When data is too big to fit into memory, one approach is to break it into smaller pieces, operate on each piece, and then join the results back together. Here’s how to do that to calculate rolling mean pairwise correlations of a large stock universe.

## Background

We’ve been using the problem of calculating mean rolling correlations of ETF constituents as a test case for solving in-memory computation limitations in R.

We’re interested in this calculation as a research input to a statistical arbitrage strategy that leverages ETF-driven trading in the constituents. We wrote about an early foray into this trade.

Previously, we introduced this problem along with the concept of profiling code for performance bottlenecks here. We can do the calculation in-memory without any trouble for a regular ETF, say XLF (the SPDR financial sector ETF), but we quickly run into problems if we want to look at SPY.

In this post, we’re going to explore one workaround for R’s in-memory limitations by splitting the problem into smaller pieces and recombining them to get our desired result.

### The problem

When we performed this operation on the constituents of the XLF ETF, our largest intermediate dataframe consisted of around 3-million rows, easily within the capabilities of modern laptops.

XLF currently holds 68 constituent stocks. So for any day, we have \( \frac{68*67}{2} = 2,278 \) correlations to estimate (67 because we don’t want the diagonal of the correlation matrix, take half as we only need its upper or lower triangle).

We calculated five years of rolling correlations, so we had \( 5*250*2,278 = 2,847,500 \) correlations in total.

*Piece of cake.*

The problem gets a lot more interesting if we consider the SPY ETF and its 500 constituents.

For any day, we’d have \( \frac{500*499}{2} = 124,750 \) correlations to estimate. On five years of data, that’s \( 5*250*124,750 = 155,937,500 \) correlations in total.

I tried to do all of that at once in memory on my laptop…and failed.

So our original problem of designing the data wrangling pipeline to achieve our goal has now morphed into a problem of overcoming performance barriers. Let’s see what we can do about that.

## Load data and functions

First, we load some libraries and data (you can get the data used in this post from our github repo:

library(tidyverse) library(lubridate) library(glue) library(here) library(microbenchmark) library(profvis) theme_set(theme_bw()) load(here::here("data", "spxprices_2015.RData")) spx_prices <- spx_prices %>% filter(inSPX == TRUE)

Next, load the functions in our pipeline (we explored these in more detail in the last post):

# pad any missing values pad_missing <- function(df) { df %>% complete(ticker, date) } # calculate returns to each stock get_returns <- function(df) { df %>% group_by(ticker) %>% arrange(date, .by_group = TRUE) %>% mutate(return_simple = close / dplyr::lag(close) - 1) %>% select(date, ticker, return_simple) } # full join on date fjoin_on_date <- function(df) { df %>% full_join(df, by = "date") } # ditch corr matrix diagonal, one half wrangle_combos <- function(combinations_df) { combinations_df %>% ungroup() %>% # drop diagonal filter(ticker.x != ticker.y) %>% # remove duplicate pairs (eg A-AAL, AAL-A) mutate(tickers = ifelse(ticker.x < ticker.y, glue("{ticker.x}, {ticker.y}"), glue("{ticker.y}, {ticker.x}"))) %>% distinct(date, tickers, .keep_all = TRUE) } pairwise_corrs <- function(combination_df, period) { combination_df %>% group_by(tickers) %>% arrange(date, .by_group = TRUE) %>% mutate(rollingcor = slider::slide2_dbl( .x = return_simple.x, .y = return_simple.y, .f = ~cor(.x, .y), .before = (wdw-1), # resulting window size is before + current element .complete = TRUE) ) %>% select(date, tickers, rollingcor) } mean_pw_cors <- function(correlations_df) { correlations_df %>% group_by(date) %>% summarise(mean_pw_corr = mean(rollingcor, na.rm = TRUE)) }

For completeness, here’s our full pipeline with and without the intermediate objects:

spx_prices <- spx_prices %>% pad_missing() returns_df <- spx_prices %>% get_returns() combos_df <- returns_df %>% fjoin_on_date() wrangled_combos_df <- combos_df %>% wrangle_combos() corr_df <- wrangled_combos_df %>% pairwise_corrs(period = 60) meancorr_df <- corr_df %>% mean_pw_cors() meancorr_df <- spx_prices %>% pad_missing() %>% get_returns() %>% fjoin_on_date() %>% wrangle_combos() %>% pairwise_corrs(period = 60) %>% mean_pw_cors()

## Chunking our data

We know that the bottleneck is the rolling pairwise correlations calculation. But the prior steps can also blow our memory limits, particularly if we’ve got other objects in our environment. So we’ll split the entire pipeline into chunks.

But first, let’s talk about *why* it’s valid to split our pipeline into chunks.

We can chunk our data in this case because the output – the mean of the rolling pairwise correlations – is only dependent on the window of returns data over which those correlations are calculated.

For example, if our window used 20 periods, we could calculate today’s value from the matrix of returns for our stock universe over the last 20 periods. The calculation has no other dependencies.

The implication is that we could do all of those 20-period mean correlation calculations independently, then jam all the individual outputs together and get the correct answer.

Sweet!

But there are a couple of things to consider.

### Unintended consequences of chunking

Every time we calculate a window of returns, the first value of the window will be `NA`

: we need yesterday’s price to calculate the (close to close) return today.

Those prices exist in our raw data, but by extracting each window, we’re artificially dropping them for our calculation. We can see this if we take a few slices of our prices:

wdw <- 5 spx_prices <- spx_prices %>% pad_missing() %>% group_by(ticker) %>% arrange(date) spx_prices %>% slice(1:wdw) %>% get_returns() %>% pivot_wider(id_cols = date, names_from = ticker, values_from = return_simple) %>% select(starts_with("A")) spx_prices %>% slice((1+wdw):(wdw+wdw)) %>% get_returns() %>% pivot_wider(id_cols = date, names_from = ticker, values_from = return_simple) %>% select(starts_with("A"))

If you run that code, you can see that the first row of each slice is NA.

One solution would be to do that return calculation on all the data upfront, which we can do in memory, but isn’t really in the spirit of what we’re trying to demonstrate here.

Instead, we’ll extract more price data than we need for our return windows such that each window is complete.

That implies that we could process our data one chunk at a time where each chunk was a minimum size of `wdw + 1`

. Let’s test that out.

wdw <- 60 test <- spx_prices %>% slice(1:(wdw+1)) system.time({ test_corr <- test %>% get_returns() %>% fjoin_on_date() %>% wrangle_combos() %>% na.omit() %>% # comfortable omitting NA here as we've been careful about alignment via padding missing values pairwise_corrs(period = wdw) %>% na.omit() %>% # this na.omit removes NA in prior window due to complete = TRUE requirement mean_pw_cors() }) # user system elapsed # 50.64 1.14 51.84

Hmmm. That took the best part of a minute. And that’s a single calculation! Clearly that’s not going to be feasible when we have hundreds or thousands of calculations to perform.

### Increase chunk size

Let’s use a larger chunk size this time so that from one chunk we can do five rolling window calculations instead of one. Is processing time additive, or are there speedups to be had by going for scale?

wdw <- 60 test <- spx_prices %>% slice(1:(wdw+1+5)) system.time({ test_corrs <- test %>% get_returns() %>% fjoin_on_date() %>% wrangle_combos() %>% na.omit() %>% # comfortable omitting NA here as we've been careful about alignment via padding missing values pairwise_corrs(period = wdw) %>% na.omit() %>% # this na.omit removes NA in prior window due to complete = TRUE requirement mean_pw_cors() }) # user system elapsed # 76.01 1.90 78.37

That’s interesting. Doing five windows didn’t result in five times the computation time.

This would be partly explained by the execution path being “hot” following the first calculation, but there does seem to be some significant increase in efficiency when we do several calculations in a single chunk.

Let’s use profvis to see if we can figure out what’s going on.

### Profiling different chunking approaches

First, we profile the single window case:

library(profvis) profvis({ test <- spx_prices %>% slice(1:(wdw+1)) returns_df <- test %>% get_returns() combos_df <- returns_df %>% fjoin_on_date() wrangled_combos_df <- combos_df %>% wrangle_combos() %>% na.omit() corr_df <- wrangled_combos_df %>% pairwise_corrs(period = 60) %>% na.omit() meancorr_df <- corr_df %>% mean_pw_cors() })

Next, let’s profile the multi-window case:

profvis({ test <- spx_prices %>% slice(1:(wdw+1+5)) returns_df <- test %>% get_returns() combos_df <- returns_df %>% fjoin_on_date() wrangled_combos_df <- combos_df %>% wrangle_combos() %>% na.omit() corr_df <- wrangled_combos_df %>% pairwise_corrs(period = 60) %>% na.omit() meancorr_df <- corr_df %>% mean_pw_cors() })

In both cases, we spend a comparable amount of time in each step, except for the pairwise correlation step, in which the five-window case saw a 50% increase in time spent – which seems like a very good deal!

The time spent calculating the mean of the correlations doubled, but this accounts for only a negligible amount of the total time spent so it’s not worth worrying about.

This suggests that doing as many windows as possible in each chunk will likely give us the biggest bang for our buck.

Notice also in the previous `profiz`

output that the largest amount of memory is allocated at the `wrangle_combos`

step (this is not the *slowest* step, but it produces the largest dataframe). This step will give us a good proxy for estimating a sensible chunk size.

But first, we need to know how much memory we can actually allocate.

## How much memory can we use?

It depends on your operating system and your machine specs. I’m on Windows 10 and my machine has 32 GB of RAM.

I can use `memory.limit()`

and `pryr::mem_used`

to see the memory status of my machine:

library(pryr) memory.limit() mem_used() # 32537 # 4.76 GB

Cool – apparently R can max out my RAM (not that you’d let it…) and R has used 4.76 GB. That’s mostly because I’ve got a bunch of large objects in memory from things I did previously, and which I’ll remove before I do anything serious.

We can’t get too precise when it comes to estimating how much memory an R object might require.

The memory allocation of R objects doesn’t grow linearly with size, as R requests oversized blocks of memory and then manages those blocks, rather than incrementally asking the operating system for more each time something is created.

There are also memory overheads with the data structures themselves, such as metadata and pointers to other objects in memory.

Let’s make a dataframe of combinations for 250 days of stock data (about one year), calculate the size of the object, and use that to estimate whether we might be able to cope with that chunk size:

test <- spx_prices %>% slice(1:250) %>% get_returns() %>% fjoin_on_date() %>% wrangle_combos() test %>% object_size() # 2.28 GB

OK – we should be able to process our data a year at a time with a bit of wiggle room to overlap our chunks (we must overlap our chunks because the first value calculated in a new chunk needs the previous `wdw+1`

values for the first calculation). Let’s give it a shot.

## Performing the operation in chunks

# helper functions get_slice <- function(df, idx, chunk_size) { df %>% slice(idx:(idx+chunk_size)) } process_chunk <- function(chunk) { chunk %>% get_returns() %>% fjoin_on_date() %>% wrangle_combos() %>% na.omit() %>% # comfortable omitting NA here as we've been careful about alignment via padding missing values pairwise_corrs(period = wdw) %>% na.omit() %>% # this na.omit removes NA in prior window due to complete = TRUE requirement mean_pw_cors() } # set up sequential chunk processing wdw <- 50 chunk_days <- 250 num_days <- spx_prices %>% ungroup() %>% pull(date) %>% n_distinct(na.rm = TRUE) - wdw num_chunks <- ceiling(num_days / (chunk_days + wdw + 1)) + 1 corr_list <- list() system.time({ for(i in c(1, c(1:num_chunks)*chunk_days)) { corr_list[[i]] <- spx_prices %>% get_slice(i, (chunk_days+wdw+1)) %>% process_chunk() } }) # user system elapsed # 3670 109.8 3819

Result! That takes quite a long time, but at least we’ve managed to get the job done.

Let’s check out the final product:

bind_rows(corr_list) %>% filter(date >= "2017-01-01", date <= "2019-01-01") %>% ggplot(aes(x = date, y = mean_pw_corr)) + geom_line() + labs( x = 'Date', y = 'Mean pairwise correlation', title = 'Rolling Mean Pairwise Correlation', subtitle = 'SPX constituents' ) + theme_bw()

Looks good!

## Next steps

As well as testing the output for correctness, the next step would be to consider getting the job done faster by farming the chunks out to individual workers so that they could be processed in parallel.

We can do this because each chunk is independent of the other chunks – it’s simply a matter of splitting our data appropriately, performing the calculation on each chunk, and combining the results back together.

There are a bunch of other potentially cheap optimisations that we mentioned in the introductory post) – we’ll explore these as well.

## Conclusion

In this post, we split our big data problem of calculating mean rolling pairwise correlations of a large universe of stocks into manageable chunks and processed the entire job in memory using an everyday laptop.

The main obstacles that we needed to think about were the introduction of NA values upon calculation of returns from prices, correct alignment of our data by date, and the need for overlapping chunks due to the rolling nature of the operation being performed.