Davidson Missouri W/Mathematical Modeling

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<center>[[Davidson Missouri W| <span style="color:red">Home</span>]] | [[Davidson Missouri W/Background Information| <span style="color:red">Background Information</span>]] | [[Davidson Missouri W/Solving the HPP in vivo| <span style="color:red">Current Project: Solving the Hamiltonian Path Problem ''in vivo''</span>]] | [[Davidson Missouri W/Mathematical Modeling| <span style="color:black">Mathematical Modeling</span>]] | [[Davidson Missouri W/Gene splitting| <span style="color:red"> Gene Splitting </span>]] | [[Davidson Missouri W/Traveling Salesperson Problem| <span style="color:red">Traveling Salesperson Problem</span> ]] | [[Davidson Missouri W/Software|<span style="color:red">Software</span>]] | [[Davidson Missouri W/Resources and Citations|<span style="color:red">Resources and Citations</span>]]</center>
+
<center>[[Davidson Missouri W| <span style="color:red">Home</span>]] | [[Davidson Missouri W/Background Information| <span style="color:red">Background Information</span>]] | [[Davidson Missouri W/Solving the HPP in vivo| <span style="color:red">Current Project: Solving the Hamiltonian Path Problem ''in vivo''</span>]] | [[Davidson Missouri W/Mathematical Modeling| <span style="color:black">Mathematical Modeling</span>]] | [[Davidson Missouri W/Gene splitting| <span style="color:red"> Gene Splitting </span>]] | [[Davidson Missouri W/Results| <span style="color:red">Results</span>]] | [[Davidson Missouri W/Traveling Salesperson Problem| <span style="color:red">Traveling Salesperson Problem</span> ]] | [[Davidson Missouri W/Software|<span style="color:red">Software</span>]] | [[Davidson Missouri W/Resources and Citations|<span style="color:red">Resources and Citations</span>]]</center>
<hr>
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=Markov Chain Model For Flipping=
=Markov Chain Model For Flipping=
 +
Before spending time and resources in the wet lab, we wanted to model the biological systems ''in silico'' to better our understanding.  A relatively simple, yet powerful, mathematical method is the use of Markov chains.
-
'''Does starting orientation matter?'''
+
==Does starting orientation matter?==
-
Question: Will the order in which we place the edges in our E.coli computers affect the probability of us detecting a Hamiltonian Path?
+
[[Image:MarkovChaingraph4nod3edg.jpg|frame|right|Analysis of starting orientations.  As the number of flips increases, the probability of finding a Hamiltonian path solution converges.  Each colored line represents a different starting orientation.]]
-
Our mathematical reasoning relies on the fact that the Hin/''hixC'' DNA recombination system makes a large number of flips and for any one flip there is some element of chance that dictates which Hix sites are chosen to flip.
+
'''Question''': Will the initial order in which we place the edges in our ''E.coli'' computers affect the probability of us detecting a Hamiltonian Path?
-
Answer: We developed a Markov Chain model that takes every signed permutation of edges as a state and the connections between these states are found and categorized by our Matlab programs by both the number of edges that need to be flipped to move between those states and categorized by which specific edge needed to be flipped. Further, our Matlab programs creates a transition matrix based on the weights a user enters to bias the probabilities of making different kinds of flips.  
+
Our mathematical reasoning relies on the fact that the Hin/''hixC'' DNA recombination system makes a large number of flips and for any one flip there is some element of chance that dictates which ''hixC'' sites are chosen to flip.  
-
Once this transition matrix is generated we then compute, for each possible starting state, the probability of that starting state transitioning to any of the solved states after x flips given the biasing of different types of flipsWhen our Markov Chain model is run for more than 2 edges we always observe a convergence of these probabilities no matter what bias we apply. This shows that starting orientation does not matter since the probability of being solved after a large number of flips is the same for each starting orientation.
+
'''Answer''': We developed a Markov Chain model in Matlab that takes every signed permutation of edges as a state.  Matlab then finds the connections between these states and categorizes them using both the number of edges that need to be flipped to move between those states, and which specific edges needed to be flippedFurther, our Matlab program creates a transition matrix based on user-inputted weights to bias the probabilities of making flips of varying sizes.
-
This begs the question: what is a large number of flips? Mostly we observe that this convergence occurs in the first 20 flips for three edges and only for very extreme probabilities does this convergence occur after 60 flips.  However, as the number of edges increase we can expect this convergence to occur later. This does not appear to be a problem. Our flipping mechanism flips edges very quickly, so quickly that it is hard to measure. Additionally, the diameter of our transition diagrams increases linearly with the number of edges so there is no reason to believe that the number of flips to convergence would increase at more than a linear pace.
+
Once this transition matrix is generated, we compute, for each possible starting state, the probability that the starting state will transition to any of the solved states after ''x'' flips, given the biasing of different types of flips.  When our Markov Chain model is run for more than 2 edges we always observe a convergence of these probabilities, regardless of any initial bias. This shows that starting orientation does not matter, since the probability of being solved after a large number of flips is the same for each starting orientation.
-
 
+
-
[[Image:MarkovChaingraph4nod3edg.jpg|frame|none|Analysis of starting orientations.  As the number of flips increases, the probability of finding a Hamiltonian path solution converges.  Each colored line represents a different starting orientation.]]
+
 +
This begs the question: what is a large number of flips?  Mostly we observe that this convergence occurs in the first 20 flips for three edges, and only for extreme probabilities does this convergence occur after 60 flips.  However, as the number of edges increases, we can expect this convergence to occur later. This does not appear to be a problem ''in vivo''. Our flipping mechanism flips edges quickly, so quickly that its speed is hard to measure. Additionally, the diameter of our transition diagrams increases linearly with the number of edges, so there is no reason to believe that the number of flips to convergence would increase at more than a linear pace.
MATLAB programs that we developed using Markov Chains:
MATLAB programs that we developed using Markov Chains:
Line 27: Line 27:
#[http://gcat.davidson.edu/iGEM07/For_Missouri_Western/biasweighter.m Bias Weighter]
#[http://gcat.davidson.edu/iGEM07/For_Missouri_Western/biasweighter.m Bias Weighter]
-
= True Positives / False Positives =
+
=Poisson Model For the Number of Plasmids=
-
[[Image:5_node-wiki.jpg|frame|none|Highlighted in solid red lines is a Hamiltonian path.  The dashed line shows where a false-positive can occur.]] 
+
'''How many E.coli computers?'''
-
True positives are the Hamiltonian Path represented in the above graph with solid red arrows and below as the PCR Fragment Length where the edges after the Hamiltonian path can be arranged in any order.
+
'''Question''': If a Hamiltonian Path exists in the graph, how many E.coli computers do we need to have working on the problem to be highly confident that at least k of our computers will find it? 
-
[[Image:True_positive.jpg|frame|none|The plasmid orientation for a true solution.]]
+
Our mathematical reasoning relies on the fact that the Hin/''hixC'' DNA recombination system makes a large number of flips, and for any one flip there is some element of chance that dictates which ''hixC'' sites are chosen to flip.  Note: This has now been shown to be true, but it needs to be stated that our math relies on it.  
-
A false positive is represented in the graph above with the solid red arrows and the dotted red arrow showing a "teleport" which causes this path to be a false positive. Teleporting occurs when an edge entering a node is followed by an edge leaving from a different node. Below you can see an extra edge that is highlighted, representing the teleport. In this example, the edge from 4 to 3 is followed by the edge from 4 to 5.
+
If this is the case, then our ''E. coli'' computers will be equally likely to have flipped into each possible signed ordering of edges by the time we screen our results. For more details on this see the above section about [[Davidson Missouri W/Mathematical Modeling#Does_starting_orientation_matter.3F | starting orientations]].
-
[[Image:Fpospic.jpg|frame|none|The plasmid orientation for a false-positive solution.]]
+
We can use a '''cumulative Poisson distribution''' to answer this question.  
-
         
+
Poisson is an appropriate distribution to use since:
-
MATLAB programs used to find the false positives:
+
-
#[http://gcat.davidson.edu/iGEM07/For_Missouri_Western/fpVTgraph.m False positives for the 8 edge / 5 node graph shown above]. true positives = 384, false positives = 1,362
+
-
#[http://gcat.davidson.edu/iGEM07/For_Missouri_Western/davidson85Iweighter.m Counter Program for the 8 edge / 5 node graph]
+
-
#[http://gcat.davidson.edu/iGEM07/For_Missouri_Western/adelman147.m Adelman's false positives with 14 edges / 7 nodes]. true positives = 10,321,920, false positives = 168,006,848
+
-
#[http://gcat.davidson.edu/iGEM07/For_Missouri_Western/fpweighter.m Counter Program for Adelman's False Positives]
+
 +
1) The number of plasmids finding a solved state is a discrete random variable.
 +
2) The probability of a plasmid finding a solved state is known and constant after convergence.
 +
3) The probability of a plasmid finding a solved state is independent of other plasmids finding solved states. 
-
Below are two graphs that show how quickly graphs can be complex so it makes the Hamiltonian Path harder to detect. Also, from these graphs you can tell that even though the graphs are very complex that there is still a 5% change of detecting a Hamiltonian Path or a true positive.  
+
'''Answer''': The Poisson distribution gives the probability of k occurrences when the expected number of occurrences is λ as (e^-λ)*(λ^k)/(k!). By taking 1 minus the sum of the Poisson probabilities from 0 to k-1 we get the cumulative probability of at least k occurrences.  
-
[[Image:Positivenodedg.jpg|frame|none|The total number of positive solutions for several different graphs, whose edge and node counts are shown.]]
+
For example, examine the Adleman graph. It has 14 edges and 7 nodes. This means that a Hamiltonian Path is represented in our plasmids by 6 edges in the proper order and orientation followed by 8 edges in any order and in any orientation.  
-
[[Image:Truetotalpos.jpg|frame|none|The ratio of true positive solutions to all positive solutions for the same graphs.]]
+
Thus there are 8!*2^8 orderings that are true positives and 14!*2^14 possible orderings. The probability of any one plasmid holding a true positive is then 8!*2^8 / 14!*2^14. If we have m plasmids, λ= m*8!*2^8 / 14!*2^14.
-
=Possion Model For the Number of Plasmids=
+
If we want to know the probability of at least one true positive in m plasmids then k = 1, and the expression becomes 1-(e^-(m*8!*2^8/14!*2^14)), as the summand collapses to the case where k=0 and (λ^0)/(0!) then equals 1.
 +
We can then solve for m depending on how confident we want to be. If we use
 +
m=1 billion plasmids, then we expect to find the Hamiltonian Path when one exists 99.92% of the time.
-
'''How many E.coli computers?'''  
+
1 billion ''E. coli'' computers fit conveniently into 1 mL of culture.
-
Question: If a Hamiltonian Path exists in the graph, how many E.coli computers do we need to have working on the problem to be highly confident that at least k of our computers will find it? 
+
Using this statistical method we made a chart showing the probability of finding true positives based on the number of plasmids for the Adleman graph.
-
Our mathematical reasoning relies on the fact that the Hin/HixC DNA recombination system makes a large number of flips and for any one flip there is some element of chance that dictates which Hix sites are chosen to flip.  Note: This has now been shown to be true but it needs to be stated that our math relies on it.
 
-
If this is the case then our E.coli computers will be equally likely to have flipped into each possible signed ordering of edges by the time we screen our results. For more details on this see our section answering the question does starting orientation matter?
+
<center>
 +
===Probability of at least ''k'' solutions on ''m'' plasmids for Adelman's 14-edge graph===
 +
</center>
 +
{| border="1" cellpadding="5" cellspacing="0" align="center" width="100%"
 +
|-
 +
! style="color: black; background-color: white;" |
 +
! style="color: black; background-color: white;" | k=1
 +
! style="color: black; background-color: white;" | 5
 +
! style="color: black; background-color: white;" | 10
 +
! style="color: black; background-color: white;" | 20
 +
|-
-
We can use a cumulative Poisson distribution to answer this question.  
+
|''m'' = 10,000,000
 +
|0.0697
 +
|0
 +
|0
 +
|0
 +
|-
-
Answer: The Poisson distribution gives the probability of k occurrences when the expected number of occurrences is λ as (e^-λ)*(λ^k)/(k!). By taking 1 minus the sum of the Poisson probabilities from 0 to k-1 we get the cumulative probability of at least k occurrences.  
+
|50,000,000
 +
|0.3032
 +
|0.00004
 +
|0
 +
|0
 +
|-
-
For example examine the Adleman graph. It has 14 edges and 7 nodes. This means that a Hamiltonian Path is represented in our plasmids by 6 edges in the proper order and orientation followed by 8 edges in any order and in any orientation.  
+
|100,000,000
 +
|0.5145
 +
|0.0009
 +
|0
 +
|0
 +
|-
-
Thus there are 8!*2^8 orderings that are true positives and 14!*2^14 possible orderings. The probability of any one plasmid holding a true positive is then 8!*2^8 / 14!*2^14. If we have m plasmids, λ= m*8!*2^8 / 14!*2^14.
+
|200,000,000
 +
|0.7643
 +
|0.0161
 +
|0.000003
 +
|0
 +
|-
-
Now if we want to know the probability of at least one true positive in m plasmids, we use k =1 and the expression becomes 1-(e^-(m*8!*2^8/14!*2^14)) as the summand collapses to the case where k=0 and (λ^0)/(0!) then equals 1.  
+
|500,000,000
 +
|0.973
 +
|0.2961
 +
|0.0041
 +
|0
 +
|-
-
We can then solve for m depending on how confident we want to be. If we use
+
|1,000,000,000
-
m=1 billion plasmids then we expect to find the Hamiltonian Path when one exists 99.92% of the time.  
+
|0.9992
 +
|0.8466
 +
|0.1932
 +
|0.00007
 +
|-
-
1 billion E.coli computers fit conveniently into 1 mL of culture.
+
|}
 +
[[Image:Formula_one.png]], where ''n'' equals the number of solutions and κ equals actual number of occurrences. λ is the expected number of occurrences, which equals [[Image:Formula_2.png]].
-
Using this statistical method we made a chart showing the probability of finding true positives based on the number of plasmids.
+
''m'' = number of plasmids, ''s'' = number of solved permutations of edges, ''p'' = number of permutations of edges.
-
[[Image:Plasmidsneeded.jpg]]
+
=True Positives and False Positives=
 +
 
 +
The crux of our experimental design is the simplicity of detecting answers phenotypically.  Unfortunately, there exists the possibility that a phenotypically-correct colony may have an incorrect genotype.  Although it looks like it has computed a solution, in fact it has not; this is known as a false-positive.  We estimated the ratio of false-positives to true positives using Matlab simulations for several graphs.
 +
 
 +
[[Image:5_node-wiki.jpg|frame|none|Highlighted in solid red lines is a Hamiltonian path.  The dashed line shows where a false-positive can occur.]] 
 +
 
 +
An example of a Hamiltonian Path in the above graph is shown with solid red arrows; below, its equivalent on a plasmid.  There are in fact many possible true positive answers.  Their PCR fragment lengths are identical, but the edges after the Hamiltonian path, following the transcriptional terminator node 5, can be arranged in any order.  Some of these true positives might show the same path, whereas others may be different.
 +
 
 +
[[Image:True_positive.jpg|frame|none|The plasmid orientation for a true solution.]]
 +
 
 +
A false positive is represented in the graph above with the the dotted red arrow, which shows a "teleport". Teleporting occurs when an edge entering a node is followed by an edge leaving from a different node. Below you can see an extra edge that is highlighted, representing the teleport. In this example, the edge from 4 to 3 is followed by the edge from 4 to 5.
 +
 
 +
[[Image:Fpospic.jpg|frame|none|The plasmid orientation for a false-positive solution.]]
 +
 
 +
         
 +
MATLAB programs used to find the false positives:
 +
#[http://gcat.davidson.edu/iGEM07/For_Missouri_Western/fpVTgraph.m False positives for the 8 edge / 5 node graph shown above]. true positives = 384, false positives = 1,362
 +
#[http://gcat.davidson.edu/iGEM07/For_Missouri_Western/davidson85Iweighter.m Counter Program for the 8 edge / 5 node graph]
 +
#[http://gcat.davidson.edu/iGEM07/For_Missouri_Western/adelman147.m Adelman's false positives with 14 edges / 7 nodes]. true positives = 10,321,920, false positives = 168,006,848
 +
#[http://gcat.davidson.edu/iGEM07/For_Missouri_Western/fpweighter.m Counter Program for Adelman's False Positives]
 +
 
 +
 
 +
Below are two graphs that show how quickly the number of possible orderings of edges grows when the instance of the Hamiltonian Path Problem increases in the number of edges and nodes. This makes a Hamiltonian Paths harder to detect. Also, from these graphs you can tell that even though the graphs are becoming complex, there is still at least a 1 true positive for every 20 positives.
 +
 
 +
[[Image:Positivenodedg.jpg|frame|none|The total number of positive solutions for several different graphs, whose edge and node counts are shown.]]
 +
 
 +
[[Image:Truetotalpos.jpg|frame|none|The ratio of true positive solutions to all positive solutions for the same graphs.]]
 +
 
 +
 
 +
 
 +
 
 +
<hr>
 +
<center>
 +
<[[Davidson Missouri W/Solving the HPP in vivo | Previous Section]] | [[Davidson Missouri W/Gene splitting | Next Section>]]
 +
</center>

Latest revision as of 16:45, 25 October 2007

Home | Background Information | Current Project: Solving the Hamiltonian Path Problem in vivo | Mathematical Modeling | Gene Splitting | Results | Traveling Salesperson Problem | Software | Resources and Citations


Contents

Markov Chain Model For Flipping

Before spending time and resources in the wet lab, we wanted to model the biological systems in silico to better our understanding. A relatively simple, yet powerful, mathematical method is the use of Markov chains.

Does starting orientation matter?

Analysis of starting orientations. As the number of flips increases, the probability of finding a Hamiltonian path solution converges. Each colored line represents a different starting orientation.

Question: Will the initial order in which we place the edges in our E.coli computers affect the probability of us detecting a Hamiltonian Path?

Our mathematical reasoning relies on the fact that the Hin/hixC DNA recombination system makes a large number of flips and for any one flip there is some element of chance that dictates which hixC sites are chosen to flip.

Answer: We developed a Markov Chain model in Matlab that takes every signed permutation of edges as a state. Matlab then finds the connections between these states and categorizes them using both the number of edges that need to be flipped to move between those states, and which specific edges needed to be flipped. Further, our Matlab program creates a transition matrix based on user-inputted weights to bias the probabilities of making flips of varying sizes.

Once this transition matrix is generated, we compute, for each possible starting state, the probability that the starting state will transition to any of the solved states after x flips, given the biasing of different types of flips. When our Markov Chain model is run for more than 2 edges we always observe a convergence of these probabilities, regardless of any initial bias. This shows that starting orientation does not matter, since the probability of being solved after a large number of flips is the same for each starting orientation.

This begs the question: what is a large number of flips? Mostly we observe that this convergence occurs in the first 20 flips for three edges, and only for extreme probabilities does this convergence occur after 60 flips. However, as the number of edges increases, we can expect this convergence to occur later. This does not appear to be a problem in vivo. Our flipping mechanism flips edges quickly, so quickly that its speed is hard to measure. Additionally, the diameter of our transition diagrams increases linearly with the number of edges, so there is no reason to believe that the number of flips to convergence would increase at more than a linear pace.

MATLAB programs that we developed using Markov Chains:

  1. [http://gcat.davidson.edu/iGEM07/For_Missouri_Western/fliplength.m Flip Length]
  2. [http://gcat.davidson.edu/iGEM07/For_Missouri_Western/pureflip.m Pure Flip]
  3. [http://gcat.davidson.edu/iGEM07/For_Missouri_Western/biasweighter.m Bias Weighter]

Poisson Model For the Number of Plasmids

How many E.coli computers?

Question: If a Hamiltonian Path exists in the graph, how many E.coli computers do we need to have working on the problem to be highly confident that at least k of our computers will find it?

Our mathematical reasoning relies on the fact that the Hin/hixC DNA recombination system makes a large number of flips, and for any one flip there is some element of chance that dictates which hixC sites are chosen to flip. Note: This has now been shown to be true, but it needs to be stated that our math relies on it.

If this is the case, then our E. coli computers will be equally likely to have flipped into each possible signed ordering of edges by the time we screen our results. For more details on this see the above section about starting orientations.

We can use a cumulative Poisson distribution to answer this question.

Poisson is an appropriate distribution to use since:

1) The number of plasmids finding a solved state is a discrete random variable. 2) The probability of a plasmid finding a solved state is known and constant after convergence. 3) The probability of a plasmid finding a solved state is independent of other plasmids finding solved states.

Answer: The Poisson distribution gives the probability of k occurrences when the expected number of occurrences is λ as (e^-λ)*(λ^k)/(k!). By taking 1 minus the sum of the Poisson probabilities from 0 to k-1 we get the cumulative probability of at least k occurrences.

For example, examine the Adleman graph. It has 14 edges and 7 nodes. This means that a Hamiltonian Path is represented in our plasmids by 6 edges in the proper order and orientation followed by 8 edges in any order and in any orientation.

Thus there are 8!*2^8 orderings that are true positives and 14!*2^14 possible orderings. The probability of any one plasmid holding a true positive is then 8!*2^8 / 14!*2^14. If we have m plasmids, λ= m*8!*2^8 / 14!*2^14.

If we want to know the probability of at least one true positive in m plasmids then k = 1, and the expression becomes 1-(e^-(m*8!*2^8/14!*2^14)), as the summand collapses to the case where k=0 and (λ^0)/(0!) then equals 1.

We can then solve for m depending on how confident we want to be. If we use m=1 billion plasmids, then we expect to find the Hamiltonian Path when one exists 99.92% of the time.

1 billion E. coli computers fit conveniently into 1 mL of culture.

Using this statistical method we made a chart showing the probability of finding true positives based on the number of plasmids for the Adleman graph.


Probability of at least k solutions on m plasmids for Adelman's 14-edge graph

k=1 5 10 20
m = 10,000,000 0.0697 0 0 0
50,000,000 0.3032 0.00004 0 0
100,000,000 0.5145 0.0009 0 0
200,000,000 0.7643 0.0161 0.000003 0
500,000,000 0.973 0.2961 0.0041 0
1,000,000,000 0.9992 0.8466 0.1932 0.00007

Formula one.png, where n equals the number of solutions and κ equals actual number of occurrences. λ is the expected number of occurrences, which equals Formula 2.png.

m = number of plasmids, s = number of solved permutations of edges, p = number of permutations of edges.

True Positives and False Positives

The crux of our experimental design is the simplicity of detecting answers phenotypically. Unfortunately, there exists the possibility that a phenotypically-correct colony may have an incorrect genotype. Although it looks like it has computed a solution, in fact it has not; this is known as a false-positive. We estimated the ratio of false-positives to true positives using Matlab simulations for several graphs.

Highlighted in solid red lines is a Hamiltonian path. The dashed line shows where a false-positive can occur.

An example of a Hamiltonian Path in the above graph is shown with solid red arrows; below, its equivalent on a plasmid. There are in fact many possible true positive answers. Their PCR fragment lengths are identical, but the edges after the Hamiltonian path, following the transcriptional terminator node 5, can be arranged in any order. Some of these true positives might show the same path, whereas others may be different.

The plasmid orientation for a true solution.

A false positive is represented in the graph above with the the dotted red arrow, which shows a "teleport". Teleporting occurs when an edge entering a node is followed by an edge leaving from a different node. Below you can see an extra edge that is highlighted, representing the teleport. In this example, the edge from 4 to 3 is followed by the edge from 4 to 5.

The plasmid orientation for a false-positive solution.


MATLAB programs used to find the false positives:

  1. [http://gcat.davidson.edu/iGEM07/For_Missouri_Western/fpVTgraph.m False positives for the 8 edge / 5 node graph shown above]. true positives = 384, false positives = 1,362
  2. [http://gcat.davidson.edu/iGEM07/For_Missouri_Western/davidson85Iweighter.m Counter Program for the 8 edge / 5 node graph]
  3. [http://gcat.davidson.edu/iGEM07/For_Missouri_Western/adelman147.m Adelman's false positives with 14 edges / 7 nodes]. true positives = 10,321,920, false positives = 168,006,848
  4. [http://gcat.davidson.edu/iGEM07/For_Missouri_Western/fpweighter.m Counter Program for Adelman's False Positives]


Below are two graphs that show how quickly the number of possible orderings of edges grows when the instance of the Hamiltonian Path Problem increases in the number of edges and nodes. This makes a Hamiltonian Paths harder to detect. Also, from these graphs you can tell that even though the graphs are becoming complex, there is still at least a 1 true positive for every 20 positives.

The total number of positive solutions for several different graphs, whose edge and node counts are shown.
The ratio of true positive solutions to all positive solutions for the same graphs.




< Previous Section | Next Section>