From 9a55b5129ffd41728800f169413db97048a99211 Mon Sep 17 00:00:00 2001 From: youainti Date: Tue, 11 May 2021 16:57:05 -0700 Subject: [PATCH] added work on Survial Analysis --- .../sections/03_SurvivalAnalysis.tex | 104 +++++++++++------- 1 file changed, 63 insertions(+), 41 deletions(-) diff --git a/CurrentWriting/sections/03_SurvivalAnalysis.tex b/CurrentWriting/sections/03_SurvivalAnalysis.tex index 10d8307..f74dde0 100644 --- a/CurrentWriting/sections/03_SurvivalAnalysis.tex +++ b/CurrentWriting/sections/03_SurvivalAnalysis.tex @@ -4,58 +4,80 @@ \begin{document} In his dissertation \cite{RaoDissertation} briefly examines the "survival rates" of a satellite constellation. I've applied this to my model and extended the results. +This approach allows us to construct a elasticity of survival and satellite additions, i.e. an elasticity +of risk. +%I should probably look up how to analyze changes in risk level and quantitative representations etc. % Marginal survival. -The survival rate for a constellation $i$ is defined as $R_i = 1-l^i(\cdot)$, the proportion of satellites- +The survival rate for a constellation $i$ is defined as $R^i = 1-l^i(\cdot)$, i.e. the proportion of satellites- that were not lost (degraded nor destroyed) between period $t$ and $t+1$. Thus the marginal survival rate represents the additional loss of satellites due to a slightly larger constellation or fleet stock. -Let $S_t = \sum^n_{j=1} s^j_t$. -Then the survival rates for a constellation and for society's fleet are respectively defined as: +To extend this definition to all fleets, we can measure the total number of +satellites that survive. +This can be calculated as the weighted sum of survival rates. \begin{align} - R_i =& \frac{s^i_{t+1}- x^i_t}{s^i_t} = 1- l^i(s^i_t,S_t,D_t) \\ - R =& \frac{S_{t+1}- X_t}{S_t} = \frac{\sum_{i=1}^N s^i_t[1-l^i(s^i_t,S_t,D_t)] }{S_t} \label{EQ:socsurv} + R =& \frac{\sum_{i=1}^n s^i_t R^i}{\sum_{i=1}^n s^i_t} \end{align} -In this case, the fleet survival rate \cref{EQ:socsurv}, represents the proportion of satellites- -in period $t+1$ that survived from period $t$. -The marginal survival rates when a given constellation $i$ changes size are: +\subsubsection{Marginal Survival Rates} + +We can find the marginal survival rate with respect to a given constellation $s^i_t$ as: + \begin{align} - \parder{R_i}{s^i_t}{} =& -\left(\parder{l^i}{s^i_t}{} + \parder{l^i}{S_t}{}\parder{S_t}{s^i_t}{} \right) - = - \parder{l^i}{s^i_t}{} - \parder{l^i}{S_t}{} \label{EQ:iii} \\ - \parder{R}{s^i_t}{} =& \frac{S_t \sum_{i=1}^N \left( [1-l^i(s^i_t,S_t,D_t)] - + s^i_t [ -\parder{l^i}{s^i_t}{} -\parder{l^i}{S_t}{}\parder{S_t}{s^i_t}{}] \right) - - \left( \sum_{i=1}^N s^i_t[1-l^i(s^i_t,S_t,D_t)] \right)} - {(S_t)^2} \notag{}\\ - =& \sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t} - +\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{} \label{EQ:i} + \parder{R}{s^i_t}{} =& \parder{}{s^i_t}{}\frac{\sum_{j=1}^n s^j_t R^i}{\sum_{j=1}^n s^i_t} \\ + =& \left(\frac{1}{\sum_{j=1}^n s^j_t}\right)^2 + \left[ + \left(\sum^n_{j=1}s^j_t\right) \left(\parder{}{s^i_t}{}\sum^n_{j=1} s^j_t R^j\right) + - \sum^n_{j=1} s^j_t R^j + \right] \\ + =& \left(\frac{1}{\sum_{j=1}^n s^j_t}\right) + \left[ + \left(\sum^n_{j \neq i} s^j_t \parder{R^j}{s^i_t}{}\right) + + \left( R^i + s^i_t \parder{R^i}{s^i_t}{}\right) + - R + \right] + \\ + =& + \left(\sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) \parder{R^j}{s^i_t}{}\right) + + \left(\frac{ R^i - R }{\sum_{j=1}^n s^j_t}\right) + \\ + \parder{R}{s^i_t}{} + =& + \sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) \parder{R^j}{s^i_t}{} + + \left(\frac{ R^i - R }{\sum_{j=1}^n s^j_t}\right) \label{EQ:MarginalSurvivalRelation} \end{align} -Note that $ \sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{}$ is the weighted, average marginal survival -rate across constellation operators. -The derivation of \cref{EQ:i} is in Appendix \ref{APX:Derivations:SurvivalRates}. -Direct comparison between the marginal survival rates of an individual operator and the social planner's fleet -cannot proceed further without specifying the functional loss forms $l^i(\cdot)$ -and specifying which firm will be compared to society. -In spite of this, conditions on the average effects can be developted as follows. - -The marginal survival rate of the fleet is less than the weighted, arithmetic mean of marginal survival rates- -of the constellations when: +This can also be written in differential form as \begin{align} - \sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t} - +\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{} - \leq& \sum_{i=1}^N \frac{s^i_t}{S_t} \parder{R_i}{s^i_t}{} \\ - \sum_{i=1}^N R_i - R \leq& 0\\ - \sum_{i=1}^N [1- l^i(s^i_t,S_t,D_t)] - \sum_{i=1}^N s^i_t [1- l^i(s^i_t,S_t,D_t)] \leq& 0\\ - \sum_{i=1}^N (1 - s^i_t) [1- l^i(s^i_t,S_t,D_t)] \leq& 0 \label{EQ:ii} + d{R} + =& + \sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) d{R^j} + + \left(\frac{ R^i - R }{\sum_{j=1}^n s^j_t}\right) d{s^i_t}\label{EQ:differentialSurvivalRelation} \end{align} -which is true if every constellation has at least one satellite. -As any constellation of interest has at least one satellite -and $\parder{R_i}{s^i_t}{} < 0$ from the assumption on collision mechanics that $\der{l^i}{s_t^i}{}>0$, -we conclude that the marginal survival rate of the entire satellite fleet is lower -than the weighted arithmetic mean of marginal survival rates across constellations. -Note that it is possible for some constellations to have a lower marginal survival rate than the fleet, -but the survival rate for many operators must be higher than the societal rate. -Consequently, we would expect many operators to underestimate the impact of their behaviors on others -if they use their own observed or expected risk factors to estimate the risk they impose on others. + +From \cref{EQ:MarginalSurvivalRelation,EQ:differentialSurvivalRelation}, +we can see that the fleetwide marginal survival rate +is made up of two components. +\begin{itemize} + \item $\sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) \parder{R^j}{s^i_t}{}$ + represents the effect on each satellite constellation, and is always negative because + $\parder{R^j}{s^i_t}{} < 0$ by assumption. + Thus each constellations' survival rate will decrease as satellites are added to + any constellation. + \item $\frac{ R^i - R }{\sum_{j=1}^n s^j_t}$, + represents the effect of averaging out marginal survival rates. + Intuitively, when a constellation has a higher survival rate + than the fleet's survival rate, adding a satellite to that fleet contributes + less colision risk than if it were given to another + Note that it is positive but only when $R^i > R$. + Additionally, it disappears quickly as the total number of satellites increase. + Thus when there are a large number of satellites in orbit, regardless of who + owns them, it is almost certain that any increase in satellite stocks will + lead to a reduction in the survival rate. + \footnote{I believe Rao makes this an assumption, I show it is a result} +\end{itemize} + +Consequently, we can see that in many cases, the marginal survival rate will be negative. + \end{document}