Source: https://christophm.github.io/interpretable-ml-book

Initialize Notebook

Load required packages.

library(tidyverse)
library(ggthemr)
library(caret)
library(ranger)
library(Metrics)
library(iml)

Set up workspace, i.e., remove all existing data from working memory, initialize the random number generator, turn of scientific notation of large numbers, set a standard theme for plotting. Finally, we activate parallel processing on multiple CPU cores.

rm(list=ls())
set.seed(42)
knitr::opts_chunk$set(echo = TRUE, warning = FALSE)
options(scipen=10000)
ggthemr('fresh')
doParallel::registerDoParallel(cores = 4)

Problem Description

Ask a home buyer to describe their dream house, and they probably won’t begin with the height of the basement ceiling or the proximity to an east-west railroad. But this dataset proves that much more influences price negotiations than the number of bedrooms or a white-picket fence. With 76 explanatory variables describing (almost) every aspect of residential homes in Ames, Iowa, this dataset challenges you to predict the final price of each home. More: https://www.kaggle.com/c/house-prices-advanced-regression-techniques

Prepare Data

Load data from CSV file.

data <- read_csv("ameshousing.csv")
Rows: 2930 Columns: 77
── Column specification ───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
Delimiter: ","
chr (42): MSSubClass, MSZoning, Street, Alley, LotShape, LandContour, Utilities, LotConfig, LandSlope, Neighborhood, Condition1, Condition2, Bl...
dbl (35): LotFrontage, LotArea, OverallQual, OverallCond, YearBuilt, YearRemodAdd, MasVnrArea, BsmtFinSF1, BsmtFinSF2, BsmtUnfSF, TotalBsmtSF, ...

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

Fix some data type issues, i.e., turn character variables into factors and remove some irrelevant predictors.

data <- data %>% 
  mutate_if(is.character,as.factor) %>% 
  select(-MSSubClass, -MSZoning, -Exterior1st, -Exterior2nd, -SaleType, -SaleCondition, -MiscVal, -MoSold)

Create train/test split.

training_rows <- createDataPartition(y=data$SalePrice, p = 0.8, list = FALSE)
data_training <- slice(data, training_rows)
data_test <- slice(data, -training_rows)

Modelling

Linear Regression

Fit a linear model on all features.

lm_fit <- lm(SalePrice ~ ., data = data_training)

Display regression coefficients.

summary(lm_fit)

Call:
lm(formula = SalePrice ~ ., data = data_training)

Residuals:
    Min      1Q  Median      3Q     Max 
-406788   -9957      82    9436  179529 

Coefficients: (7 not defined because of singularities)
                        Estimate   Std. Error t value             Pr(>|t|)    
(Intercept)          131577.1307  823477.1409   0.160             0.873068    
LotFrontage              21.1493      19.2855   1.097             0.272921    
LotArea                   0.6323       0.1044   6.059   0.0000000016150362 ***
StreetPave            17669.5855    9961.6567   1.774             0.076245 .  
Alleynone               664.3902    3231.7793   0.206             0.837138    
AlleyPave             -1009.9021    4752.0618  -0.213             0.831723    
LotShapeIR2            6831.6885    3676.5826   1.858             0.063283 .  
LotShapeIR3            6309.4710    7700.1506   0.819             0.412652    
LotShapeReg            2303.3460    1360.6998   1.693             0.090645 .  
LandContourHLS        18860.8978    4241.3481   4.447   0.0000091568778017 ***
LandContourLow         -626.4191    5340.9399  -0.117             0.906644    
LandContourLvl        11579.7587    3026.4122   3.826             0.000134 ***
UtilitiesNoSeWa      -21033.6294   26284.6161  -0.800             0.423669    
UtilitiesNoSewr        8691.6846   18899.5029   0.460             0.645642    
LotConfigCulDSac       6226.8024    2762.1547   2.254             0.024276 *  
LotConfigFR2          -6808.8515    3486.7761  -1.953             0.050978 .  
LotConfigFR3          -2140.7264    7739.4546  -0.277             0.782115    
LotConfigInside         413.6771    1465.1482   0.282             0.777706    
LandSlopeMod           7804.2392    3341.1421   2.336             0.019594 *  
LandSlopeSev         -31034.1293   10484.7794  -2.960             0.003111 ** 
NeighborhoodBlueste   -5485.4272   11660.0476  -0.470             0.638084    
NeighborhoodBrDale     2992.7338    8921.1018   0.335             0.737306    
NeighborhoodBrkSide  -14142.0168    7572.8645  -1.867             0.061974 .  
NeighborhoodClearCr  -13686.0612    8156.3040  -1.678             0.093498 .  
NeighborhoodCollgCr  -13819.5256    6415.7289  -2.154             0.031351 *  
NeighborhoodCrawfor    -636.4693    7223.6304  -0.088             0.929798    
NeighborhoodEdwards  -25393.1955    6875.4032  -3.693             0.000227 ***
NeighborhoodGilbert  -16353.1014    6744.9275  -2.425             0.015412 *  
NeighborhoodGreens     2018.6115   12355.1098   0.163             0.870233    
NeighborhoodGrnHill  135468.7260   25809.4200   5.249   0.0000001681958060 ***
NeighborhoodIDOTRR   -23153.1479    7648.0992  -3.027             0.002497 ** 
NeighborhoodLandmrk   -3438.2105   25670.0615  -0.134             0.893464    
NeighborhoodMeadowV   -8747.9193    8307.0341  -1.053             0.292426    
NeighborhoodMitchel  -19888.1334    6933.5747  -2.868             0.004166 ** 
NeighborhoodNAmes    -19753.4041    6786.1205  -2.911             0.003642 ** 
NeighborhoodNoRidge   29954.3669    7335.8275   4.083   0.0000460322345043 ***
NeighborhoodNPkVill    6201.8221    8705.8592   0.712             0.476311    
NeighborhoodNridgHt   22499.3828    6642.5035   3.387             0.000719 ***
NeighborhoodNWAmes   -21241.2745    6937.1943  -3.062             0.002226 ** 
NeighborhoodOldTown  -25484.0482    7312.7278  -3.485             0.000502 ***
NeighborhoodSawyer   -15200.5130    7001.7400  -2.171             0.030044 *  
NeighborhoodSawyerW  -14440.6336    6749.3701  -2.140             0.032504 *  
NeighborhoodSomerst    9993.7009    6591.3260   1.516             0.129619    
NeighborhoodStoneBr   27723.7535    7379.3163   3.757             0.000177 ***
NeighborhoodSWISU    -17289.0401    8333.2353  -2.075             0.038133 *  
NeighborhoodTimber   -11170.0960    7138.1658  -1.565             0.117768    
NeighborhoodVeenker   -7217.5111    8833.9136  -0.817             0.414006    
Condition1Feedr        1991.8526    4013.3951   0.496             0.619733    
Condition1Norm        11266.3971    3340.7666   3.372             0.000758 ***
Condition1PosA        12330.6113    8100.3956   1.522             0.128101    
Condition1PosN        23102.0009    5746.0530   4.020   0.0000600849749277 ***
Condition1RRAe        -3763.9418    6704.7203  -0.561             0.574593    
Condition1RRAn         6165.6291    5488.3071   1.123             0.261389    
Condition1RRNe         4079.6186   13297.7044   0.307             0.759032    
Condition1RRNn        -2623.3735   12946.6537  -0.203             0.839444    
Condition2Feedr       -4248.8419   15564.9445  -0.273             0.784899    
Condition2Norm        -3459.8301   13383.5384  -0.259             0.796035    
Condition2PosA        78630.5657   23461.0517   3.352             0.000818 ***
Condition2PosN      -129662.8328   20498.3421  -6.326   0.0000000003063132 ***
Condition2RRAe       -15203.8113   37809.1587  -0.402             0.687636    
Condition2RRAn          711.0949   28502.2368   0.025             0.980098    
Condition2RRNn        -2967.0777   22542.8465  -0.132             0.895298    
BldgType2fmCon        -8547.9570    4550.9686  -1.878             0.060480 .  
BldgTypeDuplex        -8755.8864    5468.0373  -1.601             0.109461    
BldgTypeTwnhs        -33035.0180    4409.9702  -7.491   0.0000000000000994 ***
BldgTypeTwnhsE       -24890.3681    2913.8486  -8.542 < 0.0000000000000002 ***
HouseStyle1.5Unf      11889.0049    7209.0554   1.649             0.099258 .  
HouseStyle1Story      12194.6960    2964.4228   4.114   0.0000404115535015 ***
HouseStyle2.5Fin     -29459.2362   13721.9210  -2.147             0.031915 *  
HouseStyle2.5Unf       2051.1682    6548.8036   0.313             0.754150    
HouseStyle2Story      -5257.2460    2585.2720  -2.034             0.042121 *  
HouseStyleSFoyer       4686.7154    4677.1713   1.002             0.316437    
HouseStyleSLvl         2645.9104    3828.8869   0.691             0.489616    
OverallQual            6810.0108     802.1970   8.489 < 0.0000000000000002 ***
OverallCond            5897.9892     706.0844   8.353 < 0.0000000000000002 ***
YearBuilt               222.6986      59.0092   3.774             0.000165 ***
YearRemodAdd             66.7363      44.2448   1.508             0.131615    
RoofStyleGable        15187.2265   13251.3750   1.146             0.251887    
RoofStyleGambrel      20295.8921   14646.0087   1.386             0.165964    
RoofStyleHip          16784.8075   13341.9707   1.258             0.208512    
RoofStyleMansard       1021.3063   15907.5396   0.064             0.948815    
RoofStyleShed         16392.6544   26663.8659   0.615             0.538759    
RoofMatlCompShg      551847.8645   30490.4352  18.099 < 0.0000000000000002 ***
RoofMatlMetal        606736.5639   42883.4945  14.148 < 0.0000000000000002 ***
RoofMatlRoll         561775.1857   39666.7182  14.162 < 0.0000000000000002 ***
RoofMatlTar&Grv      561213.2952   32529.6657  17.252 < 0.0000000000000002 ***
RoofMatlWdShake      550848.6415   32194.8551  17.110 < 0.0000000000000002 ***
RoofMatlWdShngl      674279.2205   36009.3200  18.725 < 0.0000000000000002 ***
MasVnrTypeBrkFace      3993.9865    6063.9130   0.659             0.510192    
MasVnrTypeCBlock     -59517.4286   35715.7384  -1.666             0.095776 .  
MasVnrTypenone         5251.7401    8386.0737   0.626             0.531221    
MasVnrTypeNone         7143.2877    6073.4108   1.176             0.239663    
MasVnrTypeStone        6188.0979    6322.8162   0.979             0.327842    
MasVnrArea               10.8310       5.0305   2.153             0.031425 *  
ExterQualFa          -23516.2234    7694.8111  -3.056             0.002270 ** 
ExterQualGd          -25890.4792    3959.4028  -6.539   0.0000000000772073 ***
ExterQualTA          -29047.2925    4435.8143  -6.548   0.0000000000726051 ***
ExterCondFa           -2872.1944   10018.2995  -0.287             0.774374    
ExterCondGd            2971.7605    9151.6533   0.325             0.745422    
ExterCondPo           -7081.4842   20613.4749  -0.344             0.731228    
ExterCondTA            3888.4895    9065.0701   0.429             0.668001    
FoundationCBlock        269.0556    2485.1273   0.108             0.913795    
FoundationPConc        4521.8863    2741.0044   1.650             0.099147 .  
FoundationSlab         3433.3545    7200.5323   0.477             0.633539    
FoundationStone        8865.5328    9724.5331   0.912             0.362047    
FoundationWood        -4568.5388   15150.2535  -0.302             0.763025    
BsmtQualFa           -16660.3977    4828.9512  -3.450             0.000571 ***
BsmtQualGd           -20156.7648    2729.6226  -7.384   0.0000000000002180 ***
BsmtQualnone          -2134.3874   24987.6290  -0.085             0.931937    
BsmtQualPo           -12960.4448   21303.6750  -0.608             0.543009    
BsmtQualTA           -18874.8026    3409.2095  -5.536   0.0000000346568150 ***
BsmtCondFa             3292.2658   18142.3220   0.181             0.856017    
BsmtCondGd             3377.1412   18064.3948   0.187             0.851717    
BsmtCondnone                  NA           NA      NA                   NA    
BsmtCondPo            24991.7197   26369.2947   0.948             0.343359    
BsmtCondTA             2364.6003   17861.9327   0.132             0.894694    
BsmtExposureGd        10646.4239    2440.3778   4.363   0.0000134617837265 ***
BsmtExposureMn        -9390.1293    2475.6480  -3.793             0.000153 ***
BsmtExposureNo        -8802.0734    1857.7668  -4.738   0.0000023001189487 ***
BsmtExposurenone     -15132.7717   14391.0465  -1.052             0.293129    
BsmtFinType1BLQ          23.3527    2302.8564   0.010             0.991910    
BsmtFinType1GLQ        1963.5544    2071.6028   0.948             0.343316    
BsmtFinType1LwQ       -3916.5990    2894.5383  -1.353             0.176167    
BsmtFinType1none              NA           NA      NA                   NA    
BsmtFinType1Rec        -476.2670    2378.8807  -0.200             0.841338    
BsmtFinType1Unf         113.6525    2361.6674   0.048             0.961622    
BsmtFinSF1               28.0607       3.8994   7.196   0.0000000000008526 ***
BsmtFinType2BLQ       -2285.8789    5490.6914  -0.416             0.677218    
BsmtFinType2GLQ        5099.4234    6887.1331   0.740             0.459122    
BsmtFinType2LwQ       -4243.8420    5305.6623  -0.800             0.423875    
BsmtFinType2none              NA           NA      NA                   NA    
BsmtFinType2Rec       -1543.1538    5207.5363  -0.296             0.767006    
BsmtFinType2Unf        2931.1412    5261.1981   0.557             0.577501    
BsmtFinSF2               31.5440       6.7718   4.658   0.0000033861408377 ***
BsmtUnfSF                14.2267       3.5784   3.976   0.0000724945122263 ***
TotalBsmtSF                   NA           NA      NA                   NA    
HeatingGasA           -6717.9275   25975.3311  -0.259             0.795948    
HeatingGasW           -1361.2956   26497.4473  -0.051             0.959032    
HeatingGrav          -14307.8783   27869.7289  -0.513             0.607736    
HeatingOthW          -43210.6266   31869.7380  -1.356             0.175289    
HeatingWall            7397.1416   29323.7992   0.252             0.800866    
HeatingQCFa            -591.4595    3675.0768  -0.161             0.872157    
HeatingQCGd           -2057.1082    1676.3982  -1.227             0.219920    
HeatingQCPo          -51472.7460   44881.7962  -1.147             0.251571    
HeatingQCTA           -3533.9748    1651.5354  -2.140             0.032483 *  
CentralAirY            -311.8612    2906.1391  -0.107             0.914552    
ElectricalFuseF        1887.8064    4708.7534   0.401             0.688523    
ElectricalFuseP        -277.9348    9880.4680  -0.028             0.977561    
ElectricalMix           794.8354   37147.4445   0.021             0.982931    
ElectricalSBrkr        -408.7491    2361.6989  -0.173             0.862609    
FirstFlrSF               42.9278       4.0350  10.639 < 0.0000000000000002 ***
SecondFlrSF              62.8987       4.4237  14.218 < 0.0000000000000002 ***
LowQualFinSF             44.0808      13.3068   3.313             0.000939 ***
GrLivArea                     NA           NA      NA                   NA    
BsmtFullBath           3367.4572    1559.0900   2.160             0.030892 *  
BsmtHalfBath          -5191.9846    2399.1910  -2.164             0.030570 *  
FullBath               6883.5741    1773.6297   3.881             0.000107 ***
HalfBath               3943.0530    1679.9289   2.347             0.019008 *  
BedroomAbvGr          -2268.1559    1080.6592  -2.099             0.035946 *  
KitchenAbvGr         -15268.9538    4960.0325  -3.078             0.002108 ** 
KitchenQualFa        -17906.5841    4982.9569  -3.594             0.000334 ***
KitchenQualGd        -18639.9689    2858.5923  -6.521   0.0000000000870358 ***
KitchenQualTA        -20054.5711    3198.6458  -6.270   0.0000000004361825 ***
TotRmsAbvGrd           -455.9306     744.4026  -0.612             0.540286    
FunctionalMaj2       -17853.0922   12080.8024  -1.478             0.139607    
FunctionalMin1          690.3888    7823.2064   0.088             0.929687    
FunctionalMin2         1834.0859    7798.2481   0.235             0.814082    
FunctionalMod           792.2803    8428.6469   0.094             0.925119    
FunctionalSal         -5456.4197   27160.8608  -0.201             0.840802    
FunctionalSev        -24940.2358   20232.5183  -1.233             0.217830    
FunctionalTyp          9667.7856    7029.2693   1.375             0.169163    
Fireplaces             7786.0593    2056.6131   3.786             0.000157 ***
FireplaceQuFa        -14339.0443    5917.7659  -2.423             0.015473 *  
FireplaceQuGd         -7969.7391    4800.0345  -1.660             0.096990 .  
FireplaceQunone       -6146.1535    5378.2358  -1.143             0.253257    
FireplaceQuPo        -13134.7656    6493.3906  -2.023             0.043219 *  
FireplaceQuTA        -10801.4757    4929.1155  -2.191             0.028533 *  
GarageTypeAttchd      11893.0112    6831.1709   1.741             0.081829 .  
GarageTypeBasment     12514.0352    8579.8356   1.459             0.144838    
GarageTypeBuiltIn      9505.1865    7321.4981   1.298             0.194339    
GarageTypeCarPort      6327.3163   10176.9946   0.622             0.534187    
GarageTypeDetchd      11543.0317    6812.9192   1.694             0.090356 .  
GarageTypenone        23715.7690   19446.3665   1.220             0.222771    
GarageFinishnone      -1929.3846   24439.4893  -0.079             0.937083    
GarageFinishRFn       -3739.4538    1598.3264  -2.340             0.019396 *  
GarageFinishUnf       -1549.9336    1928.8635  -0.804             0.421747    
GarageCars             3689.9879    1841.8736   2.003             0.045261 *  
GarageArea               22.2360       6.2100   3.581             0.000350 ***
GarageQualFa         -36930.9691   23900.1038  -1.545             0.122440    
GarageQualGd         -20919.8202   23460.4248  -0.892             0.372650    
GarageQualnone                NA           NA      NA                   NA    
GarageQualPo         -42456.5374   30125.1635  -1.409             0.158880    
GarageQualTA         -37901.2894   23681.2020  -1.600             0.109640    
GarageCondFa          35808.4419   22174.7204   1.615             0.106494    
GarageCondGd          34813.8686   22949.3341   1.517             0.129417    
GarageCondnone                NA           NA      NA                   NA    
GarageCondPo          38257.9514   24277.1162   1.576             0.115200    
GarageCondTA          39464.5159   21798.4667   1.810             0.070370 .  
PavedDriveP           -2524.1790    4312.2529  -0.585             0.558374    
PavedDriveY            1469.3221    2691.9176   0.546             0.585241    
 [ reached getOption("max.print") -- omitted 11 rows ]
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 24410 on 2142 degrees of freedom
Multiple R-squared:  0.9152,    Adjusted R-squared:  0.9072 
F-statistic: 113.9 on 203 and 2142 DF,  p-value: < 0.00000000000000022

Make predictions on test set and calculate RMSE.

lm_preds <- predict(lm_fit, data = data_test)
rmse(data_test$SalePrice, lm_preds)
[1] 109610.3

Random Forest

Fit a random forest model on all features using the ranger package.

ranger_fit <- ranger(SalePrice ~ ., data = data_training)

Make predictions on test set and calculate RMSE.

ranger_preds <- predict(ranger_fit, data = data_test)
ranger_preds <- ranger_preds$predictions

rmse(data_test$SalePrice, ranger_preds)
[1] 21455.41

Interpretation

Source: https://christophm.github.io/interpretable-ml-book

Note

The following content is based on Chapter 16 of “Hands-On Machine Learning with R” by Boehmke and Greenwell (https://bradleyboehmke.github.io/HOML/iml.html).

Create IML Object

The iml package implements various model-agnostic post-hoc explainability methods. However, it needs a bit of setup.

# define a wrapper around the predict function of the ranger package
ranger_predict <- function(model, newdata)  {
  results <- predict(model, newdata)
  results <- as.data.frame(results$predictions)
  return(results)
}

# store the features and targets in separate variables
X <- data  %>% select(-SalePrice)
y <- data %>% select(SalePrice)

# create IML predictor object for ranger
ranger_predictor <- Predictor$new(
  model = ranger_fit, 
  data = X, 
  y = y, 
  predict.fun = ranger_predict,
  class = "regression"
  )

Global Explanations: Permutation-based Feature Importance

Permutation-based feature importance measures a feature’s importance by calculating the increase of the model’s prediction error after permuting the values of a given feature. The idea is that if we randomly permute the values of an important feature in the training data, the training performance would degrade (since permuting the values of a feature effectively destroys any relationship between that feature and the target variable). The permutation approach uses the difference (or ratio) between some baseline performance measure (e.g., RMSE) and the same performance measure obtained after permuting the values of a particular feature in the training data.

Algorithm 1: A simple algorithm for computing permutation-based variable importance for the feature set X.

1. Compute accuracy of the original model
2. For feature i in {1,...,p} do
     | Permute values of feature i
     | Make predictions for all observations in the training set
     | Compute accuracy of permuted predicitons
     | Calculate feature importance, i.e., difference or ratio 
       between permuted and original accuracy
   End
3. Sort variables by descending feature importance 

Calculate and plot permutation-based feature importance.

imp <- FeatureImp$new(ranger_predictor, loss = "rmse", compare = "difference",
                      n.repetitions = 1)
plot(imp)

Global Explanations: Partial Dependence Plot (PDP)

To create a PDP we will split the feature of interest into j equally spaced values. For example, the GrLivArea feature ranges from 334 to 5095 square feet. Say the user selects j=20. We will then first create an evenly spaced grid consisting of 20 values across the distribution of GrLivArea (e.g., 334.00, 584.58,… , 5095.00). Then we will make 20 copies of the original training data (one copy for each value in the grid). The algorithm will then set GrLivArea for all observations in the first copy to 334, 585 in the second copy, 835 in the third copy, …, and finally to 5095 in the 20-th copy (all other features remain unchanged). The algorithm then predicts the outcome for each observation in each of the 20 copies, and then averages the predicted values for each set. These averaged predicted values are known as partial dependence values and are plotted against the 20 evenly spaced values for GrLivArea.

Algorithm 2: A simple algorithm for constructing the partial dependence plot of the response on a single predictor x.

For a selected predictor (x)
1. Construct a grid of j evenly spaced values across the distribution
   of x: {x1, x2, ..., xj}
2. For i in {1,...,j} do
     | Copy the training data and replace all original values of x 
       with the constant xi
     | Compute predictions for all observations
     | Average predictions
   End
3. Plot the averaged predictions against x1, x2, ..., xj 

Calculate and plot a partial dependence plot (PDP) with the IML package.

pdp <- FeatureEffect$new(ranger_predictor, method = "pdp", feature = "GrLivArea", grid.size = 20)
plot(pdp)

Local Explanations: Individual Conditional Expectation (ICE) Plots

An ICE plot visualizes the dependence of the predicted response on a feature for each instance separately, resulting in multiple lines, one for each observation, compared to one line in partial dependence plots. A PDP is the average of the lines of an ICE plot. Note that the following algorithm is the same as the PDP algorithms except for the last line where PDPs averaged the predicted values.

Algorithm 3: A simple algorithm for constructing the individual conditional expectation of the response on a single predictor x.

For a selected predictor (x)
1. Construct a grid of j evenly spaced values across the distribution 
   of x: {x1, x2, ..., xj}
2. For i in {1,...,j} do
     | Copy the training data and replace the original values of x 
       with the constant xi
     | Compute predictions for all observations
   End
3. Plot the predictions against x1, x2, ..., xj with lines connecting 
   oberservations that correspond to the same row number in the original 
   training data

Calculate and plot ICE plot for all observations in the test set.

ice <- FeatureEffect$new(ranger_predictor, method = "pdp+ice", 
                         feature = "GrLivArea")
plot(ice)

Local Explanations: Shapley Values

Another method for explaining individual predictions borrows ideas from game theory to produce whats called Shapley values.

The concept of Shapley values is based on the idea that the feature values of an individual observation work together to cause a change in the model’s prediction with respect to the model’s expected output, and it divides this total change in prediction among the feature values in a way that is “fair” to their contributions across all possible subsets of features.

Source: https://christophm.github.io/interpretable-ml-book

To calculate Shapley values, we assess every combination of predictor values to determine each predictor value’s impact. Focusing on feature xj and its value for the observation of interest, the approach will test how adding the feature value of xj to every possible combination of values of the other features changes the prediction on average.

For complex models, computing exact Shapley values is computationally expensive and most implementations use sampling-based heuristics to calculate approximate Shapley values. The example below shows a one sample repetition of estimating the impact of “cat banned” to the coalition of “park nearby”, “50sqm”, and “1st floor”. For this sample, the impact of “cat banned” is €-10.000. Repeating this process multiple times by holding the feature value “cat banned” fixed and combining it with other simulated coalitions yields an approximation of the Shapley value of “cat banned”.

Source: https://christophm.github.io/interpretable-ml-book

Calculate and plot Shapley values for the first observation in the training set.

shapley <- Shapley$new(ranger_predictor, x.interest = data_training[1,], sample.size = 10)
plot(shapley)

---
title: "Data Science for Business - Week 10: Interpretable Machine Learning"
author: "Oliver Mueller"
output: html_notebook
---

![](comic_01.png) Source: <https://christophm.github.io/interpretable-ml-book>

## Initialize Notebook

Load required packages.

```{r load packages, warning=FALSE, message=FALSE}
library(tidyverse)
library(ggthemr)
library(caret)
library(ranger)
library(Metrics)
library(iml)

```

Set up workspace, i.e., remove all existing data from working memory, initialize the random number generator, turn of scientific notation of large numbers, set a standard theme for plotting. Finally, we activate parallel processing on multiple CPU cores.

```{r setup}
rm(list=ls())
set.seed(42)
knitr::opts_chunk$set(echo = TRUE, warning = FALSE)
options(scipen=10000)
ggthemr('fresh')
doParallel::registerDoParallel(cores = 4)

```

## Problem Description

Ask a home buyer to describe their dream house, and they probably won't begin with the height of the basement ceiling or the proximity to an east-west railroad. But this dataset proves that much more influences price negotiations than the number of bedrooms or a white-picket fence. With 76 explanatory variables describing (almost) every aspect of residential homes in Ames, Iowa, this dataset challenges you to predict the final price of each home. More: <https://www.kaggle.com/c/house-prices-advanced-regression-techniques>

## Prepare Data

Load data from CSV file.

```{r}
data <- read_csv("ameshousing.csv")

```

Fix some data type issues, i.e., turn character variables into factors and remove some irrelevant predictors.

```{r}
data <- data %>% 
  mutate_if(is.character,as.factor) %>% 
  select(-MSSubClass, -MSZoning, -Exterior1st, -Exterior2nd, -SaleType, -SaleCondition, -MiscVal, -MoSold)

```

Create train/test split.

```{r}
training_rows <- createDataPartition(y=data$SalePrice, p = 0.8, list = FALSE)
data_training <- slice(data, training_rows)
data_test <- slice(data, -training_rows)

```

## Modelling

### Linear Regression

Fit a linear model on all features.

```{r}
lm_fit <- lm(SalePrice ~ ., data = data_training)

```

Display regression coefficients.

```{r}
summary(lm_fit)

```

Make predictions on test set and calculate RMSE.

```{r, warning=FALSE}
lm_preds <- predict(lm_fit, data = data_test)
rmse(data_test$SalePrice, lm_preds)

```

### Random Forest

Fit a random forest model on all features using the `ranger` package.

```{r}
ranger_fit <- ranger(SalePrice ~ ., data = data_training)

```

Make predictions on test set and calculate RMSE.

```{r}
ranger_preds <- predict(ranger_fit, data = data_test)
ranger_preds <- ranger_preds$predictions

rmse(data_test$SalePrice, ranger_preds)

```

## Interpretation

![](comic_02.png) Source: <https://christophm.github.io/interpretable-ml-book>

### Note

The following content is based on Chapter 16 of "Hands-On Machine Learning with R" by Boehmke and Greenwell (<https://bradleyboehmke.github.io/HOML/iml.html>).

### Create IML Object

The `iml` package implements various model-agnostic post-hoc explainability methods. However, it needs a bit of setup.

```{r}
# define a wrapper around the predict function of the ranger package
ranger_predict <- function(model, newdata)  {
  results <- predict(model, newdata)
  results <- as.data.frame(results$predictions)
  return(results)
}

# store the features and targets in separate variables
X <- data  %>% select(-SalePrice)
y <- data %>% select(SalePrice)

# create IML predictor object for ranger
ranger_predictor <- Predictor$new(
  model = ranger_fit, 
  data = X, 
  y = y, 
  predict.fun = ranger_predict,
  class = "regression"
  )

```

### Global Explanations: Permutation-based Feature Importance

Permutation-based feature importance measures a feature's importance by calculating the increase of the model's prediction error after permuting the values of a given feature. The idea is that if we randomly permute the values of an important feature in the training data, the training performance would degrade (since permuting the values of a feature effectively destroys any relationship between that feature and the target variable). The permutation approach uses the difference (or ratio) between some baseline performance measure (e.g., RMSE) and the same performance measure obtained after permuting the values of a particular feature in the training data.

Algorithm 1: A simple algorithm for computing permutation-based variable importance for the feature set X.

    1. Compute accuracy of the original model
    2. For feature i in {1,...,p} do
         | Permute values of feature i
         | Make predictions for all observations in the training set
         | Compute accuracy of permuted predicitons
         | Calculate feature importance, i.e., difference or ratio 
           between permuted and original accuracy
       End
    3. Sort variables by descending feature importance 

Calculate and plot permutation-based feature importance.

```{r}
imp <- FeatureImp$new(ranger_predictor, loss = "rmse", compare = "difference",
                      n.repetitions = 1)
plot(imp)

```

### Global Explanations: Partial Dependence Plot (PDP)

To create a PDP we will split the feature of interest into j equally spaced values. For example, the `GrLivArea` feature ranges from 334 to 5095 square feet. Say the user selects j=20. We will then first create an evenly spaced grid consisting of 20 values across the distribution of `GrLivArea` (e.g., 334.00, 584.58,... , 5095.00). Then we will make 20 copies of the original training data (one copy for each value in the grid). The algorithm will then set `GrLivArea` for all observations in the first copy to 334, 585 in the second copy, 835 in the third copy, ..., and finally to 5095 in the 20-th copy (all other features remain unchanged). The algorithm then predicts the outcome for each observation in each of the 20 copies, and then averages the predicted values for each set. These averaged predicted values are known as partial dependence values and are plotted against the 20 evenly spaced values for `GrLivArea`.

Algorithm 2: A simple algorithm for constructing the partial dependence plot of the response on a single predictor `x`.

    For a selected predictor (x)
    1. Construct a grid of j evenly spaced values across the distribution
       of x: {x1, x2, ..., xj}
    2. For i in {1,...,j} do
         | Copy the training data and replace all original values of x 
           with the constant xi
         | Compute predictions for all observations
         | Average predictions
       End
    3. Plot the averaged predictions against x1, x2, ..., xj 

Calculate and plot a partial dependence plot (PDP) with the IML package.

```{r}
pdp <- FeatureEffect$new(ranger_predictor, method = "pdp", feature = "GrLivArea", grid.size = 20)
plot(pdp)

```

### Local Explanations: Individual Conditional Expectation (ICE) Plots

An ICE plot visualizes the dependence of the predicted response on a feature for each instance separately, resulting in multiple lines, one for each observation, compared to one line in partial dependence plots. A PDP is the average of the lines of an ICE plot. Note that the following algorithm is the same as the PDP algorithms except for the last line where PDPs averaged the predicted values.

Algorithm 3: A simple algorithm for constructing the individual conditional expectation of the response on a single predictor `x`.

    For a selected predictor (x)
    1. Construct a grid of j evenly spaced values across the distribution 
       of x: {x1, x2, ..., xj}
    2. For i in {1,...,j} do
         | Copy the training data and replace the original values of x 
           with the constant xi
         | Compute predictions for all observations
       End
    3. Plot the predictions against x1, x2, ..., xj with lines connecting 
       oberservations that correspond to the same row number in the original 
       training data

Calculate and plot ICE plot for all observations in the test set.

```{r}
ice <- FeatureEffect$new(ranger_predictor, method = "pdp+ice", 
                         feature = "GrLivArea")
plot(ice)

```

### Local Explanations: Shapley Values

Another method for explaining individual predictions borrows ideas from game theory to produce whats called Shapley values.

The concept of Shapley values is based on the idea that the feature values of an individual observation work together to cause a change in the model's prediction with respect to the model's expected output, and it divides this total change in prediction among the feature values in a way that is "fair" to their contributions across all possible subsets of features.

![](cat_ban_01.png) Source: <https://christophm.github.io/interpretable-ml-book>

To calculate Shapley values, we assess every combination of predictor values to determine each predictor value's impact. Focusing on feature `xj` and its value for the observation of interest, the approach will test how adding the feature value of `xj` to every possible combination of values of the other features changes the prediction on average.

For complex models, computing exact Shapley values is computationally expensive and most implementations use sampling-based heuristics to calculate approximate Shapley values. The example below shows a one sample repetition of estimating the impact of "cat banned" to the coalition of "park nearby", "50sqm", and "1st floor". For this sample, the impact of "cat banned" is €-10.000. Repeating this process multiple times by holding the feature value "cat banned" fixed and combining it with other simulated coalitions yields an approximation of the Shapley value of "cat banned".

![](cat_ban_02.png) Source: <https://christophm.github.io/interpretable-ml-book>

Calculate and plot Shapley values for the first observation in the training set.

```{r}
shapley <- Shapley$new(ranger_predictor, x.interest = data_training[1,], sample.size = 10)
plot(shapley)

```
