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Multinomial Classification

Using vtreat with Multinomial Classification Problems

Nina Zumel and John Mount October 2019

Note this is a description of the R version of vtreat, the same example for the Python version of vtreat can be found here.

Preliminaries

Load modules/packages.

library(rqdatatable)
## Loading required package: wrapr

## Loading required package: rquery

library(vtreat)
packageVersion('vtreat')
## [1] '1.6.0'

suppressPackageStartupMessages(library(ggplot2))
library(WVPlots)

Generate example data.

  • y is a noisy sinusoidal function of the variable x
  • yc is the output to be predicted: y‘s quantized value as ’large’, ‘liminal’, or ‘small’.
  • Input xc is a categorical variable that represents a discretization of y, along some NAs
  • Input x2 is a pure noise variable with no relationship to the output
set.seed(2020)

make_data <- function(nrows) {
    d <- data.frame(x = 5*rnorm(nrows))
    d['y'] = sin(d['x']) + 0.1*rnorm(n = nrows)
    d[4:10, 'x'] = NA                  # introduce NAs
    d['xc'] = paste0('level_', 5*round(d$y/5, 1))
    d['x2'] = rnorm(n = nrows)
    d[d['xc']=='level_-1', 'xc'] = NA  # introduce a NA level
    d['yc'] = ifelse(d[['y']] > 0.5, 'large', ifelse(d[['y']] < -0.5, 'small', 'liminal'))
    return(d)
}

d = make_data(500)

d %.>%
  head(.) %.>%
  knitr::kable(.)
x y xc x2 yc
1.884861 1.0717646 level_1 0.0046504 large
1.507742 0.9958029 level_1 -1.2287497 large
-5.490116 0.8315705 level_1 -0.1405980 large
NA 0.6007655 level_0.5 -0.2073270 large
NA -0.8339836 NA -0.9215306 small
NA -0.5329006 level_-0.5 0.3604742 small

Some quick data exploration

Check how many levels xc has, and their distribution (including NA)

unique(d['xc'])
##             xc
## 1      level_1
## 4    level_0.5
## 5         <NA>
## 6   level_-0.5
## 27     level_0
## 269 level_-1.5

table(d$xc, useNA = 'ifany')
## 
## level_-0.5 level_-1.5    level_0  level_0.5    level_1       <NA> 
##         94          1         85         98        103        119

Show the distribution of yc

table(d$yc, useNA = 'ifany')
## 
##   large liminal   small 
##     162     164     174

Build a transform appropriate for classification problems.

Now that we have the data, we want to treat it prior to modeling: we want training data where all the input variables are numeric and have no missing values or NAs.

First create the data treatment transform design object, in this case a treatment for a multinomial classification problem.

We use the training data d to fit the transform and the return a treated training set: completely numeric, with no missing values.

unpack[
  transform = treat_m,
  d_prepared = cross_frame,
  score_frame = score_frame
  ] <- vtreat::mkCrossFrameMExperiment(
    d,                                    # data to learn transform from
    setdiff(colnames(d), c('y', 'yc')),   # columns to transform
    'yc'                                  # outcome variable
  )

Notice that d_prepared now only includes derived variables and the outcome yc. The derived variables will be discussed below.

d_prepared %.>%
  head(.) %.>%
  knitr::kable(.)
x x_isBAD xc_catP x2 xc_lev_NA xc_lev_x_level_minus_0_5 xc_lev_x_level_0 xc_lev_x_level_0_5 xc_lev_x_level_1 large_xc_catB liminal_xc_catB small_xc_catB yc
1.8848606 0 0.206 0.0046504 0 0 0 0 1 14.169093 -12.5759659 -12.8410398 large
1.5077419 0 0.206 -1.2287497 0 0 0 0 1 14.181970 -12.8643483 -12.6639435 large
-5.4901159 0 0.206 -0.1405980 0 0 0 0 1 14.169093 -12.5759659 -12.8410398 large
-0.2704873 1 0.196 -0.2073270 0 0 0 1 0 1.096309 0.3616142 -12.7584224 large
-0.2704873 1 0.238 -0.9215306 1 0 0 0 0 -12.940744 -12.7354342 14.1933572 small
-0.2704873 1 0.188 0.3604742 0 1 0 0 0 -12.773690 0.5879950 0.8699281 small

Note that for the training data d: crossFrame is not the same as prepare(transform, d); the second call can lead to nested model bias in some situations, and is not recommended. For other, later data, not seen during transform design transform.preprare(o) is an appropriate step.

vtreat version 1.5.1 and newer issue a warning if you call the incorrect transform pattern on your original training data:

d_prepared_wrong <- prepare(transform, d)
## Warning in prepare.multinomial_plan(transform, d): possibly called prepare() on
## same data frame as designTreatments*()/mkCrossFrame*Experiment(), this can lead
## to over-fit. To avoid this, please use mkCrossFrameMExperiment$crossFrame.

The Score Frame

Now examine the score frame, which gives information about each new variable, including its type, which original variable it is derived from, its (cross-validated) correlation with the outcome, and its (cross-validated) significance as a one-variable linear model for the outcome.

# only show a subset of the columns
cols = c("varName", "origName", "code", "outcome_level", "rsq", "sig", "varMoves", "default_threshold", "recommended")
knitr::kable(score_frame[,cols])
varName origName code outcome_level rsq sig varMoves default_threshold recommended
x x clean large 0.0005756 0.5470919 TRUE 0.03333333 FALSE
x_isBAD x isBAD large 0.0000771 0.8255885 TRUE 0.06666667 FALSE
x2 x2 clean large 0.0026075 0.2000083 TRUE 0.03333333 FALSE
xc_catP xc catP large 0.0002361 0.6997734 TRUE 0.06666667 FALSE
xc_lev_NA xc lev large 0.1750095 0.0000000 TRUE 0.01333333 TRUE
xc_lev_x_level_0 xc lev large 0.1185254 0.0000000 TRUE 0.01333333 TRUE
xc_lev_x_level_0_5 xc lev large 0.0644178 0.0000000 TRUE 0.01333333 TRUE
xc_lev_x_level_1 xc lev large 0.4701626 0.0000000 TRUE 0.01333333 TRUE
xc_lev_x_level_minus_0_5 xc lev large 0.1328708 0.0000000 TRUE 0.01333333 TRUE
x x clean liminal 0.0022396 0.2338771 TRUE 0.03333333 FALSE
x_isBAD x isBAD liminal 0.0020122 0.2591571 TRUE 0.06666667 FALSE
x2 x2 clean liminal 0.0023614 0.2215731 TRUE 0.03333333 FALSE
xc_catP xc catP liminal 0.4410001 0.0000000 TRUE 0.06666667 TRUE
xc_lev_NA xc lev liminal 0.1769596 0.0000000 TRUE 0.01333333 TRUE
xc_lev_x_level_0 xc lev liminal 0.3615209 0.0000000 TRUE 0.01333333 TRUE
xc_lev_x_level_0_5 xc lev liminal 0.0041777 0.1039765 TRUE 0.01333333 FALSE
xc_lev_x_level_1 xc lev liminal 0.1492650 0.0000000 TRUE 0.01333333 TRUE
xc_lev_x_level_minus_0_5 xc lev liminal 0.0076508 0.0277896 TRUE 0.01333333 FALSE
x x clean small 0.0048286 0.0773262 TRUE 0.03333333 FALSE
x_isBAD x isBAD small 0.0022777 0.2250506 TRUE 0.06666667 FALSE
x2 x2 clean small 0.0000045 0.9569844 TRUE 0.03333333 FALSE
xc_catP xc catP small 0.3117712 0.0000000 TRUE 0.06666667 TRUE
xc_lev_NA xc lev small 0.5132320 0.0000000 TRUE 0.01333333 TRUE
xc_lev_x_level_0 xc lev small 0.1265119 0.0000000 TRUE 0.01333333 TRUE
xc_lev_x_level_0_5 xc lev small 0.1488474 0.0000000 TRUE 0.01333333 TRUE
xc_lev_x_level_1 xc lev small 0.1576977 0.0000000 TRUE 0.01333333 TRUE
xc_lev_x_level_minus_0_5 xc lev small 0.0387613 0.0000006 TRUE 0.01333333 TRUE
large_xc_catB xc catB large 0.7904963 0.0000000 TRUE 0.06666667 TRUE
liminal_xc_catB xc catB liminal 0.5786498 0.0000000 TRUE 0.06666667 TRUE
small_xc_catB xc catB small 0.7914976 0.0000000 TRUE 0.06666667 TRUE

Note that the variable xc has been converted to multiple variables:

  • an indicator variable for each possible level, plus NA (xc_lev_*)
  • the value of a (cross-validated) one-variable model for each level of yc as a function of xc (*_xc_catB)
  • a variable that returns how prevalent this particular value of xc is in the training data (xc_catP)

The variable x has been converted to two new variables:

  • a clean version of x that has no missing values or NaNs
  • a variable indicating when x was NA in the original data (x_isBAD).

Any or all of these new variables are available for downstream modeling.

Note that unlike mkCrossFrameCExperiment (binomial classification), mkCrossFrameMExperiment produces multiple catB variables for a single categorical variable: one for each possible outcome class. Each catB variable is the output of a one-variable regularized logistic regression of the original variable against a single target outcome class. In other words, mkCrossFrameMExperiment treats multiclass classification as multiple “one against rest” classification problems. For a more detailed discussion of the catB variables, see the binary classification example.

Similarly, the rsq and sig columns report the variable’s cross-validated correlation and its cross-validated significance as a one-variable linear model for each target outcome.

This means that for multiclass classification problems, the score frame has multiple rows per new variable. In this example, the score frame has 30 rows for 12 variables.

The recommended column indicates which variables are non constant (varMoves == TRUE) and have a significance value smaller than default_threshold. See the section Deriving the Default Thresholds below for the reasoning behind such a default threshold. Recommended columns are intended as advice about which variables appear to be most likely to be useful in a downstream model. This advice attempts to be conservative, to reduce the possibility of mistakenly eliminating variables that may in fact be useful (although, obviously, it can still mistakenly eliminate variables that have a real but non-linear relationship to the output, as is the case with x, in our example).

Since each variable has multiple recommendations, one can consider a variable to be recommended if it is recommended for any of the outcome targets: an OR of all the recommendations.

Examining variables

To select variables we either make our selection in terms of new variables as follows.

good_new_variables = unique(score_frame[score_frame[['recommended']], 'varName', drop = TRUE])
good_new_variables
## [1] "xc_lev_NA"                "xc_lev_x_level_0"        
## [3] "xc_lev_x_level_0_5"       "xc_lev_x_level_1"        
## [5] "xc_lev_x_level_minus_0_5" "xc_catP"                 
## [7] "large_xc_catB"            "liminal_xc_catB"         
## [9] "small_xc_catB"

Or in terms of original variables as follows.

good_original_variables = unique(score_frame[score_frame[['recommended']], 'origName', drop = TRUE])
good_original_variables
## [1] "xc"

In each case we must call unique() as each variable (derived or original) is potentially qualified against each possible outcome.

Using the Prepared Data in a Model

Of course, what we really want to do with the prepared training data is to fit a model jointly with all the (recommended) variables. Let’s try fitting a logistic regression model to d_prepared.

library(glmnet)
## Loading required package: Matrix

## 
## Attaching package: 'Matrix'

## The following object is masked _by_ '.GlobalEnv':
## 
##     unpack

## The following object is masked from 'package:wrapr':
## 
##     unpack

## Loaded glmnet 3.0-2

# use only the recommended variables for this example
model_vars <- good_new_variables

model <- glmnet(x = as.matrix(d_prepared[, model_vars, drop=FALSE]), 
               y = d_prepared[['yc']],
               family = 'multinomial')
# convenience functions for predicting and adding predictions to original data frame

add_predictions <- function(d_prepared, model_vars, model) {
  preds <- predict(
    model, 
    newx = as.matrix(d_prepared[, model_vars, drop=FALSE]),
    s = min(model$lambda),  # in practice we would cross-validated for a good s
    type = 'response')
  preds <- as.data.frame(preds[, , 1])
  preds$prob_on_predicted_class <- apply(preds, 1, max)
  preds$predict <- NA_character_
  for(col in colnames(preds)) {
    alter = is.na(preds$predict) & preds[[col]] >= preds$prob_on_predicted_class
    preds$predict[alter] <- col
  }
  d_prepared <- cbind(d_prepared, preds)
  d_prepared
}

add_value_by_column <- function(d_prepared, name_column, new_column) {
  vals = unique(d_prepared[[name_column]])
  d_prepared[[new_column]] <- NA_real_
  for(col in vals) {
    match <- d_prepared[[name_column]] == col
    d_prepared[[new_column]][match] <- d_prepared[match, col, drop=TRUE]
  }
  d_prepared
}
# now predict
d_prepared <- add_predictions(d_prepared, model_vars, model)
d_prepared <- add_value_by_column(d_prepared, 'yc', 'prob_on_correct_class')

to_print <- c('yc', 'predict', 'large','liminal','small', 'prob_on_predicted_class', 'prob_on_correct_class')
d_prepared[, to_print, drop = FALSE] %.>%
  head(.) %.>%
  knitr::kable(.)
yc predict large liminal small prob_on_predicted_class prob_on_correct_class
large large 0.9999509 0.0000491 0.0000000 0.9999509 0.9999509
large large 0.9999472 0.0000528 0.0000000 0.9999472 0.9999472
large large 0.9999509 0.0000491 0.0000000 0.9999509 0.9999509
large large 0.5977609 0.4020513 0.0001879 0.5977609 0.5977609
small small 0.0000012 0.0000695 0.9999292 0.9999292 0.9999292
small small 0.0001415 0.3623118 0.6375467 0.6375467 0.6375467 
table(truth = d_prepared$yc, prediction = d_prepared$predict)
##          prediction
## truth     large liminal small
##   large     162       0     0
##   liminal    39     104    21
##   small       0      10   164

In the above confusion matrix, the entry [row, column] gives the number of true items of class[row] that also have prediction of class[column]. In other words, the entry ['large', 'liminal'] gives the number of ‘large’ items predicted to be ‘liminal’.

Now apply the model to new data.

# create the new data
dtest <- make_data(450)

# prepare the new data with vtreat
dtest_prepared = prepare(transform, dtest)
# dtest %.>% transform is an alias for prepare(transform, dtest)

dtest_prepared <- add_predictions(dtest_prepared, model_vars, model)
dtest_prepared <- add_value_by_column(dtest_prepared, 'yc', 'prob_on_correct_class')

dtest_prepared[, to_print, drop = FALSE] %.>%
  head(.) %.>%
  knitr::kable(.)
yc predict large liminal small prob_on_predicted_class prob_on_correct_class
small small 0.0001459 0.4512153 0.5486388 0.5486388 0.5486388
small small 0.0001459 0.4512153 0.5486388 0.5486388 0.5486388
small small 0.0000021 0.0001456 0.9998523 0.9998523 0.9998523
liminal liminal 0.0000054 0.9996821 0.0003125 0.9996821 0.9996821
large large 0.9999566 0.0000434 0.0000000 0.9999566 0.9999566
liminal large 0.6084001 0.3914308 0.0001691 0.6084001 0.3914308 
table(truth = dtest_prepared$yc, prediction = dtest_prepared$predict)
##          prediction
## truth     large liminal small
##   large     150       0     0
##   liminal    38      67    38
##   small       0       0   157

Parameters for mkCrossFrameMExperiment

We’ve tried to set the defaults for all parameters so that vtreat is usable out of the box for most applications.

suppressPackageStartupMessages(library(printr))
args("mkCrossFrameMExperiment")
## function (dframe, varlist, outcomename, ..., weights = c(), minFraction = 0.02, 
##     smFactor = 0, rareCount = 0, rareSig = 1, collarProb = 0, 
##     codeRestriction = NULL, customCoders = NULL, scale = FALSE, 
##     doCollar = FALSE, splitFunction = vtreat::kWayCrossValidation, 
##     ncross = 3, forceSplit = FALSE, catScaling = FALSE, y_dependent_treatments = c("catB"), 
##     verbose = FALSE, parallelCluster = NULL, use_parallel = TRUE, 
##     missingness_imputation = NULL, imputation_map = NULL) 
## NULL

Some parameters of note include:

codeRestriction: The types of synthetic variables that vtreat will (potentially) produce. By default, all possible applicable types will be produced. See Types of prepared variables below.

minFraction (default: 0.02): For categorical variables, indicator variables (type lev) are only produced for levels that are present at least minFraction of the time. A consequence of this is that 1/minFraction is the maximum number of indicators that will be produced for a given categorical variable. To make sure that all possible indicator variables are produced, set minFraction = 0

splitFunction: The cross validation method used by vtreat. Most people won’t have to change this.

ncross (default: 3): The number of folds to use for cross-validation

missingness_imputation: The function or value that vtreat uses to impute or “fill in” missing numerical values. The default is mean. To change the imputation function or use different functions/values for different columns, see the Imputation example.

customCoders: For passing in user-defined transforms for custom data preparation. Won’t be needed in most situations, but see here for an example of applying a GAM transform to input variables.

Types of prepared variables

clean: Produced from numerical variables: a clean numerical variable with no NAs or missing values

lev: Produced from categorical variables, one for each (common) level: for each level of the variable, indicates if that level was “on”

catP: Produced from categorical variables: indicates how often each level of the variable was “on” (its prevalence)

catB: Produced from categorical variables: score from a one-dimensional model of the centered output as a function of the variable

isBAD: Produced for numerical variables: an indicator variable that marks when the original variable was missing or NaN.

More on the coding types can be found here.

Example: Produce only a subset of variable types

In this example, suppose you only want to use indicators and continuous variables in your model; in other words, you only want to use variables of types (clean, isBAD, and lev), and no catB or catP variables.

unpack[
  transform_thin = treat_m,
  d_prepared_thin = cross_frame,
  score_frame_thin = score_frame
  ] <- vtreat::mkCrossFrameMExperiment(
    d,                                    # data to learn transform from
    setdiff(colnames(d), c('y', 'yc')),   # columns to transform
    'yc',                                 # outcome variable
    codeRestriction = c('lev',            # transforms we want
                        'clean',
                        'isBAD')
  )

d_prepared_thin %.>%
  head(.) %.>%
  knitr::kable(.)
x x_isBAD x2 xc_lev_NA xc_lev_x_level_minus_0_5 xc_lev_x_level_0 xc_lev_x_level_0_5 xc_lev_x_level_1 yc
1.8848606 0 0.0046504 0 0 0 0 1 large
1.5077419 0 -1.2287497 0 0 0 0 1 large
-5.4901159 0 -0.1405980 0 0 0 0 1 large
-0.2704873 1 -0.2073270 0 0 0 1 0 large
-0.2704873 1 -0.9215306 1 0 0 0 0 small
-0.2704873 1 0.3604742 0 1 0 0 0 small 
# no catB or catP
knitr::kable(score_frame_thin[,cols])
varName origName code outcome_level rsq sig varMoves default_threshold recommended
x x clean large 0.0005756 0.5470919 TRUE 0.05555556 FALSE
x_isBAD x isBAD large 0.0000771 0.8255885 TRUE 0.11111111 FALSE
x2 x2 clean large 0.0026075 0.2000083 TRUE 0.05555556 FALSE
xc_lev_NA xc lev large 0.1750095 0.0000000 TRUE 0.02222222 TRUE
xc_lev_x_level_0 xc lev large 0.1185254 0.0000000 TRUE 0.02222222 TRUE
xc_lev_x_level_0_5 xc lev large 0.0644178 0.0000000 TRUE 0.02222222 TRUE
xc_lev_x_level_1 xc lev large 0.4701626 0.0000000 TRUE 0.02222222 TRUE
xc_lev_x_level_minus_0_5 xc lev large 0.1328708 0.0000000 TRUE 0.02222222 TRUE
x x clean liminal 0.0022396 0.2338771 TRUE 0.05555556 FALSE
x_isBAD x isBAD liminal 0.0020122 0.2591571 TRUE 0.11111111 FALSE
x2 x2 clean liminal 0.0023614 0.2215731 TRUE 0.05555556 FALSE
xc_lev_NA xc lev liminal 0.1769596 0.0000000 TRUE 0.02222222 TRUE
xc_lev_x_level_0 xc lev liminal 0.3615209 0.0000000 TRUE 0.02222222 TRUE
xc_lev_x_level_0_5 xc lev liminal 0.0041777 0.1039765 TRUE 0.02222222 FALSE
xc_lev_x_level_1 xc lev liminal 0.1492650 0.0000000 TRUE 0.02222222 TRUE
xc_lev_x_level_minus_0_5 xc lev liminal 0.0076508 0.0277896 TRUE 0.02222222 FALSE
x x clean small 0.0048286 0.0773262 TRUE 0.05555556 FALSE
x_isBAD x isBAD small 0.0022777 0.2250506 TRUE 0.11111111 FALSE
x2 x2 clean small 0.0000045 0.9569844 TRUE 0.05555556 FALSE
xc_lev_NA xc lev small 0.5132320 0.0000000 TRUE 0.02222222 TRUE
xc_lev_x_level_0 xc lev small 0.1265119 0.0000000 TRUE 0.02222222 TRUE
xc_lev_x_level_0_5 xc lev small 0.1488474 0.0000000 TRUE 0.02222222 TRUE
xc_lev_x_level_1 xc lev small 0.1576977 0.0000000 TRUE 0.02222222 TRUE
xc_lev_x_level_minus_0_5 xc lev small 0.0387613 0.0000006 TRUE 0.02222222 TRUE

Deriving the Default Thresholds

While machine learning algorithms are generally tolerant to a reasonable number of irrelevant or noise variables, too many irrelevant variables can lead to serious overfit; see this article for an extreme example, one we call “Bad Bayes”. The default threshold is an attempt to eliminate obviously irrelevant variables early.

Imagine that you have a pure noise dataset, where none of the n inputs are related to the output. If you treat each variable as a one-variable model for the output, and look at the significances of each model, these significance-values will be uniformly distributed in the range [0:1]. You want to pick a weakest possible significance threshold that eliminates as many noise variables as possible. A moment’s thought should convince you that a threshold of 1/n allows only one variable through, in expectation.

This leads to the general-case heuristic that a significance threshold of 1/n on your variables should allow only one irrelevant variable through, in expectation (along with all the relevant variables). Hence, 1/n used to be our recommended threshold, when we originally developed the R version of vtreat.

We noticed, however, that this biases the filtering against numerical variables, since there are at most two derived variables (of types clean and isBAD) for every numerical variable in the original data. Categorical variables, on the other hand, are expanded to many derived variables: several indicators (one for every common level), plus a catB and a catP. So we now reweight the thresholds.

Suppose you have a (treated) data set with ntreat different types of vtreat variables (clean, lev, etc). There are nT variables of type T. Then the default threshold for all the variables of type T is 1/(ntreat nT). This reweighting helps to reduce the bias against any particular type of variable. The heuristic is still that the set of recommended variables will allow at most one noise variable into the set of candidate variables.

As noted above, because vtreat estimates variable significances using linear methods by default, some variables with a non-linear relationship to the output may fail to pass the threshold. In this case, you may not wish to filter the variables to be used in the models to only recommended variables (as we did in the main example above), but instead use all the variables, or select the variables to use by your own criteria.

Conclusion

In all cases (classification, regression, unsupervised, and multinomial classification) the intent is that vtreat transforms are essentially one liners.

The preparation commands are organized as follows:

These current revisions of the examples are designed to be small, yet complete. So as a set they have some overlap, but the user can rely mostly on a single example for a single task type.