interest earned during that period.
A planar graph is a graph that can be drawn in the plane. A planar graph can always be drawn so that no two edges cross.
price to earnings ratio of a share
The price to earnings ratio of a share (P/E ratio) is defined as:
`P/E ratio= (Market price per share)/(An n ual e a rni ngs per share)`
An algorithm for determining a minimum spanning tree in a connected weighted graph.
The square of the hypotenuse of a right-angled triangle equals the sum of the squares of the lengths of the other two sides. As a rule: `c^2=a^2+b^2`, where c is the length of the hypotenuse.
reasoned argument or conclusion
One that is sound, well-groomed, considered and thought out
A recurrence relation is an equation that recursively defines a sequence; that is, once one or more initial terms are given, each further term of the sequence is defined as a function of the preceding terms.
The repeated application of a recursive procedure or definition.
reducing balance depreciation
In reducing balance depreciation, the value of an asset is depreciated by a fixed percentage of its value each year, until the asset has no residual value after a defined number of years. Reducing balance depreciation is sometimes called diminishing value depreciation.
reducing balance loan
A reducing balance loan is a compound interest loan where the loan is repaid by making regular payments and the interest paid is calculated on the amount still owing, the reducing balance of loan, after each payment is made.
The difference between the observed value and the value predicted by a statistical model; for example, by a least-squares line.
A residual plot is a scatterplot with the residual values shown on the vertical axis and the explanatory variable shown on the horizontal axis. Residual plots are useful in assessing the fit of the statistical model; for example, by a least‐squares line.
Also known as the dependent variable; its value is dependent on the value of the explanatory, or independent, variable.
round-robin sporting competition
A single round robin sporting competition is a competition in which each competitor plays each other competitor once only.
A number that scales, or multiplies, some quantity. In the equation `y = kx`, `k` is the scale factor for `x`; if two or more figures are similar, their sizes can be compared. The scale factor is the ratio of the length of one side on one figure to the length of the corresponding side on the other figure. It is a measure of magnification; the change of size.
A two-dimensional data plot using Cartesian co-ordinates to display the values of two variables in a bivariate data set.
A term used to describe a time series from which periodic variations due to seasonal effects have been removed.
seasonal index, indices
The seasonal index can be used to remove seasonality from data. An index value is attached to each period of the time series within a year. For the seasons of the year, Summer, Autumn, Winter, Spring, there are four separate seasonal indices; for months, there are 12 separate seasonal indices, one for each month, and so on. There are several methods for determining seasonal indices.
A regular rise and fall in the time series that recurs each year. Seasonal variation is measured in terms of a seasonal index.
A sequence is an ordered list of numbers, or objects.
For example, 1, 3, 5, 7 is a sequence of numbers that differs from the sequence 3, 1, 7, 5 as order matters.
A sequence maybe finite; for example, 1, 3, 5, 7, the sequence of the first four odd numbers, or infinite; for example, 1, 3, 5, … the sequence of all odd numbers.
Two geometric figures are similar if they are of the same shape but not necessarily of the same size.
A simple graph has no loops or multiple edges.
In any right-angled triangle,
`sinθ= (opposite )/(hypoten use)`
For a triangle of side lengths `a`, `b` and `c` and angles `A`, `B` and `C`, the sine rule states that:
`a/(sinA) =b/(sinB) =c/(sinC)`
Simple interest is the interest (`I`) accumulated when the interest payment in each period is a fixed fraction of the principal, e.g. if the principle (`P`) earns simple interest at the rate (`R`) expressed as a percentage per period, then after (`T`) periods the accumulated simple interest is:
When plotted on a graph, the total amount accrued is shown to grow linearly.
A spanning tree is a subgraph of a connected graph that connects all vertices and is also a tree.
The standard deviation is a measure of the variability or spread of a data set. It gives an indication of the degree to which the individual data values are spread around their mean.
The standard deviation of 𝑛 observations `x_1,x_2,…,x_n` is:
`s=sqrt((Σ(x_i-x ̅ )^2)/(n-1))`
In any right-angled triangle,
`tanθ = (opposite)/(adjacent)`
Values of a variable recorded, usually at regular intervals, over a period of time. The observed movement and fluctuations of many such series comprise long‐term trend, seasonal variation, and irregular variation or noise.
time series plot
The graph of a time series with time plotted on the horizontal axis.
Trend, time series
Trend is the term used to describe the general direction of a time series, increasing or decreasing, over a long period of time.
A trail is a walk in which no edge is repeated.
A tree is a connected graph with no cycles.
True bearings are measured in degrees in a clockwise direction from the north line. Three figures are used to specify the direction. Thus, north is specified as 000°T, east is specified as 090°T, south-east is specified as 135°T.
two-way frequency table
A two-way frequency table is commonly used for displaying the two-way frequency distribution that arises when a group of individuals or objects are categorised according to two criteria.
The row and column totals represent the total number of observations in each row and column and are sometimes called row sums or column sums.
If the table is ‘percentaged’ using row sums the resulting percentages are called row percentages. If the table is ‘percentaged’ using column sums the resulting percentages are called column percentages.
unit cost depreciation
In unit cost depreciation, the value of an asset is depreciated by an amount related to the number of units produced by the asset during the year.
walk in a graph
A walk in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. A walk that starts and finishes at different vertices is said to be an open walk. A walk that starts and finishes at the same vertex is said to be closed walk.
A weighted graph is a graph in which each edge is labelled with a number used to represent some quantity associated with the edge. For example, if the vertices represent towns, the weights on the edges may represent the distances in kilometres between the towns.