24 Регрессионные модели: SVR

24.1 Регрессионные алгоритмы

В машинном обучении проблемы, связанные с количественным откликом, называют проблемами регрессии, а проблемы, связанные с качественным откликом, проблемами классификации. В прошлом уроке мы познакомились с простой и множественной регрессией.

Задание

Объясните своими словами, что такое множественная регрессия?

Но регрессионных алгоритмов великое множество. Вот лишь некоторые из них.

полиномиальная регрессия: расширение линейной регрессии, позволяющее учитывать нелинейные зависимости.
логистическая регрессия: используется для прогнозирования категориальных (бинарных) откликов.
регрессия на опорных векторах (SVM): ищет гиперплоскость, позволяющую минимизировать ошибку в многомерном пространстве.
деревья регрессии: строят иерархическую древовидную модель, последовательно разбивая данные на подгруппы.
случайный лес: комбинирует предсказания множества деревьев для повышения точности и устойчивости.

Кроме того, существуют методы регуляризации линейных моделей, позволяющие существенно улучшить их качество на данных большой размерности (т.е. с большим количеством предкторов). К таким алгоритмам относятся гребневая регрессия и метод лассо. О них мы поговорим в одном из следующих уроков.

О математической стороне дела см. Г. Джеймс, Д. Уиттон, Т. Хасти, Р. Тибришани (2017). В этом уроке мы начнем работать с различными регрессионными алгоритмами, используя библиотеку {tidymodels}.

24.2 Библиотека `tidymodels`

Библиотека tidymodels позволяет обучать модели и оценивать их эффективность с использованием принципов опрятных данных. Она представляет собой набор пакетов R, которые разработаны для работы с машинным обучением и совместима с экосистемой {tidyverse}.

Вот некоторые из ключевых пакетов, входящих в состав {tidymodels}:

{parsnip} - универсальный интерфейс для различных моделей машинного обучения, который упрощает переключение между разными типами моделей;
{recipes} - фреймворк для создания и управления “рецептами” предварительной обработки данных перед тренировкой модели;
{rsample} - инструменты для разделения данных на обучающую и тестовую выборки, а также для кросс-валидации;
{tune} - функции для оптимизации гиперпараметров моделей машинного обучения;
{yardstick} - инструменты для оценки производительности моделей;
{workflow} позволяет объединить различные компоненты модели в единый объект: препроцессинг данных, модель машинного обучения, настройку гиперпараметров.

Мы также будем использовать пакет {textrecipes}, который представляет собой аналог {recipes} для текстовых данных.

library(tidyverse)
library(tidymodels)
library(textrecipes)

24.3 Данные

Датасет для этого урока хранит данные о названиях, рейтингах, жанре, цене и числе отзывов на некоторые книги с Amazon. Мы попробуем построить регресионную модель, которая будет предсказывать цену книги. Скачайте данные по ссылке и прочитайте в окружение.

books  <- readxl::read_xlsx("../files/AmazonBooks.xlsx")
books

Данные не очень опрятны, и прежде всего их надо тайдифицировать.

colnames(books) <- tolower(colnames(books))

books <- books |> 
  rename(rating = `user rating`)

На графике ниже видно, что сильной корреляции между количественными переменными не прослеживается, так что задача перед нами стоит незаурядная. Посмотрим, что можно сделать в такой ситуации.

books |> 
  select_if(is.numeric) |> 
  cor() |> 
  corrplot::corrplot(method = "ellipse")

Мы видим, что количественные предикторы объясняют лишь ничтожную долю дисперсии (чуть более информативен жанр).

summary(lm(price ~ reviews + year + rating + genre, data  = books))


Call:
lm(formula = price ~ reviews + year + rating + genre, data = books)

Residuals:
    Min      1Q  Median      3Q     Max 
-16.472  -5.050  -1.841   2.307  89.686 

Coefficients:
                   Estimate Std. Error t value Pr(>|t|)    
(Intercept)       8.987e+02  2.734e+02   3.287  0.00107 ** 
reviews           7.779e-07  3.181e-05   0.024  0.98050    
year             -4.324e-01  1.370e-01  -3.156  0.00168 ** 
rating           -3.655e+00  1.933e+00  -1.891  0.05909 .  
genreNon Fiction  3.920e+00  8.669e-01   4.522 7.41e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 10.16 on 595 degrees of freedom
Multiple R-squared:  0.06903,   Adjusted R-squared:  0.06277 
F-statistic: 11.03 on 4 and 595 DF,  p-value: 1.235e-08

Посмотрим, можно ли как-то улучшить этот результат. Но сначала оценим визуально связь между ценой, с одной стороны, и годом и жанром, с другой.

g1 <- books |> 
  ggplot(aes(year, price, color = genre, group = genre)) + 
  geom_jitter(show.legend = FALSE, alpha = 0.7) + 
  geom_smooth(method = "lm", se = FALSE) +
  theme_minimal()

g2 <- books |> 
  ggplot(aes(genre, price, color = genre)) + 
  geom_boxplot() + 
  theme_minimal()

gridExtra::grid.arrange(g1, g2, nrow = 1)

24.4 Обучающая и контрольная выборка

Вы уже знаете, при обучении модели мы стремимся к минимизации среднеквадратичной ошибки (MSE), однако в большинстве случаев нас интересует не то, как метод работает на обучающих данных, а то, как он покажет себя на контрольных данных. Чтобы избежать переобучения, очень важно в самом начале разделить доступные наблюдения на две группы.

set.seed(080426)
books_split <- books |> 
  initial_split()

books_train <- training(books_split)
books_test <- testing(books_split)

24.5 Определение модели

Определение модели включает следующие шаги:

указывается тип модели на основе ее математической структуры (например, линейная регрессия, случайный лес, KNN и т. д.);
указывается механизм для подгонки модели – чаще всего это программный пакет, который должен быть использован, например {glmnet}. Это самостоятельные модели, и {parsnip} обеспечивает согласованные интерфейсы, используя их в качестве движков для моделирования.
при необходимости объявляется режим модели. Режим отражает тип прогнозируемого результата. Для числовых результатов режимом является регрессия, для качественных - классификация. Если алгоритм модели может работать только с одним типом результатов прогнозирования, например, линейной регрессией, режим уже задан.

На заметку

В R существует много разных пакетов и отдельных функций для построения моделей. У каждого пакета свой синтаксис и свои названия параметров. Например, в {ranger}, чтобы задать количество деревьев, пишут num.trees, а в {xgboost} — nrounds. Это неудобно: каждый раз нужно вспоминать правила конкретного пакета. {parsnip} создает единый, унифицированный интерфейс для всех моделей. Это позволяет переключаться между разными пакетами, не переучивая синтаксис.

24.6 Регрессия на опорных векторах

Support Vector Regression — это метод машинного обучения, основанный на идеях метода опорных векторов (SVM), но адаптированный к задаче регрессии, а не классификации (о чем см. следующий урок).

Вместо поиска разделяющей гиперплоскости между классами (как в классификации), SVR старается найти функцию, которая:

игнорирует небольшие отклонения внутри некоторого допустимого порога ε (эпсилон),
акцентирует внимание на точках, которые лежат вне этой “трубы”, — это и есть опорные векторы.

В этом заключается отличие от обычной регрессии, которая старается проложить прямую, наиболее близкую ко всем точкам и “наказывает” любое отклонение.

SVR тоже строит линию (или кривую в случае нелинейного ядра), но с другим подходом. Она “довольна”, если предсказание находится в пределах допустимой ошибки ε (эпсилон) от настоящего значения.

SVR концентрируется только на тех точках, что выходят за эту “зону безразличия” или лежат на ее границе — они называются опорными векторами. Именно они определяют форму и положение модели. Остальные точки (в пределах ε) никак не влияют на модель.

24.7 Линейная SVR

Функция translate() позволяет понять, как parsnip переводит пользовательский код на язык пакета.

svm_spec <- svm_linear() |>
  set_engine("LiblineaR") |> 
  set_mode("regression")

# svm_spec |> 
#   translate()

Пока это просто спецификация модели без данных и без формулы. Добавим ее к воркфлоу.

svm_wflow <- workflow() |> 
  add_model(svm_spec)

svm_wflow

══ Workflow ════════════════════════════════════════════════════════════════════
Preprocessor: None
Model: svm_linear()

── Model ───────────────────────────────────────────────────────────────────────
Linear Support Vector Machine Model Specification (regression)

Computational engine: LiblineaR

24.8 Дизайн переменных

Теперь нам нужен препроцессор. За него отвечает пакет {recipes}. Если вы не уверены, какие шаги необходимы на этом этапе, можно заглянуть в шпаргалку. В случае с линейной регрессией это может быть логарифмическая трансформация, нормализация, отсев переменных с нулевой дисперсией (zero variance), добавление (impute) недостающих значений или удаление переменных, которые коррелируют с другими переменными.

Вот так выглядит наш первый рецепт. Обратите внимание, что формула записывается так же, как мы это делали ранее внутри функции lm(). Порядок шагов важен (поэтому применяем нормализацию в самом конце)!

books_rec <- recipe(price ~ year + genre + name + reviews + rating, 
                    data = books_train) |> 
  step_dummy(genre)  |> 
  step_tokenize(name)  |> 
  step_tokenfilter(name, max_tokens = 1000)  |> 
  step_tfidf(name) |> 
  step_normalize(all_predictors())

При желании можно посмотреть на результат предобработки.

Для этого сначала передаем функции prep(), которая отвечает за подготовку (обучение рецепта). На этом этапе компьютер внимательно изучает ваши тренировочные данные, чтобы вычислить параметры для каждого шага, например среднее и стандартное отклонение для step_normalize(), словарь для step_tokenfilter() и т.д. Рецепт “запоминает” эти значения, чтобы потом применить их точно так же к новым данным. Сами данные на этом этапе еще не меняются.

Команда bake() примененяет рецепт к данным, т.е. берет таблицу с данными и прогоняет её через все шаги, используя параметры из prep(). В результате вы получаете готовую широкую матрицу с числами, которую уже можно “скармливать” алгоритму SVR.

prep(books_rec, books_train) |> 
  bake(new_data = NULL) |> 
  head(5) |> 
  gt::gt()

year	reviews	rating	price	genre_Non.Fiction	tfidf_name_1	tfidf_name_1,111	tfidf_name_10	tfidf_name_100	tfidf_name_11	tfidf_name_12	tfidf_name_150	tfidf_name_17	tfidf_name_1936	tfidf_name_2	tfidf_name_2.0	tfidf_name_20	tfidf_name_2016	tfidf_name_22	tfidf_name_3	tfidf_name_30	tfidf_name_300	tfidf_name_4	tfidf_name_451	tfidf_name_49	tfidf_name_5	tfidf_name_500	tfidf_name_5000	tfidf_name_52	tfidf_name_5th	tfidf_name_6	tfidf_name_60	tfidf_name_6th	tfidf_name_7	tfidf_name_700	tfidf_name_8	tfidf_name_a	tfidf_name_about	tfidf_name_absurd	tfidf_name_account	tfidf_name_achieving	tfidf_name_acid	tfidf_name_activity	tfidf_name_adult	tfidf_name_adults	tfidf_name_advanced	tfidf_name_adventures	tfidf_name_adversity	tfidf_name_afterlife	tfidf_name_aftermath	tfidf_name_against	tfidf_name_ages	tfidf_name_agreements	tfidf_name_ahead	tfidf_name_alaska	tfidf_name_alchemist	tfidf_name_alex	tfidf_name_all	tfidf_name_allâ	tfidf_name_allegiant	tfidf_name_almost	tfidf_name_alphabet	tfidf_name_am	tfidf_name_amateur	tfidf_name_amazing	tfidf_name_america	tfidf_name_america's	tfidf_name_american	tfidf_name_americans	tfidf_name_an	tfidf_name_ancient	tfidf_name_and	tfidf_name_animal	tfidf_name_animals	tfidf_name_answers	tfidf_name_antidote	tfidf_name_antiracist	tfidf_name_apologizing	tfidf_name_approach	tfidf_name_are	tfidf_name_arguing	tfidf_name_art	tfidf_name_as	tfidf_name_asians	tfidf_name_assassination	tfidf_name_association	tfidf_name_at	tfidf_name_athena	tfidf_name_atomic	tfidf_name_attitude	tfidf_name_audacious	tfidf_name_autobiography	tfidf_name_awesome	tfidf_name_azkaban	tfidf_name_b	tfidf_name_back	tfidf_name_bad	tfidf_name_badass	tfidf_name_ball	tfidf_name_ballad	tfidf_name_barefoot	tfidf_name_be	tfidf_name_bear	tfidf_name_beasts	tfidf_name_beautiful	tfidf_name_beck's	tfidf_name_become	tfidf_name_becoming	tfidf_name_bed	tfidf_name_bee	tfidf_name_beginner's	tfidf_name_beginners	tfidf_name_believing	tfidf_name_belly	tfidf_name_berlin	tfidf_name_between	tfidf_name_big	tfidf_name_bill	tfidf_name_billionaires	tfidf_name_bin	tfidf_name_birth	tfidf_name_blitz	tfidf_name_blood	tfidf_name_blue	tfidf_name_boat	tfidf_name_body	tfidf_name_bombers	tfidf_name_book	tfidf_name_books	tfidf_name_boom	tfidf_name_born	tfidf_name_bossypants	tfidf_name_boxed	tfidf_name_boy	tfidf_name_boys	tfidf_name_brain	tfidf_name_brain's	tfidf_name_brave	tfidf_name_brawl	tfidf_name_break	tfidf_name_breaking	tfidf_name_bree	tfidf_name_brigance	tfidf_name_brink	tfidf_name_broke	tfidf_name_brothers	tfidf_name_brown	tfidf_name_brush	tfidf_name_build	tfidf_name_building	tfidf_name_business	tfidf_name_buy	tfidf_name_by	tfidf_name_cabin	tfidf_name_cakes	tfidf_name_called	tfidf_name_calligraphy	tfidf_name_calling	tfidf_name_calm	tfidf_name_camelot	tfidf_name_can	tfidf_name_can't	tfidf_name_cannot	tfidf_name_capital	tfidf_name_captain	tfidf_name_carbs	tfidf_name_case	tfidf_name_caste	tfidf_name_catching	tfidf_name_caterpillar	tfidf_name_cats	tfidf_name_cauldron	tfidf_name_century	tfidf_name_chamber	tfidf_name_champion	tfidf_name_change	tfidf_name_changed	tfidf_name_changing	tfidf_name_chaos	tfidf_name_chicka	tfidf_name_choose	tfidf_name_christian	tfidf_name_chronicles	tfidf_name_churchill	tfidf_name_ck	tfidf_name_clash	tfidf_name_class	tfidf_name_classic	tfidf_name_clintons	tfidf_name_club	tfidf_name_collected	tfidf_name_collection	tfidf_name_college	tfidf_name_color	tfidf_name_coloring	tfidf_name_colors	tfidf_name_columbus's	tfidf_name_comfort	tfidf_name_commander	tfidf_name_comments	tfidf_name_common	tfidf_name_companies	tfidf_name_complete	tfidf_name_confession	tfidf_name_confident	tfidf_name_confronts	tfidf_name_constitution	tfidf_name_construction	tfidf_name_contessa	tfidf_name_control	tfidf_name_cookbook	tfidf_name_cookbooks	tfidf_name_cooker	tfidf_name_cooking	tfidf_name_cooks	tfidf_name_cooling	tfidf_name_counterintuitive	tfidf_name_crawdads	tfidf_name_crayons	tfidf_name_crazy	tfidf_name_create	tfidf_name_created	tfidf_name_creating	tfidf_name_creative	tfidf_name_creator	tfidf_name_crest	tfidf_name_crisis	tfidf_name_cross	tfidf_name_crows	tfidf_name_cultivate	tfidf_name_culture	tfidf_name_cups	tfidf_name_cutting	tfidf_name_cycle	tfidf_name_da	tfidf_name_dad	tfidf_name_dance	tfidf_name_dangerous	tfidf_name_danny	tfidf_name_dare	tfidf_name_darker	tfidf_name_dawn	tfidf_name_day	tfidf_name_days	tfidf_name_dead	tfidf_name_dear	tfidf_name_death	tfidf_name_decades	tfidf_name_decision	tfidf_name_deckled	tfidf_name_decluttering	tfidf_name_defiance	tfidf_name_definitive	tfidf_name_defy	tfidf_name_delicious	tfidf_name_delivering	tfidf_name_deluxe	tfidf_name_designed	tfidf_name_designs	tfidf_name_devotional	tfidf_name_diagnostic	tfidf_name_diary	tfidf_name_die	tfidf_name_diet	tfidf_name_difference	tfidf_name_difficult	tfidf_name_dirt	tfidf_name_discontents	tfidf_name_discovery	tfidf_name_disease	tfidf_name_disorders	tfidf_name_divergent	tfidf_name_divine	tfidf_name_do	tfidf_name_doctor's	tfidf_name_documents	tfidf_name_dog	tfidf_name_dome	tfidf_name_don't	tfidf_name_donkey	tfidf_name_doomsday	tfidf_name_double	tfidf_name_doubting	tfidf_name_down	tfidf_name_dragon	tfidf_name_dragons	tfidf_name_dream	tfidf_name_drive	tfidf_name_dsm	tfidf_name_duck	tfidf_name_dungeons	tfidf_name_during	tfidf_name_dysfunctions	tfidf_name_earth	tfidf_name_easy	tfidf_name_eat	tfidf_name_eater's	tfidf_name_echoed	tfidf_name_eclipse	tfidf_name_edge	tfidf_name_edition	tfidf_name_educated	tfidf_name_effective	tfidf_name_electric	tfidf_name_elegance	tfidf_name_elegy	tfidf_name_elements	tfidf_name_elephants	tfidf_name_embracing	tfidf_name_end	tfidf_name_english	tfidf_name_enjoying	tfidf_name_enough	tfidf_name_epic	tfidf_name_essential	tfidf_name_eternity	tfidf_name_everyday	tfidf_name_everything	tfidf_name_everywhere	tfidf_name_exceptional	tfidf_name_f	tfidf_name_fable	tfidf_name_face	tfidf_name_facing	tfidf_name_facts	tfidf_name_fahrenheit	tfidf_name_faith	tfidf_name_families	tfidf_name_family	tfidf_name_fast	tfidf_name_fat	tfidf_name_fault	tfidf_name_fbi	tfidf_name_fear	tfidf_name_feast	tfidf_name_featuring	tfidf_name_feel	tfidf_name_feeling	tfidf_name_fetch	tfidf_name_fever	tfidf_name_fey	tfidf_name_fifty	tfidf_name_find	tfidf_name_finding	tfidf_name_fire	tfidf_name_fires	tfidf_name_first	tfidf_name_firsthand	tfidf_name_fish	tfidf_name_five	tfidf_name_flap	tfidf_name_fleas	tfidf_name_flower	tfidf_name_flowers	tfidf_name_food	tfidf_name_foolproof	tfidf_name_for	tfidf_name_forâ	tfidf_name_forever	tfidf_name_found	tfidf_name_four	tfidf_name_fox	tfidf_name_fragility	tfidf_name_freakonomics	tfidf_name_free	tfidf_name_freed	tfidf_name_freedom	tfidf_name_french	tfidf_name_friends	tfidf_name_from	tfidf_name_frontier	tfidf_name_frozen	tfidf_name_full	tfidf_name_fully	tfidf_name_fun	tfidf_name_future	tfidf_name_game	tfidf_name_games	tfidf_name_garden	tfidf_name_gathering	tfidf_name_gatsby	tfidf_name_general	tfidf_name_gentleman	tfidf_name_geographic	tfidf_name_george	tfidf_name_get	tfidf_name_getaway	tfidf_name_gifts	tfidf_name_giraffes	tfidf_name_girl	tfidf_name_girls	tfidf_name_give	tfidf_name_giving	tfidf_name_glenn	tfidf_name_glitter	tfidf_name_global	tfidf_name_glory	tfidf_name_go	tfidf_name_goals	tfidf_name_goblet	tfidf_name_god	tfidf_name_going	tfidf_name_gold	tfidf_name_gone	tfidf_name_good	tfidf_name_goodnight	tfidf_name_government	tfidf_name_grades	tfidf_name_grain	tfidf_name_gratitude	tfidf_name_great	tfidf_name_greatness	tfidf_name_greenlights	tfidf_name_grey	tfidf_name_grime	tfidf_name_guardians	tfidf_name_guernsey	tfidf_name_guide	tfidf_name_guided	tfidf_name_gut	tfidf_name_guts	tfidf_name_habit	tfidf_name_habits	tfidf_name_half	tfidf_name_hamilton	tfidf_name_hand	tfidf_name_handbook	tfidf_name_handmaid's	tfidf_name_hands	tfidf_name_happened	tfidf_name_happiness	tfidf_name_happy	tfidf_name_harbinger	tfidf_name_hard	tfidf_name_hardcover	tfidf_name_harry	tfidf_name_hate	tfidf_name_haul	tfidf_name_have	tfidf_name_haven	tfidf_name_hayek	tfidf_name_head	tfidf_name_heal	tfidf_name_healing	tfidf_name_health	tfidf_name_healthy	tfidf_name_healthyâ	tfidf_name_heat	tfidf_name_heaven	tfidf_name_hedgehog	tfidf_name_help	tfidf_name_henna	tfidf_name_henrietta	tfidf_name_her	tfidf_name_heroes	tfidf_name_higher	tfidf_name_highly	tfidf_name_hillbilly	tfidf_name_his	tfidf_name_historia	tfidf_name_history	tfidf_name_hitler's	tfidf_name_holds	tfidf_name_homebody	tfidf_name_honey	tfidf_name_hornet's	tfidf_name_horse	tfidf_name_hour	tfidf_name_house	tfidf_name_how	tfidf_name_human	tfidf_name_humanity	tfidf_name_humans	tfidf_name_hunger	tfidf_name_hungry	tfidf_name_hurricane	tfidf_name_hurt	tfidf_name_hypothetical	tfidf_name_hyrule	tfidf_name_i	tfidf_name_ice	tfidf_name_icons	tfidf_name_ideas	tfidf_name_idiots	tfidf_name_if	tfidf_name_ii	tfidf_name_ii's	tfidf_name_illustrated	tfidf_name_imagination	tfidf_name_immortal	tfidf_name_in	tfidf_name_inches	tfidf_name_includes	tfidf_name_incredible	tfidf_name_inferno	tfidf_name_influence	tfidf_name_inheritance	tfidf_name_inside	tfidf_name_inspired	tfidf_name_instant	tfidf_name_insurance	tfidf_name_insurgent	tfidf_name_intimate	tfidf_name_into	tfidf_name_introverts	tfidf_name_irreverent	tfidf_name_is	tfidf_name_it	tfidf_name_it's	tfidf_name_iv	tfidf_name_jackson	tfidf_name_jake	tfidf_name_japan	tfidf_name_japanese	tfidf_name_jesus	tfidf_name_jobs	tfidf_name_jokes	tfidf_name_journal	tfidf_name_journey	tfidf_name_joy	tfidf_name_k	tfidf_name_kane	tfidf_name_keeps	tfidf_name_kennedy	tfidf_name_keto	tfidf_name_ketogenic	tfidf_name_kicked	tfidf_name_kid	tfidf_name_kids	tfidf_name_kill	tfidf_name_killed	tfidf_name_killers	tfidf_name_killing	tfidf_name_kings	tfidf_name_kissed	tfidf_name_kitchen	tfidf_name_kitties	tfidf_name_knickerbocker	tfidf_name_knock	tfidf_name_know	tfidf_name_knowledge	tfidf_name_lacks	tfidf_name_laden	tfidf_name_land	tfidf_name_languages	tfidf_name_last	tfidf_name_lasts	tfidf_name_laugh	tfidf_name_lead	tfidf_name_leadership	tfidf_name_lean	tfidf_name_leap	tfidf_name_learn	tfidf_name_leave	tfidf_name_lectin	tfidf_name_lecture	tfidf_name_left	tfidf_name_legacy	tfidf_name_legend	tfidf_name_lego	tfidf_name_leonardo	tfidf_name_less	tfidf_name_lessons	tfidf_name_lettering	tfidf_name_lies	tfidf_name_life	tfidf_name_lifestyle	tfidf_name_lifetime	tfidf_name_lift	tfidf_name_light	tfidf_name_like	tfidf_name_lincoln	tfidf_name_literary	tfidf_name_litigators	tfidf_name_little	tfidf_name_live	tfidf_name_living	tfidf_name_lonely	tfidf_name_long	tfidf_name_looking	tfidf_name_lord	tfidf_name_lose	tfidf_name_losing	tfidf_name_loss	tfidf_name_lost	tfidf_name_loud	tfidf_name_love	tfidf_name_loved	tfidf_name_loyalty	tfidf_name_luck	tfidf_name_machine	tfidf_name_magic	tfidf_name_magical	tfidf_name_magnolia	tfidf_name_make	tfidf_name_man	tfidf_name_man's	tfidf_name_mandalas	tfidf_name_manual	tfidf_name_mark	tfidf_name_martian	tfidf_name_master	tfidf_name_mastering	tfidf_name_matter	tfidf_name_maze	tfidf_name_mccain	tfidf_name_me	tfidf_name_meals	tfidf_name_meant	tfidf_name_memoir	tfidf_name_mental	tfidf_name_midnight	tfidf_name_milk	tfidf_name_millennium	tfidf_name_mind	tfidf_name_minds	tfidf_name_miracles	tfidf_name_mission	tfidf_name_mockingbird	tfidf_name_mockingjay	tfidf_name_modern	tfidf_name_mole	tfidf_name_mon	tfidf_name_moon	tfidf_name_more	tfidf_name_moscow	tfidf_name_most	tfidf_name_mother	tfidf_name_motivates	tfidf_name_mountains	tfidf_name_much	tfidf_name_mueller	tfidf_name_murders	tfidf_name_my	tfidf_name_mystery	tfidf_name_national	tfidf_name_navy	tfidf_name_need	tfidf_name_neptune	tfidf_name_nest	tfidf_name_neurosurgeon's	tfidf_name_never	tfidf_name_new	tfidf_name_night	tfidf_name_nightingale	tfidf_name_nine	tfidf_name_no	tfidf_name_not	tfidf_name_novel	tfidf_name_novella	tfidf_name_now	tfidf_name_numbers	tfidf_name_nutrient	tfidf_name_o'reilly's	tfidf_name_obama	tfidf_name_odds	tfidf_name_of	tfidf_name_official	tfidf_name_oh	tfidf_name_old	tfidf_name_olympian	tfidf_name_olympians	tfidf_name_olympics	tfidf_name_olympus	tfidf_name_on	tfidf_name_one	tfidf_name_ones	tfidf_name_oprah's	tfidf_name_option	tfidf_name_or	tfidf_name_organizing	tfidf_name_origins	tfidf_name_orphan	tfidf_name_osage	tfidf_name_osama	tfidf_name_others	tfidf_name_our	tfidf_name_out	tfidf_name_outliers	tfidf_name_ove	tfidf_name_over	tfidf_name_overwhelmed	tfidf_name_p	tfidf_name_pacific	tfidf_name_paine	tfidf_name_paint	tfidf_name_paisley	tfidf_name_paisleyâ	tfidf_name_palin	tfidf_name_paperback	tfidf_name_paradox	tfidf_name_paris	tfidf_name_passion	tfidf_name_passions	tfidf_name_pastimes	tfidf_name_path	tfidf_name_patriot's	tfidf_name_patriotic	tfidf_name_patriots	tfidf_name_patterns	tfidf_name_patton	tfidf_name_peace	tfidf_name_peel	tfidf_name_people	tfidf_name_percy	tfidf_name_performers	tfidf_name_personal	tfidf_name_pet	tfidf_name_pictures	tfidf_name_pie	tfidf_name_pilgrims	tfidf_name_pioneer	tfidf_name_places	tfidf_name_plan	tfidf_name_planet	tfidf_name_plant	tfidf_name_played	tfidf_name_player	tfidf_name_player's	tfidf_name_point	tfidf_name_points	tfidf_name_pokã	tfidf_name_politics	tfidf_name_portrait	tfidf_name_pot	tfidf_name_potato	tfidf_name_potter	tfidf_name_potty	tfidf_name_pounds	tfidf_name_pout	tfidf_name_power	tfidf_name_powerful	tfidf_name_practical	tfidf_name_prayer	tfidf_name_pre	tfidf_name_preschool	tfidf_name_presence	tfidf_name_press	tfidf_name_pressure	tfidf_name_preventing	tfidf_name_prisoner	tfidf_name_profits	tfidf_name_program	tfidf_name_promised	tfidf_name_promote	tfidf_name_proof	tfidf_name_prostitutes	tfidf_name_proven	tfidf_name_psychological	tfidf_name_publication	tfidf_name_punishment	tfidf_name_purpose	tfidf_name_put	tfidf_name_pyramid	tfidf_name_quest	tfidf_name_questions	tfidf_name_quiet	tfidf_name_quintet	tfidf_name_quit	tfidf_name_race	tfidf_name_racing	tfidf_name_racism	tfidf_name_racketeer	tfidf_name_radical	tfidf_name_rain	tfidf_name_rapid	tfidf_name_readingâ	tfidf_name_ready	tfidf_name_recipes	tfidf_name_reckoning	tfidf_name_red	tfidf_name_redemption	tfidf_name_references	tfidf_name_relentless	tfidf_name_relief	tfidf_name_relieving	tfidf_name_report	tfidf_name_resilience	tfidf_name_restore	tfidf_name_results	tfidf_name_reusable	tfidf_name_revere	tfidf_name_revised	tfidf_name_revolution	tfidf_name_revolutionary	tfidf_name_rich	tfidf_name_riddles	tfidf_name_right	tfidf_name_ring	tfidf_name_rising	tfidf_name_road	tfidf_name_rogue	tfidf_name_rolls	tfidf_name_room	tfidf_name_routines	tfidf_name_row	tfidf_name_rules	tfidf_name_run	tfidf_name_runner	tfidf_name_rush	tfidf_name_sacred	tfidf_name_saga	tfidf_name_sagas	tfidf_name_salt	tfidf_name_sat	tfidf_name_save	tfidf_name_saved	tfidf_name_says	tfidf_name_school	tfidf_name_scientific	tfidf_name_score	tfidf_name_scripture	tfidf_name_seal	tfidf_name_second	tfidf_name_secret	tfidf_name_secrets	tfidf_name_see	tfidf_name_sense	tfidf_name_serfdom	tfidf_name_series	tfidf_name_serious	tfidf_name_serpent's	tfidf_name_sesame	tfidf_name_set	tfidf_name_seuss	tfidf_name_sex	tfidf_name_sh	tfidf_name_shack	tfidf_name_shades	tfidf_name_shadow	tfidf_name_shame	tfidf_name_shapes	tfidf_name_shocking	tfidf_name_short	tfidf_name_should	tfidf_na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На заметку

Список доступных рецептов ищите здесь: https://www.tidymodels.org/find/recipes/.

Доработаем наш рецепт.

books_rec <- recipe(price ~ year + genre + name + reviews + rating, 
                    data = books_train) |> 
  step_mutate(name = str_replace_all(name, "\\d+([\\.,]\\d+)?", "0")) |> 
  step_dummy(genre)  |> 
  step_tokenize(name)  |> 
  step_tokenfilter(name, max_tokens = 1000)  |> 
  step_tfidf(name) |> 
  step_normalize(all_predictors())

Добавляем препроцессор в воркфлоу.

svm_wflow <- svm_wflow |> 
  add_recipe(books_rec)  # update_recipe(), если надо заменить

svm_wflow

══ Workflow ════════════════════════════════════════════════════════════════════
Preprocessor: Recipe
Model: svm_linear()

── Preprocessor ────────────────────────────────────────────────────────────────
6 Recipe Steps

• step_mutate()
• step_dummy()
• step_tokenize()
• step_tokenfilter()
• step_tfidf()
• step_normalize()

── Model ───────────────────────────────────────────────────────────────────────
Linear Support Vector Machine Model Specification (regression)

Computational engine: LiblineaR

24.9 Подгонка модели

Подгоним модель на обучающих данных.

svm_fit <- svm_wflow |>
  fit(data = books_train)

Пакет {broom} позволяет тайдифицировать модель. Посмотрим на переменные, которые приводят к “удорожанию” книг. Видно, что в начале списка – слова, связанные с научными публикациями, что не лишено смысла.

Задание

Самостоятельно изучите конец списка и интерпретируйте результат.

svm_fit |> 
  tidy() |> 
  arrange(-estimate) |> 
  head(6) |> 
  gt::gt()

term	estimate
Bias	13.227513
tfidf_name_collection	3.460221
tfidf_name_alchemist	1.448187
tfidf_name_edition	1.347852
tfidf_name_watchmen	1.269329
tfidf_name_dragons	1.021878

На заметку

Столбец Bias (смещение) в контексте SVR – это то же самое, что Intercept (свободный член) в обычной линейной регрессии.

Оценим модель на контрольных данных. Для оценки используем корень из среднеквадратичной ошибки, коэффициент детерминации и среднюю абсолютную ошибку.

24.10 Контрольные данные

pred_data <- tibble(truth = books_test$price,
                    estimate = predict(svm_fit, books_test)$.pred)

books_metrics <- metric_set(rmse, rsq, mae)

books_metrics(pred_data, truth = truth,  estimate = estimate) |> 
  gt::gt()

.metric	.estimator	.estimate
rmse	standard	9.87292855
rsq	standard	0.07380427
mae	standard	4.95101282

Можно заметить, что \(R^2\) лишь немного отличается от простой линейной модели. При использовании линейного ядра в методе опорных векторов (svm_linear()) итоговые метрики качества часто оказываются сопоставимы с результатами классической линейной регрессии. Это объясняется тем, что оба алгоритма нацелены на поиск линейных закономерностей в пространстве признаков.

На заметку

Средняя абсолютная ошибка (MAE) представляет собой среднее арифметическое модулей разностей между предсказанными и фактическими значениями целевой переменной. В отличие от среднеквадратичной ошибки (MSE), MAE измеряется в тех же единицах, что и сами данные, что делает её интуитивно понятной для интерпретации: она показывает, на сколько единиц в среднем ошибается модель. Важной особенностью этой метрики является её устойчивость к выбросам, поскольку при расчёте все ошибки учитываются линейно, а не возводятся в квадрат.

24.11 Нелинейная SVR

Нелинейные модели опорных векторов (SVR) используют специальный математический прием, называемый «ядерным переходом» (kernel trick), для проецирования исходных данных в пространство более высокой размерности. В этом новом пространстве алгоритм строит линейную гиперплоскость, которая при обратном проецировании в исходные координаты превращается в сложную нелинейную кривую или поверхность.

Наиболее популярное радиально-базисное ядро (RBF) позволяет модели учитывать локальные особенности данных: оно придает больший вес точкам, находящимся вблизи предсказываемого значения, и плавно снижает влияние удаленных наблюдений. Простыми словами: при оценке стоимости конкретной книги из контрольных данных модель в первую очередь смотрит на её ближайших соседей из тренировочного набора. Те книги, которые максимально похожи на оцениваемую (по году, жанру или словам в названии), имеют решающий “вес”, в то время как непохожие издания почти не учитываются. Таким образом, модель не просто выводит общую формулу для всего набора, а ищет уникальные локальные закономерности для каждого наблюдения.

Другое решение - полиномиальное ядро, когда пространство предикторов расширяется за счет полиномов 2-й, 3-й и т.д. степеней. Такие модели, однако, легко переобучить: они идеально описывают случайные совпадения в тренировочном наборе, но пасуют перед новыми данными.

Сначала ставим пакет {kernlab}, так как он является основным движком для RBF-ядра в R.

# Определение нелинейной модели SVR с RBF-ядром
svm_rbf_spec <- svm_rbf(
  cost = 1,          # Штраф за ошибки (можно подбирать через tune())
  rbf_sigma = 0.01   # Параметр "ширины" влияния каждой точки (gamma)
) |>
  set_engine("kernlab") |> 
  set_mode("regression")

# Создание нового workflow
svm_rbf_wflow <- workflow() |> 
  add_recipe(books_rec) |> 
  add_model(svm_rbf_spec)

# Обучение
svm_rbf_fit <- fit(svm_rbf_wflow, data = books_train)

# Оценка
pred_data <- tibble(truth = books_test$price,
                    estimate = predict(svm_rbf_fit, books_test)$.pred)

books_metrics <- metric_set(rmse, rsq, mae)

books_metrics(pred_data, truth = truth,  estimate = estimate) |> 
  gt::gt()

.metric	.estimator	.estimate
rmse	standard	6.6614073
rsq	standard	0.2009881
mae	standard	3.7919709

24.12 Повторные выборки

Чтобы не распечатывать каждый раз тестовые данные (в идеале мы их используем один, максимум два раза!), задействуется ряд методов, позволяющих оценить ошибку путем исключения части обучающих наблюдений из процесса подгонки модели и последующего применения этой модели к исключенным наблюдениям.

В пакете {rsample} из библиотеки {tidymodels} реализованы, среди прочего, следующие методы повторных выборок для оценки производительности моделей машинного обучения:

Метод проверочной выборки – набор наблюдений делится на обучающую и проверочную, или удержанную, выборку (validation set): для этого используется initial_validation_split().
K-кратная перекрестная проверка – наблюдения разбиваются на k групп примерно одинакового размера, первый блок служит в качестве проверочной выборки, а модель подгоняется по остальным k-1 блокам; процедура повторяется k раз: функция vfold_cv().
Перекрестная проверка Монте-Карло – в отличие от предыдущего метода, создается множество случайных разбиений данных на обучающую и тестовую выборки: функция mc_cv().
Бутстреп – отбор наблюдений выполняется с возвращением, т.е. одно и то же наблюдение может встречаться несколько раз: функция bootstraps().
Перекрестная проверка по отдельным наблюдениям (leave-one-out сross-validation): одно наблюдение используется в качестве контрольного, а остальные составляют обучающую выборку; модель подгоняется по n-1 наблюдениям, что повторяется n раз: функция loo_cv().

Эти методы повторных выборок позволяют получить надежные оценки производительности моделей машинного обучения, избегая переобучения и обеспечивая репрезентативность тестовых выборок.

set.seed(05102024)
books_folds <- vfold_cv(books_train, v = 10) 

svm_rbf_rs <- fit_resamples(
  svm_rbf_wflow,
  books_folds,
  control = control_resamples(save_pred = TRUE)
)

В vfold_cv() модель обучается и тестируется 10 раз на разных частях данных. Итоговый — это среднее арифметическое результатов всех 10 попыток. Среднее значение обычно выглядит “оптимистичнее” и стабильнее, так как оно сглаживает случайные провалы:

collect_metrics(svm_rbf_rs) |> 
  gt::gt()

.metric	.estimator	mean	n	std_err	.config
rmse	standard	7.5597222	10	1.5046226	Preprocessor1_Model1
rsq	standard	0.5224215	10	0.0670075	Preprocessor1_Model1

svm_rbf_rs |> 
  collect_predictions() |> 
  ggplot(aes(price, .pred, color = id)) +
  geom_jitter(alpha = 0.3) +
  geom_abline(lty = 2, color = "grey80") + 
  theme_minimal() +
  coord_cartesian(xlim = c(0,50), ylim = c(0,50))

Что значит крест: для огромного количества книг модель просто выдает среднее значение цены (или близкое к нему). Когда SVR (или любая другая модель) не находит в признаках (словах, рейтинге) четкого сигнала для конкретной книги, она «минимизирует риски» и предсказывает среднее по выборке. Это плохо.

Задание

Проведите кросс-валидацию с использованием линейной SVR. Соберите метрики, постройте график. Как отличается коэффициент детерминации? Сравните два графика.

24.13 Нулевая модель

Кстати, проверим, какой результат даст нулевая модель.

null_reg <- null_model() |> 
  set_engine("parsnip") |> 
  set_mode("regression")

null_wflow <- workflow() |> 
    add_model(null_reg) |> 
    add_recipe(books_rec)

null_rs <- fit_resamples(
  null_wflow,
  books_folds,
  control = control_resamples(save_pred = TRUE)
  )

collect_metrics(null_rs) |> 
  gt::gt()

.metric	.estimator	mean	n	std_err	.config
rmse	standard	10.65831	10	1.281173	Preprocessor1_Model1
rsq	standard	NaN	0	NA	Preprocessor1_Model1

\(R^2\) в таком случае должен быть NaN.

24.14 Удаление стопслов

Изменим рецепт приготовления данных.

stopwords_rec <- function(stopwords_name) {
  recipe(price ~ year + genre + name + reviews + rating, 
                    data = books_train) |> 
  step_mutate(name = str_replace_all(name, "\\d+([\\.,]\\d+)?", "0")) |> 
  step_dummy(genre)  |> 
  step_tokenize(name)  |> 
  step_stopwords(name, stopword_source = stopwords_name) |> 
  step_tokenfilter(name, max_tokens = 1000)  |> 
  step_tfidf(name) |> 
  step_normalize(all_predictors())
}

Создадим новый воркфлоу.

svm_rbf_wflow <- workflow() |> 
  add_model(svm_rbf_spec)

И снова проведем перекрестную проверку, на этот раз с разными списками стоп-слов.

set.seed(123)
snowball_rs <- fit_resamples(
  svm_rbf_wflow |>  
    add_recipe(stopwords_rec("snowball")),
  books_folds
)

set.seed(234)
smart_rs <- fit_resamples(
  svm_rbf_wflow |> 
    add_recipe(stopwords_rec("smart")),
  books_folds
)

set.seed(345)
stopwords_iso_rs <- fit_resamples(
  svm_rbf_wflow |>
    add_recipe(stopwords_rec("stopwords-iso")),
  books_folds
)

collect_metrics(smart_rs)  |> 
  gt::gt()

.metric	.estimator	mean	n	std_err	.config
rmse	standard	7.7466013	10	1.48713642	Preprocessor1_Model1
rsq	standard	0.4913497	10	0.07223065	Preprocessor1_Model1

collect_metrics(snowball_rs) |> 
  gt::gt()

.metric	.estimator	mean	n	std_err	.config
rmse	standard	7.651848	10	1.51263320	Preprocessor1_Model1
rsq	standard	0.507731	10	0.06971969	Preprocessor1_Model1

collect_metrics((stopwords_iso_rs)) |> 
  gt::gt()

.metric	.estimator	mean	n	std_err	.config
rmse	standard	7.6764663	10	1.48085099	Preprocessor1_Model1
rsq	standard	0.5005151	10	0.07010337	Preprocessor1_Model1

В нашем случае удаление стоп-слов заметного положительного эффекта не имело.

word_counts <- tibble(name = c("snowball", "smart", "stopwords-iso")) %>%
  mutate(words = map_int(name, ~length(stopwords::stopwords(source = .))))

list(snowball = snowball_rs,
     smart = smart_rs,
     `stopwords-iso` = stopwords_iso_rs)  |> 
  map_dfr(show_best, metric = "rmse", .id = "name")  |> 
  left_join(word_counts, by = "name")  |> 
  mutate(name = paste0(name, " (", words, " words)"),
         name = fct_reorder(name, words))  |> 
  ggplot(aes(name, mean, color = name)) +
  geom_crossbar(aes(ymin = mean - std_err, ymax = mean + std_err), alpha = 0.6) +
  geom_point(size = 3, alpha = 0.8) +
  theme(legend.position = "none") + 
  theme_minimal()

24.15 Настройки числа n-grams

ngram_rec <- function(ngram_options) {
  recipe(price ~ year + genre + name + reviews + rating, data = books_train) |> 
  step_mutate(name = str_replace_all(name, "\\d+([\\.,]\\d+)?", "0")) |>
  step_dummy(genre)  |> 
  step_tokenize(name, token = "ngrams", options = ngram_options)  |> 
  step_tokenfilter(name, max_tokens = 1000)  |> 
  step_tfidf(name) |> 
  step_normalize(all_predictors())
}

fit_ngram <- function(ngram_options) {
  fit_resamples(
    svm_rbf_wflow  |> 
    add_recipe(ngram_rec(ngram_options)),
    books_folds
  )
}

set.seed(123)
unigram_rs <- fit_ngram(list(n = 1))

set.seed(234)
bigram_rs <- fit_ngram(list(n = 2, n_min = 1))

set.seed(345)
trigram_rs <- fit_ngram(list(n = 3, n_min = 1))

collect_metrics(unigram_rs) |> 
  gt::gt()

.metric	.estimator	mean	n	std_err	.config
rmse	standard	7.5597222	10	1.5046226	Preprocessor1_Model1
rsq	standard	0.5224215	10	0.0670075	Preprocessor1_Model1

collect_metrics(bigram_rs) |> 
  gt::gt()

.metric	.estimator	mean	n	std_err	.config
rmse	standard	7.5147223	10	1.61280063	Preprocessor1_Model1
rsq	standard	0.5181945	10	0.08163889	Preprocessor1_Model1

collect_metrics(trigram_rs) |> 
  gt::gt()

.metric	.estimator	mean	n	std_err	.config
rmse	standard	7.5709675	10	1.61781943	Preprocessor1_Model1
rsq	standard	0.5113992	10	0.08112386	Preprocessor1_Model1

Визуализируем результат:

list(`1` = unigram_rs,
     `1 and 2` = bigram_rs,
     `1, 2, and 3` = trigram_rs) |> 
  map_dfr(collect_metrics, .id = "name")  |> 
  filter(.metric == "rmse")  |> 
  ggplot(aes(name, mean, color = name)) +
  geom_crossbar(aes(ymin = mean - std_err, ymax = mean + std_err), 
                alpha = 0.6) +
  geom_point(size = 3, alpha = 0.8) +
  theme(legend.position = "none") +
  labs(
    y = "RMSE"
  ) + 
  theme_minimal()

24.16 Финализация и оценка

# books_split — это объект, который вы получили в самом начале через initial_split()
final_res <- last_fit(svm_rbf_wflow |> 
                        add_recipe(books_rec), 
                      books_split)

# 1. Посмотреть метрики на контрольных данных (RMSE, RSQ)
collect_metrics(final_res) |> 
  gt::gt()

.metric	.estimator	.estimate	.config
rmse	standard	6.6614073	Preprocessor1_Model1
rsq	standard	0.2009881	Preprocessor1_Model1

# 2. Посмотреть предсказания для тестовой выборки
final_preds <- collect_predictions(final_res)

# 3. Визуализировать результат (тот самый график Predicted vs Actual)
final_preds |> 
  ggplot(aes(price, .pred)) +
  geom_point(alpha = 0.4, color = "midnightblue") +
  geom_abline(lty = 2, color = "gray") +
  theme_minimal() +
  labs(title = "Результат на контрольных данных")

Также посмотрим, какие предикторы больше всего влияют на цену. Проще всего это сделать через линейную аппроксимацию.

lm_fit <- workflow() |> 
  add_recipe(books_rec) |> 
  add_model(svm_spec) |> 
  fit(data = training(books_split))

lm_fit |> 
  tidy() |> 
  filter(term != "year") |> 
  filter(term != "Bias") |> 
  filter(!str_detect(term, "genre")) |> 
  mutate(sign = case_when(estimate > 0 ~ "дороже",
                          .default = "дешевле"),
         estimate = abs(estimate), 
         term = str_remove_all(term, "tfidf_name_")) |> 
  group_by(sign) |> 
  top_n(20, estimate) |> 
  ungroup() |> 
  ggplot(aes(x = estimate, y = fct_reorder(term, estimate),
             fill = sign)) +
  geom_col(show.legend = FALSE) +
  scale_x_continuous(expand = c(0,0)) +
  facet_wrap(~sign, scales = "free") +
  labs(y = NULL, 
       title = "Связь слов с ценой книг") +
  theme_minimal()

24.17 Видео

Видео 2025 г.
Видео 2026 г.

Г. Джеймс, Д. Уиттон, Т. Хасти, Р. Тибришани. 2017. Введение в статистическое обучение с примерами на языке R. ДМК.

24.1 Регрессионные алгоритмы

24.2 Библиотека tidymodels

24.3 Данные

24.4 Обучающая и контрольная выборка

24.5 Определение модели

24.6 Регрессия на опорных векторах

24.7 Линейная SVR

24.8 Дизайн переменных

24.9 Подгонка модели

24.10 Контрольные данные

24.11 Нелинейная SVR

24.12 Повторные выборки

24.13 Нулевая модель

24.14 Удаление стопслов

24.15 Настройки числа n-grams

24.16 Финализация и оценка

24.17 Видео

24.2 Библиотека `tidymodels`