Cubist Regresion Models
Cubist is an R port of the Cubist GPL C code released by RuleQuest at http://rulequest.com/cubist-info.html. See the last section of this document for information on the porting. The other parts describes the functionality of the R package.
Model Trees
Cubist is a rule-based model that is an extension of Quinlan’s M5 model tree. A tree is grown where the terminal leaves contain linear regression models. These models are based on the predictors used in previous splits. Also, there are intermediate linear models at each step of the tree. A prediction is made using the linear regression model at the terminal node of the tree, but is “smoothed” by taking into account the prediction from the linear model in the previous node of the tree (which also occurs recursively up the tree). The tree is reduced to a set of rules, which initially are paths from the top of the tree to the bottom. Rules are eliminated via pruning and/or combined for simplification.
This is explained better in Quinlan (1992). Wang and Witten (1997) attempted to recreate this model using a “rational reconstruction” of Quinlan (1992) that is the basis for the M5P model in Weka (and the R package RWeka).
An example of a model tree can be illustrated using the Ames housing data in the modeldata package.
library(Cubist)
data(ames, package = "modeldata")
# model the data on the log10 scale
ames$Sale_Price <- log10(ames$Sale_Price)
set.seed(11)
in_train_set <- sample(1:nrow(ames), floor(.8*nrow(ames)))
predictors <-
c("Lot_Area", "Alley", "Lot_Shape", "Neighborhood", "Bldg_Type",
"Year_Built", "Total_Bsmt_SF", "Central_Air", "Gr_Liv_Area",
"Bsmt_Full_Bath", "Bsmt_Half_Bath", "Full_Bath", "Half_Bath",
"TotRms_AbvGrd", "Year_Sold", "Longitude", "Latitude")
ames$Sale_Price <- log10(ames$Sale_Price)
train_pred <- ames[ in_train_set, predictors]
test_pred <- ames[-in_train_set, predictors]
train_resp <- ames$Sale_Price[ in_train_set]
test_resp <- ames$Sale_Price[-in_train_set]
model_tree <- cubist(x = train_pred, y = train_resp)
model_tree
##
## Call:
## cubist.default(x = train_pred, y = train_resp)
##
## Number of samples: 2344
## Number of predictors: 17
##
## Number of committees: 1
## Number of rules: 11
##
## Call:
## cubist.default(x = train_pred, y = train_resp)
##
##
## Cubist [Release 2.07 GPL Edition] Sat Feb 5 08:18:18 2022
## ---------------------------------
##
## Target attribute `outcome'
##
## Read 2344 cases (18 attributes) from undefined.data
##
## Model:
##
## Rule 1: [107 cases, mean 0.6922308, range 0.6135074 to 0.725238, est err 0.0086873]
##
## if
## Year_Built <= 1952
## Central_Air = N
## Gr_Liv_Area <= 1692
## then
## outcome = 2.8339713 + 2.89e-05 Gr_Liv_Area - 0.0011 Year_Sold
## + 2e-05 Year_Built + 1.1e-06 Total_Bsmt_SF
## + 0.0003 Bsmt_Full_Bath
##
## Rule 2: [333 cases, mean 0.7065404, range 0.6720027 to 0.7540954, est err 0.0054064]
##
## if
## Neighborhood in {Old_Town, Edwards, Gilbert, Sawyer, Sawyer_West,
## Brookside, Iowa_DOT_and_Rail_Road, Timberland,
## South_and_West_of_Iowa_State_University}
## Year_Built <= 1952
## Central_Air = Y
## then
## outcome = 0.4666239 + 1.87e-05 Gr_Liv_Area + 0.00011 Year_Built
## + 3.4e-06 Total_Bsmt_SF + 0.0009 Bsmt_Full_Bath
## + 7e-08 Lot_Area
##
## Rule 3: [176 cases, mean 0.7106560, range 0.6802335 to 0.7269588, est err 0.0030744]
##
## if
## Neighborhood in {College_Creek, Edwards, Somerset, Northridge_Heights,
## Gilbert, Sawyer_West, Mitchell, Iowa_DOT_and_Rail_Road,
## Timberland, Clear_Creek, Meadow_Village, Briardale,
## Blueste, Landmark}
## Year_Built > 1952
## Total_Bsmt_SF <= 765
## Gr_Liv_Area <= 1692
## then
## outcome = 0.01018 + 0.00034 Year_Built + 1.79e-05 Gr_Liv_Area
## + 2.2e-06 Total_Bsmt_SF + 0.0006 Bsmt_Full_Bath
##
## Rule 4: [87 cases, mean 0.7109181, range 0.6841727 to 0.7294574, est err 0.0049527]
##
## if
## Neighborhood in {North_Ames, Crawford}
## Year_Built <= 1952
## Gr_Liv_Area <= 1692
## then
## outcome = 0.6804143 + 1.45e-05 Gr_Liv_Area + 7e-06 Year_Built
## + 4e-07 Total_Bsmt_SF
##
## Rule 5: [35 cases, mean 0.7117702, range 0.6949773 to 0.7293048, est err 0.0047720]
##
## if
## Neighborhood in {North_Ames, Old_Town, Sawyer, Northwest_Ames, Crawford,
## Green_Hills}
## Year_Built > 1952
## Total_Bsmt_SF <= 765
## Gr_Liv_Area <= 1692
## then
## outcome = 0.2370069 + 0.000232 Year_Built + 1.16e-05 Gr_Liv_Area
## + 8.4e-06 Total_Bsmt_SF + 0.0013 Bsmt_Full_Bath
##
## Rule 6: [1428 cases, mean 0.7131603, range 0.6135074 to 0.7540954, est err 0.0043013]
##
## if
## Neighborhood in {North_Ames, College_Creek, Old_Town, Edwards, Gilbert,
## Sawyer, Northwest_Ames, Sawyer_West, Mitchell,
## Iowa_DOT_and_Rail_Road, Timberland,
## South_and_West_of_Iowa_State_University,
## Meadow_Village, Veenker}
## Bldg_Type in {OneFam, TwoFmCon, TwnhsE}
## Year_Built <= 2004
## then
## outcome = 0.3519752 + 1.26e-05 Gr_Liv_Area + 0.000171 Year_Built
## + 7.2e-06 Total_Bsmt_SF + 1.8e-07 Lot_Area
##
## Rule 7: [54 cases, mean 0.7134706, range 0.6808683 to 0.7362626, est err 0.0044510]
##
## if
## Bldg_Type in {Duplex, Twnhs}
## Gr_Liv_Area > 1692
## then
## outcome = 0.3791743 + 1.48e-05 Gr_Liv_Area + 0.000151 Year_Built
## + 4.7e-06 Total_Bsmt_SF + 1.3e-07 Lot_Area
## + 0.0004 Bsmt_Full_Bath
##
## Rule 8: [997 cases, mean 0.7173781, range 0.6792599 to 0.74771, est err 0.0034352]
##
## if
## Year_Built > 1952
## Total_Bsmt_SF > 765
## Gr_Liv_Area <= 1692
## then
## outcome = 0.3008866 + 1.41e-05 Gr_Liv_Area + 0.000195 Year_Built
## + 9e-06 Total_Bsmt_SF + 0.0026 Bsmt_Full_Bath + 5e-08 Lot_Area
##
## Rule 9: [166 cases, mean 0.7354736, range 0.711188 to 0.7687976, est err 0.0044621]
##
## if
## Neighborhood in {Somerset, Northridge_Heights, Brookside, Crawford,
## Northridge, Stone_Brook, Clear_Creek}
## Year_Built <= 2004
## Gr_Liv_Area > 1692
## then
## outcome = 0.451118 + 1.11e-05 Gr_Liv_Area + 0.000126 Year_Built
## + 7.5e-06 Total_Bsmt_SF + 4e-08 Lot_Area
##
## Rule 10: [128 cases, mean 0.7408611, range 0.7227189 to 0.7608459, est err 0.0035555]
##
## if
## Year_Built > 2004
## Total_Bsmt_SF <= 2024
## Gr_Liv_Area > 1692
## then
## outcome = 0.1972512 + 0.000248 Year_Built + 1.44e-05 Gr_Liv_Area
## + 9.1e-06 Total_Bsmt_SF + 1.7e-07 Lot_Area
## + 0.0018 Bsmt_Full_Bath
##
## Rule 11: [20 cases, mean 0.7468034, range 0.7163473 to 0.7624165, est err 0.0068615]
##
## if
## Year_Built > 2004
## Total_Bsmt_SF > 2024
## then
## outcome = 0.7741479 - 1e-05 Gr_Liv_Area
##
##
## Evaluation on training data (2344 cases):
##
## Average |error| 0.0045926
## Relative |error| 0.40
## Correlation coefficient 0.89
##
##
## Attribute usage:
## Conds Model
##
## 98% 99% Year_Built
## 63% Neighborhood
## 50% 100% Gr_Liv_Area
## 42% Bldg_Type
## 38% 99% Total_Bsmt_SF
## 12% Central_Air
## 88% Lot_Area
## 52% Bsmt_Full_Bath
## 3% Year_Sold
##
##
## Time: 0.0 secs
There is no formula method for cubist(); the predictors are specified as matrix or data frame, The outcome is a numeric vector.
There is a predict method for the model:
model_tree_pred <- predict(model_tree, test_pred)
## Test set RMSE
sqrt(mean((model_tree_pred - test_resp)^2))
## [1] 0.00635
## Test set R^2
cor(model_tree_pred, test_resp)^2
## [1] 0.813
Ensembles By Committees
The Cubist model can also use a boosting-like scheme called committees where iterative model trees are created in sequence. The first tree follows the procedure described in the last section. Subsequent trees are created using adjusted versions to the training set outcome: if the model over-predicted a value, the response is adjusted downward for the next model (and so on, see this blog post). Unlike traditional boosting, stage weights for each committee are not used to average the predictions from each model tree; the final prediction is a simple average of the predictions from each model tree.
The committee option can be used to control number of model trees:
set.seed(1)
com_model <- cubist(x = train_pred, y = train_resp, committees = 3)
summary(com_model)
##
## Call:
## cubist.default(x = train_pred, y = train_resp, committees = 3)
##
##
## Cubist [Release 2.07 GPL Edition] Sat Feb 5 08:18:18 2022
## ---------------------------------
##
## Target attribute `outcome'
##
## Read 2344 cases (18 attributes) from undefined.data
##
## Model 1:
##
## Rule 1/1: [107 cases, mean 0.6922308, range 0.6135074 to 0.725238, est err 0.0086873]
##
## if
## Year_Built <= 1952
## Central_Air = N
## Gr_Liv_Area <= 1692
## then
## outcome = 2.8339713 + 2.89e-05 Gr_Liv_Area - 0.0011 Year_Sold
## + 2e-05 Year_Built + 1.1e-06 Total_Bsmt_SF
## + 0.0003 Bsmt_Full_Bath
##
## Rule 1/2: [333 cases, mean 0.7065404, range 0.6720027 to 0.7540954, est err 0.0054064]
##
## if
## Neighborhood in {Old_Town, Edwards, Gilbert, Sawyer, Sawyer_West,
## Brookside, Iowa_DOT_and_Rail_Road, Timberland,
## South_and_West_of_Iowa_State_University}
## Year_Built <= 1952
## Central_Air = Y
## then
## outcome = 0.4666239 + 1.87e-05 Gr_Liv_Area + 0.00011 Year_Built
## + 3.4e-06 Total_Bsmt_SF + 0.0009 Bsmt_Full_Bath
## + 7e-08 Lot_Area
##
## Rule 1/3: [176 cases, mean 0.7106560, range 0.6802335 to 0.7269588, est err 0.0030744]
##
## if
## Neighborhood in {College_Creek, Edwards, Somerset, Northridge_Heights,
## Gilbert, Sawyer_West, Mitchell, Iowa_DOT_and_Rail_Road,
## Timberland, Clear_Creek, Meadow_Village, Briardale,
## Blueste, Landmark}
## Year_Built > 1952
## Total_Bsmt_SF <= 765
## Gr_Liv_Area <= 1692
## then
## outcome = 0.01018 + 0.00034 Year_Built + 1.79e-05 Gr_Liv_Area
## + 2.2e-06 Total_Bsmt_SF + 0.0006 Bsmt_Full_Bath
##
## Rule 1/4: [87 cases, mean 0.7109181, range 0.6841727 to 0.7294574, est err 0.0049527]
##
## if
## Neighborhood in {North_Ames, Crawford}
## Year_Built <= 1952
## Gr_Liv_Area <= 1692
## then
## outcome = 0.6804143 + 1.45e-05 Gr_Liv_Area + 7e-06 Year_Built
## + 4e-07 Total_Bsmt_SF
##
## Rule 1/5: [35 cases, mean 0.7117702, range 0.6949773 to 0.7293048, est err 0.0047720]
##
## if
## Neighborhood in {North_Ames, Old_Town, Sawyer, Northwest_Ames, Crawford,
## Green_Hills}
## Year_Built > 1952
## Total_Bsmt_SF <= 765
## Gr_Liv_Area <= 1692
## then
## outcome = 0.2370069 + 0.000232 Year_Built + 1.16e-05 Gr_Liv_Area
## + 8.4e-06 Total_Bsmt_SF + 0.0013 Bsmt_Full_Bath
##
## Rule 1/6: [1428 cases, mean 0.7131603, range 0.6135074 to 0.7540954, est err 0.0043013]
##
## if
## Neighborhood in {North_Ames, College_Creek, Old_Town, Edwards, Gilbert,
## Sawyer, Northwest_Ames, Sawyer_West, Mitchell,
## Iowa_DOT_and_Rail_Road, Timberland,
## South_and_West_of_Iowa_State_University,
## Meadow_Village, Veenker}
## Bldg_Type in {OneFam, TwoFmCon, TwnhsE}
## Year_Built <= 2004
## then
## outcome = 0.3519752 + 1.26e-05 Gr_Liv_Area + 0.000171 Year_Built
## + 7.2e-06 Total_Bsmt_SF + 1.8e-07 Lot_Area
##
## Rule 1/7: [54 cases, mean 0.7134706, range 0.6808683 to 0.7362626, est err 0.0044510]
##
## if
## Bldg_Type in {Duplex, Twnhs}
## Gr_Liv_Area > 1692
## then
## outcome = 0.3791743 + 1.48e-05 Gr_Liv_Area + 0.000151 Year_Built
## + 4.7e-06 Total_Bsmt_SF + 1.3e-07 Lot_Area
## + 0.0004 Bsmt_Full_Bath
##
## Rule 1/8: [997 cases, mean 0.7173781, range 0.6792599 to 0.74771, est err 0.0034352]
##
## if
## Year_Built > 1952
## Total_Bsmt_SF > 765
## Gr_Liv_Area <= 1692
## then
## outcome = 0.3008866 + 1.41e-05 Gr_Liv_Area + 0.000195 Year_Built
## + 9e-06 Total_Bsmt_SF + 0.0026 Bsmt_Full_Bath + 5e-08 Lot_Area
##
## Rule 1/9: [166 cases, mean 0.7354736, range 0.711188 to 0.7687976, est err 0.0044621]
##
## if
## Neighborhood in {Somerset, Northridge_Heights, Brookside, Crawford,
## Northridge, Stone_Brook, Clear_Creek}
## Year_Built <= 2004
## Gr_Liv_Area > 1692
## then
## outcome = 0.451118 + 1.11e-05 Gr_Liv_Area + 0.000126 Year_Built
## + 7.5e-06 Total_Bsmt_SF + 4e-08 Lot_Area
##
## Rule 1/10: [128 cases, mean 0.7408611, range 0.7227189 to 0.7608459, est err 0.0035555]
##
## if
## Year_Built > 2004
## Total_Bsmt_SF <= 2024
## Gr_Liv_Area > 1692
## then
## outcome = 0.1972512 + 0.000248 Year_Built + 1.44e-05 Gr_Liv_Area
## + 9.1e-06 Total_Bsmt_SF + 1.7e-07 Lot_Area
## + 0.0018 Bsmt_Full_Bath
##
## Rule 1/11: [20 cases, mean 0.7468034, range 0.7163473 to 0.7624165, est err 0.0068615]
##
## if
## Year_Built > 2004
## Total_Bsmt_SF > 2024
## then
## outcome = 0.7741479 - 1e-05 Gr_Liv_Area
##
## Model 2:
##
## Rule 2/1: [31 cases, mean 0.6767582, range 0.6135074 to 0.7025505, est err 0.0169966]
##
## if
## Central_Air = N
## Gr_Liv_Area <= 832
## then
## outcome = -60.1255459 - 0.649 Longitude + 3.1e-06 Gr_Liv_Area
## + 1.6e-06 Total_Bsmt_SF + 1.7e-05 Year_Built + 5e-08 Lot_Area
##
## Rule 2/2: [91 cases, mean 0.6914554, range 0.6135074 to 0.7112178, est err 0.0085649]
##
## if
## Gr_Liv_Area <= 832
## then
## outcome = -7.0404897 + 3.32e-05 Total_Bsmt_SF + 2.71e-05 Gr_Liv_Area
## + 1.81e-06 Lot_Area + 0.000148 Year_Built - 0.079 Longitude
##
## Rule 2/3: [126 cases, mean 0.7014069, range 0.6629523 to 0.7343875, est err 0.0089801]
##
## if
## Central_Air = N
## Gr_Liv_Area > 832
## then
## outcome = -18.0819225 + 0.585 Latitude + 1.28e-05 Gr_Liv_Area
## - 0.0029 Year_Sold - 0.00035 TotRms_AbvGrd
##
## Rule 2/4: [389 cases, mean 0.7057415, range 0.6629523 to 0.7540954, est err 0.0063773]
##
## if
## Neighborhood in {Old_Town, Edwards, Gilbert, Sawyer_West, Mitchell,
## Iowa_DOT_and_Rail_Road,
## South_and_West_of_Iowa_State_University}
## Year_Built <= 1966
## Gr_Liv_Area > 832
## then
## outcome = 1.8797961 + 1.97e-05 Gr_Liv_Area - 0.00211 TotRms_AbvGrd
## - 0.0006 Year_Sold + 9e-06 Year_Built + 4e-07 Total_Bsmt_SF
##
## Rule 2/5: [615 cases, mean 0.7130445, range 0.6807946 to 0.7433388, est err 0.0040729]
##
## if
## Neighborhood in {North_Ames, Sawyer, Northwest_Ames, Brookside,
## Crawford, Timberland, Clear_Creek}
## Gr_Liv_Area > 832
## Latitude > 42.03093
## then
## outcome = 0.4372397 + 1.06e-05 Gr_Liv_Area + 0.000134 Year_Built
## - 0.00198 TotRms_AbvGrd + 5.1e-07 Lot_Area
## + 6e-06 Total_Bsmt_SF
##
## Rule 2/6: [339 cases, mean 0.7155511, range 0.6858544 to 0.7433388, est err 0.0042090]
##
## if
## Neighborhood in {North_Ames, Old_Town, Edwards, Sawyer_West, Mitchell,
## Iowa_DOT_and_Rail_Road, Meadow_Village, Briardale,
## Bloomington_Heights, Landmark}
## Year_Built > 1966
## Total_Bsmt_SF <= 2220
## Gr_Liv_Area > 832
## then
## outcome = -2.3923159 + 0.000258 Year_Built + 1.29e-05 Gr_Liv_Area
## + 6.6e-07 Lot_Area + 0.006 Bsmt_Full_Bath
## + 6e-06 Total_Bsmt_SF + 0.061 Latitude - 0.00026 TotRms_AbvGrd
##
## Rule 2/7: [63 cases, mean 0.7158888, range 0.7042104 to 0.7362626, est err 0.0022518]
##
## if
## Lot_Area <= 3612
## Neighborhood in {College_Creek, Somerset, Northridge_Heights, Gilbert,
## Sawyer, Northwest_Ames, Crawford, Timberland,
## Northridge, Stone_Brook,
## South_and_West_of_Iowa_State_University, Clear_Creek,
## Veenker, Northpark_Villa, Blueste, Greens, Green_Hills}
## then
## outcome = 0.2224039 + 0.000239 Year_Built + 9.8e-06 Gr_Liv_Area
## + 2.3e-06 Total_Bsmt_SF + 0.0013 Bsmt_Full_Bath
## + 6e-08 Lot_Area
##
## Rule 2/8: [21 cases, mean 0.7226428, range 0.7021946 to 0.747059, est err 0.0093020]
##
## if
## Year_Built > 1966
## Total_Bsmt_SF <= 2220
## Bsmt_Full_Bath > 1
## then
## outcome = 0.6899718 + 1.92e-05 Gr_Liv_Area
##
## Rule 2/9: [1326 cases, mean 0.7238501, range 0.6858544 to 0.7631194, est err 0.0037527]
##
## if
## Year_Built > 1966
## Total_Bsmt_SF <= 2220
## Bsmt_Full_Bath <= 1
## then
## outcome = -0.0413273 + 0.000249 Year_Built + 1.34e-05 Gr_Liv_Area
## + 7.8e-06 Total_Bsmt_SF + 0.0041 Bsmt_Full_Bath
## + 1.1e-07 Lot_Area - 0.00029 TotRms_AbvGrd + 0.019 Latitude
## + 0.006 Longitude
##
## Rule 2/10: [199 cases, mean 0.7242572, range 0.6844585 to 0.7503841, est err 0.0053956]
##
## if
## Neighborhood in {North_Ames, Sawyer, Northwest_Ames, Brookside,
## Crawford, Timberland, Clear_Creek}
## Latitude <= 42.03093
## then
## outcome = 2.9085627 + 1.59e-05 Gr_Liv_Area + 2.7e-06 Total_Bsmt_SF
## - 0.00064 TotRms_AbvGrd - 0.049 Latitude + 2.5e-05 Year_Built
## + 1.3e-07 Lot_Area - 0.0001 Year_Sold
##
## Rule 2/11: [494 cases, mean 0.7320032, range 0.691652 to 0.7625942, est err 0.0039935]
##
## if
## Lot_Area > 3612
## Neighborhood in {Somerset, Northridge_Heights, Northwest_Ames, Crawford,
## Timberland, Stone_Brook,
## South_and_West_of_Iowa_State_University, Clear_Creek,
## Veenker, Blueste, Greens, Green_Hills}
## Year_Built > 1966
## then
## outcome = -0.0693572 + 0.000386 Year_Built + 9.8e-06 Gr_Liv_Area
## + 0.0062 Bsmt_Full_Bath + 7.1e-06 Total_Bsmt_SF
## + 1.5e-07 Lot_Area
##
## Rule 2/12: [20 cases, mean 0.7484384, range 0.7163473 to 0.7687976, est err 0.0111278]
##
## if
## Total_Bsmt_SF > 2220
## then
## outcome = -1.0179687 + 0.000906 Year_Built - 1.49e-05 Total_Bsmt_SF
##
## Model 3:
##
## Rule 3/1: [104 cases, mean 0.6924329, range 0.6135074 to 0.7139776, est err 0.0101724]
##
## if
## Gr_Liv_Area <= 845
## then
## outcome = 0.6422932 + 7.43e-05 Gr_Liv_Area
##
## Rule 3/2: [92 cases, mean 0.6954694, range 0.6146095 to 0.724802, est err 0.0076345]
##
## if
## Neighborhood in {Iowa_DOT_and_Rail_Road, Meadow_Village, Briardale}
## Total_Bsmt_SF <= 800
## then
## outcome = -21.4519142 + 1.8e-05 Gr_Liv_Area + 2.01e-05 Total_Bsmt_SF
## - 0.215 Longitude + 7.1e-05 Year_Built + 0.044 Latitude
##
## Rule 3/3: [431 cases, mean 0.7078069, range 0.6135074 to 0.7294574, est err 0.0054390]
##
## if
## Neighborhood in {North_Ames, College_Creek, Old_Town, Edwards, Somerset,
## Gilbert, Sawyer, Northwest_Ames, Sawyer_West, Mitchell,
## Brookside, Crawford, Timberland, Northridge,
## South_and_West_of_Iowa_State_University, Clear_Creek,
## Veenker, Blueste, Green_Hills}
## Year_Built <= 2005
## Total_Bsmt_SF <= 800
## Gr_Liv_Area <= 1832
## then
## outcome = 0.5911446 + 2.12e-05 Gr_Liv_Area + 1.12e-05 Total_Bsmt_SF
## + 4.2e-05 Year_Built + 0.00027 TotRms_AbvGrd
##
## Rule 3/4: [170 cases, mean 0.7106175, range 0.6802335 to 0.7362626, est err 0.0053771]
##
## if
## Bldg_Type in {Duplex, Twnhs}
## Gr_Liv_Area > 845
## then
## outcome = 0.2154946 + 0.000239 Year_Built + 9.3e-06 Gr_Liv_Area
## + 9.7e-06 Total_Bsmt_SF
##
## Rule 3/5: [1411 cases, mean 0.7132381, range 0.6135074 to 0.7540954, est err 0.0043492]
##
## if
## Neighborhood in {North_Ames, College_Creek, Old_Town, Edwards, Gilbert,
## Sawyer, Northwest_Ames, Sawyer_West,
## Iowa_DOT_and_Rail_Road, Timberland,
## South_and_West_of_Iowa_State_University,
## Meadow_Village, Bloomington_Heights, Northpark_Villa}
## Bldg_Type in {OneFam, TwoFmCon, TwnhsE}
## Year_Built <= 2005
## then
## outcome = 2.7340427 + 1.35e-05 Gr_Liv_Area + 0.000196 Year_Built
## + 9e-06 Total_Bsmt_SF + 0.0014 Half_Bath + 1.2e-07 Lot_Area
## + 0.026 Longitude
##
## Rule 3/6: [52 cases, mean 0.7157766, range 0.6927507 to 0.7362626, est err 0.0056324]
##
## if
## Bldg_Type in {Duplex, Twnhs}
## Total_Bsmt_SF > 1221
## then
## outcome = 0.242855 - 2.58e-05 Total_Bsmt_SF + 0.000254 Year_Built
## + 5.3e-06 Gr_Liv_Area
##
## Rule 3/7: [30 cases, mean 0.7163877, range 0.6906617 to 0.7467062, est err 0.0104669]
##
## if
## Year_Built <= 1960
## Total_Bsmt_SF <= 800
## Gr_Liv_Area > 1832
## then
## outcome = -1.2544615 + 0.000342 Year_Built + 3e-06 Gr_Liv_Area
## + 2.5e-06 Total_Bsmt_SF - 0.009 Longitude + 0.011 Latitude
##
## Rule 3/8: [991 cases, mean 0.7212871, range 0.6807946 to 0.7687976, est err 0.0040766]
##
## if
## Neighborhood in {North_Ames, College_Creek, Old_Town, Edwards, Somerset,
## Gilbert, Sawyer, Northwest_Ames, Sawyer_West, Mitchell,
## Brookside, Crawford, Timberland, Northridge,
## South_and_West_of_Iowa_State_University, Clear_Creek,
## Veenker, Blueste, Green_Hills}
## Bldg_Type in {OneFam, TwoFmCon, TwnhsE}
## Year_Built > 1960
## Year_Built <= 2005
## then
## outcome = -0.4926223 + 1.38e-05 Gr_Liv_Area + 1.3e-06 Total_Bsmt_SF
## + 1.1e-05 Year_Built - 0.008 Longitude + 0.01 Latitude
##
## Rule 3/9: [682 cases, mean 0.7277318, range 0.6879801 to 0.7687976, est err 0.0049696]
##
## if
## Neighborhood in {Somerset, Northridge_Heights, Mitchell, Brookside,
## Crawford, Northridge, Stone_Brook, Clear_Creek,
## Veenker, Blueste, Greens, Green_Hills}
## Gr_Liv_Area > 845
## then
## outcome = 0.4310254 + 1.22e-05 Gr_Liv_Area + 1.11e-05 Total_Bsmt_SF
## + 0.000132 Year_Built + 0.0041 Half_Bath + 5e-08 Lot_Area
## - 0.00015 TotRms_AbvGrd
##
## Rule 3/10: [235 cases, mean 0.7322826, range 0.6925699 to 0.7608459, est err 0.0041824]
##
## if
## Year_Built > 2005
## Total_Bsmt_SF <= 2006
## then
## outcome = -2.1757049 + 0.001422 Year_Built + 2.14e-05 Gr_Liv_Area
## + 1.46e-05 Total_Bsmt_SF
##
## Rule 3/11: [20 cases, mean 0.7467262, range 0.7163473 to 0.7624165, est err 0.0108212]
##
## if
## Year_Built > 2005
## Total_Bsmt_SF > 2006
## then
## outcome = 0.7768462 - 3e-05 Gr_Liv_Area + 1.88e-05 Total_Bsmt_SF
##
##
## Evaluation on training data (2344 cases):
##
## Average |error| 0.0043261
## Relative |error| 0.38
## Correlation coefficient 0.90
##
##
## Attribute usage:
## Conds Model
##
## 80% 97% Year_Built
## 69% Neighborhood
## 42% 100% Gr_Liv_Area
## 36% Bldg_Type
## 34% 98% Total_Bsmt_SF
## 12% 35% Bsmt_Full_Bath
## 7% 27% Latitude
## 5% Central_Air
## 5% 73% Lot_Area
## 36% TotRms_AbvGrd
## 35% Longitude
## 18% Half_Bath
## 7% Year_Sold
##
##
## Time: 0.1 secs
For this model:
com_pred <- predict(com_model, test_pred)
## RMSE
sqrt(mean((com_pred - test_resp)^2))
## [1] 0.00592
## R^2
cor(com_pred, test_resp)^2
## [1] 0.837
Instance-Based Corrections
Another innovation in Cubist using nearest-neighbors to adjust the predictions from the rule-based model. First, a model tree (with or without committees) is created. Once a sample is predicted by this model, Cubist can find it’s nearest neighbors and determine the average of these training set points. See Quinlan (1993a) for the details of the adjustment as well as this blog post.
The development of rules and committees is independent of the choice of using instances. The original C code allowed the program to choose whether to use instances, not use them or let the program decide. Our approach is to build a model with the cubist() function that is ignorant to the decision about instances. When samples are predicted, the argument neighbors can be used to adjust the rule-based model predictions (or not).
We can add instances to the previously fit committee model:
inst_pred <- predict(com_model, test_pred, neighbors = 5)
## RMSE
sqrt(mean((inst_pred - test_resp)^2))
## [1] 0.00565
## R^2
cor(inst_pred, test_resp)^2
## [1] 0.851
Note that the previous models used the implicit default of neighbors = 0 for their predictions.
It may also be useful to see how the different models fit a single predictor. Here is the test set data for a model with one predictor (Gr_Liv_Area), 100 committees, and various values of neighbors:
After the initial use of the instance-based correction, there is very little change in the mainstream of the data.
Model tuning
R modeling packages such as caret, tidymodels, and mlr3 can be used to tune the model. See the examples here for more details.
It should be noted that this variable importance measure does not capture the influence of the predictors when using the instance-based correction.
Tidy rules
Rules from a Cubist model can be viewed using summary as follows:
##
## Call:
## cubist.default(x = train_pred, y = train_resp)
##
##
## Cubist [Release 2.07 GPL Edition] Sat Feb 5 08:18:18 2022
## ---------------------------------
##
## Target attribute `outcome'
##
## Read 2344 cases (18 attributes) from undefined.data
##
## Model:
##
## Rule 1: [107 cases, mean 0.6922308, range 0.6135074 to 0.725238, est err 0.0086873]
##
## if
## Year_Built <= 1952
## Central_Air = N
## Gr_Liv_Area <= 1692
## then
## outcome = 2.8339713 + 2.89e-05 Gr_Liv_Area - 0.0011 Year_Sold
## + 2e-05 Year_Built + 1.1e-06 Total_Bsmt_SF
## + 0.0003 Bsmt_Full_Bath
##
## Rule 2: [333 cases, mean 0.7065404, range 0.6720027 to 0.7540954, est err 0.0054064]
##
## if
## Neighborhood in {Old_Town, Edwards, Gilbert, Sawyer, Sawyer_West,
## Brookside, Iowa_DOT_and_Rail_Road, Timberland,
## South_and_West_of_Iowa_State_University}
## Year_Built <= 1952
## Central_Air = Y
## then
## outcome = 0.4666239 + 1.87e-05 Gr_Liv_Area + 0.00011 Year_Built
## + 3.4e-06 Total_Bsmt_SF + 0.0009 Bsmt_Full_Bath
## + 7e-08 Lot_Area
##
## Rule 3: [176 cases, mean 0.7106560, range 0.6802335 to 0.7269588, est err 0.0030744]
##
## if
## Neighborhood in {College_Creek, Edwards, Somerset, Northridge_Heights,
## Gilbert, Sawyer_West, Mitchell, Iowa_DOT_and_Rail_Road,
## Timberland, Clear_Creek, Meadow_Village, Briardale,
## Blueste, Landmark}
## Year_Built > 1952
## Total_Bsmt_SF <= 765
## Gr_Liv_Area <= 1692
## then
## outcome = 0.01018 + 0.00034 Year_Built + 1.79e-05 Gr_Liv_Area
## + 2.2e-06 Total_Bsmt_SF + 0.0006 Bsmt_Full_Bath
##
## Rule 4: [87 cases, mean 0.7109181, range 0.6841727 to 0.7294574, est err 0.0049527]
##
## if
## Neighborhood in {North_Ames, Crawford}
## Year_Built <= 1952
## Gr_Liv_Area <= 1692
## then
## outcome = 0.6804143 + 1.45e-05 Gr_Liv_Area + 7e-06 Year_Built
## + 4e-07 Total_Bsmt_SF
##
## Rule 5: [35 cases, mean 0.7117702, range 0.6949773 to 0.7293048, est err 0.0047720]
##
## if
## Neighborhood in {North_Ames, Old_Town, Sawyer, Northwest_Ames, Crawford,
## Green_Hills}
## Year_Built > 1952
## Total_Bsmt_SF <= 765
## Gr_Liv_Area <= 1692
## then
## outcome = 0.2370069 + 0.000232 Year_Built + 1.16e-05 Gr_Liv_Area
## + 8.4e-06 Total_Bsmt_SF + 0.0013 Bsmt_Full_Bath
##
## Rule 6: [1428 cases, mean 0.7131603, range 0.6135074 to 0.7540954, est err 0.0043013]
##
## if
## Neighborhood in {North_Ames, College_Creek, Old_Town, Edwards, Gilbert,
## Sawyer, Northwest_Ames, Sawyer_West, Mitchell,
## Iowa_DOT_and_Rail_Road, Timberland,
## South_and_West_of_Iowa_State_University,
## Meadow_Village, Veenker}
## Bldg_Type in {OneFam, TwoFmCon, TwnhsE}
## Year_Built <= 2004
## then
## outcome = 0.3519752 + 1.26e-05 Gr_Liv_Area + 0.000171 Year_Built
## + 7.2e-06 Total_Bsmt_SF + 1.8e-07 Lot_Area
##
## Rule 7: [54 cases, mean 0.7134706, range 0.6808683 to 0.7362626, est err 0.0044510]
##
## if
## Bldg_Type in {Duplex, Twnhs}
## Gr_Liv_Area > 1692
## then
## outcome = 0.3791743 + 1.48e-05 Gr_Liv_Area + 0.000151 Year_Built
## + 4.7e-06 Total_Bsmt_SF + 1.3e-07 Lot_Area
## + 0.0004 Bsmt_Full_Bath
##
## Rule 8: [997 cases, mean 0.7173781, range 0.6792599 to 0.74771, est err 0.0034352]
##
## if
## Year_Built > 1952
## Total_Bsmt_SF > 765
## Gr_Liv_Area <= 1692
## then
## outcome = 0.3008866 + 1.41e-05 Gr_Liv_Area + 0.000195 Year_Built
## + 9e-06 Total_Bsmt_SF + 0.0026 Bsmt_Full_Bath + 5e-08 Lot_Area
##
## Rule 9: [166 cases, mean 0.7354736, range 0.711188 to 0.7687976, est err 0.0044621]
##
## if
## Neighborhood in {Somerset, Northridge_Heights, Brookside, Crawford,
## Northridge, Stone_Brook, Clear_Creek}
## Year_Built <= 2004
## Gr_Liv_Area > 1692
## then
## outcome = 0.451118 + 1.11e-05 Gr_Liv_Area + 0.000126 Year_Built
## + 7.5e-06 Total_Bsmt_SF + 4e-08 Lot_Area
##
## Rule 10: [128 cases, mean 0.7408611, range 0.7227189 to 0.7608459, est err 0.0035555]
##
## if
## Year_Built > 2004
## Total_Bsmt_SF <= 2024
## Gr_Liv_Area > 1692
## then
## outcome = 0.1972512 + 0.000248 Year_Built + 1.44e-05 Gr_Liv_Area
## + 9.1e-06 Total_Bsmt_SF + 1.7e-07 Lot_Area
## + 0.0018 Bsmt_Full_Bath
##
## Rule 11: [20 cases, mean 0.7468034, range 0.7163473 to 0.7624165, est err 0.0068615]
##
## if
## Year_Built > 2004
## Total_Bsmt_SF > 2024
## then
## outcome = 0.7741479 - 1e-05 Gr_Liv_Area
##
##
## Evaluation on training data (2344 cases):
##
## Average |error| 0.0045926
## Relative |error| 0.40
## Correlation coefficient 0.89
##
##
## Attribute usage:
## Conds Model
##
## 98% 99% Year_Built
## 63% Neighborhood
## 50% 100% Gr_Liv_Area
## 42% Bldg_Type
## 38% 99% Total_Bsmt_SF
## 12% Central_Air
## 88% Lot_Area
## 52% Bsmt_Full_Bath
## 3% Year_Sold
##
##
## Time: 0.0 secs
The tidyRules function in the tidyrules package returns rules in a tibble (an extension of data frames) with one row per rule. The tibble provides these information about the rule: support, mean, min, max, error, LHS, RHS and committee. The values in LHS and RHS columns are strings which can be parsed as R expressions. These can be pasted inside the parenthesis of dplyr::filter() to obtain the rows of the data corresponding to the rule and evaluate the response variable.
library(tidyrules)
tr <- tidyRules(model_tree)
tr
## # A tibble: 11 × 9
## id LHS RHS support mean min max error committee
## <int> <chr> <chr> <int> <dbl> <dbl> <dbl> <dbl> <int>
## 1 1 Year_Built <= 1952 &… (2.8… 107 0.692 0.614 0.725 0.00869 1
## 2 2 Neighborhood %in% c(… (0.4… 333 0.707 0.672 0.754 0.00541 1
## 3 3 Neighborhood %in% c(… (0.0… 176 0.711 0.680 0.727 0.00307 1
## 4 4 Neighborhood %in% c(… (0.6… 87 0.711 0.684 0.729 0.00495 1
## 5 5 Neighborhood %in% c(… (0.2… 35 0.712 0.695 0.729 0.00477 1
## 6 6 Neighborhood %in% c(… (0.3… 1428 0.713 0.614 0.754 0.00430 1
## 7 7 Bldg_Type %in% c('Du… (0.3… 54 0.713 0.681 0.736 0.00445 1
## 8 8 Year_Built > 1952 & … (0.3… 997 0.717 0.679 0.748 0.00344 1
## 9 9 Neighborhood %in% c(… (0.4… 166 0.735 0.711 0.769 0.00446 1
## 10 10 Year_Built > 2004 & … (0.1… 128 0.741 0.723 0.761 0.00356 1
## 11 11 Year_Built > 2004 & … (0.7… 20 0.747 0.716 0.762 0.00686 1
## # A tibble: 11 × 2
## LHS RHS
## <chr> <chr>
## 1 Year_Built <= 1952 & Central_Air == 'N' & Gr_Liv_Area <= 1692 (2.8…
## 2 Neighborhood %in% c('Old_Town', 'Edwards', 'Gilbert', 'Sawyer', 'Sawye… (0.4…
## 3 Neighborhood %in% c('College_Creek', 'Edwards', 'Somerset', 'Northridg… (0.0…
## 4 Neighborhood %in% c('North_Ames', 'Crawford') & Year_Built <= 1952 & G… (0.6…
## 5 Neighborhood %in% c('North_Ames', 'Old_Town', 'Sawyer', 'Northwest_Ame… (0.2…
## 6 Neighborhood %in% c('North_Ames', 'College_Creek', 'Old_Town', 'Edward… (0.3…
## 7 Bldg_Type %in% c('Duplex', 'Twnhs') & Gr_Liv_Area > 1692 (0.3…
## 8 Year_Built > 1952 & Total_Bsmt_SF > 765 & Gr_Liv_Area <= 1692 (0.3…
## 9 Neighborhood %in% c('Somerset', 'Northridge_Heights', 'Brookside', 'Cr… (0.4…
## 10 Year_Built > 2004 & Total_Bsmt_SF <= 2024 & Gr_Liv_Area > 1692 (0.1…
## 11 Year_Built > 2004 & Total_Bsmt_SF > 2024 (0.7…
# lets look at 7th rule
tr$LHS[7]
## [1] "Bldg_Type %in% c('Duplex', 'Twnhs') & Gr_Liv_Area > 1692"
## [1] "(0.3791743) + (1.48e-05 * Gr_Liv_Area) + (0.000151 * Year_Built) + (4.7e-06 * Total_Bsmt_SF) + (1.3e-07 * Lot_Area) + (0.0004 * Bsmt_Full_Bath)"
These results can be used to look at specific parts of the data. For example, the 7th rule predictions are:
library(dplyr)
library(rlang)
char_to_expr <- function(x, index = 1, model = TRUE) {
x <- x %>% dplyr::slice(index)
if (model) {
x <- x %>% dplyr::pull(RHS) %>% rlang::parse_expr()
} else {
x <- x %>% dplyr::pull(LHS) %>% rlang::parse_expr()
}
x
}
rule_expr <- char_to_expr(tr, 7, model = FALSE)
model_expr <- char_to_expr(tr, 7, model = TRUE)
# filter the data corresponding to rule 7
ames %>%
dplyr::filter(eval_tidy(rule_expr, ames)) %>%
dplyr::mutate(sale_price_pred = eval_tidy(model_expr, .)) %>%
dplyr::select(Sale_Price, sale_price_pred)
## # A tibble: 59 × 2
## Sale_Price sale_price_pred
## <dbl> <dbl>
## 1 0.703 0.707
## 2 0.693 0.712
## 3 0.715 0.715
## 4 0.710 0.714
## 5 0.711 0.712
## 6 0.716 0.716
## 7 0.727 0.713
## 8 0.719 0.719
## 9 0.711 0.709
## 10 0.716 0.722
## # … with 49 more rows
Variable Importance
The summary() method for Cubist shows the usage of each variable in either the rule conditions or the (terminal) linear model. In actuality, many more linear models are used in prediction that are shown in the output. Because of this, the variable usage statistics shown at the end of the output of the summary() function will probably be inconsistent with the rules also shown in the output. At each split of the tree, Cubist saves a linear model (after feature selection) that is allowed to have terms for each variable used in the current split or any split above it. Quinlan (1992) discusses a smoothing algorithm where each model prediction is a linear combination of the parent and child model along the tree. As such, the final prediction is a function of all the linear models from the initial node to the terminal node. The percentages shown in the Cubist output reflects all the models involved in prediction (as opposed to the terminal models shown in the output).
The raw usage statistics are contained in a data frame called usage in the cubist object.
The caret and vip packages have general variable importance functions caret::varImp() and vip::vi(). When using this function on a cubist argument, the variable importance is a linear combination of the usage in the rule conditions and the model.
For example, to compute the scores:
caret::varImp(model_tree)
# or
vip::vi(model_tree)
Current Limitations
There are a few features in the C code that are not yet operational in the R package:
- only continuous and categorical predictors can be used (the original source code allows for other data types)
- there is an option to let the
C code decide on using instances or not. The choice is more explicit in this package
- non-standard predictor names are not currently checked/fixed
- the
C code supports binning of predictors