Modeling "This Time is Different" in R

***WHOOPS Made some mistakes in analysis. Update HERE***


Check out PDF for Equations

And Code to run yourself

Introduction:

“This Time is different” by Reinhart and Rogoff is an empirical history of financial crises and panics.  It describes many economic events such as; inflation, bubbles, and defaults.

The theme that ties these events together is the idea that people have an unrealistic expectation of the future because they believe “This time is different”. With this new belief, they act in a way as to cause the next crisis.

While the book dealt with the empirical investigation of the matter, it didn’t describe the theme itself in intricate detail. This post will attempt to fill that void by giving a mathematical interpretation of the theme.

There are 3 aspects of the model I’d like to capture,

1.     prior beliefs act guide and tell us how to make decisions

2.     decisions have an effect on the future

3.     people update our beliefs when new information becomes available

Economic Maximization Problem:

I came up with a simple driving example that describes these properties.  Suppose we’d like to maximize the speed of a car and minimize the probability of a crash (or maximize probability of not crashing). However, the probability of a car crashing is dependent on the speed. This can be described as a utility maximization problem described below, where we maximize subject to a constraint.

Maximize:

Utility(Speed,Probability of Not Crashing) = (Speed^a)*(Probability of not crashing ^(1-a))

Subject to:

Probability of Not Crashing = B*log(1/speed-1) + C

Where B and  C  are the constant and slope parameters of an inverse logistic function and a is tradeoff for utility function between the two goods (Probability of not crashing and Speed).

(Look in PDF for optimal solution)

Bayesian Updating Problem:

Described so far the problem is a fairly straightforward maximization problem. However, in this model, the parameters B and C are unknown to the individual. While there is a true function for the relationship between Speed and probability of not crashing  (that is a true B and C) the individual doesn’t know it.

Bayesian statistics seems like a natural to model the individual’s beliefs over these parameters. With this prior belief over parameters, he maximizes his utility subject to it and makes a decision in each time period. When new information becomes he updates his prior information and uses this to make a new decision. In this sense, the system is dynamic and a time series can be produced.

Unfortunately, I don’t believe there is a closed form solution to updating Bayesian logistic regression. Therefore, I used RSTAN to simulate from the posterior distributions (Code shown at end).

I set the model with parameters of with, a=.05, B=1, and C=100. The initial prior probabilities are parameters C~N(100,5) and B~N(1,.05) ,  where N(x,y) is a normal distribution with mean x and stdev y.  Below is one example of a simulated time series.

 

As you can see, the individual is learning the true parameters as he goes through time. When a crash occurs, he dramatically reduces his decision and if a crash doesn’t occur, he slightly increased his decision. This can be seen as a smoothed over “this time is different” where the actor adjusts his decision more optimistically until a negative event happen, in which case, he changes his decision more abruptly.

The optimum decision is shown in the blue horizontal line, and while the actor doesn’t achieve it in this example, he will eventually (… I think) as this is the equilibrium.

Conclusion:

The purpose of this post was to describe a mathematical interpretation of “This Time is Different”. I did this by solving a maximization problem, with unknown parameters, and then updating those parameters using the Bayesian Framework.

While the example I used was only relation to speed and car crashes, I don’t think its much of a stretch to use this model to describe economic events. An example of might be that a bank wants to have an optimal amount of amount of capital requirements vs event of bankruptcy for a bank. (The smaller capital requirements, the higher the expected profits, and higher probability of bankruptcy.)

Modeling "This Time is Different"

***WHOOPS Made some mistakes in analysis. Update HERE***

Check out PDF for Equations

And Code to run yourself

Introduction:

“This Time is different” by Reinhart and Rogoff is an empirical history of financial crises and panics.  It describes many economic events such as; inflation, bubbles, and defaults.

The theme that ties these events together is the idea that people have an unrealistic expectation of the future because they believe “This time is different”. With this new belief, they act in a way as to cause the next crisis.

While the book dealt with the empirical investigation of the matter, it didn’t describe the theme itself in intricate detail. This post will attempt to fill that void by giving a mathematical interpretation of the theme.

There are 3 aspects of the model I’d like to capture,

1.     prior beliefs act guide and tell us how to make decisions

2.     decisions have an effect on the future

3.     people update our beliefs when new information becomes available

Economic Maximization Problem:

I came up with a simple driving example that describes these properties.  Suppose we’d like to maximize the speed of a car and minimize the probability of a crash (or maximize probability of not crashing). However, the probability of a car crashing is dependent on the speed. This can be described as a utility maximization problem described below, where we maximize subject to a constraint.

Maximize:

Utility(Speed,Probability of Not Crashing) = (Speed^a)*(Probability of not crashing ^(1-a))

Subject to:

Probability of Not Crashing = B*log(1/speed-1) + C

Where B and  C  are the constant and slope parameters of an inverse logistic function and a is tradeoff for utility function between the two goods (Probability of not crashing and Speed).

(Look in PDF for optimal solution)

Bayesian Updating Problem:

Described so far the problem is a fairly straightforward maximization problem. However, in this model, the parameters B and C are unknown to the individual. While there is a true function for the relationship between Speed and probability of not crashing  (that is a true B and C) the individual doesn’t know it.

Bayesian statistics seems like a natural to model the individual’s beliefs over these parameters. With this prior belief over parameters, he maximizes his utility subject to it and makes a decision in each time period. When new information becomes he updates his prior information and uses this to make a new decision. In this sense, the system is dynamic and a time series can be produced.

Unfortunately, I don’t believe there is a closed form solution to updating Bayesian logistic regression. Therefore, I used RSTAN to simulate from the posterior distributions (Code shown at end).

I set the model with parameters of with, a=.05, B=1, and C=100. The initial prior probabilities are parameters C~N(100,5) and B~N(1,.05) ,  where N(x,y) is a normal distribution with mean x and stdev y.  Below is one example of a simulated time series.

 

As you can see, the individual is learning the true parameters as he goes through time. When a crash occurs, he dramatically reduces his decision and if a crash doesn’t occur, he slightly increased his decision. This can be seen as a smoothed over “this time is different” where the actor adjusts his decision more optimistically until a negative event happen, in which case, he changes his decision more abruptly.

The optimum decision is shown in the blue horizontal line, and while the actor doesn’t achieve it in this example, he will eventually (… I think) as this is the equilibrium.

Conclusion:

The purpose of this post was to describe a mathematical interpretation of “This Time is Different”. I did this by solving a maximization problem, with unknown parameters, and then updating those parameters using the Bayesian Framework.

While the example I used was only relation to speed and car crashes, I don’t think its much of a stretch to use this model to describe economic events. An example of might be that a bank wants to have an optimal amount of amount of capital requirements vs event of bankruptcy for a bank. (The smaller capital requirements, the higher the expected profits, and higher probability of bankruptcy.)

Economist Year in Review: Part 3

Finally, I want to show a network graph of countries based on the correlations using LDA. I used qgraph() in R. Kind of interesting how Switzerland and Norway are outside the main European Cluster. 

Economist Year in Review: Part 2

I like how The Economist focuses on Geo Political issues and I want to capture that in my analysis. I therefore labelled each article by the countries/demonyns included in the text (If either “United States” or “American” was included in text then it would label that particular article “United States”. Of course, an article can be labelled many countries).

I then removed all instances of Country Names from the text. I did so because I wanted to use LDA again, and having country names in the text would create a dependancy I don't want.

So, for this part of analysis I have two matrices. One involves LDA matrix of 3440 articles by 100 topics that describes each article as a percentage of 100 topics (with the country names and demonyms removed from the text). The second matrix is 3440 by 193 which represents each article and which countries are mentioned in the text of the article.

The question I want to address is; what are the main groups of international affairs? For example; we expect Syria, Iran, and US to be in a cluster and China, Japan, and US to a cluster as well.

To do this, I used K-means clustering on the articles by countries matrix to group articles based on the countries included in the text. Then to understand what these groups discuss, I averaged the topics of each classified group.

Country Assigning Code

Lets just see how many times each country is mentioned at least once in an article

load("/Users/sweiss/Google Drive/countrymatrix.rdata")
country.sums = colSums(country.mat)
barplot(sort(country.sums, decreasing = TRUE)[1:50], las = 3, cex.names = 0.7, 
    ylab = "Number of Times a country name or Demonym occured at least once in an article", 
    main = "Number of times a Country was Identified in an article (Economist 2013)")

USA is number 1 with UK in a distance 2nd. Obviously there's some UK bias because this is an English newspaper.

Below are the clusters of articles based on the countries named and Topics most associated with those clusters. I chose 10 clusters using the elbow graph method (not shown).

library(topicmodels)
load("/users/sweiss/google drive/economistnocountryld100.Rda")
theta = posterior(ld.100)$topics
theta.average = colMeans(theta)
top.5.factors = names(sort(theta.average, decreasing = TRUE))
load("/Users/sweiss/Google Drive/countrymatrix.rdata")

kmeans.10 <- kmeans(x = country.mat, centers = 10)


cluster.10 = kmeans.10$cluster
for (i in 1:10) {
    theta.average.cluster.1 = colMeans(theta[which(cluster.10 == i), ])
    top.5.factors.cluster.1 = names(sort(theta.average.cluster.1, decreasing = TRUE))
    print(paste("Cluster", i))
    print("Top 10 Countries in Cluster")
    print(sort(colSums(country.mat[which(cluster.10 == i), ]), decreasing = TRUE)[1:10])
    print("Average Number of Countries in Article")
    print(mean(rowSums(country.mat[which(cluster.10 == i), ])))
    print("Number of Articles in Cluster")
    print(sum(rowSums(country.mat[which(cluster.10 == i), ])))
    print(terms(ld.100, 10)[, as.numeric(top.5.factors.cluster.1)[1:10]])
}

## [1] "Cluster 1"
## [1] "Top 10 Countries in Cluster"
## United Kingdom         France          Japan         Russia          Italy 
##            254             73             60             46             44 
##          Spain         Israel         Turkey    Netherlands         Canada 
##             34             33             33             32             31 
## [1] "Average Number of Countries in Article"
## [1] 1.224
## [1] "Number of Articles in Cluster"
## [1] 1567
##       Topic 33    Topic 19   Topic 97   Topic 79   Topic 66    
##  [1,] "rate"      "elect"    "cell"     "labour"   "minist"    
##  [2,] "price"     "parti"    "research" "cameron"  "govern"    
##  [3,] "economist" "vote"     "scienc"   "tori"     "prime"     
##  [4,] "interest"  "voter"    "work"     "parti"    "polit"     
##  [5,] "index"     "poll"     "human"    "britain"  "leader"    
##  [6,] "market"    "polit"    "brain"    "conserv"  "parliament"
##  [7,] "trade"     "campaign" "one"      "polit"    "parti"     
##  [8,] "commod"    "seat"     "cancer"   "miliband" "opposit"   
##  [9,] "job"       "win"      "might"    "mps"      "elect"     
## [10,] "balanc"    "candid"   "use"      "david"    "berlusconi"
##       Topic 40     Topic 16  Topic 26   Topic 58  Topic 53 
##  [1,] "presid"     "polic"   "protest"  "test"    "local"  
##  [2,] "polit"      "crime"   "govern"   "time"    "town"   
##  [3,] "elect"      "prison"  "street"   "ask"     "park"   
##  [4,] "power"      "crimin"  "erdogan"  "peopl"   "build"  
##  [5,] "leader"     "say"     "polic"    "experi"  "council"
##  [6,] "presidenti" "sentenc" "support"  "think"   "centr"  
##  [7,] "constitut"  "drug"    "call"     "word"    "place"  
##  [8,] "countri"    "murder"  "demonstr" "suggest" "peopl"  
##  [9,] "year"       "peopl"   "polit"    "person"  "plan"   
## [10,] "run"        "jail"    "opposit"  "relat"   "new"    
## [1] "Cluster 2"
## [1] "Top 10 Countries in Cluster"
##         France        Germany          Italy          Spain United Kingdom 
##            128            118            116            103             58 
##  United States         Greece    Netherlands        Ireland         Russia 
##             55             53             52             38             32 
## [1] "Average Number of Countries in Article"
## [1] 8.138
## [1] "Number of Articles in Cluster"
## [1] 1237
##       Topic 9    Topic 28   Topic 88  Topic 62   Topic 83  Topic 14 
##  [1,] "euro"     "european" "bank"    "bond"     "price"   "economi"
##  [2,] "zone"     "europ"    "financi" "debt"     "market"  "growth" 
##  [3,] "european" "union"    "loan"    "rate"     "cost"    "econom" 
##  [4,] "countri"  "commiss"  "lend"    "yield"    "rise"    "gdp"    
##  [5,] "bank"     "countri"  "capit"   "investor" "year"    "invest" 
##  [6,] "crisi"    "nation"   "deposit" "govern"   "demand"  "export" 
##  [7,] "bail"     "want"     "credit"  "interest" "low"     "account"
##  [8,] "imf"      "brussel"  "crisi"   "market"   "increas" "product"
##  [9,] "market"   "treati"   "borrow"  "borrow"   "like"    "year"   
## [10,] "economi"  "like"     "asset"   "default"  "high"    "busi"   
##       Topic 29   Topic 17   Topic 91   Topic 52   
##  [1,] "worker"   "firm"     "mrs"      "left"     
##  [2,] "job"      "market"   "merkel"   "holland"  
##  [3,] "work"     "industri" "left"     "right"    
##  [4,] "labour"   "product"  "parti"    "presid"   
##  [5,] "employ"   "big"      "coalit"   "polit"    
##  [6,] "wage"     "new"      "green"    "socialist"
##  [7,] "unemploy" "profit"   "centr"    "now"      
##  [8,] "skill"    "busi"     "govern"   "put"      
##  [9,] "pay"      "sale"     "democrat" "yet"      
## [10,] "factori"  "compani"  "social"   "fran"     
## [1] "Cluster 3"
## [1] "Top 10 Countries in Cluster"
##  United States United Kingdom         Canada         France         Russia 
##            686            115             49             46             37 
##          Japan    Afghanistan      Australia          Spain        Georgia 
##             34             27             24             23             21 
## [1] "Average Number of Countries in Article"
## [1] 2.257
## [1] "Number of Articles in Cluster"
## [1] 1548
##       Topic 87     Topic 72 Topic 19   Topic 90   Topic 63   Topic 47    
##  [1,] "republican" "court"  "elect"    "insur"    "fund"     "state"     
##  [2,] "obama"      "law"    "parti"    "health"   "investor" "feder"     
##  [3,] "democrat"   "case"   "vote"     "will"     "return"   "california"
##  [4,] "senat"      "legal"  "voter"    "plan"     "invest"   "governor"  
##  [5,] "polit"      "judg"   "poll"     "care"     "asset"    "year"      
##  [6,] "congress"   "rule"   "polit"    "feder"    "equiti"   "say"       
##  [7,] "hous"       "right"  "campaign" "obamacar" "share"    "one"       
##  [8,] "bill"       "lawyer" "seat"     "mani"     "manag"    "texa"      
##  [9,] "america"    "suprem" "win"      "exchang"  "profit"   "back"      
## [10,] "barack"     "justic" "candid"   "state"    "money"    "also"      
##       Topic 97   Topic 64    Topic 62   Topic 88 
##  [1,] "cell"     "compani"   "bond"     "bank"   
##  [2,] "research" "firm"      "debt"     "financi"
##  [3,] "scienc"   "busi"      "rate"     "loan"   
##  [4,] "work"     "deal"      "yield"    "lend"   
##  [5,] "human"    "share"     "investor" "capit"  
##  [6,] "brain"    "billion"   "govern"   "deposit"
##  [7,] "one"      "sharehold" "interest" "credit" 
##  [8,] "cancer"   "buy"       "market"   "crisi"  
##  [9,] "might"    "stake"     "borrow"   "borrow" 
## [10,] "use"      "privat"    "default"  "asset"  
## [1] "Cluster 4"
## [1] "Top 10 Countries in Cluster"
##        Germany  United States United Kingdom         France          China 
##            278            112             94             70             44 
##    Netherlands    Switzerland         Russia          Japan         Greece 
##             32             29             27             26             21 
## [1] "Average Number of Countries in Article"
## [1] 4.324
## [1] "Number of Articles in Cluster"
## [1] 1202
##       Topic 91   Topic 28   Topic 9    Topic 88  Topic 17   Topic 64   
##  [1,] "mrs"      "european" "euro"     "bank"    "firm"     "compani"  
##  [2,] "merkel"   "europ"    "zone"     "financi" "market"   "firm"     
##  [3,] "left"     "union"    "european" "loan"    "industri" "busi"     
##  [4,] "parti"    "commiss"  "countri"  "lend"    "product"  "deal"     
##  [5,] "coalit"   "countri"  "bank"     "capit"   "big"      "share"    
##  [6,] "green"    "nation"   "crisi"    "deposit" "new"      "billion"  
##  [7,] "centr"    "want"     "bail"     "credit"  "profit"   "sharehold"
##  [8,] "govern"   "brussel"  "imf"      "crisi"   "busi"     "buy"      
##  [9,] "democrat" "treati"   "market"   "borrow"  "sale"     "stake"    
## [10,] "social"   "like"     "economi"  "asset"   "compani"  "privat"   
##       Topic 76  Topic 66     Topic 29   Topic 15  
##  [1,] "billion" "minist"     "worker"   "investig"
##  [2,] "year"    "govern"     "job"      "claim"   
##  [3,] "will"    "prime"      "work"     "case"    
##  [4,] "cost"    "polit"      "labour"   "alleg"   
##  [5,] "also"    "leader"     "employ"   "charg"   
##  [6,] "worth"   "parliament" "wage"     "report"  
##  [7,] "total"   "parti"      "unemploy" "trial"   
##  [8,] "last"    "opposit"    "skill"    "said"    
##  [9,] "make"    "elect"      "pay"      "former"  
## [10,] "estim"   "berlusconi" "factori"  "accus"   
## [1] "Cluster 5"
## [1] "Top 10 Countries in Cluster"
##         Syria United States          Iraq          Iran        Israel 
##           121            88            80            73            66 
##        Turkey       Lebanon        Russia         Egypt  Saudi Arabia 
##            57            42            40            38            37 
## [1] "Average Number of Countries in Article"
## [1] 7.39
## [1] "Number of Articles in Cluster"
## [1] 1005
##       Topic 46  Topic 48     Topic 81      Topic 92   Topic 21   
##  [1,] "rebel"   "america"    "muslim"      "forc"     "attack"   
##  [2,] "assad"   "obama"      "ian"         "armi"     "kill"     
##  [3,] "regim"   "presid"     "islam"       "defenc"   "war"      
##  [4,] "war"     "nuclear"    "islamist"    "militari" "bomb"     
##  [5,] "govern"  "polici"     "brother"     "arm"      "group"    
##  [6,] "forc"    "relat"      "now"         "secur"    "terrorist"
##  [7,] "north"   "intern"     "morsi"       "general"  "dead"     
##  [8,] "arm"     "weapon"     "brotherhood" "war"      "violenc"  
##  [9,] "group"   "washington" "back"        "soldier"  "terror"   
## [10,] "western" "foreign"    "includ"      "troop"    "drone"    
##       Topic 89      Topic 26   Topic 66     Topic 37    Topic 7  
##  [1,] "white"       "protest"  "minist"     "deal"      "polit"  
##  [2,] "black"       "govern"   "govern"     "trade"     "putin"  
##  [3,] "palestinian" "street"   "prime"      "talk"      "anti"   
##  [4,] "king"        "erdogan"  "polit"      "negoti"    "now"    
##  [5,] "relat"       "polic"    "leader"     "agreement" "also"   
##  [6,] "arab"        "support"  "parliament" "agre"      "may"    
##  [7,] "say"         "call"     "parti"      "side"      "kremlin"
##  [8,] "west"        "demonstr" "opposit"    "free"      "soviet" 
##  [9,] "state"       "polit"    "elect"      "sign"      "western"
## [10,] "two"         "opposit"  "berlusconi" "two"       "power"  
## [1] "Cluster 6"
## [1] "Top 10 Countries in Cluster"
##          Niger        Nigeria  United States         France United Kingdom 
##             55             52             20             14             13 
##           Mali   South Africa          China          Ghana       Cameroon 
##             12             12             10              9              7 
## [1] "Average Number of Countries in Article"
## [1] 7.182
## [1] "Number of Articles in Cluster"
## [1] 395
##       Topic 60  Topic 46  Topic 21    Topic 22  Topic 40     Topic 5     
##  [1,] "africa"  "rebel"   "attack"    "money"   "presid"     "food"      
##  [2,] "ship"    "assad"   "kill"      "pay"     "polit"      "farm"      
##  [3,] "african" "regim"   "war"       "servic"  "elect"      "farmer"    
##  [4,] "port"    "war"     "bomb"      "save"    "power"      "product"   
##  [5,] "region"  "govern"  "group"     "charg"   "leader"     "say"       
##  [6,] "contain" "forc"    "terrorist" "cost"    "presidenti" "produc"    
##  [7,] "intern"  "north"   "dead"      "card"    "constitut"  "agricultur"
##  [8,] "world"   "arm"     "violenc"   "fee"     "countri"    "meat"      
##  [9,] "dubai"   "group"   "terror"    "payment" "year"       "rice"      
## [10,] "countri" "western" "drone"     "account" "run"        "year"      
##       Topic 92   Topic 88  Topic 49  Topic 81     
##  [1,] "forc"     "bank"    "immigr"  "muslim"     
##  [2,] "armi"     "financi" "border"  "ian"        
##  [3,] "defenc"   "loan"    "migrant" "islam"      
##  [4,] "militari" "lend"    "mani"    "islamist"   
##  [5,] "arm"      "capit"   "peopl"   "brother"    
##  [6,] "secur"    "deposit" "say"     "now"        
##  [7,] "general"  "credit"  "illeg"   "morsi"      
##  [8,] "war"      "crisi"   "year"    "brotherhood"
##  [9,] "soldier"  "borrow"  "countri" "back"       
## [10,] "troop"    "asset"   "issu"    "includ"     
## [1] "Cluster 7"
## [1] "Top 10 Countries in Cluster"
##          China  United States          Japan United Kingdom         Russia 
##            409            187             91             60             40 
##         France      Australia         Canada        Vietnam         Taiwan 
##             39             29             24             24             23 
## [1] "Average Number of Countries in Article"
## [1] 3.399
## [1] "Number of Articles in Cluster"
## [1] 1390
##       Topic 85    Topic 24 Topic 17   Topic 95    Topic 4      Topic 14 
##  [1,] "offici"    "south"  "firm"     "project"   "parti"      "economi"
##  [2,] "beij"      "north"  "market"   "mine"      "polit"      "growth" 
##  [3,] "said"      "korea"  "industri" "water"     "power"      "econom" 
##  [4,] "recent"    "asia"   "product"  "govern"    "leader"     "gdp"    
##  [5,] "report"    "island" "big"      "build"     "nation"     "invest" 
##  [6,] "one"       "region" "new"      "river"     "politician" "export" 
##  [7,] "govern"    "relat"  "profit"   "say"       "member"     "account"
##  [8,] "ministri"  "east"   "busi"     "construct" "congress"   "product"
##  [9,] "public"    "sea"    "sale"     "one"       "support"    "year"   
## [10,] "communist" "also"   "compani"  "plan"      "constitut"  "busi"   
##       Topic 86 Topic 88  Topic 48     Topic 74 
##  [1,] "open"   "bank"    "america"    "foreign"
##  [2,] "also"   "financi" "obama"      "govern" 
##  [3,] "will"   "loan"    "presid"     "countri"
##  [4,] "mani"   "lend"    "nuclear"    "local"  
##  [5,] "hong"   "capit"   "polici"     "make"   
##  [6,] "can"    "deposit" "relat"      "control"
##  [7,] "anoth"  "credit"  "intern"     "abroad" 
##  [8,] "kong"   "crisi"   "weapon"     "intern" 
##  [9,] "argu"   "borrow"  "washington" "last"   
## [10,] "one"    "asset"   "foreign"    "may"    
## [1] "Cluster 8"
## [1] "Top 10 Countries in Cluster"
##  United States         Brazil         Mexico          Spain          Chile 
##            124             80             78             23             21 
##          China      Venezuela      Argentina United Kingdom       Colombia 
##             18             16             15             15             14 
## [1] "Average Number of Countries in Article"
## [1] 5.233
## [1] "Number of Articles in Cluster"
## [1] 675
##       Topic 11  Topic 40     Topic 17   Topic 16  Topic 87     Topic 82
##  [1,] "countri" "presid"     "firm"     "polic"   "republican" "reform"
##  [2,] "world"   "polit"      "market"   "crime"   "obama"      "govern"
##  [3,] "global"  "elect"      "industri" "prison"  "democrat"   "will"  
##  [4,] "america" "power"      "product"  "crimin"  "senat"      "polici"
##  [5,] "develop" "leader"     "big"      "say"     "polit"      "chang" 
##  [6,] "rich"    "presidenti" "new"      "sentenc" "congress"   "system"
##  [7,] "emerg"   "constitut"  "profit"   "drug"    "hous"       "plan"  
##  [8,] "intern"  "countri"    "busi"     "murder"  "bill"       "polit" 
##  [9,] "latin"   "year"       "sale"     "peopl"   "america"    "public"
## [10,] "accord"  "run"        "compani"  "jail"    "barack"     "need"  
##       Topic 14  Topic 37    Topic 75  Topic 47    
##  [1,] "economi" "deal"      "number"  "state"     
##  [2,] "growth"  "trade"     "america" "feder"     
##  [3,] "econom"  "talk"      "sinc"    "california"
##  [4,] "gdp"     "negoti"    "time"    "governor"  
##  [5,] "invest"  "agreement" "less"    "year"      
##  [6,] "export"  "agre"      "rate"    "say"       
##  [7,] "account" "side"      "year"    "one"       
##  [8,] "product" "free"      "declin"  "texa"      
##  [9,] "year"    "sign"      "rise"    "back"      
## [10,] "busi"    "two"       "increas" "also"      
## [1] "Cluster 9"
## [1] "Top 10 Countries in Cluster"
##          India          China  United States United Kingdom          Japan 
##            292            123            114             78             51 
##       Pakistan         Russia         Brazil      Australia      Indonesia 
##             43             39             33             32             31 
## [1] "Average Number of Countries in Article"
## [1] 5.014
## [1] "Number of Articles in Cluster"
## [1] 1464
##       Topic 4      Topic 19   Topic 2   Topic 17   Topic 24 Topic 66    
##  [1,] "parti"      "elect"    "govern"  "firm"     "south"  "minist"    
##  [2,] "polit"      "parti"    "nation"  "market"   "north"  "govern"    
##  [3,] "power"      "vote"     "peopl"   "industri" "korea"  "prime"     
##  [4,] "leader"     "voter"    "ethnic"  "product"  "asia"   "polit"     
##  [5,] "nation"     "poll"     "local"   "big"      "island" "leader"    
##  [6,] "politician" "polit"    "villag"  "new"      "region" "parliament"
##  [7,] "member"     "campaign" "one"     "profit"   "relat"  "parti"     
##  [8,] "congress"   "seat"     "just"    "busi"     "east"   "opposit"   
##  [9,] "support"    "win"      "group"   "sale"     "sea"    "elect"     
## [10,] "constitut"  "candid"   "countri" "compani"  "also"   "berlusconi"
##       Topic 21    Topic 95    Topic 11  Topic 39 
##  [1,] "attack"    "project"   "countri" "one"    
##  [2,] "kill"      "mine"      "world"   "world"  
##  [3,] "war"       "water"     "global"  "argu"   
##  [4,] "bomb"      "govern"    "america" "blog"   
##  [5,] "group"     "build"     "develop" "histori"
##  [6,] "terrorist" "river"     "rich"    "long"   
##  [7,] "dead"      "say"       "emerg"   "great"  
##  [8,] "violenc"   "construct" "intern"  "view"   
##  [9,] "terror"    "one"       "latin"   "point"  
## [10,] "drone"     "plan"      "accord"  "much"   
## [1] "Cluster 10"
## [1] "Top 10 Countries in Cluster"
##     Tanzania        Kenya       Rwanda       Uganda South Africa 
##           17           16           15           15           14 
##        China        Niger        India      Nigeria       Angola 
##           13           12           11           11           10 
## [1] "Average Number of Countries in Article"
## [1] 11.87
## [1] "Number of Articles in Cluster"
## [1] 273
##       Topic 60  Topic 49  Topic 46  Topic 35 Topic 13 Topic 2   Topic 92  
##  [1,] "africa"  "immigr"  "rebel"   "peopl"  "store"  "govern"  "forc"    
##  [2,] "ship"    "border"  "assad"   "mani"   "retail" "nation"  "armi"    
##  [3,] "african" "migrant" "regim"   "fire"   "shop"   "peopl"   "defenc"  
##  [4,] "port"    "mani"    "war"     "now"    "sale"   "ethnic"  "militari"
##  [5,] "region"  "peopl"   "govern"  "caus"   "chain"  "local"   "arm"     
##  [6,] "contain" "say"     "forc"    "need"   "sell"   "villag"  "secur"   
##  [7,] "intern"  "illeg"   "north"   "damag"  "buy"    "one"     "general" 
##  [8,] "world"   "year"    "arm"     "also"   "custom" "just"    "war"     
##  [9,] "dubai"   "countri" "group"   "help"   "can"    "group"   "soldier" 
## [10,] "countri" "issu"    "western" "miss"   "good"   "countri" "troop"   
##       Topic 14  Topic 95    Topic 63  
##  [1,] "economi" "project"   "fund"    
##  [2,] "growth"  "mine"      "investor"
##  [3,] "econom"  "water"     "return"  
##  [4,] "gdp"     "govern"    "invest"  
##  [5,] "invest"  "build"     "asset"   
##  [6,] "export"  "river"     "equiti"  
##  [7,] "account" "say"       "share"   
##  [8,] "product" "construct" "manag"   
##  [9,] "year"    "one"       "profit"  
## [10,] "busi"    "plan"      "money"

The easy clusters to understand are numbers 1 (Super Powers), 4 (Euro Zone), 5 (Mideast Conflict), 7 (Asia), and 10 (South America). Cluster 3 seems to be about Regional Politics around India. Clusters 2 and 6 have low average number of countries per article so they are intranational articles.