Implementing Undirected Graphs in Python

Anirudh Technical , , , , , , ,

There are 2 popular ways of representing an undirected graph.

Adjacency List
Each list describes the set of neighbors of a vertex in the graph.

adjacencyList

Adjacency Matrix
The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.

adjacencyMatrix

Here’s an implementation of the above in Python:

class Vertex:
def __init__(self, vertex):
self.name = vertex
self.neighbors = []
def add_neighbor(self, neighbor):
if isinstance(neighbor, Vertex):
if neighbor.name not in self.neighbors:
self.neighbors.append(neighbor.name)
neighbor.neighbors.append(self.name)
self.neighbors = sorted(self.neighbors)
neighbor.neighbors = sorted(neighbor.neighbors)
else:
return False
def add_neighbors(self, neighbors):
for neighbor in neighbors:
if isinstance(neighbor, Vertex):
if neighbor.name not in self.neighbors:
self.neighbors.append(neighbor.name)
neighbor.neighbors.append(self.name)
self.neighbors = sorted(self.neighbors)
neighbor.neighbors = sorted(neighbor.neighbors)
else:
return False
def __repr__(self):
return str(self.neighbors)
class Graph:
def __init__(self):
self.vertices = {}
def add_vertex(self, vertex):
if isinstance(vertex, Vertex):
self.vertices[vertex.name] = vertex.neighbors
def add_vertices(self, vertices):
for vertex in vertices:
if isinstance(vertex, Vertex):
self.vertices[vertex.name] = vertex.neighbors
def add_edge(self, vertex_from, vertex_to):
if isinstance(vertex_from, Vertex) and isinstance(vertex_to, Vertex):
vertex_from.add_neighbor(vertex_to)
if isinstance(vertex_from, Vertex) and isinstance(vertex_to, Vertex):
self.vertices[vertex_from.name] = vertex_from.neighbors
self.vertices[vertex_to.name] = vertex_to.neighbors
def add_edges(self, edges):
for edge in edges:
self.add_edge(edge[0],edge[1])
def adjacencyList(self):
if len(self.vertices) >= 1:
return [str(key) + ":" + str(self.vertices[key]) for key in self.vertices.keys()]
else:
return dict()
def adjacencyMatrix(self):
if len(self.vertices) >= 1:
self.vertex_names = sorted(g.vertices.keys())
self.vertex_indices = dict(zip(self.vertex_names, range(len(self.vertex_names))))
import numpy as np
self.adjacency_matrix = np.zeros(shape=(len(self.vertices),len(self.vertices)))
for i in range(len(self.vertex_names)):
for j in range(i, len(self.vertices)):
for el in g.vertices[self.vertex_names[i]]:
j = g.vertex_indices[el]
self.adjacency_matrix[i,j] = 1
return self.adjacency_matrix
else:
return dict()
def graph(g):
""" Function to print a graph as adjacency list and adjacency matrix. """
return str(g.adjacencyList()) + '\n' + '\n' + str(g.adjacencyMatrix())
###################################################################################
a = Vertex('A')
b = Vertex('B')
c = Vertex('C')
d = Vertex('D')
e = Vertex('E')
a.add_neighbors([b,c,e])
b.add_neighbors([a,c])
c.add_neighbors([b,d,a,e])
d.add_neighbor(c)
e.add_neighbors([a,c])
g = Graph()
print(graph(g))
print()
g.add_vertices([a,b,c,d,e])
g.add_edge(b,d)
print(graph(g))
view raw graphUndirected.py hosted with ❤ by GitHub

Output:

{}
{}
["A:['B', 'C', 'E']", "C:['A', 'B', 'D', 'E']", "B:['A', 'C', 'D']", "E:['A', 'C']", "D:['B', 'C']"]
[[ 0. 1. 1. 0. 1.]
[ 1. 0. 1. 1. 0.]
[ 1. 1. 0. 1. 1.]
[ 0. 1. 1. 0. 0.]
[ 1. 0. 1. 0. 0.]]
view raw graphUndirected_output.txt hosted with ❤ by GitHub

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Deterministic Selection Algorithm Python Code

Anirudh Technical , , , , , , ,

Through this post, I’m sharing Python code implementing the median of medians algorithm, an algorithm that resembles quickselect, differing only in the way in which the pivot is chosen, i.e, deterministically, instead of at random.

Its best case complexity is O(n) and worst case complexity O(nlog2n)

I don’t have a formal education in CS, and came across this algorithm while going through Tim Roughgarden’s Coursera MOOC on the design and analysis of algorithms. Check out my implementation in Python.

def merge_tuple(a,b):
""" Function to merge two arrays of tuples """
c = []
while len(a) != 0 and len(b) != 0:
if a[0][0] < b[0][0]:
c.append(a[0])
a.remove(a[0])
else:
c.append(b[0])
b.remove(b[0])
if len(a) == 0:
c += b
else:
c += a
return c
def mergesort_tuple(x):
""" Function to sort an array using merge sort algorithm """
if len(x) == 0 or len(x) == 1:
return x
else:
middle = len(x)/2
a = mergesort_tuple(x[:middle])
b = mergesort_tuple(x[middle:])
return merge_tuple(a,b)
def lol(x,k):
""" Function to divide a list into a list of lists of size k each. """
return [x[i:i+k] for i in range(0,len(x),k)]
def preprocess(x):
""" Function to assign an index to each element of a list of integers, outputting a list of tuples"""
return zip(x,range(len(x)))
def partition(x, pivot_index = 0):
""" Function to partition an unsorted array around a pivot"""
i = 0
if pivot_index !=0: x[0],x[pivot_index] = x[pivot_index],x[0]
for j in range(len(x)-1):
if x[j+1] < x[0]:
x[j+1],x[i+1] = x[i+1],x[j+1]
i += 1
x[0],x[i] = x[i],x[0]
return x,i
def ChoosePivot(x):
""" Function to choose pivot element of an unsorted array using 'Median of Medians' method. """
if len(x) <= 5:
return mergesort_tuple(x)[middle_index(x)]
else:
lst = lol(x,5)
lst = [mergesort_tuple(el) for el in lst]
C = [el[middle_index(el)] for el in lst]
return ChoosePivot(C)
def DSelect(x,k):
""" Function to """
if len(x) == 1:
return x[0]
else:
xpart = partition(x,ChoosePivot(preprocess(x))[1])
x = xpart[0] # partitioned array
j = xpart[1] # pivot index
if j == k:
return x[j]
elif j > k:
return DSelect(x[:j],k)
else:
k = k - j - 1
return DSelect(x[(j+1):], k)
arr = range(100,0,-1)
print DSelect(arr,50)
%timeit DSelect(arr,50)
view raw DSelect.py hosted with ❤ by GitHub

I get the following output:

51
100 loops, best of 3: 2.38 ms per loop

Note that on the same input, quickselect is faster, giving us:

1000 loops, best of 3: 254 µs per loop

scikit-learn Linear Regression Example

Anirudh Technical , , , , , , , , , ,

Here’s a quick example case for implementing one of the simplest of learning algorithms in any machine learning toolbox – Linear Regression. You can download the IPython / Jupyter notebook here so as to play around with the code and try things out yourself.

I’m doing a series of posts on scikit-learn. Its documentation is vast, so unless you’re willing to search for a needle in a haystack, you’re better off NOT jumping into the documentation right away. Instead, knowing chunks of code that do the job might help.

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view raw linear_regression.ipynb hosted with ❤ by GitHub

Sharing IPython / Jupyter Notebooks via WordPress

Anirudh Technical , ,

In order to share (a static version of) your IPython / Jupyter notebook on your WordPress site, follow three straightforward steps.

Step 1: Let’s say your Jupyter Notebook looks like this:

blog_item_20160718_01

Open this notebook in a text editor and copy the content which may look like so:

blog_item_20160718_02

Step 2: Ctrl + A and Ctrl + C this content. Then Ctrl + V this to a GitHub Gist that you should create, like so:

blog_item_20160718_03

Step 3: Now simply Create public gist and embed the gist like you always embed gists on WordPress, viz., go to the HTML editor and add like so:

blog_item_20160718_04

I followed the exact steps that I’ve mentioned above to get the following result:

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view raw knn.ipynb hosted with ❤ by GitHub

 

Randomized Selection Algorithm (Quickselect) – Python Code

Anirudh Technical , , ,

Find the kth smallest element in an array without sorting.

That’s basically what this algorithm does. It piggybacks on the partition subroutine from the Quick Sort. If you don’t know what that is, you can check out more about the Quick Sort algorithm here and here, and understand the usefulness of partitioning an unsorted array around a pivot.

Selecting_quickselect_frames
Animated visualization of the randomized selection algorithm selecting the 22nd
 smallest value

Python Implementation

from random import randrange
def partition(x, pivot_index = 0):
i = 0
if pivot_index !=0: x[0],x[pivot_index] = x[pivot_index],x[0]
for j in range(len(x)-1):
if x[j+1] < x[0]:
x[j+1],x[i+1] = x[i+1],x[j+1]
i += 1
x[0],x[i] = x[i],x[0]
return x,i
def RSelect(x,k):
if len(x) == 1:
return x[0]
else:
xpart = partition(x,randrange(len(x)))
x = xpart[0] # partitioned array
j = xpart[1] # pivot index
if j == k:
return x[j]
elif j > k:
return RSelect(x[:j],k)
else:
k = k - j - 1
return RSelect(x[(j+1):], k)
x = [3,1,8,4,7,9]
for i in range(len(x)):
print RSelect(x,i),
view raw RSelect.py hosted with ❤ by GitHub

 

Related Posts
Quick Sort Python Code
Computing Work Done (Total Pivot Comparisons) by Quick Sort

Computing Work Done (Total Pivot Comparisons) by Quick Sort

Anirudh Technical , , , , ,

A key aspect of the Quick Sort algorithm is how the pivot element is chosen. In my earlier post on the Python code for Quick Sort, my implementation takes the first element of the unsorted array as the pivot element.

However with some mathematical analysis it can be seen that such an implementation is O(n2) in complexity while if a pivot is randomly chosen, the Quick Sort algorithm is O(nlog2n).

To witness this in action, one can measure the work done by the algorithm comparing two cases, one with a randomized pivot choice – and one with a fixed pivot choice, say the first element of the array (or the last element of the array).

Implementation

A decent proxy for the amount of work done by the algorithm would be the number of pivot comparisons. These comparisons needn’t be computed one-by-one, rather when there is a recursive call on a subarray of length m, you should simply add m−1 to your running total of comparisons.

3 Cases

To put things in perspective, let’s look at 3 cases. (This is basically straight out of a homework assignment from Tim Roughgarden’s course on the Design and Analysis of Algorithms).
Case I with the pivot being the first element.
Case II with the pivot being the last element.
Case III using the “median-of-three” pivot rule. The primary motivation behind this rule is to do a little bit of extra work to get much better performance on input arrays that are nearly sorted or reverse sorted.

Median-of-Three Pivot Rule

Consider the first, middle, and final elements of the given array. (If the array has odd length it should be clear what the “middle” element is; for an array with even length 2k, use the kth element as the “middle” element. So for the array 4 5 6 7, the “middle” element is the second one —- 5 and not 6! Identify which of these three elements is the median (i.e., the one whose value is in between the other two), and use this as your pivot.

Python Code

This file contains all of the integers between 1 and 10,000 (inclusive, with no repeats) in unsorted order. The integer in the ith row of the file gives you the ith entry of an input array. I downloaded this file and named it QuickSort_List.txt

You can run the code below and see for yourself that the number of comparisons for Case III are 138,382 compared to 162,085 and 164,123 for Case I and Case II respectively. You can play around with the code in an IPython / Jupyter notebook here.

#!/usr/bin/env
# Case I
# First element of the unsorted array is chosen as pivot element for sorting using Quick Sort
def countComparisonsWithFirst(x):
""" Counts number of comparisons while using Quick Sort with first element of unsorted array as pivot """
global count_pivot_first
if len(x) == 1 or len(x) == 0:
return x
else:
count_pivot_first += len(x)-1
i = 0
for j in range(len(x)-1):
if x[j+1] < x[0]:
x[j+1],x[i+1] = x[i+1], x[j+1]
i += 1
x[0],x[i] = x[i],x[0]
first_part = countComparisonsWithFirst(x[:i])
second_part = countComparisonsWithFirst(x[i+1:])
first_part.append(x[i])
return first_part + second_part
# Case II
# Last element of the unsorted array is chosen as pivot element for sorting using Quick Sort
def countComparisonsWithLast(x):
""" Counts number of comparisons while using Quick Sort with last element of unsorted array as pivot """
global count_pivot_last
if len(x) == 1 or len(x) == 0:
return x
else:
count_pivot_last += len(x)-1
x[0],x[-1] = x[-1],x[0]
i = 0
for j in range(len(x)-1):
if x[j+1] < x[0]:
x[j+1],x[i+1] = x[i+1], x[j+1]
i += 1
x[0],x[i] = x[i],x[0]
first_part = countComparisonsWithLast(x[:i])
second_part = countComparisonsWithLast(x[i+1:])
first_part.append(x[i])
return first_part + second_part
# Case III
# Median-of-three method used to choose pivot element for sorting using Quick Sort
def middle_index(x):
""" Returns the index of the middle element of an array """
if len(x) % 2 == 0:
middle_index = len(x)/2 - 1
else:
middle_index = len(x)/2
return middle_index
def median_index(x,i,j,k):
""" Returns the median index of three when passed an array and indices of any 3 elements of that array """
if (x[i]-x[j])*(x[i]-x[k]) < 0:
return i
elif (x[j]-x[i])*(x[j]-x[k]) < 0:
return j
else:
return k
def countComparisonsMedianOfThree(x):
""" Counts number of comparisons while using Quick Sort with median-of-three element is chosen as pivot """
global count_pivot_median
if len(x) == 1 or len(x) == 0:
return x
else:
count_pivot_median += len(x)-1
k = median_index(x, 0, middle_index(x), -1)
if k != 0: x[0], x[k] = x[k], x[0]
i = 0
for j in range(len(x)-1):
if x[j+1] < x[0]:
x[j+1],x[i+1] = x[i+1], x[j+1]
i += 1
x[0],x[i] = x[i],x[0]
first_part = countComparisonsMedianOfThree(x[:i])
second_part = countComparisonsMedianOfThree(x[i+1:])
first_part.append(x[i])
return first_part + second_part
#####################################################################
# initializing counts
count_pivot_first = 0; count_pivot_last = 0; count_pivot_median = 0
#####################################################################
# Cast I
# Read the contents of the file into a Python list
NUMLIST_FILENAME = "QuickSort_List.txt"
inFile = open(NUMLIST_FILENAME, 'r')
with inFile as f: numList = [int(integers.strip()) for integers in f.readlines()]
# call functions to count comparisons
countComparisonsWithFirst(numList)
#####################################################################
# Read the contents of the file into a Python list
NUMLIST_FILENAME = "QuickSort_List.txt"
inFile = open(NUMLIST_FILENAME, 'r')
with inFile as f: numList = [int(integers.strip()) for integers in f.readlines()]
# call functions to count comparisons
countComparisonsWithLast(numList)
#####################################################################
# Read the contents of the file into a Python list
NUMLIST_FILENAME = "QuickSort_List.txt"
inFile = open(NUMLIST_FILENAME, 'r')
with inFile as f: numList = [int(integers.strip()) for integers in f.readlines()]
# call functions to count comparisons
countComparisonsMedianOfThree(numList)
#####################################################################
print count_pivot_first, count_pivot_last, count_pivot_median
view raw countComparisons.py hosted with ❤ by GitHub

Quick Sort Python Code

Anirudh Technical , , ,

Sorting_quicksort_anim

Yet another post for the crawlers to better index my site for algorithms and as a repository for Python code. The quick sort algorithm is well explained in the topmost Google search result for ‘Quick Sort Python Code’, but the code is unnecessarily convoluted. Instead, go with the code below.

In it, I assume the pivot to be the first element. You can easily add a function to  randomize selection of the pivot. Choosing a random pivot minimizes the chance that you will encounter worst-case O(n2) performance. Always choosing first or last would cause worst-case performance for nearly-sorted or nearly-reverse-sorted data.

def quicksort(x):
if len(x) == 1 or len(x) == 0:
return x
else:
pivot = x[0]
i = 0
for j in range(len(x)-1):
if x[j+1] < pivot:
x[j+1],x[i+1] = x[i+1], x[j+1]
i += 1
x[0],x[i] = x[i],x[0]
first_part = quicksort(x[:i])
second_part = quicksort(x[i+1:])
first_part.append(x[i])
return first_part + second_part
alist = [54,26,93,17,77,31,44,55,20]
quicksort(alist)
print(alist)
view raw quicksort.py hosted with ❤ by GitHub

Also read:
Computing Work Done (Total Pivot Comparisons) by Quick Sort
Karatsuba Multiplication Algorithm – Python Code
Merge Sort

How to become a Data Scientist in 6 months

Anirudh Non Technical , , , ,

Disclaimer: I’m not a data scientist yet. That’s still work in progress, but I’d recommend this excellent talk given by  Tetiana Ivanova to put an enthusiast’s data science journey in perspective.

MITx 15.071x (Analytics Edge) – 2016

Anirudh Non Technical , , , , , ,

I am auditing this course currently and just completed its 2nd assignment. It’s probably one of the best courses out there to learn R in a way that you go beyond the syntax with an objective in mind – to do analytics and run machine learning algorithms to derive insight from data. This course is different from machine learning courses by say, Andrew Ng in that this course won’t focus on coding the algorithm and rather would emphasize on diving right into the implementation of those algorithms using libraries that the R programming language already equips us with.

Take a look at the course logistics. And hey, they’ve got a Kaggle competition!

AnalyticsEdgeLogistics

There’s still time to enroll and grab a certificate (or simply audit). The course is offered once a year. I met a bunch of people who did well at a data hackathon I had gone to recently, who had learned the ropes in data science thanks to Analytics Edge.

Detecting Structural Breaks in China’s FX Regime

Anirudh Technical , , , , , , , , ,
Edit: This post is in its infancy. Work is still ongoing as far as deriving insight from the data is concerned. More content and economic insight is expected to be added to this post as and when progress is made in that direction.

This is an attempt to detect structural breaks in China’s FX regime using Frenkel Wei regression methodology (this was later improved by Perron and Bai). I came up with the motivation to check for these structural breaks while attending a guest lecture on FX regimes by Dr. Ajay Shah delivered at IGIDR. This is work that I and two other classmates are working on as a term paper project under the supervision of Dr. Rajeswari Sengupta.

The code below can be replicated and run as is, to get same results.

## if fxregime or strucchange package is absent from installed packages, download it and load it
if(!require('fxregime')){
install.packages("fxregime")
}
if(!require('strucchange')){
install.packages("strucchange")
}
## load packages
library("fxregime")
library('strucchange')
# load the necessary data related to exchange rates - 'FXRatesCHF'
# this dataset treats CHF as unit currency
data("FXRatesCHF", package = "fxregime")
## compute returns for CNY (and explanatory currencies)
## since China abolished fixed USD regime
cny <- fxreturns("CNY", frequency = "daily",
start = as.Date("2005-07-25"), end = as.Date("2010-02-12"),
other = c("USD", "JPY", "EUR", "GBP"))
## compute all segmented regression with minimal segment size of
## h = 100 and maximal number of breaks = 10
regx <- fxregimes(CNY ~ USD + JPY + EUR + GBP,
data = cny, h = 100, breaks = 10, ic = "BIC")
## Print summary of regression results
summary(regx)
## minimum BIC is attained for 2-segment (1-break) model
plot(regx)
round(coef(regx), digits = 3)
sqrt(coef(regx)[, "(Variance)"])
## inspect associated confidence intervals
cit <- confint(regx, level = 0.9)
cit
breakdates(cit)
## plot LM statistics along with confidence interval
flm <- fxlm(CNY ~ USD + JPY + EUR + GBP, data = cny)
scus <- gefp(flm, fit = NULL)
plot(scus, functional = supLM(0.1))
## add lines related to breaks to your plot
lines(cit)
view raw FrankelWei_2005_2010.r hosted with ❤ by GitHub

As can be seen in the figure below, the structural breaks correspond to the vertical bars. We are still working on understanding the motivations of China’s central bank in varying the degree of the managed float exchange rate.

strucchange_china_2006_2010

EDIT (May 16, 2016):

The code above uses data provided by the package itself. If you wished to replicate this analysis on data after 2010, you will have to use your own data. We used Quandl, which lets you get 10 premium datasets for free. An API key (for only 10 calls on premium datasets) is provided if you register there. Foreign exchange rate data (2000 onward till date) apparently, is premium data. You can find these here.

Here are the (partial) results and code to work the same methodology on the data from 2010 to 2016:

20102016

## if fxregime is absent from installed packages, download it and load it
if(!require('fxregime')){
install.packages("fxregime")
}
## load package
library("fxregime")
# load the necessary data related to exchange rates - 'FXRatesCHF'
# this dataset treats CHF as unit currency
# install / load Quandl
if(!require('Quandl')){
install.packages("Quandl")
}
library(Quandl)
# Extract and load currency data series with respect to CHF from Quandl
# Extract data series from Quandl. Each Quandl user will have unique api_key
# upon signing up. The freemium version allows access up to 10 fx rate data sets
# USDCHF <- Quandl("CUR/CHF", api_key="p2GsFxccPGFSw7n1-NF9")
# write.csv(USDCHF, file = "USDCHF.csv")
# USDCNY <- Quandl("CUR/CNY", api_key="p2GsFxccPGFSw7n1-NF9")
# write.csv(USDCNY, file = "USDCNY.csv")
# USDJPY <- Quandl("CUR/JPY", api_key="p2GsFxccPGFSw7n1-NF9")
# write.csv(USDJPY, file = "USDJPY.csv")
# USDEUR <- Quandl("CUR/EUR", api_key="p2GsFxccPGFSw7n1-NF9")
# write.csv(USDEUR, file = "USDEUR.csv")
# USDGBP <- Quandl("CUR/GBP", api_key="p2GsFxccPGFSw7n1-NF9")
# write.csv(USDGBP, file = "USDGBP.csv")
# load the data sets into R
USDCHF <- read.csv("G:/China's Economic Woes/USDCHF.csv")
USDCHF <- USDCHF[,2:3]
USDCNY <- read.csv("G:/China's Economic Woes/USDCNY.csv")
USDCNY <- USDCNY[,2:3]
USDEUR <- read.csv("G:/China's Economic Woes/USDEUR.csv")
USDEUR <- USDEUR[,2:3]
USDGBP <- read.csv("G:/China's Economic Woes/USDGBP.csv")
USDGBP <- USDGBP[,2:3]
USDJPY <- read.csv("G:/China's Economic Woes/USDJPY.csv")
USDJPY <- USDJPY[,2:3]
start = 1 # corresponds to 2016-05-12
end = 2272 # corresponds to 2010-02-12
dates <- as.Date(USDCHF[start:end,1])
USD <- 1/USDCHF[start:end,2]
CNY <- USDCNY[start:end,2]/USD
JPY <- USDJPY[start:end,2]/USD
EUR <- USDEUR[start:end,2]/USD
GBP <- USDGBP[start:end,2]/USD
# reverse the order of the vectors to reflect dates from 2005 - 2010 instead of
# the other way around
USD <- USD[length(USD):1]
CNY <- CNY[length(CNY):1]
JPY <- JPY[length(JPY):1]
EUR <- EUR[length(EUR):1]
GBP <- GBP[length(GBP):1]
dates <- dates[length(dates):1]
df <- data.frame(CNY, USD, JPY, EUR, GBP)
df$weekday <- weekdays(dates)
row.names(df) <- dates
df <- subset(df, weekday != 'Sunday')
df <- subset(df, weekday != 'Saturday')
df <- df[,1:5]
zoo_df <- as.zoo(df)
# Code to replicate analysis
cny_rep <- fxreturns("CNY", data = zoo_df, frequency = "daily",
other = c("USD", "JPY", "EUR", "GBP"))
time(cny_rep) <- as.Date(row.names(df)[2:1627])
regx_rep <- fxregimes(CNY ~ USD + JPY + EUR + GBP,
data = cny_rep, h = 100, breaks = 10, ic = "BIC")
summary(regx_rep)
## minimum BIC is attained for 2-segment (5-break) model
plot(regx_rep)
round(coef(regx_rep), digits = 3)
sqrt(coef(regx_rep)[, "(Variance)"])
## inspect associated confidence intervals
cit_rep <- confint(regx_rep, level = 0.9)
breakdates(cit_rep)
## plot LM statistics along with confidence interval
flm_rep <- fxlm(CNY ~ USD + JPY + EUR + GBP, data = cny_rep)
scus_rep <- gefp(flm_rep, fit = NULL)
plot(scus_rep, functional = supLM(0.1))
## add lines related to breaks to your plot
lines(cit_rep)
apply(cny_rep,1,function(x) sum(is.na(x)))
view raw FrankelWei_2010_2016.r hosted with ❤ by GitHub

We got breaks in 2010 and in 2015 (when China’s stock markets crashed). We would have hoped for more breaks (we can still get them), but that would depend on the parameters chosen for our regression.