The authors of the article "The Fate of Priority Areas for...

The authors of the article "The Fate of Priority Areas for Conservation in Protected Areas: A Fine-Scale Markov Chain Approach" (Envir. Mgmnt., $2011 : 263-278$ ) postulated the following model for landscape changes in the forest regions of Italy. Each "pixel" on a map is classified as forested $(F)$ or non-forested $(N F) .$ For any specific pixel, $X_{n}$ represents its status $n$ years after 2000 $($ so $X_{1}$ corresponds to $2001, X_{2}$ to $2002,$ and so on \right$) .$ Their analysis showed that a pixel has a 90$\%$ chance of being forested next year if it is forested this year and an 11$\%$ chance of being forested next year if it non-forested this year; moreover, data in the twenty-first century are consistent with the assumptions of a Markov chain. $$ \begin{array}{l}{\text { (a) Construct the one-step transition matrix for this chain, with states } 1=F \text { and } 2=N F \text { . }} \\ {\text { (b) If a map pixel was forested in the year } 2000, \text { what is the probability it was still forested in }} \\ {2002 ? 2013 ?}\end{array} $$ $$ \begin{array}{l}{\text { (c) If a map pixel was non-forested in the year } 2000, \text { what is the probability it was still }} \\ {\text { non-forested in } 2002 ? 2013 ?} \\ {\text { (d) The article's authors use this model to project forested status for several Italian regions in }} \\ {\text { the years } 2050 \text { and } 2100 . \text { Comment on the assumptions required for these projections to be }} \\ {\text { valid. }}\end{array} $$

The authors of the article "The Fate of Priority Areas for Conservation in Protected Areas: A Fine-Scale Markov Chain Approach" (Envir. Mgmnt., $2011 : 263-278$ ) postulated the following model for landscape changes in the forest regions of Italy. Each "pixel" on a map is classified as
forested $(F)$ or non-forested $(N F) .$ For any specific pixel, $X_{n}$ represents its status $n$ years after 2000 $($ so $X_{1}$ corresponds to $2001, X_{2}$ to $2002,$ and so on \right$) .$ Their analysis showed that a pixel has a 90$\%$ chance of being forested next year if it is forested this year and an 11$\%$ chance of being
forested next year if it non-forested this year; moreover, data in the twenty-first century are
consistent with the assumptions of a Markov chain.
$$
\begin{array}{l}{\text { (a) Construct the one-step transition matrix for this chain, with states } 1=F \text { and } 2=N F \text { . }} \\ {\text { (b) If a map pixel was forested in the year } 2000, \text { what is the probability it was still forested in }} \\ {2002 ? 2013 ?}\end{array}
$$
$$
\begin{array}{l}{\text { (c) If a map pixel was non-forested in the year } 2000, \text { what is the probability it was still }} \\ {\text { non-forested in } 2002 ? 2013 ?} \\ {\text { (d) The article's authors use this model to project forested status for several Italian regions in }} \\ {\text { the years } 2050 \text { and } 2100 . \text { Comment on the assumptions required for these projections to be }} \\ {\text { valid. }}\end{array}
$$

Key Concepts

Markov Chain

A Markov chain is a stochastic process in which the future state depends solely on the present state and not on the sequence of events that preceded it. This memoryless property simplifies the modeling of systems that evolve over time by considering only the current status to determine future transitions.

Transition Matrix

The transition matrix is a key tool in representing a Markov chain. It is a square matrix where each entry corresponds to the probability of transitioning from one state to another in a single time step. This matrix forms the basis for computing the evolution of the system and understanding dynamics between different states.

Multi-Step Transition Probabilities

Multi-step transition probabilities are obtained by multiplying the one-step transition matrix by itself multiple times. This process, often involving matrix exponentiation, allows for the assessment of the system's state after several time periods, thus facilitating predictions about long-term behavior or the system's evolution over time.

Assumptions for Long-Term Projections

For long-term projections based on a Markov chain, it is assumed that the process is time-homogeneous, meaning that the transition probabilities do not change over time. Additionally, the validity of these projections relies on the assumption that past transitions accurately reflect future dynamics, with no significant external changes or new factors affecting the system.

Transcript

00:01 In this problem, we are told that x sub n is the state of an area, whether it's forested or non -forested.

00:13 And that is n years after the year 2000.

00:23 So we have a state space that is either forested or non -forested.

00:35 For part a, we are asked to construct the one -step transition matrix for this chain by denoting state f as 1 and state nf as 2.

00:49 In the question, we are told that probability of going from a forested state to a forested state in the next step is equal to 90%.

01:03 0 .90.

01:07 And we can also denote this small p sub 1 -1, which is the single -step transition probability for going from state 1 to state 1.

01:22 Based on that, we can also say that the transition probability from state one to state two is 0 .1, because there is only two possibilities.

01:32 You're either going to a forested state or you are going to a non -forested state.

01:37 We are also told that if it is forested this year, the probability of it being non -if it is not forested this year, the probability of it being forested the next year is 11 % or 0 %.

01:52 0 .11, which means that if it is not forced it this year, the probability of it being not forsted in the next year is equal to 0 .89, which is the complement of 11%.

02:11 Now the transition matrix, the one -step transition matrix, is simply these entries.

02:27 So if we substitute in the actual values for these probabilities, we get following.

02:33 For part b, we're asked if the area was forested in the year 2000, what is the probability that it was forested in 2002 and in 2013? so if the starting point is 2000, the year 2002 implies n is equal to 2, which means that there would be two transitions from the year 2000 until the year 2002.

03:17 So what we're looking for is the two -step probability from forested to forested.

03:42 Now to find the two -step transition probabilities, we can multiply the one -step transition probabilities matrix by itself using matrix multiplication.

03:56 You can do this by hand for small matrices like this one, or you can use software, but it comes out to these entries.

04:04 0 .821, 0 .179, 0 .1969, and 0 .8031.

04:31 And so now the two -step probability of transitioning from forested to forested is found by looking at entry 1 -1 in the matrix, which is this entry here...

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