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COMP226 Assignment 2: Strategy

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The latest version of this document can be found here:

https://www2.csc.liv.ac.uk/~rahul/teaching/comp226/_downloads/a2.pdf

Continuous

Assessment Number

2 (of 2)

Weighting 10%

Assignment Circulated Thursday 18 March 2020

Deadline 17:00 Friday 1st May 2020

Submission Mode Electronic only

Learning Outcomes

Assessed

This assignment will address the following learning outcomes:

• Understand the spectrum of computer-based trading

applications and techniques, from profit-seeking trading

strategies to execution algorithms.

• Be able to design trading strategies and evaluate critically

their historical performance and robustness.

• Understand the common pitfalls in developing trading

strategies with historical data.

• Understand methods for measuring risk and diversification

at the portfolio level.

Summary of

Assessment

The goal of this assignment is to implement and optimize a

well-defined trading strategy within the backtester_v5.3

framework. The assignment will be assessed via the testing of 6

functions that you need to implement. The input and output

behaviour of each function is fully specified and a code template

is provided as a starting point.

Marking Criteria Individual marks are attributed for each of 6 functions that

should be implemented. If all 6 function implementations pass

all the automated tests then a mark of 100% will be achieved.

Partial credit for a function may be awarded if some but not all

automated tests for that function are passed. The marks

available for each function are given below.

Submission necessary

in order to satisfy

module requirements

No

Late Submission

Penalty

Standard UoL policy; note that no resubmissions after the

deadline will be considered.

Expected time taken Roughly 8 hours

Introduction: the backtester framework

You will write a strategy that should run in the backtester framework, which is available

from

http://www2.csc.liv.ac.uk/~rahul/teaching/comp226/bt.html#backtester

The first thing you should do is download and unzip backtester_v5.3.zip, which will create

a directory backtester_v5.3 on your hard drive. Here is a listing of the zip file contents:

backtester_v5.3

├── DATA

│ ├── A2

│ │ ├── 01.csv

│ │ └── 02.csv

│ └── EXAMPLE

│ ├── 01.csv

│ ├── 02.csv

│ ├── 03.csv

│ ├── 04.csv

│ └── 05.csv

├── example_strategies.R

├── framework

│ ├── backtester.R

│ ├── data.R

│ ├── processResults.R

│ └── utilities.R

├── in-sample_period.R

├── main.R

├── main_optimize.R

├── main_template.R

└── strategies

├── a2_template.R

├── bankrupt.R

├── bbands_contrarian.R

├── bbands_holding_period.R

├── bbands_trend_following.R

├── big_spender.R

├── copycat.R

├── extreme_limit.R

├── fixed.R

├── random.R

├── rsi_contrarian.R

└── simple_limit.R

5 directories, 28 files

Next you should open R and make sure that the working directory is the backtester_v5.3

directory on your hard drive (you can use setwd if required). You can now try the example

code as follows:

source('main.R')

If this doesn't work, first make sure you are have set the working directory correctly, and

then make sure you have installed all the required packages (see the error messages you

get to figure out what the problem is). When it works it will produce a plot like the following:

Active on 100 % of days; PD ratio = −153.44

Jan Apr Jul

999400

999600

999800

1000000

05 : PD ratio = 3.88 / 13.7 = 0.28

03 : PD ratio = −0.19 04 : PD ratio = 23.02 / 138 = 0.17

01 : PD ratio = 0.06 / 0.03 = 1.97 02 : PD ratio = −180.2

Jan Apr Jul

Jan Apr Jul Jan Apr Jul

Jan Apr Jul Jan Apr Jul

−600

−400

−200

0

−50

0

50

100

0.00

0.02

0.04

0.06

−0.5

−0.4

−0.3

−0.2

−0.1

0.0

−5

0

5

There is one equity curve for each series in the data (5 of them in this case), and one

aggregate equity curve.

Let's go through main.R and see what the individual parts do.

Sourcing the framework and example strategies

First we source the framework itself.

source('framework/data.R')

source('framework/backtester.R')

source('framework/processResults.R')

source('framework/utilities.R')

Then we load example_strategies.R, which provides an easy way to run several examples,

and which we will return to shortly.

source('example_strategies.R')

Loading data

Next, we load in data that we will use via the function getData, which is defined in

framework/data.R. This function returns a list of xts objects. These will be passed to the

function backtester, though we may first change the start and end dates of the xts objects

(which you will need to do for assignment 2).

# load data

dataList <- getData(directory="EXAMPLE")

There are 5 series in the directory backtester_5.3/DATA/EXAMPLE/, and therefore the list

dataList has 5 elements too.

> length(dataList)

[1] 5

Each element is an xts:

> for (x in dataList) print(class(x))

[1] "xts" "zoo"

[1] "xts" "zoo"

[1] "xts" "zoo"

[1] "xts" "zoo"

[1] "xts" "zoo"

All the series have the same start and end dates:

> for (x in dataList) print(paste(start(x),end(x)))

[1] "1970-01-02 1973-01-05"

[1] "1970-01-02 1973-01-05"

[1] "1970-01-02 1973-01-05"

[1] "1970-01-02 1973-01-05"

[1] "1970-01-02 1973-01-05"

The individual series contain Open, High, Low, Close, and Volume columns:

> head(dataList[[1]])

Open High Low Close Volume

1970-01-02 0.7676 0.7698 0.7667 0.7691 3171

1970-01-03 0.7689 0.7737 0.7683 0.7729 6311

1970-01-04 0.7725 0.7748 0.7718 0.7732 4317

1970-01-05 0.7739 0.7756 0.7739 0.7751 3409

1970-01-06 0.7760 0.7770 0.7754 0.7757 2904

1970-01-07 0.7738 0.7744 0.7728 0.7743 3514

Loading a strategy

We will now load a strategy using the load_strategy function that is defined in

example_strategies.R. We can pick a strategy from a list of examples strategies that is

specified at the start of example_strategies.R:

example_strategies <- c("fixed",

"big_spender",

"bankrupt",

"copycat",

"random",

"rsi_contrarian",

"bbands_trend_following",

"bbands_contrarian",

"bbands_holding_period",

"simple_limit")

Returning now to main.R we see that we have picked one of these (and then checked that it

was a valid choice with)

# choose strategy from example_strategies

strategy <- "fixed"

# check that the choice is valid

is_valid_example_strategy <- function(strategy) {

strategy %in% example_strategies

}

stopifnot(is_valid_example_strategy(strategy))

Now we actually "load the strategy".

# load in strategy and params

load_strategy(strategy) # function from example_strategies.R

We used the function load_strategy from example_strategies.R. This function sources

the strategy file, in this case backtester_v5.3/strategies/fixed.R, and sets a variable

params using example_params from example_strategies.R:

load_strategy <- function(strategy) {

# load strategy

strategyFile <- file.path('strategies', paste0(strategy,'.R'))

cat("Sourcing",strategyFile,"\n")

source(strategyFile) # load in getOrders

# set params via global assignment

params <<- example_params[[strategy]]

print("Parameters:")

print(params)

}

The structure of a strategy

Here is the contents of the strategy file backtester_v5.3/strategies/fixed.R:

# This strategy only uses market orders

# params$sizes specifies a fixed number of contracts per series

# We take the corresponding long/short position in each series

# by placing a market order on the 1st iteration

# No further orders are placed by getOrders

# The backtester automatically exits all positions

# as market orders at the end when the data runs out

getOrders <- function(store, newRowList, currentPos, info, params) {

allzero <- rep(0,length(newRowList))

marketOrders <- allzero

if (is.null(store)) {

# take position during first iteration and hold

marketOrders <- params$sizes

store <- 1 # not null

}

return(list(store=store,marketOrders=marketOrders,

limitOrders1=allzero,

limitPrices1=allzero,

limitOrders2=allzero,

limitPrices2=allzero))

}

The backtester framework runs a strategy by calling getOrders. The arguments to

getOrders are fixed, i.e., they are the same for all strategies. In the example strategy

fixed.R, getOrders is the only function. The arguments to getOrders are as follows:

getOrders <- function(store, newRowList, currentPos, info, params) {

• store: contains all data you choose to save from one period to the next

• newRowList: new day's data (a list of single rows from the series)

• currentPos: the vector of current positions in each series

• params: a list of parameters that are sent to the function

Here's how the strategy fixed.R works. In the very first period the backtester always (for

every strategy) passes store to getOrders with NULL as its value. Thus in the first period,

and the first period only, the vector marketOrders will be set to the parameter

params$sizes, which should be a vector of positions with length equal to the number of

series, which is 5 in this case. In example_strategies.R we see this parameter

params$sizes, which is the only parameter for fixed.R, set as follows:

list(sizes=rep(1,5))

With these sizes, we buy and hold one unit in every series.

Changing the parameters

We can change the parameters and take positions in only some series and go short in some

series, e.g., with:

params <- list(sizes=c(1,2,0,0,-1))

We can set this either in example_strategies.R, or in main.R, as long as it comes after we

have called load_strategy.

Active on 100 % of days; PD ratio = −364.37

Jan Apr Jul

998500

999000

999500

1000000

05 : PD ratio = −4.02

03 : PD ratio = 0 04 : PD ratio = 0

01 : PD ratio = 0.06 / 0.03 = 1.97 02 : PD ratio = −360.4

Jan Apr Jul

Jan Apr Jul Jan Apr Jul

Jan Apr Jul Jan Apr Jul

−1500

−1000

−500

0

−0.050

−0.025

0.000

0.025

0.050

0.00

0.02

0.04

0.06

−0.050

−0.025

0.000

0.025

0.050

−5

0

5

Compare the new equity curves with the ones above. Note that for series 1 they are the

same, for series 2 the new one is scaled by 2, for series 3 and 4 we no longer trade, and for

series 5 we now take a short position, so the new series 5 is a reflection (in the profit and

loss axis) of the old series 5 equity curve.

Market orders

The fixed example strategy enters position at the first opportunity using market orders. The

backester framework supports market and limit orders and some of the example strategies

use limit orders. However, we will not use limit orders for assignment 2, only market orders.

Recall that market orders specify volume and direction (but not price), and limit orders

specify price, volume, and direction. In the backtester framework, trading decisions are

made after the close of day k, and trades are executed on day k+1. For each day, the

framework supports one market order for each series, and two limit orders for each

series. These orders are returned from getOrders as follows:

return(list(store=store,marketOrders=marketOrders,

limitOrders1=limitOrders1,

limitPrices1=limitPrices1,

limitOrders2=limitOrders2,

limitPrices2=limitPrices2))

Market orders will be executed at the open on day k+1. The sizes and directions of

market orders are encoded in the vector marketOrders of the return list of getOrders. For

example, the vector

c(0,-5,0,1,0)

means place a market order for 5 units short in series 2, and 1 unit long in series 4.

To repeat, we will not use limit orders for assignment 2, so you can leave limitOrders1,

limitPrices1, limitOrders2, limitPrices2 as zero vectors when you do assignment 2.

Subsetting the data

The next thing in main.R is a subsetting of the time period for the backtest as follows:

inSampDays <- 200 # in-sample period 1:inSampDays

dataList <- lapply(dataList, function(x) x[1:inSampDays])

So we are only using the first 200 days.

Hint

You should adapt this use of lapply on dataList in order to define the in-sample

period in assignment 2.

Running the backtest

Finally we actually do the backtest and plot the results as follows:

# Do backtest

results <- backtest(dataList,getOrders,params,sMult=0.2)

pfolioPnL <- plotResults(dataList,results)

The arguments to the function backtest are the following:

• dataList - list of (daily) xts objects (with identical indexes)

• getOrders - the strategy

• params - the parameters for the strategy

• sMult - slippage multiplier (proportion of overnight gap)

Results for individual series are available in results$pnlList. The portfolio results are

available in pfolioPnL, which is produced by plotResults(dataList,results). This

function also automatically plots individual and aggregate equity curves, and computes

variant of the Calmar Ratio that we call the Profit Drawdown Ratio (PD ratio for short) - it is

the final profit divided by the maximum drawdown in terms of profit and loss, or if the

strategy makes a loss overall the PD ratio is just that loss (which is negative). You do not

need to write code to compute this, since it has already been done for you. In assignment 2

we will optimize the aggregate PD ratio - this value is stored in pfolioPnL$fitAgg, e.g., for

our first example we have:

> print(pfolioPnL$fitAgg)

[1] -153.44

This matches up with the PD ratio that appears at the top of the aggregate equity curve

produced by plotResults.

Parameter optimization

Before we move on to assignment 2, we will briefly look at an example of parameter

optimization that will be useful for assignment 2. To make it easier to carry out parameter

optimizations, getOrders takes an argument params. This can be used to pass a parameter

combination to a strategy. This in turn can be used to do a parameter optimization as

main_optimize.R demonstrates. Here is the source code for main_optmize.R, which uses

the example strategy bbands_contrarian, which is an implementation of the "BBands

Overbought/Oversold" strategy from the slides:

source('framework/data.R'); source('framework/backtester.R')

source('framework/processResults.R'); source('strategies/bbands_contrarian.R')

numOfDays <- 200

dataList <- getData(directory="EXAMPLE")

dataList <- lapply(dataList, function(x) x[1:numOfDays])

sMult <- 0.2 # slippage multiplier

lookbackSeq <- seq(from=20,to=40,by=10)

sdParamSeq <- seq(from=1.5,to=2,by=0.5)

paramsList <- list(lookbackSeq,sdParamSeq)

numberComb <- prod(sapply(paramsList,length))

resultsMatrix <- matrix(nrow=numberComb,ncol=3)

colnames(resultsMatrix) <- c("lookback","sdParam","PD Ratio")

pfolioPnLList <- vector(mode="list",length=numberComb)

count <- 1

for (lb in lookbackSeq) {

for (sdp in sdParamSeq) {

params <- list(lookback=lb,sdParam=sdp,series=1:5,posSizes=rep(1,5))

results <- backtest(dataList, getOrders, params, sMult)

pfolioPnL <- plotResults(dataList,results)

resultsMatrix[count,] <- c(lb,sdp,pfolioPnL$fitAgg)

pfolioPnLList[[count]]<- pfolioPnL

cat("Just completed",count,"out of",numberComb,"\n")

#print(resultsMatrix[count,])

count <- count + 1

}

}

print(resultsMatrix[order(resultsMatrix[,"PD Ratio"]),])

The code template and data for assignment 2

You are now ready to start working on assignment 2. To do so you should read and work

through the rest of this document very carefully.

As a first step, try to run main_template.R, which is setup to use the right data for the

assignment and to load in the template strategy code strategies/a2_template.R. If you

try to source main_template.R you will get an error as follows:

Error in if (store$iter > params$lookbacks$long) { :

argument is of length zero

If you read on you will see that the final strategy requires a parameter called lookbacks.

Read on to see what form this parameter should take.

The code template contains templates for the 6 functions that you need to complete. These

functions are:

1. getTMA

2. getPosSignFromTMA

3. getPosSize

4. getOrders

5. getInSampleResult

6. getInSampleOptResult

The rest of the document is split into two parts. The first part describes the function

requirements and marking criteria for the first 4 functions, which relate to the strategy

implementation. The second part describes the function requirements and marking criteria

for the last 2 functions. Hints are given on how best to implement things, so read carefully.

For all 6 functions, example outputs are provided so that you can test whether you

have implemented the functions correctly.

Note

You can develop the first three functions without running the backtester, which may

be easier.

Part 1: strategy implementation

The overall goal of the assignment is the implementation and optimization of a triple moving

average crossover (TMA) trading strategy. The specification of the strategy and the functions

that it should comprise are given in full detail, so the correctness of your code can and will

be checked automatically.

The TMA strategy you will implement is related to Example 1 in COMP226 slides 17.

However, long and short positions are swapped as compared to that example (so you will

here implement a mean-reversion as opposed to a trend following type strategy).

The strategy uses three moving averages with three different lookbacks (window lengths).

The short lookback should be smaller than the medium window, which in turn should be

smaller than the long lookback. In every trading period, the strategy will compute the value

of these three moving averages. You will achieve this be completing the implementation of

the function getTMA.

The following table indicates the position that the strategy will take depending on the relative

values of the three moving averages (MAs). You will compute this position (sign, but not

size) by completing the function getPosSignFromTMA. The system is out of the market (i.e.,

flat) when the relationship between the short moving average and the medium moving

average does not match the relationship between the medium moving avergage and long

moving average.

MA MA MA Position

short > medium > long short

short < medium < long long

The function getPosSignFromTMA should use a function getTMA. The position size, i.e., the

number of units to be long or short, will be determined by the function getPosSize. Finally,

as usual in the backtester framework for COMP226 and COMP396, the position sizes are

given to the backtester in the function getOrders. Here are the detailed specification and

marks available for these first 4 functions.

Function

name

Input parameters Expected behaviour Marks available for a

correct implementation

getTMA close_prices;

lookbacks. The

specific form that

these arguments

should take is

specified in the

template code via

the 6 checks that

you need to

implement.

You should first implement

the checks as described in

the template. Hints of how

to implement them are

given below.

The function should return

a list with three named

elements (named short,

medium, and long). Each

element should be equal to

the value of a simple

moving average with the

respective window size as

defined by lookbacks. The

windows should all end in

the same period, which

should be the final row of

close_prices.

18% (3% per check) for

the checks; 12% for a

correct return

getPosSign

FromTMA

tma_list is a list

with three named

elements, short,

medium, and long.

These correspond

to the simple

moving averages

as returned by

getTMA.

Note: You do not

need to check the

validity of the

function argument

in this case, or for

the remaining

functions either.

This function should return

either 0, 1, or -1.

If the short value of

tma_list is less than the

medium value, and the

medium value is less than

the long value, it should

return 1 (indicating a long

position).

If the short value of

tma_list is greater than

the medium value, and the

medium value is greater

than the long value, it

should return -1 (indicating

a short position).

Otherwise, the return value

should be 0 (indicating a

flat position).

15%

getPosSize current_close:

this is the current

close for one of

the series.

constant: this

argument should

have a default

value of 1000.

The function should return

(constant divided by

current_close) rounded

down to the nearest

integer.

5%

getOrders The arguments to

this function are

always the same

for all strategies

used in the

backtester

framework.

This function should

implement the strategy

outlined above and again

below in "Strategy

specification".

20%

Strategy specification

The strategy should apply the following logic independently for both series.

The strategy does nothing until there have been params$lookbacks$long-many

periods.

In the (params$lookbacks$long+1)-th period, and in every period after, the strategy

computes three simple moving averages with window lengths equal to:

• params$lookbacks$short

• params$lookbacks$medium

• params$lookbacks$long

The corresponding windows always end in the current period. The strategy should in

this period send market orders to assume a position (make sure you take into

account positions from earlier) according to getPosSignFromTMA and getPosSize.

(Limit orders are not required at all, and can be left as all zero.)

Hints

For the checks for getTMA you may find the following functions useful:

• The operator ! means not, and can be used to negate a boolean.

• sapply allows one to apply a function element-wise to a vector or list (e.g., to

a vector list c("short","medium","long")).

• all is a function that checks if all elements of a vector are true (for example,

it can be used on the result of sapply).

• %in% can be used to check if a element exists inside a vector.

To compute the moving average in getTMA you can use SMA from the TTR package.

Note: The list returned by getTMA should work as input to the function

getPosSignFromTMA.

For getPosSize, you can use the function floor.

As in the template, use the negative of currentPos summed with the new positions

you want to take to make sure that you assume the correct position.

In order to help you check whether you have implemented the functions correctly, we next

give some examples of how correct implementations of the functions will behave. These

examples assume that you have correctly implemented the first 4 functions in

a2_template.R and sourced the resulting code to make the functions available in the R

environment.

Example output for getTMA

First you should make sure that you have correctly implemented all 6 checks on the

function arguments. Here are 3 examples of expected behaviour:

> close_prices <- c(1,2,3)

> lookbacks <- list(short=as.integer(5),medium=as.integer(10),long=as.integer(20))

> getTMA(close_prices,lookbacks) # bad close_prices

Error in getTMA(close_prices, lookbacks) :

E04: close_prices is not an xts according to is.xts()

> dataList <- getData(directory="A2")

Read 2 series from DATA/A2

> close_prices <- dataList[[1]]$Close[1:19]

> getTMA(close_prices,lookbacks) # bad close_prices; too short

Error in getTMA(close_prices, lookbacks) :

E05: close_prices does not enough rows

> lookbacks <- list(5,10,25)

> getTMA(close_prices,lookbacks) # bad lookbacks; list elements not named

Error in getTMA(close_prices, lookbacks) :

E01: At least one of "short", "medium", "long" is missing from names(lookbacks)

Here is an example where we give the function valid arguments.

> lookbacks <- list(short=as.integer(5),medium=as.integer(10),long=as.integer(20))

> close_prices <- dataList[[1]]$Close[1:20]

> getTMA(close_prices,lookbacks)

$short

[1] 169

$medium

[1] 170.4

$long

[1] 171.05

Example output for getPosSignFromTMA

> getPosSignFromTMA(list(short=10,medium=20,long=30))

[1] 1

> getPosSignFromTMA(list(short=10,medium=30,long=20))

[1] 0

> getPosSignFromTMA(list(short=30,medium=20,long=10))

[1] -1

Example output for getPosSize

> current_close <- 100.5

> getPosSize(current_close)

[1] 9

> getPosSize(current_close,constant=100.4)

[1] 0

Example output for getOrders

To check your implementation of getOrders, see part 2 for examples of correct output for

the function getInSampleResult below.

Part 2: in-sample tests

Warning

The last two functions require a working implementation of getOrders. If your

implementation of getOrders does not work, then you will receive 0 marks for these

last two functions, even if they return the correct numbers. This is a simple protection

again plagiarism and collusion.

There are two more functions that you need to implement: getInSampleResult and

getInSampleOptResult. For both functions you will need to compute your own in-sample

period, which is based on your MWS username. This ensures that for part 2 there are

different answers for different students.

To get your in-sample period you should use in-sample_period.R as follows. Source it and

run the function getInSamplePeriod with your MWS username as per the following example.

Then use the first number in the returned vector as the start of the in-sample period and the

second number as the end.

> source('in-sample_period.R')

> getInSamplePeriod('x4xz1')

[1] 230 644

So for this example username the start of the in-sample period is day 230 and the end is

644. Note: you may need to install the package digest to use this code.

Once you have your own in-sample period (and a correct implementation of getOrders), you

are ready to complete the implementation of getInSampleResult.

Function

name

Input par

ameters

Expected behaviour Marks available for a

correct implementation

getInSampleR

esult

None This function should return the

PD ratio that is achieved when

the strategy is run with short

lookback 10, medium lookback

20, and long lookback 30, on

your username-specific

in-sample period.

The function should not contain

ANY code except the return

value; it should run and

complete instantanously.

10% (0 marks will be

given if the function

contains any code other

than the return statement)

To complete the final function getInSampleOptResult you need to do an in-sample

parameter optimization using the following parameter combinations for the:

• short lookback

• medium lookback

• long lookback

You should not optimize the constant used with getPosSize, and leave it as 1000 as defined

in the template code.

The parameter combinations are defined by two things: parameter ranges and a further

restriction. Make sure you correctly use both to produce the correct set of parameter

combinations. The ranges are:

Parameter Minimum value Increment Maximum Value

short lookback 100 5 110

medium lookback 105 5 120

long lookback 110 5 130

The further restriction is the following:

Further restriction on parameter values

You should further restrict the parameter combinations as follows:

• The medium lookback should always be strictly greater than the short lookback.

• The long lookback should always be strictly greater than the medium lookback.

You need to find the best PD ratio that can be achieved one this set of parameter

combinations for the in-sample period that corresponds to your username, and set it as the

return value of getInSampleOptResult.

Hint

The correct resulting number of parameter combinations is 28.

You can adapt backtester_v5.3/main_optimize.R. It is probably easiest to use three

nested for loops in order to ensure that you only check valid parameter combinations

(where the short < medium < long for the respective window lengths).

Function

name

Input pa

rameters

Expected behaviour Marks available for a

correct implementation

getInSampleO

ptResult

None This function should return the

best PD ratio than can be

achieved with the stated

allowable parameter

combinations on your

username-specific in-sample

period.

The function should not contain

ANY code except the return

value; it should run and

complete instantanously.

20% (0 marks will be

given if the function

contains any code other

than the return statement)

Next we give some example output for these two functions.

Example output for getInSampleResult

Username Correct return value

x1xxx -747.6

x1yyy -231.6

x1zzx -639.8

Example output for getInSampleOptResult

Username Correct return value

x1xxx 4.23

x1yyy 3.42

x1zzx 4.43

Marks summary

Function Marks

getTMA 30

getPosSignFromTMA 15

getPosSize 5

getOrders 20

getInSampleResult 10 (0 if getOrders does not work)

getInSampleOptResult 20 (0 if getOrders does not work)

Submission

You need to submit a single R file that contains your implementation of 6 functions. The file

shoud be called MWS-username.R where you should replace MWS-username by your MWS

username. For example if your username is "abcd" then you should submit a file named

"abcd.R".

Submission is via the department electronic submission system:

http://www.csc.liv.ac.uk/cgi-bin/submit.pl

Warning

Your code will be put through the department's automatic plagiarism and collusion

detection system. Student's found to have plagiarized or colluded will likely receive a

mark of zero. Do not show your work to other students.

联系我们

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- 邮箱：99515681@qq.com
- 工作时间：8:00-23:00
- 微信：codinghelp2

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