added work on Survial Analysis

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 $S_t = \sum^n_{j=1} s^j_t$.
+To extend this definition to all fleets, we can measure the total number of 
-Then the survival rates for a constellation and for society's fleet are respectively defined as:
+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{\sum_{i=1}^n s^i_t R^i}{\sum_{i=1}^n s^i_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}
 \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{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} \\
-        =  - \parder{l^i}{s^i_t}{} - \parder{l^i}{S_t}{} \label{EQ:iii} \\
+        =& \left(\frac{1}{\sum_{j=1}^n s^j_t}\right)^2 
-    \parder{R}{s^i_t}{} =& \frac{S_t \sum_{i=1}^N \left(  [1-l^i(s^i_t,S_t,D_t)]
+           \left[ 
-            + s^i_t [ -\parder{l^i}{s^i_t}{} -\parder{l^i}{S_t}{}\parder{S_t}{s^i_t}{}] \right)
+                \left(\sum^n_{j=1}s^j_t\right) \left(\parder{}{s^i_t}{}\sum^n_{j=1} s^j_t R^j\right)
-            - \left( \sum_{i=1}^N s^i_t[1-l^i(s^i_t,S_t,D_t)] \right)}
+                - \sum^n_{j=1} s^j_t R^j
-        {(S_t)^2} \notag{}\\
+           \right] \\
-    =& \sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t}
+        =& \left(\frac{1}{\sum_{j=1}^n s^j_t}\right) 
-            +\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{} \label{EQ:i}
+           \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
+This can also be written in differential form as
-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:
 \begin{align}
-    \sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t}
+        d{R} 
-            +\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^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) d{R^j}
-    \sum_{i=1}^N  R_i  - R \leq& 0\\
+        + \left(\frac{ R^i - R  }{\sum_{j=1}^n s^j_t}\right) d{s^i_t}\label{EQ:differentialSurvivalRelation}
-    \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}
 \end{align}
-which is true if every constellation has at least one satellite.
+
-As any constellation of interest has at least one satellite
+From \cref{EQ:MarginalSurvivalRelation,EQ:differentialSurvivalRelation},
-and $\parder{R_i}{s^i_t}{} < 0$ from the assumption on collision mechanics that $\der{l^i}{s_t^i}{}>0$,
+we can see that the fleetwide marginal survival rate
-we conclude that the marginal survival rate of the entire satellite fleet is lower
+is made up of two components.
-than the weighted arithmetic mean of marginal survival rates across constellations.
+\begin{itemize}
-Note that it is possible for some constellations to have a lower marginal survival rate than the fleet,
+    \item $\sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) \parder{R^j}{s^i_t}{}$
-but the survival rate for many operators must be higher than the societal rate.
+    represents the effect on each satellite constellation, and is always negative because
-Consequently, we would expect many operators to underestimate the impact of their behaviors on others
+    $\parder{R^j}{s^i_t}{} < 0$ by assumption.
-if they use their own observed or expected risk factors to estimate the risk they impose on others.
+    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}