Linear Regression

Intro

The linear_regression() function can be used to fit this kind of model inside a database. It uses dplyr programming to abstract the steps needed produce a model, so that it can then be translated into SQL statements in the background.

Example setup

A lightweight SQLite database will be used for this article. Additionally, a sample data set is created.

# Open a database connection
con <- DBI::dbConnect(RSQLite::SQLite(), path = ":memory:")
RSQLite::initExtension(con)

library(dplyr)
# Copy data to the database
db_flights <- copy_to(con, nycflights13::flights, "flights")
# Create a simple sample
db_sample <- db_flights %>%
  filter(!is.na(arr_time)) %>%
  head(20000) 

Model inside the database

The linear_regression() function does not use a formula. It uses a table, and a named dependent variable. This means data preparation is needed prior to running the model. The best way to prepare the data for modeling will be using piped dplyr operations.

db_sample %>%
  select(arr_delay, dep_delay, distance) %>%
  linear_regression(arr_delay)

## # A tibble: 1 x 3
##   dep_delay distance Intercept
##       <dbl>    <dbl>     <dbl>
## 1      1.00 -0.00337    -0.659

Categorical variables

Adding a categorical a variable to a model requires prior data transformation The add_dummy_variables() appends a set of boolean variables, one for each discrete value. This function creates one-less discrete variable than the possible values. For example: if the categorical variable has three possible values, the function will append two variables. By default, add_dummy_variables() removes the original variable.

The reason for this approach is to reduce the number of database operations. Without this step, then a fitting function would have to request all of the unique values every time a new model run, which creates unnecessary processing.

db_sample %>%
  select(arr_delay, origin) %>%
  add_dummy_variables(origin, values = c("EWR", "JFK", "LGA"))

## # Source:   lazy query [?? x 3]
## # Database: sqlite 3.22.0 []
##    arr_delay origin_JFK origin_LGA
##        <dbl>      <dbl>      <dbl>
##  1       11.         0.         0.
##  2       20.         0.         1.
##  3       33.         1.         0.
##  4      -18.         1.         0.
##  5      -25.         0.         1.
##  6       12.         0.         0.
##  7       19.         0.         0.
##  8      -14.         0.         1.
##  9       -8.         1.         0.
## 10        8.         0.         1.
## # ... with more rows

In a real world scenario, the possible values are usually not known at the beginning of the analysis. So it is a good idea to load them into a vector variable so that it can be used any time that variable is added to a model. This can be easily done using the pull() command from dplyr:

origins <- db_flights %>%
  group_by(origin) %>%
  summarise() %>%
  pull()

origins

## [1] "EWR" "JFK" "LGA"

The add_dummy_variables() can be used as part of the piped code that terminates in the modeling function.

db_sample %>%
  select(arr_delay, origin) %>%
  add_dummy_variables(origin, values = origins) %>%
  linear_regression(arr_delay)

## # A tibble: 1 x 3
##   origin_JFK origin_LGA Intercept
##        <dbl>      <dbl>     <dbl>
## 1      -10.6      -7.79      9.62

Multiple linear regression

One of two arguments is needed to be set when fitting a model with three or more independent variables. The both relate to the size of the data set used for the model. So either the sample_size argument is passed, or auto_count is set to TRUE. When auto_count is set to TRUE, and no sample size is passed, then the function will do a table count as part of the model fitting. This is done in order to prevent unnecessary database operations, especially for cases when multiple models will be tested on top of the same sample data.

db_sample %>%
  select(arr_delay, arr_time, dep_delay, dep_time) %>%
  linear_regression(arr_delay, sample_size = 20000)

## # A tibble: 1 x 4
##    arr_time dep_delay dep_time Intercept
##       <dbl>     <dbl>    <dbl>     <dbl>
## 1 -0.000208      1.01 -0.00155     -1.72

Interactions

Interactions have to be handled manually prior the modeling step.

db_sample %>%
  mutate(distanceXarr_time = distance * arr_time) %>%
  select(arr_delay, distanceXarr_time) %>% 
  linear_regression(arr_delay, sample_size = 20000)

## # A tibble: 1 x 2
##   distanceXarr_time Intercept
##               <dbl>     <dbl>
## 1       -0.00000197      6.77

A more typical model would also include the two original variables:

db_sample %>%
  mutate(distanceXarr_time = distance * arr_time) %>%
  select(arr_delay, distance, arr_time, distanceXarr_time) %>% 
  linear_regression(arr_delay, sample_size = 20000)

## # A tibble: 1 x 4
##   arr_time distance distanceXarr_time Intercept
##      <dbl>    <dbl>             <dbl>     <dbl>
## 1  0.00650  0.00269       -0.00000435     -2.11

Full example

Fitting a model with regular, categorical and interaction variables will look like this:

remote_model <- db_sample %>%
  mutate(distanceXarr_time = distance * arr_time) %>%
  select(arr_delay, dep_time, distanceXarr_time, origin) %>% 
  add_dummy_variables(origin, values = origins) %>%
  linear_regression(y_var = arr_delay, sample_size = 20000)

remote_model

## # A tibble: 1 x 5
##   dep_time distanceXarr_time Intercept origin_JFK origin_LGA
##      <dbl>             <dbl>     <dbl>      <dbl>      <dbl>
## 1   0.0132       -0.00000275     -3.92      -10.1      -8.05

Run predictions with tidypredict

The as_parsed_model() function will convert the linear_regression() model output to an output that the tidypredict model can read.

parsed <- as_parsed_model(remote_model)

parsed

## # A tibble: 7 x 8
##   labels              estimate type  vals  field_1 field_2 field_3 field_4
##   <chr>                  <dbl> <chr> <chr> <chr>   <chr>   <chr>   <chr>  
## 1 labels               0.      vari~ <NA>  dep_ti~ distan~ origin~ origin~
## 2 model               NA       vari~ lm    <NA>    <NA>    <NA>    <NA>   
## 3 dep_time             1.32e-2 term  <NA>  {{:}}   <NA>    <NA>    <NA>   
## 4 distanceXarr_time   -2.75e-6 term  <NA>  <NA>    {{:}}   <NA>    <NA>   
## 5 (Intercept)         -3.92e+0 term  <NA>  <NA>    <NA>    <NA>    <NA>   
## 6 origin_JFK          -1.01e+1 term  <NA>  <NA>    <NA>    {{:}}   <NA>   
## 7 origin_LGA          -8.05e+0 term  <NA>  <NA>    <NA>    <NA>    {{:}}

To preview what the prediction SQL statement will look like use tidypredict_sql()

library(tidypredict)

tidypredict_sql(parsed, dbplyr::simulate_dbi())

## <SQL> ("dep_time") * (0.0132251596085814) + ("distanceXarr_time") * (-2.75008443762809e-06) + -3.91880281681301 + ("origin_JFK") * (-10.0912262446948) + ("origin_LGA") * (-8.04838792899506)

library(dbplot)

db_sample %>%
  mutate(distanceXarr_time = distance * arr_time) %>%
  select(arr_delay, dep_time, distanceXarr_time, origin) %>% 
  add_dummy_variables(origin, values = origins) %>%
  tidypredict_to_column(parsed) %>%
  select(fit, arr_delay) %>%
  collect() %>%   # <----- This step is only needed if working with SQLite!
  dbplot_raster(fit, arr_delay, resolution = 50)

## Warning: Removed 25 rows containing missing values (geom_raster).

db_flights %>%
  mutate(distanceXarr_time = distance * arr_time) %>%
  select(arr_delay, dep_time, distanceXarr_time, origin) %>% 
  add_dummy_variables(origin, values = origins) %>%
  tidypredict_to_column(parsed)

## # Source:   lazy query [?? x 6]
## # Database: sqlite 3.22.0 []
##    arr_delay dep_time distanceXarr_time origin_JFK origin_LGA     fit
##        <dbl>    <int>             <dbl>      <dbl>      <dbl>   <dbl>
##  1       11.      517          1162000.         0.         0.  -0.277
##  2       20.      533          1203600.         0.         1.  -8.23 
##  3       33.      542          1005147.         1.         0.  -9.61 
##  4      -18.      544          1582304.         1.         0. -11.2  
##  5      -25.      554           618744.         0.         1.  -6.34 
##  6       12.      554           532060.         0.         0.   1.94 
##  7       19.      555           972345.         0.         0.   0.747
##  8      -14.      557           162361.         0.         1.  -5.05 
##  9       -8.      557           791072.         1.         0.  -8.82 
## 10        8.      558           551949.         0.         1.  -6.11 
## # ... with more rows