# Using a genetic algorithm for the hyperparameter optimization of a SARIMA model

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## Introduction

In this blog post, I’ll use the data that I cleaned in a previous
blog post, which you can download
here. If you want to follow along,
download the monthly data. In my last blog post
I showed how to perform a grid search the “tidy” way. As an example, I looked for the right
hyperparameters of a SARIMA model. However, the goal of the post was not hyperparameter optimization
per se, so I did not bother with tuning the hyperparameters on a validation set, and used the test
set for both validation of the hyperparameters and testing the forecast. Of course, this is not great
because doing this might lead to overfitting the hyperparameters to the test set. So in this blog post
I split my data into trainig, validation and testing sets and use a genetic algorithm to look
for the hyperparameters. Again, this is not the most optimal way to go about this problem, since
the `{forecast}`

package contains the very useful `auto.arima()`

function. I just wanted to see
what kind of solution a genetic algorithm would return, and also try different cost functions.
If you’re interested, read on!

## Setup

Let’s first load some libraries and define some helper functions (the helper functions were explained in the previous blog posts):

library(tidyverse) library(forecast) library(rgenoud) library(parallel) library(lubridate) library(furrr) library(tsibble) library(brotools) ihs <- function(x){ log(x + sqrt(x**2 + 1)) } to_tibble <- function(forecast_object){ point_estimate <- forecast_object$mean %>% as_tsibble() %>% rename(point_estimate = value, date = index) upper <- forecast_object$upper %>% as_tsibble() %>% spread(key, value) %>% rename(date = index, upper80 = `80%`, upper95 = `95%`) lower <- forecast_object$lower %>% as_tsibble() %>% spread(key, value) %>% rename(date = index, lower80 = `80%`, lower95 = `95%`) reduce(list(point_estimate, upper, lower), full_join) }

Now, let’s load the data:

avia_clean_monthly <- read_csv("https://raw.githubusercontent.com/b-rodrigues/avia_par_lu/master/avia_clean_monthy.csv") ## Parsed with column specification: ## cols( ## destination = col_character(), ## date = col_date(format = ""), ## passengers = col_integer() ## )

Let’s split the data into a train set, a validation set and a test set:

avia_clean_train <- avia_clean_monthly %>% select(date, passengers) %>% filter(year(date) < 2013) %>% group_by(date) %>% summarise(total_passengers = sum(passengers)) %>% pull(total_passengers) %>% ts(., frequency = 12, start = c(2005, 1)) avia_clean_validation <- avia_clean_monthly %>% select(date, passengers) %>% filter(between(year(date), 2013, 2016)) %>% group_by(date) %>% summarise(total_passengers = sum(passengers)) %>% pull(total_passengers) %>% ts(., frequency = 12, start = c(2013, 1)) avia_clean_test <- avia_clean_monthly %>% select(date, passengers) %>% filter(year(date) >= 2016) %>% group_by(date) %>% summarise(total_passengers = sum(passengers)) %>% pull(total_passengers) %>% ts(., frequency = 12, start = c(2016, 1)) logged_test_data <- ihs(avia_clean_test) logged_validation_data <- ihs(avia_clean_validation) logged_train_data <- ihs(avia_clean_train)

I will train the models on data from 2005 to 2012, look for the hyperparameters on data from 2013
to 2016 and test the accuracy on data from 2016 to March 2018. For this kind of exercise, the ideal
situation would be to perform cross-validation. Doing this with time-series data is not obvious
because of the autocorrelation between observations, which would be broken by sampling independently
which is required by CV. Also, if for example you do leave-one-out CV,
you would end up trying to predict a point in, say, 2017, with data
from 2018, which does not make sense. So you should be careful about that. `{forecast}`

is able
to perform CV for time series and `scikit-learn`

, the
Python package, is able to perform
cross-validation of time series data
too. I will not do it in this blog post and simply focus on the genetic algorithm part.

Let’s start by defining the cost function to minimize. I’ll try several, in the first one I will minimize the RMSE:

cost_function_rmse <- function(param, train_data, validation_data, forecast_periods){ order <- param[1:3] season <- c(param[4:6], 12) model <- purrr::possibly(arima, otherwise = NULL)(x = train_data, order = order, seasonal = season, method = "ML") if(is.null(model)){ return(9999999) } else { forecast_model <- forecast::forecast(model, h = forecast_periods) point_forecast <- forecast_model$mean sqrt(mean(point_forecast - validation_data) ** 2) } }

If `arima()`

is not able to estimate a model for the given parameters, I force it to return `NULL`

,
and in that case force the cost function to return a very high cost. If a model was successfully estimated,
then I compute the RMSE.

Let’s also take a look at what `auto.arima()`

says:

starting_model <- auto.arima(logged_train_data) summary(starting_model) ## Series: logged_train_data ## ARIMA(1,0,2)(2,1,0)[12] ## ## Coefficients: ## ar1 ma1 ma2 sar1 sar2 ## 0.9754 -0.7872 0.2091 -0.7285 -0.4413 ## s.e. 0.0261 0.1228 0.1213 0.1063 0.1150 ## ## sigma^2 estimated as 0.004514: log likelihood=105.61 ## AIC=-199.22 AICc=-198.13 BIC=-184.64 ## ## Training set error measures: ## ME RMSE MAE MPE MAPE ## Training set 0.008398036 0.06095102 0.03882593 0.07009285 0.3339574 ## MASE ACF1 ## Training set 0.4425794 0.02073886

Let’s compute the cost at this vector of parameters:

cost_function_rmse(c(1, 0, 2, 2, 1, 0), train_data = logged_train_data, validation_data = logged_validation_data, forecast_periods = 65) ## [1] 0.1731473

Ok, now let’s start with optimizing the hyperparameters. Let’s help the genetic algorithm a little bit by defining where it should perform the search:

domains <- matrix(c(0, 3, 0, 2, 0, 3, 0, 3, 0, 2, 0, 3), byrow = TRUE, ncol = 2)

This matrix constraints the first parameter to lie between 0 and 3, the second one between 0 and 2, and so on.

Let’s call the `genoud()`

function from the `{rgenoud}`

package, and use 8 cores:

cl <- makePSOCKcluster(8) clusterExport(cl, c('logged_train_data', 'logged_validation_data')) tic <- Sys.time() auto_arima_rmse <- genoud(cost_function_rmse, nvars = 6, data.type.int = TRUE, starting.values = c(1, 0, 2, 2, 1, 0), # <- from auto.arima Domains = domains, cluster = cl, train_data = logged_train_data, validation_data = logged_validation_data, forecast_periods = length(logged_validation_data), hard.generation.limit = TRUE) toc_rmse <- Sys.time() - tic

`makePSOCKcluster()`

is a function from the `{parallel}`

package. I must also *export* the global
variables `logged_train_data`

or `logged_validation_data`

. If I don’t do that, the workers called
by `genoud()`

will not *know* about these variables and an error will be returned. The option
`data.type.int = TRUE`

force the algorithm to look only for integers, and `hard.generation.limit = TRUE`

forces the algorithm to stop after 100 generations.

The process took 7 minutes, which is faster than doing the grid search. What was the solution found?

auto_arima_rmse ## $value ## [1] 0.0001863039 ## ## $par ## [1] 3 2 1 1 2 1 ## ## $gradients ## [1] NA NA NA NA NA NA ## ## $generations ## [1] 11 ## ## $peakgeneration ## [1] 1 ## ## $popsize ## [1] 1000 ## ## $operators ## [1] 122 125 125 125 125 126 125 126 0

Let’s train the model using the `arima()`

function at these parameters:

best_model_rmse <- arima(logged_train_data, order = auto_arima_rmse$par[1:3], season = list(order = auto_arima_rmse$par[4:6], period = 12), method = "ML") summary(best_model_rmse) ## ## Call: ## arima(x = logged_train_data, order = auto_arima_rmse$par[1:3], seasonal = list(order = auto_arima_rmse$par[4:6], ## period = 12), method = "ML") ## ## Coefficients: ## ar1 ar2 ar3 ma1 sar1 sma1 ## -0.6999 -0.4541 -0.0476 -0.9454 -0.4996 -0.9846 ## s.e. 0.1421 0.1612 0.1405 0.1554 0.1140 0.2193 ## ## sigma^2 estimated as 0.006247: log likelihood = 57.34, aic = -100.67 ## ## Training set error measures: ## ME RMSE MAE MPE MAPE ## Training set -0.0006142355 0.06759545 0.04198561 -0.005408262 0.3600483 ## MASE ACF1 ## Training set 0.4386693 -0.008298546

Let’s extract the forecasts:

best_model_rmse_forecast <- forecast::forecast(best_model_rmse, h = 65) best_model_rmse_forecast <- to_tibble(best_model_rmse_forecast) ## Joining, by = "date" ## Joining, by = "date" starting_model_forecast <- forecast(starting_model, h = 65) starting_model_forecast <- to_tibble(starting_model_forecast) ## Joining, by = "date" ## Joining, by = "date"

and plot the forecast to see how it looks:

avia_clean_monthly %>% group_by(date) %>% summarise(total = sum(passengers)) %>% mutate(total_ihs = ihs(total)) %>% ggplot() + ggtitle("Minimization of RMSE") + geom_line(aes(y = total_ihs, x = date), colour = "#82518c") + scale_x_date(date_breaks = "1 year", date_labels = "%m-%Y") + geom_ribbon(data = best_model_rmse_forecast, aes(x = date, ymin = lower95, ymax = upper95), fill = "#666018", alpha = 0.2) + geom_line(data = best_model_rmse_forecast, aes(x = date, y = point_estimate), linetype = 2, colour = "#8e9d98") + geom_ribbon(data = starting_model_forecast, aes(x = date, ymin = lower95, ymax = upper95), fill = "#98431e", alpha = 0.2) + geom_line(data = starting_model_forecast, aes(x = date, y = point_estimate), linetype = 2, colour = "#a53031") + theme_blog()

The yellowish line and confidence intervals come from minimizing the genetic algorithm, and the
redish from `auto.arima()`

. Interesting; the point estimate is very precise, but the confidence
intervals are very wide. Low bias, high variance.

Now, let’s try with another cost function, where I minimize the BIC, similar to the `auto.arima()`

function:

cost_function_bic <- function(param, train_data, validation_data, forecast_periods){ order <- param[1:3] season <- c(param[4:6], 12) model <- purrr::possibly(arima, otherwise = NULL)(x = train_data, order = order, seasonal = season, method = "ML") if(is.null(model)){ return(9999999) } else { BIC(model) } }

Let’s take a look at the cost at the parameter values returned by `auto.arima()`

:

cost_function_bic(c(1, 0, 2, 2, 1, 0), train_data = logged_train_data, validation_data = logged_validation_data, forecast_periods = 65) ## [1] -184.6397

Let the genetic algorithm run again:

cl <- makePSOCKcluster(8) clusterExport(cl, c('logged_train_data', 'logged_validation_data')) tic <- Sys.time() auto_arima_bic <- genoud(cost_function_bic, nvars = 6, data.type.int = TRUE, starting.values = c(1, 0, 2, 2, 1, 0), # <- from auto.arima Domains = domains, cluster = cl, train_data = logged_train_data, validation_data = logged_validation_data, forecast_periods = length(logged_validation_data), hard.generation.limit = TRUE) toc_bic <- Sys.time() - tic

This time, it took 6 minutes, a bit slower than before. Let’s take a look at the solution:

auto_arima_bic ## $value ## [1] -201.0656 ## ## $par ## [1] 0 1 1 1 0 1 ## ## $gradients ## [1] NA NA NA NA NA NA ## ## $generations ## [1] 12 ## ## $peakgeneration ## [1] 1 ## ## $popsize ## [1] 1000 ## ## $operators ## [1] 122 125 125 125 125 126 125 126 0

Let’s train the model at these parameters:

best_model_bic <- arima(logged_train_data, order = auto_arima_bic$par[1:3], season = list(order = auto_arima_bic$par[4:6], period = 12), method = "ML") summary(best_model_bic) ## ## Call: ## arima(x = logged_train_data, order = auto_arima_bic$par[1:3], seasonal = list(order = auto_arima_bic$par[4:6], ## period = 12), method = "ML") ## ## Coefficients: ## ma1 sar1 sma1 ## -0.6225 0.9968 -0.832 ## s.e. 0.0835 0.0075 0.187 ## ## sigma^2 estimated as 0.004145: log likelihood = 109.64, aic = -211.28 ## ## Training set error measures: ## ME RMSE MAE MPE MAPE ## Training set 0.003710982 0.06405303 0.04358164 0.02873561 0.3753513 ## MASE ACF1 ## Training set 0.4553447 -0.03450603

And let’s plot the results:

best_model_bic_forecast <- forecast::forecast(best_model_bic, h = 65) best_model_bic_forecast <- to_tibble(best_model_bic_forecast) ## Joining, by = "date" ## Joining, by = "date" avia_clean_monthly %>% group_by(date) %>% summarise(total = sum(passengers)) %>% mutate(total_ihs = ihs(total)) %>% ggplot() + ggtitle("Minimization of BIC") + geom_line(aes(y = total_ihs, x = date), colour = "#82518c") + scale_x_date(date_breaks = "1 year", date_labels = "%m-%Y") + geom_ribbon(data = best_model_bic_forecast, aes(x = date, ymin = lower95, ymax = upper95), fill = "#5160a0", alpha = 0.2) + geom_line(data = best_model_bic_forecast, aes(x = date, y = point_estimate), linetype = 2, colour = "#208480") + geom_ribbon(data = starting_model_forecast, aes(x = date, ymin = lower95, ymax = upper95), fill = "#98431e", alpha = 0.2) + geom_line(data = starting_model_forecast, aes(x = date, y = point_estimate), linetype = 2, colour = "#a53031") + theme_blog()

The solutions are very close, both in terms of point estimates and confidence intervals. Bias increased, but variance lowered… This gives me an idea! What if I minimize the RMSE, while keeping the number of parameters low, as a kind of regularization? This is somewhat what minimising BIC does, but let’s try to do it a more “naive” approach:

cost_function_rmse_low_k <- function(param, train_data, validation_data, forecast_periods, max.order){ order <- param[1:3] season <- c(param[4:6], 12) if(param[1] + param[3] + param[4] + param[6] > max.order){ return(9999999) } else { model <- purrr::possibly(arima, otherwise = NULL)(x = train_data, order = order, seasonal = season, method = "ML") } if(is.null(model)){ return(9999999) } else { forecast_model <- forecast::forecast(model, h = forecast_periods) point_forecast <- forecast_model$mean sqrt(mean(point_forecast - validation_data) ** 2) } }

This is also similar to what `auto.arima()`

does; by default, the `max.order`

argument in `auto.arima()`

is set to 5, and is the sum of `p + q + P + Q`

. So I’ll try something similar.

Let’s take a look at the cost at the parameter values returned by `auto.arima()`

:

cost_function_rmse_low_k(c(1, 0, 2, 2, 1, 0), train_data = logged_train_data, validation_data = logged_validation_data, forecast_periods = 65, max.order = 5) ## [1] 0.1731473

Let’s see what will happen:

cl <- makePSOCKcluster(8) clusterExport(cl, c('logged_train_data', 'logged_validation_data')) tic <- Sys.time() auto_arima_rmse_low_k <- genoud(cost_function_rmse_low_k, nvars = 6, data.type.int = TRUE, starting.values = c(1, 0, 2, 2, 1, 0), # <- from auto.arima max.order = 5, Domains = domains, cluster = cl, train_data = logged_train_data, validation_data = logged_validation_data, forecast_periods = length(logged_validation_data), hard.generation.limit = TRUE) toc_rmse_low_k <- Sys.time() - tic

It took 1 minute to train this one, quite fast! Let’s take a look:

auto_arima_rmse_low_k ## $value ## [1] 0.002503478 ## ## $par ## [1] 1 2 0 3 1 0 ## ## $gradients ## [1] NA NA NA NA NA NA ## ## $generations ## [1] 11 ## ## $peakgeneration ## [1] 1 ## ## $popsize ## [1] 1000 ## ## $operators ## [1] 122 125 125 125 125 126 125 126 0

And let’s plot it:

best_model_rmse_low_k <- arima(logged_train_data, order = auto_arima_rmse_low_k$par[1:3], season = list(order = auto_arima_rmse_low_k$par[4:6], period = 12), method = "ML") summary(best_model_rmse_low_k) ## ## Call: ## arima(x = logged_train_data, order = auto_arima_rmse_low_k$par[1:3], seasonal = list(order = auto_arima_rmse_low_k$par[4:6], ## period = 12), method = "ML") ## ## Coefficients: ## ar1 sar1 sar2 sar3 ## -0.6468 -0.7478 -0.5263 -0.1143 ## s.e. 0.0846 0.1171 0.1473 0.1446 ## ## sigma^2 estimated as 0.01186: log likelihood = 57.88, aic = -105.76 ## ## Training set error measures: ## ME RMSE MAE MPE MAPE ## Training set 0.0005953302 0.1006917 0.06165919 0.003720452 0.5291736 ## MASE ACF1 ## Training set 0.6442205 -0.3706693 best_model_rmse_low_k_forecast <- forecast::forecast(best_model_rmse_low_k, h = 65) best_model_rmse_low_k_forecast <- to_tibble(best_model_rmse_low_k_forecast) ## Joining, by = "date" ## Joining, by = "date" avia_clean_monthly %>% group_by(date) %>% summarise(total = sum(passengers)) %>% mutate(total_ihs = ihs(total)) %>% ggplot() + ggtitle("Minimization of RMSE + low k") + geom_line(aes(y = total_ihs, x = date), colour = "#82518c") + scale_x_date(date_breaks = "1 year", date_labels = "%m-%Y") + geom_ribbon(data = best_model_rmse_low_k_forecast, aes(x = date, ymin = lower95, ymax = upper95), fill = "#5160a0", alpha = 0.2) + geom_line(data = best_model_rmse_low_k_forecast, aes(x = date, y = point_estimate), linetype = 2, colour = "#208480") + geom_ribbon(data = starting_model_forecast, aes(x = date, ymin = lower95, ymax = upper95), fill = "#98431e", alpha = 0.2) + geom_line(data = starting_model_forecast, aes(x = date, y = point_estimate), linetype = 2, colour = "#a53031") + theme_blog()

Looks like this was not the right strategy. There might be a better cost function than what I have tried, but looks like minimizing the BIC is the way to go.

Hope you enjoyed! If you found this blog post useful, you might want to follow me on twitter for blog post updates or buy me an espresso.

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