Hidden Equations of Threshold Risk

This paper is motivated by the dual notions of “tipping points” and “risk sensitivity” frequently arising in society’s interactions with the natural environment. On some level the problem is at least as old as the history of agriculture with farmers being concerned about rainfall falling below some acceptable level, or onset of frost; a prototypical tipping point for successful cultivation of certain crops.

More recently, concerns about adverse climate change induced global warming have focused on the level of such warming exceeding thresholds such as 1.5 or 2.0 degrees C, by the year 2030 or 2050. Similarly, in the area of sustainable management of fisheries, regulators often consider a fishery secure if the biomass of the harvested species does not fall below a certain percentage (e.g., $60\%$) of the virgin biomass. From the perspective of mathematical modelling of these concerns we first recognise two essential features:

• 1.

  often the variable that we are most interested in (e.g. harvest yield, or fish stock) depends essentially on at least one random variable;

• 2.

  the tipping point is, perhaps, most naturally represented as a “special value” of some parameter. In particular, a value such that if the variable of interest falls below (or above) that value, this is considered to be a “high risk” situation.

The use of quotation marks in that last point suggests that there is a need to make these phrases precise so as to be able to analyse them rigorously. In particular, there are already several alternative mathematical formulations of risk often stemming from actuarial science, finance or engineering. However, in this paper, we take the position that the simple threshold risk is both appropriate and already challenging in the context of the management of natural resources such as fisheries. Conceptually, this risk is modelled as a tail probability

$$P(h(\text{random variable})<\delta),$$

where $\delta$ is the threshold parameter, and $h(\cdot)$ is a given function.

At first sight, this formulation of risk may appear to correspond to a problem fully solved by mathematical statisticians and probabilists. In particular, it is a problem extensively studied in the context of extreme value theory (e.g., see [3]), financial mathematics (e.g., see [6], [7]) and large deviation theory (e.g., see [8]). These approaches focus primarily on asymptotic properties of tail probabilities of certain classes of distributions. However, our approach is essentially different in the sense that we explore the parametric sensitivity of the threshold risk induced by the algebraic form of the function of the random variable that is of interest.

Indeed, recent applied studies such as [2] and [4], indicate that the threshold risk may exhibit high sensitivity to the choice of model parameters, including the threshold parameter. The latter arose in two quite disparate contexts of hospital management and fishery population models. This leads us to a more formal definition and analysis of threshold risk and critical values of the threshold which are natural candidates for tipping points. This is taken up in the next section.

The most general one dimensional problem we will consider here is one where the random variable that is of main interest to us is actually a known rational function $h(X)$ of another random variable $X$ whose cumulative distribution function(cdf) $F(x)$ is also assumed to be known. In this paper we assume that $X$ is an absolutely continuous random variable, hence the density function $f(x)$ is well-defined (see also Remark 2 in Section 3). We begin the analysis with a simpler case where $h(X)=p(X),$ a known polynomial in $X.$

Of course, in some applications the inequality in (1) would be reversed. More generally, the threshold could be a multiple of another polynomial function $q(X)$. In such a case, the threshold risk definition is extended as follows.

Note that in both (1) and (2), we could have defined a $\delta$-perturbed polynomial $p_{\delta}(X)=p(X)-\delta q(X)$ and expressed the threshold risk as

$$R(\delta)=P\Big{(}p_{\delta}(X)<0\Big{)}.$$

Naturally, in the case of (1), $q(X)$ is identically equal to 1.

A change in risk as $\delta_{0}$ changes to $\delta_{1}$ will be measured by the ratio

$$S(\delta_{0},\delta)=\frac{|R(\delta_{0})-R(\delta)|}{|\delta_{0}-\delta|}\approx|R^{\prime}(\delta_{0})|,$$

(3)

if the derivative exists and $\delta$ is close to $\delta_{0}.$

This is related to (but not the same) as the hazard function used in demography and actuarial science. The latter considers the ratio of the probability density function at $\delta_{0}$ to the probability of exceeding that threshold.

To analyze the polynomial threshold risk in more detail it will be necessary to consider the real roots of the underlying polynomial. Let $r_{1}(\delta)\leq r_{2}(\delta)\ldots\leq r_{n_{1}}(\delta)$ be the real roots of $p_{\delta}(X)=0$ for $n_{1}\leq n$. We can partition $\mathbb{R}$ into union of following intervals

	$\displaystyle I_{0}(\delta)$	$\displaystyle=\big{(}-\infty,r_{1}(\delta)\big{)},$
	$\displaystyle I_{1}(\delta)$	$\displaystyle=\big{[}r_{1}(\delta),r_{2}(\delta)\big{)},$
	$\displaystyle I_{2}(\delta)$	$\displaystyle=\big{[}r_{2}(\delta),r_{3}(\delta)\big{)},$
	$\displaystyle\vdots$
	$\displaystyle I_{n_{1}}(\delta)$	$\displaystyle=\big{[}r_{n_{1}}(\delta),\infty\big{)},$

where we observe that the sign of the polynomial $p_{\delta}(X)$ cannot change inside any of the intervals $I_{j}$. Let $\mathcal{J}^{-}(\delta)=\{j|p_{\delta}(x)\leq 0,\;\text{if}\;\;x\in I_{j}\}$. The threshold risk can now be expressed as

$$R(\delta)=\sum_{j\in\mathcal{J^{-}}(\delta)}R(I_{j}(\delta))=\sum_{j\in\mathcal{J^{-}}(\delta)}\int_{x\in I_{j}(\delta)}f(x)dx=\sum_{j\in\mathcal{J^{-}(\delta)}}[F(r_{j+1}(\delta))-F(r_{j}(\delta))],$$

(4)

where $F(x)=\int_{-\infty}^{x}f(u)du.$ Hence its derivative with respect to $\delta$ is given by

$$R^{\prime}(\delta)=\sum_{j\in\mathcal{J^{-}}(\delta)}[f(r_{j+1}(\delta))r_{j+1}^{\prime}(\delta)-f(r_{j}(\delta))r_{j}^{\prime}(\delta)],$$

(5)

whenever the derivatives of these roots at $\delta$ exist. Let $\text{Dis}(p_{\delta}(X))$ denote the discriminant of the polynomial $p_{\delta}(X).$

Remark 1: It will be seen that the discriminant is a polynomial in $\delta$. This is what we refer to as ”the hidden polynomial”. Consider the equation

$$\text{Dis}(p_{\delta}(X))=0.$$

($*$)

In the following proposition, we show that the hidden equation ($*$ ‣ 2) is that of finding the roots of a polynomial in $\delta$ with finite order.

The next theorem shows that zeroes of the discriminant play an important role in our analysis. In particular, the theorem will show these zeroes contain the candidates of risk critical points.

Example 1: Let $p_{\delta_{0}}(X)=X^{2}-\delta_{0}$ and $X\sim\mathcal{N}(0,1^{2})$. Here $\text{Dis}(p_{\delta_{0}}(X))=4\delta_{0}$, hence $\mathcal{Z}(p_{\delta_{0}}(X))=\{0\}.$ If $\delta_{0}=0$, $r_{1}(\delta_{0})=0$ is a root with even multiplicity. By Theorem 1, it is a candidate risk critical point. If we perturb $\delta_{0}$ to $\delta=\delta_{0}+\varepsilon$, $p_{\delta}(X)$ has roots $r_{1}(\delta)=-\sqrt{\delta},r_{2}(\delta)=\sqrt{\delta}$ and $r_{1}^{\prime}(\delta)=-\frac{1}{2\sqrt{\delta}},r_{2}^{\prime}(\delta)=\frac{1}{2\sqrt{\delta}}$. Recalling the density function of standard normal distribution we see that equation (5) implies that the rate of change of the threshold risk is now given by $R^{\prime}(\delta)=\frac{1}{2\sqrt{\delta}}\frac{1}{\sqrt{2\pi}}e^{-0.5\delta}-(-\frac{1}{2\sqrt{\delta}}\frac{1}{\sqrt{2\pi}}e^{-0.5\delta})=\frac{1}{\sqrt{\delta}}\frac{1}{\sqrt{2\pi}}e^{-0.5\delta}$. Clearly, the latter diverges as $\delta\to 0$. It is now easy to see that $\delta_{0}=0$ is a risk critical point.

In the previous section, we derived a theorem which identifies candidates for risk critical points. However, since the sensitivity of the risk also depends on the coefficients of the root expansion as well as the value of the density function, we need to further decompose the intervals in (4). In particular, we shall separate the contribution to the threshold risk from repeating and non-repeating roots.

Using the above definition, (4) can be partitioned as follows

$$R(\delta)=R_{r}(\delta)+R_{n}(\delta)=\sum_{j\in J^{-}_{r}(\delta)}R(I_{j}(\delta))+\sum_{j\in J^{-}_{n}(\delta)}R(I_{j}(\delta)).$$

(7)

By Theorem 1, critical points must be among the zeroes of the discriminant of $p_{\delta}(X)$. However, the discriminant is zero only if the resultant is zero since the leading coefficient $p_{n}\neq 0$. Now the resultant is zero only if $p_{\delta}(X)$ has repeated roots. If $p_{\delta}(X)$ has no repeated roots, then the hidden equation ($*$ ‣ 2) cannot be satisfied. Therefore $J_{r}^{-}(\delta)$ is empty and $\delta$ is not a risk critical point.

The above lemma shows that in the case of a threshold $\delta_{0}$ such that the roots of polynomial $p_{\delta_{0}}(X)$ are all non-repeating, $\delta_{0}$ cannot be a risk critical point. However the case where the polynomial $p_{\delta_{0}}(X)$ has repeated roots is more complicated. Below, we first demonstrate the distinction between the case with the multiplicity of the repeated root being odd and even. Then we analyse the case where the interval $I_{j}(\delta)$ in $J_{r}^{-}(\delta)$ contains a repeated and a non-repeated root.

Remark 2: Note that the assumption that $X$ was an absolutely continuous random variable could be easily relaxed, but at the cost of more complicated notation and some additional technicalities. For instance, if $\tilde{x}$ is a discontinuity of the cdf $F(x)$ and $\delta_{0}$ is a threshold such that the $j^{th}$ root $r_{j}(\delta_{0})=\tilde{x},$ then $\delta_{0}$ is a candidate for a risk critical point.

Next we analyze the situation when the underlying function of the random variable $X$ is a rational function, namely, a ratio of two polynomials. Indeed, this was the case in the motivating study [4].

The risk sensitivity is defined as before in Definition 2. In this rational function case the roots of the denominator $q(X)$ will impact (10). To compute the threshold risk, let $\tilde{r}_{1}\leq\tilde{r}_{2}\leq\ldots\leq\tilde{r}_{m^{\prime}}$ be the real roots of polynomial $q(X)$ where $m^{\prime}\leq m$. We can factor $q(X)$ as

$$q(X)=\bigg{[}\prod_{d=1}^{m^{\prime}}\big{(}X-\tilde{r}_{d}\big{)}\bigg{]}\tilde{q}(X).$$

Now we can partition $\mathbb{R}$ by these roots as

$$\mathbb{R}=\bigcup_{j=1}^{m^{\prime}+1}\tilde{I}_{j}=(-\infty,\tilde{r}_{1})\cup[\tilde{r}_{1},\tilde{r}_{2})\cdots\cup[\tilde{r}_{m^{\prime}},\infty).$$

(11)

Some of the intervals can have zero length if there are repeated roots for $q(X)$. Next, we define the event of interest as

$$E=\bigg{\{}x\;\bigg{|}\frac{p(x)}{q(x)}\leq\delta\bigg{\}}=\bigcup_{j=1}^{m^{\prime}+1}E_{j},$$

(12)

where $E_{j}=E\cap\tilde{I}_{j}$. Then we partition the index set $J=J^{+}\cup J^{-}=\{1,2,\ldots,m^{\prime}\}$ as

	$\displaystyle J^{+}$	$\displaystyle=\big{\{}j\;|q(x)>0\;\;\;\forall x\in\tilde{I_{j}}\big{\}},$
	$\displaystyle J^{-}$	$\displaystyle=\big{\{}j\;|q(x)<0\;\;\;\forall x\in\tilde{I_{j}}\big{\}}.$

The threshold risk probability can be computed as

$$R(\delta)=P(E)=\sum_{j\in J^{+}}P(E_{j})+\sum_{j\in J^{-}}P(E_{j}).$$

(13)

For the polynomial function $h_{\delta}(X)=p(X)-\delta q(X)$, we can express the events $E_{j}$ in (13) more explicitly conditioned on the sign of $h_{\delta}(X)$

$$\begin{split}&\text{if}\;\;j\in J^{+},\;E_{j}=\big{\{}x\in\tilde{I}_{j}\big{\}}\cap\big{\{}x|h_{\delta}(x)\leq 0\big{\}},\\ &\text{if}\;\;j\in J^{-},\;E_{j}=\big{\{}x\in\tilde{I}_{j}\big{\}}\cap\big{\{}x|h_{\delta}(x)>0\big{\}}.\end{split}$$

(14)

Let $r_{1}(\delta)\leq r_{2}(\delta)\leq\cdots\leq r_{n}^{\prime}(\delta)$, where $n^{\prime}\leq n$, be the real roots of this polynomial $h_{\delta}(X)$. We can factor this polynomial as

$$h_{\delta}(X)=\bigg{[}\prod_{k=1}^{n^{\prime}}\big{(}X-r_{k}(\delta)\big{)}\bigg{]}\tilde{h}(X),$$

and similarly, we can partition $\mathbb{R}$ by these roots as

$$\mathbb{R}=\bigcup_{k=1}^{n^{\prime}+1}I_{k}(\delta)=(-\infty,r_{1}(\delta))\cup[r_{1}(\delta),r_{2}(\delta))\cdots\cup[r_{n^{\prime}}(\delta),\infty).$$

(15)

Using (14) and (15), we can sub-divide each $E_{j}$ into parts intersecting with the intervals $I_{k}(\delta)$, by defining

$$E_{jk}(\delta)=E_{j}\cap I_{k}(\delta)=E\cap\tilde{I}_{j}\cap I_{k}(\delta).$$

(16)

There are two cases to consider

$$\tilde{I}_{j}\cap I_{k}(\delta)=I_{jk}(\delta)=\begin{cases}\bigg{[}\tilde{r}_{j-1}\vee r_{k-1}(\delta),\tilde{r}_{j}\wedge r_{k}(\delta)\bigg{)},\;\;\;\text{if}\;\;\tilde{I}_{j}\cap I_{k}(\delta)\neq\emptyset,\\ \\ \emptyset,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{if}\;\;\tilde{I}_{j}\cap I_{k}(\delta)=\emptyset,\end{cases}$$

(17)

where $\tilde{r}_{j-1}\vee r_{k-1}(\delta)=\text{max}\big{(}\tilde{r}_{j-1},r_{k-1}(\delta)\big{)}$ and $\tilde{r}_{j}\wedge r_{k}(\delta)=\text{min}\big{(}\tilde{r}_{j},r_{k}(\delta)\big{)}.$ It is important to note that the sign of $h_{\delta}(X)$ remains constant on each interval $I_{k}(\delta)$ and hence also on $I_{jk}$. If $j\in J^{+}$

$$E_{jk}(\delta)=\begin{cases}I_{jk}(\delta)\;\;\;\text{if}\;\;h_{\delta}(X)\leq 0\;\text{on}\;I_{k}(\delta),\\ \\ \emptyset\;\;\;\;\;\;\;\;\text{otherwise}.\end{cases}$$

(18)

If $j\in J^{-}$

$$E_{jk}(\delta)=\begin{cases}I_{jk}(\delta)\;\;\;\text{if}\;\;h_{\delta}(X)>0\;\text{on}\;I_{k}(\delta),\\ \\ \emptyset\;\;\text{if}\;\;\;\text{otherwise}.\end{cases}$$

(19)

Hence, we can refine (13) and compute the threshold risk probability for the rational function $h(X)$ as

$$R(\delta)=\sum_{j\in J^{+}}\sum_{k=1}^{n^{\prime}+1}P(I_{jk}(\delta))+\sum_{j\in J^{-}}\sum_{k=1}^{n^{\prime}+1}P(I_{jk}(\delta)).$$

(20)

Let

	$\displaystyle K^{-}$	$\displaystyle=\bigg{\{}k\;|\;\;h_{\delta}(X)\leq 0\;\;\text{on}\;\;I_{k}(\delta)\bigg{\}}$
	$\displaystyle K^{+}$	$\displaystyle=\bigg{\{}k\;|\;\;h_{\delta}(X)>0\;\;\text{on}\;\;I_{k}(\delta)\bigg{\}}.$

Substituting (18) and (19) into (20) and rearranging, we obtain the risk probability

$$\begin{split}R(\delta)&=\sum_{j\in J^{+}}\sum_{k\in K^{-}}P(E_{jk}(\delta))+\sum_{j\in J^{-}}\sum_{k\in K^{+}}P(E_{jk}(\delta)).\\ &=\sum_{j\in J^{+}}\sum_{k\in K^{-}}\bigg{[}F\big{(}\tilde{r}_{j}\wedge r_{k}(\delta)\big{)}-F\big{(}\tilde{r}_{j-1}\vee r_{k-1}(\delta)\big{)}\bigg{]}\\ &+\sum_{j\in J^{-}}\sum_{k\in K^{+}}\bigg{[}F\big{(}\tilde{r}_{j}\wedge r_{k}(\delta)\big{)}-F\big{(}\tilde{r}_{j-1}\vee r_{k-1}(\delta)\big{)}\bigg{]},\end{split}$$

(21)

and its derivative, whenever it exists.

$$\begin{split}R^{\prime}(\delta)&=\sum_{j\in J^{+}}\sum_{k\in K^{-}}\bigg{[}f\big{(}\tilde{r}_{j}\wedge r_{k}(\delta)\big{)}\frac{d}{d\delta}\big{(}\tilde{r}_{j}\wedge r_{k}(\delta)\big{)}-f\big{(}\tilde{r}_{j-1}\vee r_{k-1}(\delta)\big{)}\frac{d}{d\delta}\big{(}\tilde{r}_{j-1}\vee r_{k-1}(\delta)\big{)}\ \bigg{]}\\ &+\sum_{j\in J^{-}}\sum_{k\in K^{+}}\bigg{[}f\big{(}\tilde{r}_{j}\wedge r_{k}(\delta)\big{)}\frac{d}{d\delta}\big{(}\tilde{r}_{j}\wedge r_{k}(\delta)\big{)}-f\big{(}\tilde{r}_{j-1}\vee r_{k-1}(\delta)\big{)}\frac{d}{d\delta}\big{(}\tilde{r}_{j-1}\vee r_{k-1}(\delta)\big{)}\bigg{]}.\end{split}$$

(22)