# riskyr

### A toolbox for rendering risk literacy more transparent

Risk-related information – like the prevalence of conditions, the
sensitivity and specificity of diagnostic tests, or the effectiveness of
interventions or treatments – can be expressed in terms of frequencies
or probabilities. By providing a toolbox of corresponding metrics and
representations, **riskyr** computes, translates, and visualizes
risk-related information in a variety of ways. Adopting multiple
complementary perspectives provides insights into the interplay between
key parameters and renders teaching and training programs on risk
literacy more transparent.

## Motivation

Solving a problem simply means representing it so as to make the solution transparent. (H.A. Simon)[1]

The goals of **riskyr** are less of a *computational* and more of a
*representational* nature: We express risk-related information in
multiple formats, facilitate the translation between them, and provide a
variety of attractive visualizations that emphasize different aspects of
risk-related scenarios. Whereas people find it difficult to understand
and compute information expressed in terms of probabilities, the same
information is easier to understand and compute when expressed in terms
of frequencies (e.g., Gigerenzer, 2002, 2014; Gigerenzer & Hoffrage,
1995). But rather than just expressing probabilities in terms of
frequencies, **riskyr** allows translating between formats and
illustrates the relationships between different representations in a
variety of ways. Switching between and interacting with different
representations fosters transparency and boosts human understanding of
risk-related information.[2]

The basic assumptions and aspirations driving the current development of
**riskyr** include:

Effective training in risk literacy requires transparent representations, smart strategies, and simple tools.

We aim to provide a set of (computational and representational) functions that facilitate various calculations, translations between formats, and enable a range of alternative views on the interplay between probabilities and frequencies.

Just as no single tool fits all tasks, no single graph illustrates all aspects of a problem. A variety of visualizations that illustrate the interplay of parameters and metrics can facilitate active and explorative learning. It is particularly helpful to view relationships from alternative perspectives and to observe the change of one parameter as a function of others.

Based on these assumptions and goals, **riskyr** provides a range of
computational and representational tools. Importantly, the objects and
functions in the **riskyr** toolbox are not isolated, but complement,
explain, and support each other. All functions and visualizations can be
used separately or explored interactively, providing immediate feedback
on the effect of changes in parameter values. By providing a variety of
customization options, users can explore and design representations of
risk-related information that fit to different tasks and meet their
personal needs and goals.

## Getting riskyr

### Installation

The current release of **riskyr** is available from
CRAN at
https://CRAN.R-project.org/package=riskyr:

```
install.packages('riskyr') # install riskyr from CRAN client
library('riskyr') # load to use the package
```

The current development version can be installed from its GitHub repository at https://github.com/hneth/riskyr/:

```
# install.packages('devtools')
devtools::install_github('hneth/riskyr')
```

### Available resources

An

**interactive online version**is available at http://riskyr.org/.The

**package documentation**is available online:- current release version: https://hneth.github.io/riskyr/
- current development version: https://hneth.github.io/riskyr/dev/

## Quick start guide

### Defining a scenario

**riskyr** is designed to address problems like the following:[3]

Screening for hustosisA screening device for detecting the clinical condition of

hustosisis developed. The current device is very good, but not perfect. We have the following information:

1. About 4% of the people of the general population suffer fromhustosis.

2. If someone suffers from hustosis, there is a chance of 80% that he or she will test positively for the condition.

3. If someone is free from hustosis, there is a chance of 5% that he or she will still test positively for the condition.Mr. and Ms. Smith have both been screened with the device:

- Mr. Smith tested positively (i.e., received a diagnosis of hustosis).

- Ms. Smith tested negatively (i.e., was judged to be free of hustosis).Please answer the following questions:

- What is the probability that Mr. Smith actually suffers from hustosis?

- What is the probability that Ms. Smith is actually free of hustosis?

#### Probabilities provided

The first challenge in solving such problems is in understanding the information that is being provided. The problem description provides three essential probabilities:

- The condition’s
*prevalence*(in the general population) is 4%:`prev = .04`

. - The device’s or diagnostic decision’s
*sensitivity*is 80%:`sens = .80`

. - The device’s or diagnostic decision’s
*false alarm rate*is 5%:`fart = .05`

, implying a*specificity*of (100% (-) 5%) = 95%:`spec = .95`

.

#### Understanding the questions asked

The second challenge here lies in understanding the questions that are
being asked — and in realizing that their answers are *not* simply the
decision’s sensitivity or specificity values. Instead, we are asked to
provide two *conditional* probabilities:

- The conditional probability of suffering from the condition given a
positive test result, aka. the
*positive predictive value*(`PPV`

). - The conditional probability of being free of the condition given a
negative test result, aka. the
*negative predictive value*(`NPV`

).

#### Translating into frequencies

One of the best tricks in risk literacy education is to translate
probabilistic information into frequencies.[4] To do this, we imagine
a representative sample of `N = 1000`

individuals. Rather than asking
about the probabilities for Mr. and Ms. Smith, we could re-frame the
questions as:

Assuming a representative sample of 1000 individuals:

- What proportion of individuals with a positive test result actually suffer from hustosis?

- What proportion of individuals with a negative test result are actually free of hustosis?

#### Using riskyr

Here is how **riskyr** allows you to view and solve such problems:

`library(riskyr) # loads the package`

#### Creating a scenario from probabilities

We define a new **riskyr** scenario (called `hustosis`

) by using the
`riskyr()`

function and entering the information provided by our problem
as its arguments:

```
hustosis <- riskyr(scen_lbl = "Example",
cond_lbl = "Hustosis",
dec_lbl = "Screening",
popu_lbl = "Sample",
N = 1000, # population size
prev = .04, sens = .80, spec = (1 - .05) # 3 probabilities
)
```

By providing the argument `N = 1000`

we define the scenario for a target
population of 1000 people. If we leave this parameter unspecified (or
`NA`

), the `riskyr()`

function will automatically pick a suitable value
of `N`

.

#### Summary

To obtain a quick overview of key parameter values, we ask for the
`summary`

of `hustosis`

:

`summary(hustosis) # summarizes key parameter values: `

The summary distinguishes between probabilities, frequencies, and
accuracy information. In `Probabilities`

we find the answer to both of
our questions that take into account all the information provided above:

The conditional probability that Mr. Smith actually suffers from hustosis given his positive test result is 40% (as

`PPV = 0.400`

).The conditional probability that Ms. Smith is actually free of hustosis given her negative test result is 99.1% (as

`NPV = 0.991`

).

If find these answers surprising, you are an ideal candidate for
additional insights into the realm of risk literacy. A key component of
**riskyr** is to analyze and view a scenario from a variety of different
perspectives. To get you started immediately, we only illustrate some
introductory commands here and focus on different types of
visualizations. (Call `riskyr.guide()`

for various vignettes that
provide more detailed information.)

#### Creating a scenario from frequencies

Rather than defining our `hustosis`

scenario by providing 3 essential
probabilities (`prev`

, `sens`

, and `spec`

), we could define the same
scenario by providing 4 essential frequencies (`hi`

, `mi`

, `fa`

, and
`cr`

) as follows:

```
hustosis_2 <- riskyr(scen_lbl = "Example",
cond_lbl = "Hustosis",
dec_lbl = "Screening",
popu_lbl = "Sample",
hi = 32, mi = 8, fa = 48, cr = 912 # 4 key frequencies
)
```

As we took the values of these frequencies from the `summary`

of
`hustosis`

, the `hustosis_2`

scenario should contain exactly the same
information as
`hustosis`

:

```
all.equal(hustosis, hustosis_2) # do both contain the same information?
#> [1] TRUE
```

### Visualizations

Various visualizations of **riskyr** scenarios can be created by a range
of plotting functions.

#### Prism plot

The default type of plot used in **riskyr** is a *prism plot* (or
network diagram) that shows key frequencies of a scenario as nodes and
key probabilities as edges linking the nodes:

```
plot(hustosis) # default plot
# => internally calls plot_prism(...) with many additional arguments:
# plot(hustosis, type = "prism", by = "cddc", area = "no", f_lbl = "num", p_lbl = "mix")
```

#### Tree diagram

A *tree diagram* is the upper half of a prism plot, which can be
obtained by plotting a scenario with 1 of 3 perspectives:

- by condition (
`by = "cd"`

), to split the population into*TRUE*vs.*FALSE*(`cond_true`

vs.`cond_false`

) cases; - by decision (
`by = "dc"`

), to split the population into*negative*vs.*positive*(`dec_neg`

vs.`dec_pos`

) decisions; - by accuracy (
`by = "ac"`

), to split the population into*correct*vs.*incorrect*(`dec_cor`

vs.`dec_err`

) decisions.

For instance, the following command plots a frequency tree by decisions:

`plot(hustosis, by = "dc") # plot a tree diagram (by decision)`

This particular tree splits the population of `N = 1000`

individuals
into two subgroups *by decision* (`by = "dc"`

) and contains the answer
to the second (frequency) version of our questions:

- The proportion of individuals with a positive test result who
actually suffer from hustosis is the frequency of “true positive”
cases (shown in darker green) divided by “decision positive” cases
(shown in purple):
`32/80 = .400`

(corresponding to our value of`PPV`

above). - The proportion of individuals with a negative test result who are
actually free from hustosis is the frequency of “true negative”
cases (shown in lighter green) divided by “decision negative” cases
(shown in blue):
`912/920 = .991`

(corresponding to our value of`NPV`

above, except for minimal differences due to rounding).

Of course, the frequencies of these ratios were already contained in the
`hustosis`

summary above. But the representation in the form of a tree
diagram makes it easier to understand the decomposition of the
population into subgroups and to see which frequencies are required to
answer a particular question.

#### Frequency net

A new type of visualization combines elements from 2x2 tables with those
of tree or double tree diagrams. The *frequency net* (Binder et al.,
2020) is similar to a 2x2 table insofar as its perspectives (shown by
arranging marginal frequencies in a vertical vs. horizontal fashion) do
not suggest an order or dependency (in contrast to trees or mosaic
plots). Additionally, the frequency net allows showing 3 kinds of
(*marginal*, *conditional*, and *joint*) probabilities:

```
plot(hustosis, type = "fnet", by = "cddc",
f_lbl = "namnum") # plot frequency net
```

See the `plot_fnet()`

function for options and details.

#### Icon array

An icon array shows the classification result for each of `N = 1000`

individuals in our population:

`plot(hustosis, type = "icons") # plot an icon array `

While this particular icon array is highly regular (as both the icons
and classification types are ordered), **riskyr** provides many
different versions of this type of graph. This allows viewing the
probability of diagnostic outcomes as either frequency, area, or density
(see `?plot_icons`

for details and examples).

#### Area plot

An area plot (or mosaic plot) offers a way of expressing classification
results as the relationship between areas. Here, the entire population
is represented as a square and the probability of its subgroups as the
size of rectangles (see `?plot_area`

for details and examples):

`plot(hustosis, type = "area") # plot an area/mosaic plot (by = "cddc")`

#### Table plot

When not scaling the size of rectangles by their relative frequencies or
probabilities, we can plot basic scenario information as a 2-by-2
confusion (or contingency) table (see `?plot_tab`

for details and
examples):

`plot(hustosis, type = "table") # plot 2x2 confusion table (by = "cddc") `

#### Bar plot

A *bar plot* allows comparing relative frequencies as the heights of
bars (see `?plot_bar`

for details and
examples):

`plot(hustosis, type = "bar", f_lbl = "abb") # plot bar chart (by "all" perspectives): `

#### Curves

By adopting a functional perspective, we can ask how the values of some
probabilities (e.g., the predictive values `PPV`

and `NPV`

) change as a
function of another (e.g., the condition’s prevalence `prev`

, see
`?plot_curve`

for details and
examples):

`plot(hustosis, type = "curve", uc = .05) # plot probability curves (by prevalence):`

#### Planes

When parameter values systematically depend on two other parameters, we
can plot this as a plane in a 3D cube. The following graph plots the
`PPV`

as a function of the sensitivity (`sens`

) and specificity (`spec`

)
of our test for a given prevalence (`prev`

, see `?plot_plane`

for
details and
examples):

`plot(hustosis, type = "plane") # plot probability plane (by sens x spec):`

The L-shape of this plane reveals a real problem with our current test:
Given a prevalence of 4% for hustosis in our target population, the
`PPV`

remains very low for the majority of the possible range of
sensitivity and specificity values. To achieve a high `PPV`

, the key
requirement for our test is an extremely high specificity. Although our
current specificity value of 95% (`spec = .95`

) may sound pretty good,
it is still not high enough to yield a `PPV`

beyond 40%.

### Using existing scenarios

As defining your own scenarios can be cumbersome and the literature is
full of risk-related problems (often referred to as “Bayesian
reasoning”), **riskyr** provides a set of — currently 24 — pre-defined
scenarios (stored in a list `scenarios`

). Here, we provide an example
that shows how you can select and explore them.

#### Selecting a scenario

Let us assume you want to learn more about the controversy surrounding
screening procedures of prostate-cancer (known as PSA screening).
Scenario 10 in our collection of `scenarios`

is from an article on this
topic (Arkes & Gaissmaier, 2012). To select a particular scenario,
simply assign it to an R object. For instance, we can assign Scenario 10
to `s10`

:

`s10 <- scenarios$n10 # assign pre-defined Scenario 10 to s10`

#### Scenario summary

Our selected scenario object `s10`

is a list with 30 elements, which
describe it in both text and numeric variables. The following commands
provide an overview of `s10`

in text form:

```
s10$scen_lbl # a descriptive label
#> [1] "PSA test (patients)"
s10$cond_lbl # the current condition
#> [1] "Prostate cancer"
s10$dec_lbl # the current decision
#> [1] "PSA-Test"
s10$popu_lbl # the current population
#> [1] "Male patients with symptoms"
s10$scen_apa # scenario source (APA)
#> [1] "Arkes, H. R., & Gaissmaier, W. (2012). Psychological research and the prostate-cancer screening controversy. Psychological Science, 23(6), 547--553."
# summary(s10) # summarises a scenario
```

Generating some **riskyr** plots allows a quick visual exploration of
the scenario. We only illustrate some selected plots and options here,
and trust that you will play with and explore the rest for yourself.

#### Prism plots

A tree diagram is a prism plot that views the population from only one
perspective, but provides a quick overview. In the following plot, the
boxes are depicted as squares with area sizes that are scaled by
relative frequencies (using the `area = "sq"`

argument):

```
plot(s10, type = "tree", by = "cd", area = "sq", # tree/prism plot with scaled squares
f_lbl = "def", f_lbl_sep = ":\n") # custom frequency labels
```

The prism plot (or network diagram) combines 2 tree diagrams to
simultaneously provide two perspectives on a population (see Wassner et
al., 2004). **riskyr** provides several variants of prism plots. To
avoid redundancy to the previous tree diagram, the following version
splits the population by accuracy and by decision (see the `by = "acdc"`

argument). In addition, the frequencies are represented as horizontal
rectangles (`area = "hr"`

) so that their relative width reflect the
number of people in the corresponding
subgroup:

```
plot(s10, type = "prism", by = "acdc", area = "hr", # prism plot with horizontal rectangles
p_lbl = "num") # numeric probability labels
```

#### Frequency net

Just like the 2x2 table, area plot, and prism plot, the frequency net
allows selecting two out of three perspectives. Additionally, the shape
and size of the frequency boxes can be adjusted by using the ```
area =
"sq"
```

option. The following example shows a frequency net by condition
and accuracy (`by = "cdac"`

) without the joint probabilities, with
custom settings for labels, links, and
colors:

```
plot(s10, type = "fnet", by = "cdac", # frequency net (by condition and accuracy)
area = "sq", joint_p = FALSE, arr_c = 2, # custom areas, links, and arrows
f_lbl = "namnum", p_lbl = "num", col_pal = pal_rgb) # custom labels and colors
```

#### Icon array

`plot(s10, type = "icons", arr_type = "shuffled") # plot a shuffled icon array `

#### Area plot

`plot(s10, type = "area", p_split = "v", p_lbl = "def") # plot an area/mosaic plot (with probabilities) `

#### Table plot

`plot(s10, type = "tab", p_split = "h", p_lbl = "def") # plot a 2x2 table (with probabilities)`

#### Curves

The following curves show the values of several conditional probabilities as a function of prevalence:

`plot(s10, type = "curve", what = "all", uc = .05) # plot all curves (by prev):`

Adding the argument `what = "all"`

also shows the proportion of positive
decisions (`ppod`

) and the decision’s overall accuracy (`accu`

) as a
function of the prevalence (`prev`

). Would you have predicted their
shape without seeing this graph?

#### Planes

The following surface shows the negative predictive value (NPV) as a function of sensitivity and specificity (for a given prevalence):

`plot(s10, type = "plane", what = "NPV") # plot plane (as a function of sens x spec):`

Hopefully, this brief overview managed to whet your appetite for visual
exploration. If so, call `riskyr.guide()`

for viewing the package
vignettes and obtaining additional information.

## About

**riskyr** originated out of a series of lectures and workshops on risk
literacy.

The current version (0.3.0, as of Mar. 23, 2021) is under active
development. Its primary designers are Hansjörg
Neth, Felix
Gaisbauer, Nico
Gradwohl, and Wolfgang
Gaissmaier, who
are researchers at the department of Social Psychology and Decision
Sciences at the University of
Konstanz, Germany.

The **riskyr** package is open source software written in
R and released under the
GPL 2 |
GPL 3
licenses.

The theoretical background of **riskyr** is illuminated further in the
following article:

- Neth, H., Gradwohl, N., Streeb, D., Keim, D.A., & Gaissmaier, W.
(2021). Perspectives on the 2x2 matrix: Solving semantically
distinct problems based on a shared structure of binary
contingencies.
*Frontiers in Psychology: Cognition*,*11*, 567817. doi: 10.3389/fpsyg.2020.567817

### Resources

The following resources and versions are currently available:

Type: | Version: | URL: |
---|---|---|

A. riskyr (R package): | Release version | https://CRAN.R-project.org/package=riskyr |

Development version | https://github.com/hneth/riskyr/ | |

B. riskyrApp (R Shiny code): | Online version | http://riskyr.org/ |

Development version | https://github.com/hneth/riskyrApp/ | |

C. Online documentation: | Release version | https://hneth.github.io/riskyr/ |

Development version | https://hneth.github.io/riskyr/dev/ |

### Contact

We appreciate your feedback, comments, or questions.

Please report any

**riskyr**-related issues at https://github.com/hneth/riskyr/issues/.Contact us at contact.riskyr@gmail.com with any comments, questions, or suggestions.

### Reference

To cite **riskyr** in derivations and publications please use:

- Neth, H., Gaisbauer, F., Gradwohl, N., & Gaissmaier, W. (2021).

riskyr: A toolbox for rendering risk literacy more transparent.

Social Psychology and Decision Sciences, University of Konstanz, Germany.

Computer software (R package version 0.3.0, Mar. 23, 2021).

Retrieved from https://CRAN.R-project.org/package=riskyr.

A BibTeX entry for LaTeX users is:

```
@Manual{riskyr,
title = {riskyr: A toolbox for rendering risk literacy more transparent},
author = {Hansjörg Neth and Felix Gaisbauer and Nico Gradwohl and Wolfgang Gaissmaier},
year = {2021},
organization = {Social Psychology and Decision Sciences, University of Konstanz},
address = {Konstanz, Germany},
note = {R package (version 0.3.0, Mar. 23, 2021)},
url = {https://CRAN.R-project.org/package=riskyr},
}
```

Calling `citation("riskyr")`

in the package also displays this
information.

### References

Arkes, H. R., & Gaissmaier, W. (2012). Psychological research and the prostate-cancer screening controversy.

*Psychological Science*,*23*, 547–553.Binder, K., Krauss, S., and Wiesner, P. (2020). A new visualization for probabilistic situations containing two binary events: The frequency net.

*Frontiers in Psychology*,*11*, 750. doi: 10.3389/fpsyg.2020.00750Garcia-Retamero, R., & Cokely, E. T. (2017). Designing visual aids that promote risk literacy: A systematic review of health research and evidence-based design heuristics.

*Human Factors*,*59*, 582–627.Gigerenzer, G. (2002).

*Reckoning with risk: Learning to live with uncertainty*. London, UK: Penguin.Gigerenzer, G. (2014).

*Risk savvy: How to make good decisions*. New York, NY: Penguin.Gigerenzer, G., & Gaissmaier, W. (2011). Heuristic decision making.

*Annual Review of Psychology*,*62*, 451–482. (Available online)Gigerenzer, G., Gaissmaier, W., Kurz-Milcke, E., Schwartz, L., & Woloshin, S. (2007). Helping doctors and patients make sense of health statistics.

*Psychological Science in the Public Interest*,*8*, 53–96. (Available online)Gigerenzer, G., & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: Frequency formats.

*Psychological Review*,*102*, 684–704.Hoffrage, U., Gigerenzer, G., Krauss, S., & Martignon, L. (2002). Representation facilitates reasoning: What natural frequencies are and what they are not.

*Cognition*,*84*, 343–352.Hoffrage, U., Krauss, S., Martignon, L., & Gigerenzer, G. (2015). Natural frequencies improve Bayesian reasoning in simple and complex inference tasks.

*Frontiers in Psychology*,*6*, 1473. doi: 10.3389/fpsyg.2015.01473 (Available online)Hoffrage, U., Lindsey, S., Hertwig, R., & Gigerenzer, G. (2000). Communicating statistical information.

*Science*,*290*, 2261–2262.Khan, A., Breslav, S., Glueck, M., & Hornbæk, K. (2015). Benefits of visualization in the mammography problem.

*International Journal of Human-Computer Studies*,*83*, 94–113.Kurzenhäuser, S., & Hoffrage, U. (2002). Teaching Bayesian reasoning: An evaluation of a classroom tutorial for medical students.

*Medical Teacher*,*24*, 516–521.Kurz-Milcke, E., Gigerenzer, G., & Martignon, L. (2008). Transparency in risk communication.

*Annals of the New York Academy of Sciences*,*1128*, 18–28.Micallef, L., Dragicevic, P., & Fekete, J.-D. (2012). Assessing the effect of visualizations on Bayesian reasoning through crowd-sourcing.

*IEEE Transactions on Visualization and Computer Graphics*,*18*, 2536–2545.Neth, H., & Gigerenzer, G. (2015). Heuristics: Tools for an uncertain world. In R. Scott & S. Kosslyn (Eds.),

*Emerging trends in the social and behavioral sciences*. New York, NY: Wiley Online Library. doi: 10.1002/9781118900772.etrds0394 (Available online)Neth, H., Gradwohl, N., Streeb, D., Keim, D.A., & Gaissmaier, W. (2021). Perspectives on the 2x2 matrix: Solving semantically distinct problems based on a shared structure of binary contingencies.

*Frontiers in Psychology*,*11*, 567817. doi: 10.3389/fpsyg.2020.567817 (Available online)Sedlmeier, P., & Gigerenzer, G. (2001). Teaching Bayesian reasoning in less than two hours.

*Journal of Experimental Psychology: General*,*130*, 380–400.Wassner, C., Martignon, L., & Biehler, R. (2004). Bayesianisches Denken in der Schule.

*Unterrichtswissenschaft*,*32*, 58–96.

(`README.Rmd`

updated on 2021-03-23 by hn.)

Simon, H.A. (1996).

*The Sciences of the Artificial*(3rd ed.). The MIT Press, Cambridge, MA. (p. 132).To clarify our notion of “risk” in this context, we need to distinguish it from its everyday usage as anything implying a chance of danger or harm.

In basic research on judgment and decision making and the more applied fields of risk perception and risk communication, the term*risk*typically refers to decisions or events for which the options and their consequences are known and probabilities for all possible outcomes can be provided.

For our present purposes, the notion of risk-related information refers to any scenario in which some events of interest are determined by probabilities. While it is important that quantitative (estimates of) probabilities are provided, their origin, reliability and validity is not questioned here. Thus, the probabilities provided can be based on clinical intuition, on recordings of extensive experience, or on statistical simulation models (e.g., repeatedly casting dice and counting the frequencies of outcomes).

This notion of*risk*is typically contrasted with the much wider notion of*uncertainty*in which options or probabilities are unknown or cannot be quantified. (See Gigerenzer and Gaissmaier, 2011, or Neth and Gigerenzer, 2015, on this conceptual distinction and corresponding decision strategies.)See Gigerenzer (2002, 2014), Gigerenzer and Hoffrage, U. (1995), Gigerenzer et al. (2007), and Hoffrage et al. (2015) for scientific background information and similar problems. See Sedlmeier and Gigerenzer (2001) and Kurzenhäuser and Hoffrage (2002) for related training programs (with remarkable results), and Micallef et al. (2012) and Khan et al. (2015) for (rather sceptical and somewhat sobering) studies on the potential benefits of static representations for solving Bayesian problems.

See Gigerenzer and Hoffrage (1995) and Hoffrage et al. (2000, 2002) on the concept of

*natural frequencies*.