* # PROBLEM 6 : NRW
### Encouraging Safe, Green and Economically Sustainable use of our Natural Environment

# Ideas
- Combine rules of thumb with behavioural models.
- Classify risky locations (or risky groups of people)?
- How does spreading of damage scale?
- Create an agent-based model of a toy scenario and see how that plays out.

# Questions to Answer
1. How do we better redistribute pressures/resources?
2. Where is best to intervene to disrupt bad behaviours?
3. How do we mitigate/balance risks of negative experiences of users against negative effects on sites?
4. What do we not know?

## Modelling directions to answer these questions:
1. What types of agents do we have? What attributes do they have? (Survey, literature)
2. How do agents flow around the network? And how does this flow affect/damage the environment? (ABM)
3. Construct data driven model linking behavioural factors with eceonomic benefit. (Regression)
4. What is the affect of a big change? (Game theory)

# Useful Links
- [Google Drive](https://drive.google.com/drive/u/0/folders/1H-AjgDnthVbv6WXNDy__BTC_r8PW7SSY)
- [Mural](https://app.mural.co/t/esgi1656450/m/esgi1656450/1617885810916/3e9985d3184c0a2e4ce6597164133b948e8f8d9d)
- [Overleaf](https://www.overleaf.com/project/606d63a311df141976788b16)
- [Natural Survey for Wales](https://naturalresources.wales/evidence-and-data/research-and-reports/national-survey-for-wales/?lang=en)
- [Welsh Outdoor Recreation Survey Key Facts for Policy and Practice: Summary Report](https://cdn.cyfoethnaturiol.cymru/media/681025/welsh-outdoor-recreation-survey-key-facts-for-policy-and-practice-2016.pdf?mode=pad&rnd=131546924000000000)

- [This study has used PPGIS, but combined with qualitative data on the reasons for route choice (eg to enjoy a viewpoint, 'usual route', practical considerations, to avoid barriers etc) Could be used to build up a model of factors that would affect route choice (and therefore overuse). ](https://core.ac.uk/download/pdf/286389513.pdf)
- [This article is a bit out of date, but shows some of the sources of data that could be used in this work ](https://www.outdoorrecreation.org.uk/stories/understanding-where-people-go-in-the-countryside/)
- [This is very 'big picture' research on tourism and recreation across a large area in europe, lacks sensitivity, but might be useful for 'all-Wales' research? ](https://www.sciencedirect.com/science/article/pii/S221204161730270X)
### Google drive summary

- Newborough behaviour insights work
- Behavioural investgation into Newborough, discusses interventions for before, on arrival, and during the visit (planning, engagement, infrastructure).
- 9 interventions available for 'promoting responsible access with planning aids', 'promoting active travel with infrastruture', and 'promoting responsible dog walking'.

- Linetop reports (Visitor Counter Data)
- `Linetop Summary Data Unpivotted Data for PowerBI.xlsx` contains monthly numbers of visitors to 14 different sites in Wales, along with their activity type (mountain biking/ pedestrian/ car park/ visitor centre). This data is summarised and presented in the various subfolders. The data is from ~2003-2017. `Linetop Summary Data with Notes.xlsx` contains the same information as far as I can make out.

- GPS Paper
- Assessing the recreation service using GPS tracks downloaded from crowd-sourced websites.
- The declared visitor presence for recreation purposes from a survey was highly spatially congruent with GPS based model outputs ($R^2 = 0.89$).
- Figure 2 at the top of Page 6 could give a structure for designing a model to estimate the number of people at a park based on proximity and attractiveness.

# What do we want to get out of the model?
- The activity patterns of human beings.
- Condense damage, versus spread damage?
- Probability of unhelpful behaviours that are cite specific
- Updatable statistcal prediction model
- see Mural board...

# Keming's study

https://www.brunel.ac.uk/research/Projects/New-regression-models-for-the-analysis-of-well-being-and-income-distribution

# Groups Sticky note

## Green - People
## Orange - Literature
## Gray - NRW intervention
## Light Blue - Characteristing locations
## Purple - Using the data available
## Blue - Types of models

# Some of Luke's feedback on day 1
- toy model rather than specific
- expert elicitation would be useful in future to parametrise these models
- the green box is very helpful
- with the interventions, there are very few levers
- the concept of resilience is interesting
- site impacts are not clear (no clear rights, but lots of wrongs)
- what can be inferred from social media is interesting
- negative feedbacks is interesting (e.g. perverse consequences)
- damage and unhelpful behaviour arising from conflicts e.g. around multi-modal users such as mountain bikers
- four measures:
- local stake holders able to influence preference between the 4 measures

## NRW measures
1. Stocks of natural resources are safeguarded and enhanced Tackling overexploitation to ensure that stocks of renewable natural resources are safeguarded and enhanced to meet the needs of current and future generations. Stocks of non-renewable natural resources are used in a sustainable manner and, where depletion is unavoidable, that substitutes are put in place to meet future needs.
2. Ecosystems are resilient to expected and unforeseen change Building ecosystem resilience to safeguard and enhance supporting ecosystem services and tackling the impacts of habitat change, climate change, pollution, invasive alien species and other identified pressures
3. Wales has healthy places for people, protected from environmental risks Environmental regulation protects people from risks, such as air, water and noise pollution, flooding etc. Regulating and cultural ecosystem services are managed to increase wellbeing and provide a healthy environment for all.
4. Contributing to a circular economy with more efficient use of natural resources Reducing the environmental impact of production and consumption and our environmental footprint within Wales and internationally, while optimising benefits of provisioning ecosystem services. (2.5 planets to 1

****Does this work cover any of the 5 SMNR pressures?****
Land-use
Resource extraction
External inputs (fertilisers, chemicals)
Emissions (pollutants and waste)
Non-native species
****Does this work cover any of the 8 broad SMNR ecosystems?****
Mountains, moorland and heaths
Semi-natural grasslands
Enclosed farmland
Woodlands
Freshwaters
Urban
Coastal margins
Marine
****Does this work address any of the 4 SMNR ecosystem service groups?****
Supporting (Ecosystem resilience)
Regulating }
Provisioning } (human wellbeing)
Cultural }
****Does this work cover any Cross-cutting Themes?****
Biodiversity (inc. Geodiversity)
INNS
Climate Change
Air Quality
Resource Efficiency: Water, Energy, Waste
Land Use & Soils
Water Quality

# Agent based modelling
- Network of sites

- Toy problem
- 'Home' source of visitors/tourists
- site nodes with capacities/thresholds for damage
- distance weighted edges between nodes
- What is the timestep? what happens if visitors attend more than one site per day or move to a quieter site
- characteristics of locations - capacity, accessibility
- anti-social behaviour occurs at large visitor numbers, filled bins/car parks lead to damage
- NetworkX Python package potentially useful
- MESA Python package also
https://mesa.readthedocs.io/en/stable/


Damage per unit area is then defined as:

<!-- $$\begin{equation}
y(x)=\left\{
\begin{aligned}
-1 \quad x<0\\
0 \quad x=0\\
1 \quad x>0\\
\end{aligned}
\right
.
\end{equation}
$$ -->

Process for an individual agent:
- Each agent has an initial location. For some time period $t$ the agent takes a random walk within the activity area (some radius of its initial location).
- Once this time period $t_p$ has expired the agent chooses a new activity location with equal probability.
- The agent decides whether to take a shorter route or more responsible route based on the probability outlined below.
-- $P(\text{responsible path taken}) = \cfrac{d_v}{d_r}R$
where
-- $d_v$ is the distance the crow flies
-- $d_r$ is the distance if agent avoids protected area
-- $R$ is the responsibility factor, computed from survey data based on actions taken to protect the environment.
- If the agent decides to take an irresponsible route, damage to the protected area is computed by the formula below
-- $D_{UA}:=\sum_{\{1,2,3\}} d_i n_i$
-- $d_i$ and $n_i$ are the percentage dog ownership and number of agents in the area of type i respectively.
- On arrival at the new activity location, the process begins again

Aim: attempt to model antisocial behaviour at various sites

To tackle the question of how the public use Wales' natural environment, and how we might model anti-social behaviour, we develop an agent based model (ABM) that simulates park users moving around and interacting with a site. To enable our modelling and also provide a tool that NRW can use and develop themselves further, we use NetLogo (ref). NetLogo is a multi-agent modelling environment used in biology, social science and mathematics.

### Pretty pictures

- Park opens and particles begin to diffuse in
 

- Agents loosely cluster based on their preferred activity (red upper right, black lower left, blue no preferred region)
 

- Park will be closing soon and agents leave
 

## Littering ABM results

| Location | Month | 2 Bins | North Bin | South Bin |
| - | - | - | - | - |
| Coed-y-Brenin | December | 74 / 12 | 50 / 32 | 43 / 39 |
| Coed-y-Brenin | August | 65 / 17 | 50 / 36 | 48 / 31 |
| Cwncarn | December | 47 / 39 | 26 / 52 | 14 / 53 |
| Cwncarn | August | 32 / 45 | 26 / 50 | 15 / 51 |
Litter in bins / litter on ground, with 100 agents with ratios given by table below. Two bins does better than one. North bin does better than south, since the exit is north of cafe.

| Location | Month | Score | % Purist | % Neutralist | % Urbanist |
| - | - | - | - | - | - |
| Coed-y-Brenin | December | 0.16 | 77 | 20 | 03 |
| Coed-y-Brenin | August | 0.24 | 69 | 25 | 06 |
| Cwncarn | December | 0.73 | 11 | 33 | 56 |
| Cwncarn | August | 0.77 | 08 | 30 | 62 |

Coed-y-Brenin is much more remote, and has a large proportion of purists.
Cwncarn is more urban, and has a large proportion of urbanists.

In December a larger portion are purists.
In August a larger portion are urbanists.

# Game Theory Group

## Effect of making big changes

Think about cost-benefit of various actions by park authorities and visitors.

Each person wants to get some benefit from visiting an outdoor recreational area. But this will cause some damage and cause a cost.

eg. Need to build a train/bridge/road in a National Park

**Cost**

* Financial
* Environmental (direct) eg. destruction of area
* Environmental (indirect) eg. larger numbers of tourists, cars
* Visual impact
* Bridge may divert tourists from other areas leading to a loss of income there

**Benefit**

* More tourism leads to more income
* Focuses tourists in one area only so avoids general damage
* More control eg. can put car parks in close to the bridge
* More information
* Income from bridge tolls, train tickets etc
* Train specials eg. steam

**Players**

* Tourists (numbers of tourists: )
* Local Residents
* Local Business ($n_{Bu}$)
* non local residents
* non local business
* Park authorities (local and non-local)
* Rangers ($n_{r}$)

## Scenario:

Two national parks eg. *Brecon Beacons* and *Wye Valley* should we build a railway line to Brecon from Cardiff?

**Cost**

Can we estimate how many more tourists per day this
would lead to

* Building the train line ($x_{bt}$)
* Maintain the train line ($x_{mt}$)
* Direct environmental damage (may be reduced by reopening old lines) ($x_{ded}$)
* More tourists leads to more damage to the environment ($f(\underline{y},\underline{n_{t}})$)

**Benefit**

How much will a typical tourist spend?
* Accomodation: £500-1500 per week ($y_{i,a}$)
* Food/pubs/restaurants: £50 per day ($y_{i,f}$)
* Car hire (this will be more likely if a train is used) ($y_{i,c}$)
* Visiting attractions
* Use of the train ($y_{i,t}$)
* Shops ($y_{i,s}$)
g

**Issues**

* Wye valley gets less tourist numbers

| Action | Cost | Benefit |
|-------------------|

------

------

# Example Build a train line

N = number of tourists **per day** coming

Consider benefits and costs as a function of $N$ noting that these have to be tuned for each park.

------

## Economic benefit

**Total Happiness** Correct utility function $U(N)$


Very likely to depend upon the cohort

--------

**$O(N) \quad \quad$ benefits**

Use of train
Use of shops
Food/restaurants
Car hire
Use of amenities

--------

**$O(N^{\alpha}) \quad \quad \alpha > 1 \quad \quad$ benefits**

(or other nonlinear function)

Accomodation (look this up)
Car parks

-------

## *Environmental and other cost*

#### *Sublinear*

Wild life disruption
Motor racing
Fly tipping

--------

**$O(N)$**

Footpath damage
General car damage
Use of resources

--------

**Piece-wise linear/step function**

General crowding
Littering (bins over flowing)
Dog poo (bins over flowing)
Fouling/sanitation (toilets overflowing)
Cars in the way (overflowing car parks)


------

**$O(N^2)$**

Disruptive behaviour
Traffic jams
Infections
Loss of WiFi (urbanist rather than purist)

------

## Other costs

Taking tourists away from other places


-------

## Long term costs and benefits

* Repeat visits
* Referrals

Function of the happiness utility function $U(N)$

-------

-------

## Equations
N is the number of cars closely correlated to the number of people
N' is the number of people not using a car. $H_i$ is the happiness of a turist
\begin{equation}
H_i(N)=K\big(\alpha\big(1-wt_i(N)\big)-\beta f_l(N) -\gamma\sum_{j\neq i}\omega(\eta_i,\eta_j) -C_i \big)
\end{equation}

where:
* $wt_i$ it's the proportion waisting time in the park due to the time needed to find a car park or in queue at the toilet or at the bar
* $f_l$ is a function measuring the impact of "littering
* $\omega$ measure the impact of the interaction
* $C_i$ is the fixed cost to visit a park
* $K,\alpha,\beta$ and $\gamma$ are the parameters

Let HP be the "happiness" of the park rangers
\begin{equation}
HP(N) = K_P\sum_{i}H_i(N) - C_e(N) - C_{int} + I(N) + R(N)
\end{equation}

where
* $C_e$ represents the cost of the environment
* $C_{int}$ represents the cost of the intervation by the authorities to provide facilities to the tourists
* $I$ measures the impact of incomes from the tourists that are used to keep the park cleaned, safe etc.
* $R$ revenue (accomodation, food,...)

_____________________________________________

# Surveys and Data
- [Technical Report](https://naturalresourceswales.gov.uk/media/4000/outdoor-recreation-and-health-in-wales-technical-report.pdf?lang=en)
- [National Survey for Wales](https://cdn.cyfoethnaturiol.cymru/media/684900/national-survey-for-wales-2016-17-key-facts-for-policy-and-practice-outdoor-recreation.pdf?mode=pad&rnd=131695533890000000)
- [Summary Report](https://cdn.cyfoethnaturiol.cymru/media/681025/welsh-outdoor-recreation-survey-key-facts-for-policy-and-practice-2016.pdf?mode=pad&rnd=131546924000000000)
- [WORS Final Report](https://cdn.cyfoethnaturiol.cymru/media/4757/wales-outdoor-recreation-survey-2014-final-report.pdf?mode=pad&rnd=131456374980000000)
## Groups and Possible Conflicts
- Walkers - More frequent users, more likely to be older.
- Problems: Littering, Parking, Conflicts with other users
- Dog walkers - Frequent users, women, midde aged.
- Problems: dog poo, uncontrolled dogs
- Picnickers/Games - More infrequent users, more women, more middle aged
- Parking, littering, BBQs
- Runners - short, infrequent visits, younger
- Conflicts with other users, parking
- Cycling - More men, younger or middle aged
- trail building, conflicts with other users
- Wild Camping
- Camping Issues, Fires, BBQs, Littering
- Motorsport - More likely to be male/under 34, increase from 2011->14
- Off road driving

relation to environmental attidues and behaviour

## Main anti-social behaviour
- direct littering/vandalism: littering, fly tipping, dog poo, vandalism
- indirect: wild/vehicle camping, fires/BBQs
- dog-related: dog poo, uncontrolled dogs
- irresponsible parking
- off-road driving, vehicle camping, wild-trail building

## Wales Outdoor recreation survey data

[Wales Outdoor Recreation Survey 2014: Final Report](https://cdn.cyfoethnaturiol.cymru/media/4757/wales-outdoor-recreation-survey-2014-final-report.pdf?mode=pad&rnd=131456374980000000)
[Data Tables](https://cdn.cyfoethnaturiol.cymru/media/4460/wors-2014-full-data-tables.pdf?mode=pad&rnd=131479742070000000)

- data on general participation in outdoor activities (last 12 months and last 4 weeks) by
- gender, age, education, car access, income, working/not working, deprived areas
- frequency of outdoor visits in last 4 weeks (non-visitors/infrequent/frequent) by
- age, car access, children, dogs
- places visited in last 12 months
- e.g. village, local park, beach, forest
- travel to and from sites
- distance travelled on most recent visit (<1mile,<5,10,20,50,>50 miles)
- main mode of transport (car, walking, bike, public transport)
- activities undertaken outdoors in last 12 months by
- all activities/main activity
- e.g. picknicking, games, wildlife watching, off-road cycling, sightseeing
- duration
- total duration of most recent visit (<1,1-3,3-6,>6hrs)
- of main activity by year (<29min,<59min,<1hr59,2hrs59,<4hrs59,>5hrs)
- by activity (<3hrs, >3hrs)
- party composition
- alone/with family/friends/organised group, number of people, number of children, dogs
- motivations for visiting the outdoors
- e.g. health/exercise, exercise dog, enjoyment, hobby, peace/quite
- awareness of biodiversity
- perception of past changes to biodiversity in Wales (no change, decrease, increase) by age and frequency, future changes, concerns about changes
- actions done to protect the environment, e.g. recycling, reducing home energy, gardening for wildlife, walking/cycling rather than car

## Forestry Footfall Data
- Forestry counters are split into Walking Trails, Mountain Bike Trails, Carparks, Visitor Centres and Shops. The spreadsheet data that we have is monthly by counter.
- The data reports are quarterly and show numbers of visitors split by month, day of the week and time of day. The reports also provide the methods used for estimating the total number of visitors from the footfall data.

# Behavioral Sciences/Classifying Agents
- Paper on visitor behaviour at locations across Iceland
- Management Impications - Accessibility affects infrastructure and visitor behaviour
- Purism Scale Model - Categorizes visitors based on preferences on infrastructure, level of use etc
- Strong purists, moderate purists, neutralists, urbanists
- Important characteristics - Distance from a city, distance from a main road, quality of road, visitor numbers, infrastructure

## Results
- Length of stay longer in difficult to access areas.

## How can we use this?
- Could use characteristics of diffeent sites to work out type of visitors (on the Purism Scale Model). These "types" of visitors could then inform a model (agent-based models).

# What types of agents do we have?

Three categories of site users: urbanists, neutralists, purists. Site users are given a score between 0 and 1 based on the number of weekly visits and distance travelled to the site.These are put into 'bins' to catergorise them into urbanists (U), neutralists (N) and purists ( P).

Definition of urbanists and purists is based on scale defined in Tverijonaite et al. 2018.

We also give each park a score, $S$, between 0 and 1 which determines the distribution of park visitors. We assume the distrubution is a normal distribution centred at $S$,
We propose the following simple formula to classify each park with a park score $S$:

\begin{equation}
\text{S} = \alpha_1 d +\alpha_2 v +\alpha_3 c,
\end{equation}
where $d$, $v$, and $f$ are all between $0$ and $1$, and represent: distance to the nearest city; estimated density of visitors on a given day; number of toilets; if shop/visitor centre exists; number of parking spaces. Weightings $\alpha_n$ can be adjusted, ensuring $\sum_n \alpha_n =1$.

$v$ is calculated using the predicted visitors for the day normalised by max daily visitors for any park (1200 visitors)

$d$ is calulated

The "Score" for each park will then be a number between $0$ and $1$, and will allow us to categorize the parks. Now, say we have 3 types of agent, Purist, Neutralist, and Urbanist, we can classify a distribution of agents for each type of park.

## park score to agent type
We plot a normal distrubution with mean $S$. Integrating over three regions gives the proportions of agents that are purists, $P$, neutralists, $N$, and urbanists, $U$.


In the example above we have a park of score 0.8. Integrating 0 to 0.3, 0.3 to 0.8, and 0.8 to 1 and normalising gives
$P = 0.007$
$N = 0.359$
$U = 0.633$

These could then be used as "inputs" for the ABM, where, for example, urbanist agents are more likely to litter etc. than purist agents.

We have used the spreadsheeet answers to the questions on how far they travelled and how many visits they undertook in the last month to seperate the visitors into Purist, Neutralist, and Urbanist. Using these clasifications we have worked out that 19% of urbanists have dogs, 28% of neutralists have dogs, and 32% of purists have dogs.

Rating how concerned they were about the variety of species in Wales from 1-5, Purists averaged 3.52, Neutralists averaged 3.47 and Urbanists 3.31. The standard deviation of all three samples was around 1.2.

Asked whether they had taken action to help the environment (gardening for wildlife,being part of a conservation group, volunteering), differing vistor types answered Yes to differing numbers of questions:

|Num Yes | Urbanist | Neutralist | Purist
|----------|----------|------------|-------
| 0 | 27% | 20% | 16%
| 1 | 52% | 52% | 47%
| 2 | 18% | 21% | 26%
| 3 | 4% | 7% | 11%

Normalised Mean of this data - Urbanist - 0.32, Neutralist - 0.38, Purist - 0.44.

We can also use the survey to examine other characteristics (or behaviours), for example we calculate the chance of littering based on how the three groups responded to a question about recycling. We estimated how keen the different groups are to move away from other people by how many said they seek the peace and quiet when going to parks. The fractions on buying food and drink, gifts and souvenirs, and owning a dog were direct questions in the survey. These results are all in the table.

Looking at the results in the tabe we can see that, using our method, urbanists are more likely to litter, are more social and buy more food and drink, whereas purists move away from others more, buy more gifts and souvenirs, and are more likely to have a dog.

|Attribute | Urbanist | Neutralist | Purist
|----------|----------|------------|-------
|Fraction who litter (estimated from recycling behaviour)| 0.027 | 0.019| 0.018
|Antisociability (drift away from high density areas) | 0.0024 | 0.0073 | 0.0068
|Mean time to spend in park | 2h 4m | 2h 23m| 3h 10m
|Fraction who buy food and drink | 0.09 | 0.05 | 0.05
|Fraction who buy gifts and sourveirs| 0.08 | 0.14 | 0.18
|Fraction who have dog| 0.19 | 0.27 | 0.32

We can also see how the time spent in the parks varies between the different groups. We can see from the graphs below that urbanists generally spent shorter amounts of time in the park, while purists stayed longer. The average time spent by the different groups can be seen in the table but the key results is the relative times between the groups.



## DISCORD chat
https://discord.gg/6CdJ9FQZ

## Friday Presentation
Slides:
https://docs.google.com/presentation/d/1Ck46onCLWDuCqr70vGd2Y0o-ztvEejefYi0mfqLNF0I/edit#slide=id.gd22f976e60_0_7

# Presentation - Natural Resources Wales (NRW)

## Our Problem
- NRW care for and advise on natural environment and resources in Wales
- Want to model human behaviour in sites (parks)
- Antisocial behaviours could include littering, conflicts, wildlife/habitat disturbance etc.
- Overcrowding can also be an issue

## Challenges and Aims

1. How do we model movement of users through a park?
2. How do we classify different users of the park?
3. Where is best to intervene to disrupt bad behaviours?
4. How do we mitigate/balance risks of negative experiences of users against negative effects on sites?

## Modelling Techniques
### 1. Agent Based Modelling of a specific site

We want to be able to predict the “type” of users that use different parks.We therefore need to classify parks and users. Parks are
classified based on a study carried out in Iceland with the key properties being distance to the nearest city, expected visitor number, and facilities/infrastructure. We give each park a score based on these properties with the aim of using this score to predict the type of park visitors.

Give each site a score between $0$ and $1$, based on distance from cities, visitor numbers and facilities and infrastructure. This score informs what precentage of different types of site users we have.

\begin{equation}
\text{S} = \alpha_1 d +\alpha_2 v +\alpha_3 f
\end{equation}

Three categories of site users: urbanists, neutralists, purists. Site users categorized using number of weekly visits and distance travelled to the site.

Assumptions:
- Agents have a destination that they move towards (which may change over time).
- Agents cannot walk through solid obstacles.
- Agents drift away from high density areas.
- Agents have a certain amount of randomness in their movements.

|Attribute | Urbanist | Neutralist | Purist
|----------|----------|------------|-------
|Fraction who litter (estimated from recycling behaviour)| 0.027 | 0.019| 0.018
|Antisociability (drift away from high density areas) | 0.0024 | 0.0073 | 0.0068
|Mean time to spend in park | 2h 4m | 2h 23m| 3h 10m
|Fraction who buy food and drink | 0.09 | 0.05 | 0.05
|Fraction who buy gifts and sourveirs| 0.08 | 0.14 | 0.18
|Fraction who have dog| 0.19 | 0.27 | 0.32

Video: Initial application of the agent based model using NetLogo to investigate litter accumulation within a park. First, a simple question: how does the number and location of bins affect litter accumulation?

Still to do with ABM: include data information from the survey team to differentiate users and investigate other anti-social behaviours within a park setting and how these change with the number of park users from each group. Attempt to combine the car park data to allow for variation with seasonality.

### 2. Regression
- Linear Regression modelling on monthly car park data and impact of bad behaviour recorded by Park Rangers.
- Car Park data is seen as the best way to estimate the number of visitors to the park.
- Initial Model: $Y_{ij}=\alpha_{i}+\beta x_{j}+\epsilon_{ij}$ with $i$ as a single type of bad behaviour and $j$ as a single car
- $Y_{ij}$ - Impact on the park
- $\alpha_{i}$ - Mean impact of bad behaviour
- $\beta$ - Regression effect due to number of cars
- $\epsilon_{ij}$ - error
- Initial Results suggest that the effects of bad car parking increase most steeply with number of cars.

- Improved the model by accounting for seasonal variation in the number of visitors as well as an increase in visitor numbers year on year.