Changes

In situations such as this model inputs may be best defined as probability distributions rather than an individual number. When more than one probability distribution is used in repeated simulations to calculate an objective outcome this is referred to as a Monte-Carlo experiment. This works by repeatedly and randomly selecting outcomes from the distributions of the various inputs and applying them to the model. Each time this is done, a possible outcome of the model is created. The more times this process is repeated the more observations from the total set of possible outcomes are calculated, until eventually enough data is collected to piece together a probability-distribution of the model output.

In situations such as this model inputs may be best defined as probability distributions rather than an individual number. When more than one probability distribution is used in repeated simulations to calculate an objective outcome this is referred to as a Monte-Carlo experiment. This works by repeatedly and randomly selecting outcomes from the distributions of the various inputs and applying them to the model. Each time this is done, a possible outcome of the model is created. The more times this process is repeated the more observations from the total set of possible outcomes are calculated, until eventually enough data is collected to piece together a probability-distribution of the model output.

[[File:Monte.jpg|center|thumb|910x910px|Monte-Carlo Simulation]]

[[File:Monte.jpg|thumb|1041x1041px|Monte-Carlo Simulation|alt=|none]]

Two of these are discrete distributions and three are continuous.  The continuous distributions used are the normal, triangle and the flat (continuous uniform).  The discrete distributions are defined (discrete) and fixed (one non-varying constant). The probability distributions need to be selected from a pull-down option from the column under the heading “'''Distribution Type'''” ('''column D''') on the '''Model Inputs''' sheet ('''Figure 2 – Area “B”''').

Two of these are discrete distributions and three are continuous.  The continuous distributions used are the normal, triangle and the flat (continuous uniform).  The discrete distributions are defined (discrete) and fixed (one non-varying constant). The probability distributions need to be selected from a pull-down option from the column under the heading “'''Distribution Type'''” ('''column D''') on the '''Model Inputs''' sheet ('''Figure 2 – Area “B”''').

[[File:Fig 2.png|thumb|774x774px|Figure 2: Setting Distribution Details|alt=|center]]

[[File:Fig 2.png|Figure 2: Setting Distribution Details|alt=|none|frame]]

The final stage in defining a distribution is entering in their parameters. These parameters differ depending on the type of distribution selected and are displayed in the table above the columns where they are entered ('''columns E – F – G). ''' When a distribution type is selected areas that do not need to be defined are blacked out.  For instance, in '''Figure 2''' the normal distribution has been selected.  As a result, in '''area “C”''' where the distribution parameters are entered the first parameter has been blackened out.  From the table above you can see that the mean of the distribution needs to be entered in '''column F''' and the variation needs to be entered in '''column G'''.

The final stage in defining a distribution is entering in their parameters. These parameters differ depending on the type of distribution selected and are displayed in the table above the columns where they are entered ('''columns E – F – G). ''' When a distribution type is selected areas that do not need to be defined are blacked out.  For instance, in '''Figure 2''' the normal distribution has been selected.  As a result, in '''area “C”''' where the distribution parameters are entered the first parameter has been blackened out.  From the table above you can see that the mean of the distribution needs to be entered in '''column F''' and the variation needs to be entered in '''column G'''.

A more detailed description of the probability distributions and the required parameters to define them follows.

A more detailed description of the probability distributions and the required parameters to define them follows.

===== '''Normal''' =====

===== '''Normal''' =====

One of the most familiar continuous probability distributions is the normal distribution as shown in Figure 2.  The normal distribution is also known as the Gaussian distribution or the bell curve. It is a very common distribution in nature and statistics and is based on the central tendency theory.  This is a situation where most items in a group tend towards the average and the further away from the average an event is the less likely it is to occur.  

One of the most familiar continuous probability distributions is the normal distribution as shown in Figure 2.  The normal distribution is also known as the Gaussian distribution or the bell curve. It is a very common distribution in nature and statistics and is based on the central tendency theory.  This is a situation where most items in a group tend towards the average and the further away from the average an event is the less likely it is to occur.  

[[File:Figure 3.svg|center|thumb|523x523px|Figure 3: Normal Distributions]]

[[File:Figure 3.svg|Figure 3: Normal Distributions|alt=|none|frame]]

To define a normal distribution only two parameters are needed: the mean and the variance (standard deviation).

To define a normal distribution only two parameters are needed: the mean and the variance (standard deviation).

[[File:Fig 4.png|center|thumb|517x517px|Figure 4: Flat (continuous uniform) Distribution]]

[[File:Fig 4.png|thumb|668x668px|Figure 4: Flat (continuous uniform) Distribution|alt=|none]]

 

As can be seen Figure 4, to define this distribution only 2 parameters are required.  They are '''a''' (the minimum possible outcome) and '''b''' (the maximum possible outcome).

As can be seen Figure 4, to define this distribution only 2 parameters are required.  They are '''a''' (the minimum possible outcome) and '''b''' (the maximum possible outcome).

===== '''Triangular''' =====

===== '''Triangular''' =====

The triangular distribution is typically used as a subjective description of a population for which there is only limited sample data, and especially in cases where the relationship between variables is known but data is scarce (possibly because of the high cost of collection). It is based on knowledge of the minimum and maximum and an "inspired guess" as to the modal value.

The triangular distribution is typically used as a subjective description of a population for which there is only limited sample data, and especially in cases where the relationship between variables is known but data is scarce (possibly because of the high cost of collection). It is based on knowledge of the minimum and maximum and an "inspired guess" as to the modal value.

[[File:Fig 5.png|center|thumb|577x577px|Figure 5; Triangular Distribution]]

[[File:Fig 5.png|thumb|648x648px|Figure 5; Triangular Distribution|alt=|none]]

The defined distribution is essentially a discrete distribution where any number of sets of outcomes and their probabilities can be created – as long as the probabilities of all outcomes sum to 1.  This can be useful in situations where there is a variable or outcome associated with a worse case scenario (pessimistic), a most likely scenario (realistic), and a best-case scenario (optimistic) as can be seen in Figure 6.  

The defined distribution is essentially a discrete distribution where any number of sets of outcomes and their probabilities can be created – as long as the probabilities of all outcomes sum to 1.  This can be useful in situations where there is a variable or outcome associated with a worse case scenario (pessimistic), a most likely scenario (realistic), and a best-case scenario (optimistic) as can be seen in Figure 6.  

[[File:Fig 6.png|center|frame|Figure 6 Discrete Defined Distribution]]

[[File:Fig 6.png|Figure 6 Discrete Defined Distribution|alt=|none|thumb]]

In the above example (Figure 7) a defined distribution has been used with a low population growth rate of 1% having a 40% probability, medium population growth rate of 3% at 50% probability, and a high population growth rate of 5% at 10% probability.  Each time a simulation is run either 1%, 3%, or 5% will be entered into cell C9.  To make it easier to reference this value in related formulas the cell is named “'''POPGrow”.''' After the cell has been named it is now possible to refer to it by its name rather than by a cell reference such as “C9”.   This will make the next stage in the process much easier and makes your spreadsheets easier to understand, both for yourself and reviewers.

In the above example (Figure 7) a defined distribution has been used with a low population growth rate of 1% having a 40% probability, medium population growth rate of 3% at 50% probability, and a high population growth rate of 5% at 10% probability.  Each time a simulation is run either 1%, 3%, or 5% will be entered into cell C9.  To make it easier to reference this value in related formulas the cell is named “'''POPGrow”.''' After the cell has been named it is now possible to refer to it by its name rather than by a cell reference such as “C9”.   This will make the next stage in the process much easier and makes your spreadsheets easier to understand, both for yourself and reviewers.

'''Note 2:  RRAT already contains several named cells that are “reserved”.  No data check is performed to preserve this reserved status but changing them is likely to result in RRAT failure these names are:'''  

'''Note 2:  RRAT already contains several named cells that are “reserved”.  No data check is performed to preserve this reserved status but changing them is likely to result in RRAT failure these names are:'''  

'''In addition, cell references can not contain a space.'''  

'''In addition, cell references can not contain a space.'''  

=== 1.1.1   Entering a Distribution by Form ===

==== Entering a Distribution by Form ====

As an alternative to manually entering a distribution directly into the spreadsheet you can add a distribution by form.  The form can be opened by pressing the “Manage Assumptions” button as in Figure 7 (below the picture of a mouse). This will open the form shown in Figure 8.

As an alternative to manually entering a distribution directly into the spreadsheet you can add a distribution by form.  The form can be opened by pressing the “Manage Assumptions” button as in Figure 7 (below the picture of a mouse). This will open the form shown in Figure 8.

 

[[File:Fig 8.png|none|thumb|624x624px|Figure 8: Adding a Distribution by Form]]

 

Figure 8: Adding a distribution by form

 

The information to define a distribution can be entered in this form and added to the spreadsheet. In addition, if an entry is added to the “Reference Name” field this name will be assigned to the value cell automatically.  Also, the titles for the parameters of the distributions will change dynamically depending on what distribution type is selected to make it more clear what data is required.

The information to define a distribution can be entered in this form and added to the spreadsheet. In addition, if an entry is added to the “Reference Name” field this name will be assigned to the value cell automatically.  Also, the titles for the parameters of the distributions will change dynamically depending on what distribution type is selected to make it more clear what data is required.

To assist the user in making a defined distribution, which can have many parameters, RRAT has an additional custom form that can be used to help (See Figure 9: Creating a Defined Distribution). This form can be used by entering an outcome/probability combination and then clicking ‘Add Outcome’ repeatedly until all discrete outcomes are added. Once the probability of all outcomes totals 1, the ‘Add Distribution’ button can be clicked and the distribution will be added to the form in the required format (outcome,probability;…).

To assist the user in making a defined distribution, which can have many parameters, RRAT has an additional custom form that can be used to help (See Figure 9: Creating a Defined Distribution). This form can be used by entering an outcome/probability combination and then clicking ‘Add Outcome’ repeatedly until all discrete outcomes are added. Once the probability of all outcomes totals 1, the ‘Add Distribution’ button can be clicked and the distribution will be added to the form in the required format (outcome,probability;…).

 

[[File:Fig 9.png|none|frame|Figure 9: Creating a Defined Distribution]]

 

Figure 9: Creating a Defined Distribution

== 1.2    Stage 2: Define Formulas for Dependent Outcomes ==

== 1.2    Stage 2: Define Formulas for Dependent Outcomes ==