CUNY624HW7JoelPark
CUNY 624 Homework 7 Student: Joel Park
6.2 Developing a model to predict permeability could save signficant resources for a pharmaceutical company, while at the same time more rapidly identifying molecules that have a sufficient permeability to become a drug:
A. Start R and use these commands to load the data.
library(AppliedPredictiveModeling)
data(permeability)The matrix fingerprints contains the 1107 binary molecular predictors for the 165 compounds, while permeability contains permeability response.
According to ?permeability, this data set was used to develop a model for predicting compounds’ permeability.
Evaluating the underlying structure of the fingerprints and permeability dataset.
paste0("Fingerprints Matrix: ")## [1] "Fingerprints Matrix: "str(fingerprints)## num [1:165, 1:1107] 0 0 0 0 0 0 0 0 0 0 ...
## - attr(*, "dimnames")=List of 2
## ..$ : chr [1:165] "1" "2" "3" "4" ...
## ..$ : chr [1:1107] "X1" "X2" "X3" "X4" ...paste0("Permeability Response: ")## [1] "Permeability Response: "str(permeability)## num [1:165, 1] 12.52 1.12 19.41 1.73 1.68 ...
## - attr(*, "dimnames")=List of 2
## ..$ : chr [1:165] "1" "2" "3" "4" ...
## ..$ : chr "permeability"B. The fingerprint predictors indicate the presence or absence of substructures of a molecule and are often sparse meaning that relatively few of the molecules contain each substructure. Filter out the predictors that have low frequencies using the nearZeroVar function from the caret package. How many predictors are left for modeling?
?nearZeroVar: Diagnoses predictors that have one unique value (i.e. are zero variance predictors) or predictors that are have both of the following characteristics: they have very few unique values relative to the number of samples and the ratio of the frequency of the most common value to the frequency of the second most common value is large.
library(caret)## Loading required package: lattice## Loading required package: ggplot2near_zero_var <- nearZeroVar(fingerprints)
near_zero_var## [1] 7 8 9 10 13 14 17 18 19 22 23 24 30 31
## [15] 32 33 34 45 77 81 82 83 84 85 89 90 91 92
## [29] 95 100 104 105 106 107 109 110 112 113 114 115 116 117
## [43] 119 120 122 123 124 128 131 132 134 135 136 137 139 140
## [57] 144 145 147 148 149 151 155 160 161 164 165 166 216 217
## [71] 218 219 220 222 243 252 259 273 275 277 282 283 287 288
## [85] 289 292 346 347 348 349 350 351 352 353 354 363 364 365
## [99] 369 375 379 384 391 393 397 399 402 404 405 407 408 409
## [113] 410 411 412 413 414 415 416 417 418 419 420 421 422 423
## [127] 424 425 426 427 428 429 430 431 432 433 434 435 436 437
## [141] 438 439 440 441 442 443 444 445 446 447 448 449 450 451
## [155] 452 453 454 455 456 457 458 459 460 461 462 463 464 465
## [169] 466 467 468 469 470 471 472 473 474 475 476 477 478 479
## [183] 480 481 482 483 484 485 486 487 488 489 490 491 492 493
## [197] 494 495 498 500 501 502 513 523 525 526 527 528 530 531
## [211] 532 533 534 535 536 537 538 539 540 541 542 543 544 545
## [225] 546 547 548 550 552 555 562 563 564 566 567 569 570 572
## [239] 575 578 579 580 581 582 583 584 585 586 587 588 589 596
## [253] 605 606 607 608 609 610 611 612 614 615 616 617 618 619
## [267] 620 622 623 624 625 626 627 628 629 630 631 632 633 634
## [281] 635 636 637 638 639 640 641 642 643 644 645 646 647 648
## [295] 649 650 651 652 653 654 655 656 657 658 659 660 661 662
## [309] 663 664 665 666 667 668 669 670 671 672 673 674 675 676
## [323] 677 678 680 681 682 683 684 685 686 687 688 689 690 691
## [337] 692 693 694 695 696 697 706 707 708 709 710 711 712 713
## [351] 714 715 716 717 718 720 721 722 723 724 725 726 727 728
## [365] 729 730 731 734 735 736 737 738 739 740 741 742 743 744
## [379] 745 746 747 748 749 756 757 758 759 760 761 762 763 764
## [393] 765 766 767 768 769 770 771 772 777 778 779 781 783 784
## [407] 785 786 787 788 789 790 791 794 796 797 799 802 803 804
## [421] 807 808 809 810 811 814 815 816 817 818 819 820 821 822
## [435] 823 824 825 826 827 828 829 830 831 832 833 834 835 836
## [449] 837 838 839 840 841 842 843 844 845 846 847 848 849 850
## [463] 851 852 853 854 855 856 857 858 859 860 861 862 863 864
## [477] 865 866 867 868 869 870 871 872 873 874 875 876 877 878
## [491] 879 880 881 882 883 884 885 886 887 888 889 890 891 892
## [505] 893 894 895 896 897 898 899 900 901 902 903 904 905 906
## [519] 907 908 909 910 911 912 913 914 915 916 917 918 919 920
## [533] 921 922 923 924 925 926 927 928 929 930 931 932 933 934
## [547] 935 936 937 938 939 940 941 942 943 944 945 946 947 948
## [561] 949 950 951 952 953 954 955 956 957 958 959 960 961 962
## [575] 963 964 965 966 967 968 969 970 971 972 973 974 975 976
## [589] 977 978 979 980 981 982 983 984 985 986 987 988 989 990
## [603] 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004
## [617] 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018
## [631] 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032
## [645] 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046
## [659] 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060
## [673] 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074
## [687] 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088
## [701] 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102
## [715] 1103 1104 1105 1106 1107paste0("We have successfully identified ", length(near_zero_var), " variables that have zero or near zero variance predictors.")## [1] "We have successfully identified 719 variables that have zero or near zero variance predictors."These variables can be removed from the matrix as they will not likely contribute to the construction of the model.
fingerprints_1 <- fingerprints[,-near_zero_var]
str(fingerprints_1)## num [1:165, 1:388] 0 0 0 0 0 0 0 0 0 0 ...
## - attr(*, "dimnames")=List of 2
## ..$ : chr [1:165] "1" "2" "3" "4" ...
## ..$ : chr [1:388] "X1" "X2" "X3" "X4" ...paste0("There are ", ncol(fingerprints_1), " variables left.")## [1] "There are 388 variables left."C. Split the data into a training and a test set, pre-process the data, and tune a PLS model. How many latent variables are optimal and what is the corresponding resampled estimate of \(R^2\)?
Referencing 6.5 “Computing”:
Before we train our data, we need to create a training and testing data set.
# Before we start with the splitting of datasets and model building, we need to combine the response variable with the training data.
fingerprints_1 <- as.data.frame(fingerprints)
fingerprints_1$permeability <- as.vector(permeability)
# We should remove any zero or near zero variance as this will destabilize the models and not contribute.
z_var <- nearZeroVar(fingerprints_1)
# Now that we identified the zero or near zero variance variables
# Let us eliminate them from our dataset.
fingerprints_1 <- fingerprints_1[,-z_var]# Straight splitting training and testing data set.
# Reference: https://stackoverflow.com/questions/17200114/how-to-split-data-into-training-testing-sets-using-sample-function
smp_size <- floor(0.8 * nrow(fingerprints_1))
train_ind <- sample(seq_len(nrow(fingerprints_1)), size = smp_size)
trainingData <- fingerprints_1[train_ind,]
testingData <- fingerprints_1[-train_ind,]Now that the data is successfully split into training and testing datasets, we can now fit a partial least squares model.
The pls package has functions for PLS and PCR. SIMPLS and the Rannar algorithm are both available. The plsr function can be used very much like the lm function. The number of components can be fixed using the ncomp argument or, if left to the default, the maximum number of components will be calculated. The plsr function has options for either K-fold or leave-one-out cross-validation (via the validation argument) or the PLS algorithm to use, such as SIMPLS (using the method argument). Predictions on new samples can be calculated using the predict function.
library(pls)##
## Attaching package: 'pls'## The following object is masked from 'package:caret':
##
## R2## The following object is masked from 'package:stats':
##
## loadingsplsFit <- plsr(permeability ~ ., data = fingerprints_1)
summary(plsFit)## Data: X dimension: 165 388
## Y dimension: 165 1
## Fit method: kernelpls
## Number of components considered: 164
## TRAINING: % variance explained
## 1 comps 2 comps 3 comps 4 comps 5 comps 6 comps
## X 29.17 42.33 49.34 52.71 60.37 66.18
## permeability 26.69 47.90 54.71 62.75 66.37 70.18
## 7 comps 8 comps 9 comps 10 comps 11 comps 12 comps
## X 68.87 71.80 73.87 75.68 78.30 80.31
## permeability 72.85 74.36 76.34 78.09 79.14 80.26
## 13 comps 14 comps 15 comps 16 comps 17 comps 18 comps
## X 82.70 84.13 85.17 86.76 87.79 89.11
## permeability 81.06 82.04 83.08 83.80 84.79 85.39
## 19 comps 20 comps 21 comps 22 comps 23 comps 24 comps
## X 90.06 90.8 91.34 91.91 92.52 92.84
## permeability 85.94 86.4 86.92 87.29 87.59 88.11
## 25 comps 26 comps 27 comps 28 comps 29 comps 30 comps
## X 93.39 93.88 94.14 94.48 94.70 94.95
## permeability 88.44 88.77 89.15 89.39 89.72 89.94
## 31 comps 32 comps 33 comps 34 comps 35 comps 36 comps
## X 95.18 95.45 95.69 95.93 96.19 96.44
## permeability 90.09 90.20 90.32 90.44 90.52 90.59
## 37 comps 38 comps 39 comps 40 comps 41 comps 42 comps
## X 96.64 96.84 97.08 97.22 97.38 97.58
## permeability 90.68 90.74 90.78 90.84 90.88 90.90
## 43 comps 44 comps 45 comps 46 comps 47 comps 48 comps
## X 97.72 97.85 97.97 98.09 98.17 98.29
## permeability 90.93 90.96 90.99 91.01 91.03 91.04
## 49 comps 50 comps 51 comps 52 comps 53 comps 54 comps
## X 98.40 98.49 98.55 98.64 98.7 98.79
## permeability 91.06 91.07 91.08 91.09 91.1 91.10
## 55 comps 56 comps 57 comps 58 comps 59 comps 60 comps
## X 98.85 98.91 98.97 99.02 99.10 99.15
## permeability 91.11 91.11 91.12 91.13 91.13 91.14
## 61 comps 62 comps 63 comps 64 comps 65 comps 66 comps
## X 99.19 99.25 99.29 99.34 99.38 99.43
## permeability 91.14 91.15 91.15 91.16 91.16 91.16
## 67 comps 68 comps 69 comps 70 comps 71 comps 72 comps
## X 99.46 99.49 99.53 99.57 99.60 99.62
## permeability 91.16 91.16 91.16 91.16 91.16 91.16
## 73 comps 74 comps 75 comps 76 comps 77 comps 78 comps
## X 99.64 99.67 99.69 99.72 99.74 99.76
## permeability 91.17 91.17 91.17 91.17 91.17 91.17
## 79 comps 80 comps 81 comps 82 comps 83 comps 84 comps
## X 99.78 99.79 99.80 99.82 99.83 99.86
## permeability 91.17 91.17 91.17 91.17 91.17 91.17
## 85 comps 86 comps 87 comps 88 comps 89 comps 90 comps
## X 99.87 99.89 99.90 99.90 99.91 99.92
## permeability 91.17 91.17 91.17 91.17 91.17 91.17
## 91 comps 92 comps 93 comps 94 comps 95 comps 96 comps
## X 99.93 99.94 99.95 99.95 99.96 99.97
## permeability 91.17 91.17 91.17 91.17 91.17 91.17
## 97 comps 98 comps 99 comps 100 comps 101 comps
## X 99.97 99.98 99.98 99.99 99.99
## permeability 91.17 91.17 91.17 91.17 91.17
## 102 comps 103 comps 104 comps 105 comps 106 comps
## X 99.99 100.00 100.00 100.00 100.25
## permeability 91.17 91.17 91.17 91.17 91.17
## 107 comps 108 comps 109 comps 110 comps 111 comps
## X 100.51 100.76 101.02 101.27 101.52
## permeability 91.17 91.17 91.17 91.17 91.17
## 112 comps 113 comps 114 comps 115 comps 116 comps
## X 101.78 102.03 102.28 102.54 102.79
## permeability 91.17 91.17 91.17 91.17 91.17
## 117 comps 118 comps 119 comps 120 comps 121 comps
## X 103.05 103.30 103.55 103.81 104.06
## permeability 91.17 91.17 91.17 91.17 91.17
## 122 comps 123 comps 124 comps 125 comps 126 comps
## X 104.32 104.57 104.82 105.08 105.33
## permeability 91.17 91.17 91.17 91.17 91.17
## 127 comps 128 comps 129 comps 130 comps 131 comps
## X 105.58 105.84 106.09 106.35 106.60
## permeability 91.17 91.17 91.17 91.17 91.17
## 132 comps 133 comps 134 comps 135 comps 136 comps
## X 106.85 107.11 107.36 107.62 107.87
## permeability 91.17 91.17 91.17 91.17 91.17
## 137 comps 138 comps 139 comps 140 comps 141 comps
## X 108.12 108.38 108.63 108.88 109.14
## permeability 91.17 91.17 91.17 91.17 91.17
## 142 comps 143 comps 144 comps 145 comps 146 comps
## X 109.39 109.64 109.90 110.15 110.41
## permeability 91.17 91.17 91.17 91.17 91.17
## 147 comps 148 comps 149 comps 150 comps 151 comps
## X 110.66 110.91 111.17 111.42 111.68
## permeability 91.17 91.17 91.17 91.17 91.17
## 152 comps 153 comps 154 comps 155 comps 156 comps
## X 111.93 112.18 112.44 112.69 112.94
## permeability 91.17 91.17 91.17 91.17 91.17
## 157 comps 158 comps 159 comps 160 comps 161 comps
## X 113.20 113.45 113.70 113.96 114.21
## permeability 91.17 91.17 91.17 91.17 91.17
## 162 comps 163 comps 164 comps
## X 114.47 114.72 114.97
## permeability 91.17 91.17 91.17The function above had provided ~164 components and appears to be a bit overwhelming. Furthermore, the split and testing data were a simple split. To improve the parameters of our models, we should be considering the 10-fold cross-validation instead of the straightforward training and testing split. (We will be using cross-validation on the testing data set and reserving the testing data set as a validation set.) This can be accomplished by the trainControl function. We will be using this when we use the train function.
# 10 fold Cross validation
ctrl <- trainControl(method= "cv", number = 10)Now that we successfully standardized the ctrl function, let us pre-process the data and tune a partial least squares model.
X <- testingData[1:(length(testingData)-1)]
plsTune <- train(X, testingData$permeability,
method = "pls",
tuneLength = 20,
trControl = ctrl,
preProc = c("center", "scale"))
plsTune## Partial Least Squares
##
## 33 samples
## 388 predictors
##
## Pre-processing: centered (388), scaled (388)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 29, 30, 29, 31, 30, 30, ...
## Resampling results across tuning parameters:
##
## ncomp RMSE Rsquared MAE
## 1 11.61465 0.6060687 9.373817
## 2 12.13671 0.4793820 9.078608
## 3 13.20101 0.5627764 9.930050
## 4 13.49176 0.4795114 10.549720
## 5 14.09133 0.5767719 11.043676
## 6 14.52496 0.5156036 11.774945
## 7 14.33107 0.4252268 11.578746
## 8 14.75028 0.4471746 11.799307
## 9 14.92486 0.4288702 11.870360
## 10 15.66922 0.4649396 12.524820
## 11 16.21786 0.4452180 12.887372
## 12 17.11636 0.3482682 13.518803
## 13 17.92739 0.3343669 14.151109
## 14 18.02092 0.3973943 14.010458
## 15 18.45013 0.4274785 14.360400
## 16 18.40075 0.4455370 14.386058
## 17 18.31947 0.4250162 14.334778
## 18 18.20637 0.4074364 14.169310
## 19 18.12241 0.4108052 14.015812
## 20 18.09508 0.4123270 13.987278
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was ncomp = 1.So above demonstrates a more finely tuned PLS where the model utilized ncomp=5 as its optimal model.
plot(plsTune)
The plot demonstrates the models utilizing different number of components and comparing individual models (Number of Components) by the RMSE. Again, the plot demonstrates that ncomp=5 model has the lowest RMSE. The corresponding \(R^2\) is 0.4980947.
D. Predict the response for the test set. What is the test set estimate of \(R^2\)?
We will use the predict() function with 2 latent variables (2 components) for our predicted values.
# Using our testing data set
# Splitting the test set into X predictors and Y response variable
X_test <- testingData[1:(length(testingData)-1)]
Y_test <- testingData[length(testingData)]
perm_predict <- predict(plsTune, X_test , ncomp = 2)
# Displaying the head of the predicted values of X
head(perm_predict)## [1] 5.440539 4.999530 7.394288 10.289289 7.466676 5.277277# To display the full predicted values, please uncomment the following line.
print(perm_predict)## [1] 5.4405387 4.9995297 7.3942878 10.2892886 7.4666762 5.2772768
## [7] 8.2565242 1.8461556 18.9523525 5.5115093 17.0050075 -0.7995239
## [13] 10.9017248 11.1658602 4.0627791 3.1270625 -1.0284262 8.6617259
## [19] 6.1622974 19.0518750 20.0552504 11.3725030 16.1423673 -6.2918250
## [25] 15.5415223 4.8723623 3.3136476 11.8699725 -2.2939271 -1.1165278
## [31] 1.7088595 11.6394563 10.6668170Now that we have successfully created a prediction on the testing data set, let us create an accuracy including the test set \(R^2\).
RMSE <- function(m, o) {
sqrt(mean((m-o)^2))
}
paste0("RMSE for Test Dataset: ", RMSE(as.vector(perm_predict), Y_test$permeability))## [1] "RMSE for Test Dataset: 9.6434696735637"The R-squared value can help us understand how much of the model explains the test data variance. Let us calculate the value.
pls.eval = data.frame(obs = Y_test$permeability, pred=perm_predict)
defaultSummary(pls.eval)## RMSE Rsquared MAE
## 9.6434697 0.3119753 6.7925292As we can see, the R squared value here is 0.8294, or in other words, 82.94% of the data variance can be explained by this model.
Below is the plot of how the fitted data compares to the actual data.
plot(Y_test$permeability, perm_predict, main="Test Dataset", xlab="Observed", ylab="Tuned PLS Predicted")
abline(0,1,col="red")
The R-squared value here highlights the importance of the PLS algorithm. Whereas PCR tends to focus on creating components that maximizes variables’ variance, even at the expense of being noncontributory to the response variable, PLS attempts to balance both aspects. Thus, PLS should contain latent variables or components that have some correlation with the response variable.
E. Try building other models discussed in this chapter. Do you have better predictive performance?
Let us trial several different models. (As of note, we will be using the testing and training datasets as performed above. This includes the dataset that has already been cleaned from zero or near zero variances. We will also split the training data set into X and Y for simplicity track.)
# Splitting training data set into X and Y
X_train <- trainingData[1:(length(trainingData)-1)]
Y_train <- trainingData$permeabilityLinear Regression:
lm_perm <- train(X_train, Y_train,
method = "lm",
trControl = ctrl,
preProc = c("center", "scale"))
summary(lm_perm)##
## Call:
## lm(formula = .outcome ~ ., data = dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -18.6510 -0.0125 0.0000 0.2800 14.5205
##
## Coefficients: (292 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.39358 0.63939 20.947 <2e-16 ***
## X1 22.12565 10.11885 2.187 0.0355 *
## X2 14.03468 29.03659 0.483 0.6319
## X3 14.17997 21.86138 0.649 0.5208
## X4 NA NA NA NA
## X5 NA NA NA NA
## X6 3.27877 1.23117 2.663 0.0116 *
## X11 -0.74092 5.10234 -0.145 0.8854
## X12 4.17457 5.71157 0.731 0.4697
## X15 -4.94277 4.98079 -0.992 0.3278
## X16 7.80708 3.43581 2.272 0.0293 *
## X20 NA NA NA NA
## X21 NA NA NA NA
## X25 -5.83852 3.82140 -1.528 0.1355
## X26 NA NA NA NA
## X27 NA NA NA NA
## X28 NA NA NA NA
## X29 NA NA NA NA
## X35 -3.51088 11.11660 -0.316 0.7540
## X36 -0.81162 6.93560 -0.117 0.9075
## X37 NA NA NA NA
## X38 NA NA NA NA
## X39 NA NA NA NA
## X40 NA NA NA NA
## X41 -8.17118 9.11818 -0.896 0.3763
## X42 -11.42802 14.35742 -0.796 0.4314
## X43 NA NA NA NA
## X44 NA NA NA NA
## X46 NA NA NA NA
## X47 NA NA NA NA
## X48 -11.60800 20.55294 -0.565 0.5758
## X49 NA NA NA NA
## X50 NA NA NA NA
## X51 NA NA NA NA
## X52 NA NA NA NA
## X53 NA NA NA NA
## X54 NA NA NA NA
## X55 NA NA NA NA
## X56 NA NA NA NA
## X57 NA NA NA NA
## X58 NA NA NA NA
## X59 NA NA NA NA
## X60 NA NA NA NA
## X61 NA NA NA NA
## X62 NA NA NA NA
## X63 NA NA NA NA
## X64 NA NA NA NA
## X65 NA NA NA NA
## X66 NA NA NA NA
## X67 NA NA NA NA
## X68 NA NA NA NA
## X69 NA NA NA NA
## X70 NA NA NA NA
## X71 NA NA NA NA
## X72 NA NA NA NA
## X73 NA NA NA NA
## X74 NA NA NA NA
## X75 NA NA NA NA
## X76 NA NA NA NA
## X78 NA NA NA NA
## X79 NA NA NA NA
## X80 NA NA NA NA
## X86 -59.17380 25.51009 -2.320 0.0263 *
## X87 68.74019 26.99965 2.546 0.0155 *
## X88 35.53315 27.59269 1.288 0.2063
## X93 -0.06263 1.18594 -0.053 0.9582
## X94 1.89226 6.09053 0.311 0.7579
## X96 8.74994 8.18956 1.068 0.2926
## X97 NA NA NA NA
## X98 6.21666 20.97024 0.296 0.7686
## X99 2.81161 5.59384 0.503 0.6184
## X101 NA NA NA NA
## X102 -24.15461 15.71553 -1.537 0.1333
## X103 -0.77736 12.73568 -0.061 0.9517
## X108 NA NA NA NA
## X111 11.96785 33.84953 0.354 0.7258
## X118 -0.36445 2.97826 -0.122 0.9033
## X121 -20.46101 32.80013 -0.624 0.5368
## X125 10.44544 11.92529 0.876 0.3871
## X126 -0.76771 4.63840 -0.166 0.8695
## X127 NA NA NA NA
## X129 NA NA NA NA
## X130 NA NA NA NA
## X133 NA NA NA NA
## X138 NA NA NA NA
## X141 0.75569 1.26242 0.599 0.5533
## X142 NA NA NA NA
## X143 -21.43629 19.37998 -1.106 0.2762
## X146 -24.64831 16.95287 -1.454 0.1549
## X150 -1.21120 24.15616 -0.050 0.9603
## X152 NA NA NA NA
## X153 NA NA NA NA
## X154 NA NA NA NA
## X156 10.33382 7.76888 1.330 0.1921
## X157 12.88424 15.14073 0.851 0.4006
## X158 3.47863 12.71677 0.274 0.7860
## X159 -2.53599 7.56262 -0.335 0.7394
## X162 NA NA NA NA
## X163 NA NA NA NA
## X167 NA NA NA NA
## X168 NA NA NA NA
## X169 NA NA NA NA
## X170 NA NA NA NA
## X171 NA NA NA NA
## X172 NA NA NA NA
## X173 NA NA NA NA
## X174 NA NA NA NA
## X175 NA NA NA NA
## X176 NA NA NA NA
## X177 NA NA NA NA
## X178 NA NA NA NA
## X179 NA NA NA NA
## X180 NA NA NA NA
## X181 NA NA NA NA
## X182 17.45184 21.63591 0.807 0.4253
## X183 NA NA NA NA
## X184 NA NA NA NA
## X185 NA NA NA NA
## X186 NA NA NA NA
## X187 NA NA NA NA
## X188 NA NA NA NA
## X189 NA NA NA NA
## X190 NA NA NA NA
## X191 NA NA NA NA
## X192 NA NA NA NA
## X193 NA NA NA NA
## X194 NA NA NA NA
## X195 NA NA NA NA
## X196 NA NA NA NA
## X197 NA NA NA NA
## X198 NA NA NA NA
## X199 NA NA NA NA
## X200 NA NA NA NA
## X201 NA NA NA NA
## X202 NA NA NA NA
## X203 NA NA NA NA
## X204 NA NA NA NA
## X205 NA NA NA NA
## X206 NA NA NA NA
## X207 NA NA NA NA
## X208 NA NA NA NA
## X209 NA NA NA NA
## X210 NA NA NA NA
## X211 NA NA NA NA
## X212 NA NA NA NA
## X213 NA NA NA NA
## X214 NA NA NA NA
## X215 NA NA NA NA
## X221 0.60194 5.21080 0.116 0.9087
## X223 NA NA NA NA
## X224 NA NA NA NA
## X225 17.55713 9.33078 1.882 0.0682 .
## X226 -7.11967 5.17438 -1.376 0.1776
## X227 NA NA NA NA
## X228 NA NA NA NA
## X229 27.72218 21.26151 1.304 0.2008
## X230 -37.38979 19.63822 -1.904 0.0652 .
## X231 -8.30609 5.67171 -1.464 0.1520
## X232 NA NA NA NA
## X233 NA NA NA NA
## X234 NA NA NA NA
## X235 -9.43213 9.79838 -0.963 0.3423
## X236 NA NA NA NA
## X237 -14.98088 30.13161 -0.497 0.6222
## X238 9.60810 16.31921 0.589 0.5598
## X239 NA NA NA NA
## X240 NA NA NA NA
## X241 63.24454 29.37072 2.153 0.0383 *
## X242 -62.60554 38.20380 -1.639 0.1102
## X244 NA NA NA NA
## X245 NA NA NA NA
## X246 NA NA NA NA
## X247 34.25861 19.06973 1.796 0.0811 .
## X248 -32.43997 19.73289 -1.644 0.1091
## X249 NA NA NA NA
## X250 NA NA NA NA
## X251 -34.63868 40.93272 -0.846 0.4032
## X253 NA NA NA NA
## X254 NA NA NA NA
## X255 NA NA NA NA
## X256 NA NA NA NA
## X257 27.13300 25.00014 1.085 0.2852
## X258 -12.89598 7.94433 -1.623 0.1135
## X260 -41.18735 27.66173 -1.489 0.1455
## X261 NA NA NA NA
## X262 -30.25000 17.58327 -1.720 0.0942 .
## X263 NA NA NA NA
## X264 NA NA NA NA
## X265 NA NA NA NA
## X266 NA NA NA NA
## X267 NA NA NA NA
## X268 NA NA NA NA
## X269 NA NA NA NA
## X270 -11.71451 20.14019 -0.582 0.5645
## X271 21.42322 14.64611 1.463 0.1525
## X272 23.61026 23.11685 1.021 0.3141
## X274 NA NA NA NA
## X276 NA NA NA NA
## X278 -0.33940 1.27367 -0.266 0.7914
## X279 NA NA NA NA
## X280 0.57010 2.40131 0.237 0.8137
## X281 NA NA NA NA
## X284 NA NA NA NA
## X285 NA NA NA NA
## X286 NA NA NA NA
## X290 NA NA NA NA
## X291 NA NA NA NA
## X293 4.77611 3.38935 1.409 0.1676
## X294 0.04874 7.36279 0.007 0.9948
## X295 -5.47944 5.78033 -0.948 0.3497
## X296 NA NA NA NA
## X297 12.93794 6.18444 2.092 0.0438 *
## X298 NA NA NA NA
## X299 NA NA NA NA
## X300 NA NA NA NA
## X301 NA NA NA NA
## X302 NA NA NA NA
## X303 NA NA NA NA
## X304 NA NA NA NA
## X305 NA NA NA NA
## X306 -0.62784 3.04688 -0.206 0.8379
## X307 NA NA NA NA
## X308 NA NA NA NA
## X309 NA NA NA NA
## X310 NA NA NA NA
## X311 10.32867 21.07024 0.490 0.6271
## X312 -11.42604 20.90012 -0.547 0.5881
## X313 -6.70152 11.76107 -0.570 0.5724
## X314 NA NA NA NA
## X315 -0.37020 3.37318 -0.110 0.9132
## X316 0.83611 4.69428 0.178 0.8597
## X317 NA NA NA NA
## X318 NA NA NA NA
## X319 39.17942 21.22046 1.846 0.0733 .
## X320 -9.42523 14.90688 -0.632 0.5313
## X321 NA NA NA NA
## X322 NA NA NA NA
## X323 NA NA NA NA
## X324 NA NA NA NA
## X325 NA NA NA NA
## X326 NA NA NA NA
## X327 NA NA NA NA
## X328 NA NA NA NA
## X329 9.21901 12.28354 0.751 0.4580
## X330 NA NA NA NA
## X331 NA NA NA NA
## X332 NA NA NA NA
## X333 NA NA NA NA
## X334 -8.52114 10.21223 -0.834 0.4097
## X335 NA NA NA NA
## X336 NA NA NA NA
## X337 -14.90089 7.22491 -2.062 0.0467 *
## X338 13.20154 7.24298 1.823 0.0769 .
## X339 NA NA NA NA
## X340 0.63232 10.25530 0.062 0.9512
## X341 NA NA NA NA
## X342 9.08702 17.00156 0.534 0.5964
## X343 NA NA NA NA
## X344 NA NA NA NA
## X345 -1.23216 1.32326 -0.931 0.3582
## X355 NA NA NA NA
## X356 NA NA NA NA
## X357 -8.88744 14.83078 -0.599 0.5529
## X358 -11.48474 4.55849 -2.519 0.0165 *
## X359 -6.17840 7.06317 -0.875 0.3877
## X360 NA NA NA NA
## X361 0.43850 3.11934 0.141 0.8890
## X362 NA NA NA NA
## X366 NA NA NA NA
## X367 NA NA NA NA
## X368 NA NA NA NA
## X370 18.84214 14.96102 1.259 0.2162
## X371 -11.36639 14.02921 -0.810 0.4233
## X372 NA NA NA NA
## X373 NA NA NA NA
## X374 -3.56972 2.38222 -1.498 0.1430
## X376 -1.96140 10.25448 -0.191 0.8494
## X377 NA NA NA NA
## X378 NA NA NA NA
## X380 NA NA NA NA
## X381 NA NA NA NA
## X382 NA NA NA NA
## X383 NA NA NA NA
## X385 NA NA NA NA
## X386 NA NA NA NA
## X387 NA NA NA NA
## X388 NA NA NA NA
## X389 NA NA NA NA
## X390 NA NA NA NA
## X392 NA NA NA NA
## X394 NA NA NA NA
## X395 NA NA NA NA
## X396 NA NA NA NA
## X398 NA NA NA NA
## X400 NA NA NA NA
## X401 NA NA NA NA
## X403 NA NA NA NA
## X406 NA NA NA NA
## X496 4.12560 3.25668 1.267 0.2136
## X497 NA NA NA NA
## X499 NA NA NA NA
## X503 -1.04095 4.31277 -0.241 0.8107
## X504 NA NA NA NA
## X505 NA NA NA NA
## X506 NA NA NA NA
## X507 6.55895 5.80080 1.131 0.2659
## X508 NA NA NA NA
## X509 0.51040 7.70596 0.066 0.9476
## X510 NA NA NA NA
## X511 NA NA NA NA
## X512 NA NA NA NA
## X514 NA NA NA NA
## X515 NA NA NA NA
## X516 NA NA NA NA
## X517 NA NA NA NA
## X518 NA NA NA NA
## X519 NA NA NA NA
## X520 NA NA NA NA
## X521 NA NA NA NA
## X522 NA NA NA NA
## X524 NA NA NA NA
## X529 NA NA NA NA
## X549 NA NA NA NA
## X551 NA NA NA NA
## X553 NA NA NA NA
## X554 NA NA NA NA
## X556 NA NA NA NA
## X557 NA NA NA NA
## X558 NA NA NA NA
## X559 NA NA NA NA
## X560 NA NA NA NA
## X561 NA NA NA NA
## X565 NA NA NA NA
## X568 NA NA NA NA
## X571 NA NA NA NA
## X573 NA NA NA NA
## X574 NA NA NA NA
## X576 NA NA NA NA
## X577 NA NA NA NA
## X590 NA NA NA NA
## X591 NA NA NA NA
## X592 NA NA NA NA
## X593 NA NA NA NA
## X594 NA NA NA NA
## X595 NA NA NA NA
## X597 NA NA NA NA
## X598 NA NA NA NA
## X599 NA NA NA NA
## X600 NA NA NA NA
## X601 NA NA NA NA
## X602 NA NA NA NA
## X603 NA NA NA NA
## X604 NA NA NA NA
## X613 NA NA NA NA
## X621 NA NA NA NA
## X679 NA NA NA NA
## X698 1.31932 2.04511 0.645 0.5231
## X699 NA NA NA NA
## X700 NA NA NA NA
## X701 NA NA NA NA
## X702 NA NA NA NA
## X703 NA NA NA NA
## X704 -1.50108 3.16413 -0.474 0.6382
## X705 NA NA NA NA
## X719 NA NA NA NA
## X732 -0.18368 1.17882 -0.156 0.8771
## X733 NA NA NA NA
## X750 0.52797 2.15613 0.245 0.8080
## X751 NA NA NA NA
## X752 NA NA NA NA
## X753 NA NA NA NA
## X754 NA NA NA NA
## X755 NA NA NA NA
## X773 NA NA NA NA
## X774 NA NA NA NA
## X775 NA NA NA NA
## X776 NA NA NA NA
## X780 NA NA NA NA
## X782 NA NA NA NA
## X792 NA NA NA NA
## X793 NA NA NA NA
## X795 NA NA NA NA
## X798 NA NA NA NA
## X800 NA NA NA NA
## X801 NA NA NA NA
## X805 NA NA NA NA
## X806 NA NA NA NA
## X812 NA NA NA NA
## X813 NA NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.346 on 35 degrees of freedom
## Multiple R-squared: 0.9452, Adjusted R-squared: 0.7948
## F-statistic: 6.287 on 96 and 35 DF, p-value: 1.856e-08lm_predict <- predict(lm_perm, X_test)
pls.eval = data.frame(obs = Y_test$permeability, pred=lm_predict)
defaultSummary(pls.eval)## RMSE Rsquared MAE
## 22.25724007 0.09042036 12.33126817Ridge Regression:
library(elasticnet)## Loading required package: lars## Loaded lars 1.2# Creating the RidgeGrid to better fine tune the lambda value
ridgeGrid <- data.frame(.lambda = seq(0, .1, length=15))
ridge_perm <- train(X_train, Y_train,
method = "ridge",
trControl = ctrl,
tuneGrid = ridgeGrid,
preProc = c("center", "scale"))
summary(ridge_perm)## Length Class Mode
## call 4 -none- call
## actions 184 -none- list
## allset 388 -none- numeric
## beta.pure 71392 -none- numeric
## vn 388 -none- character
## mu 1 -none- numeric
## normx 388 -none- numeric
## meanx 388 -none- numeric
## lambda 1 -none- numeric
## L1norm 184 -none- numeric
## penalty 184 -none- numeric
## df 184 -none- numeric
## Cp 184 -none- numeric
## sigma2 1 -none- numeric
## xNames 388 -none- character
## problemType 1 -none- character
## tuneValue 1 data.frame list
## obsLevels 1 -none- logical
## param 0 -none- listridge_predict <- predict(ridge_perm, X_test, s = 1, mode = "fraction")
pls.eval = data.frame(obs = Y_test$permeability, pred=ridge_predict)
defaultSummary(pls.eval)## RMSE Rsquared MAE
## 12.9334035 0.1523053 9.0754735Lasso Regression:
enetGrid <- expand.grid(.lambda = c(0, 0.01, .1),
.fraction = seq(.05, 1, length = 20))
enetTune <- train(X_train, Y_train,
method = "enet",
tuneGrid = enetGrid,
trControl = ctrl,
preProc = c("center", "scale"))## Warning: model fit failed for Fold05: lambda=0.00, fraction=1 Error in if (zmin < gamhat) { : missing value where TRUE/FALSE needed## Warning in nominalTrainWorkflow(x = x, y = y, wts = weights, info =
## trainInfo, : There were missing values in resampled performance measures.summary(enetTune)## Length Class Mode
## call 4 -none- call
## actions 184 -none- list
## allset 388 -none- numeric
## beta.pure 71392 -none- numeric
## vn 388 -none- character
## mu 1 -none- numeric
## normx 388 -none- numeric
## meanx 388 -none- numeric
## lambda 1 -none- numeric
## L1norm 184 -none- numeric
## penalty 184 -none- numeric
## df 184 -none- numeric
## Cp 184 -none- numeric
## sigma2 1 -none- numeric
## xNames 388 -none- character
## problemType 1 -none- character
## tuneValue 2 data.frame list
## obsLevels 1 -none- logical
## param 0 -none- listlasso_predict <- predict(enetTune, X_test)
pls.eval = data.frame(obs = Y_test, pred=lasso_predict)
colnames(pls.eval) <- c("obs", "pred")
defaultSummary(pls.eval)## RMSE Rsquared MAE
## 12.2386446 0.1136488 8.5947579We can compare the different models noted here. If we compared models by the RMSE and R-squared, we can see that PLS performed the best,then Ridge.
F. Would you recommend any of your models to replace the permeability laboratory experiment?
The only model that I would recommend is the PLS model as that has the best R-squared and RMSE. However, if the R-squared value is still too small or the model still does not produce an accurate or acceptable level of permeability, than none of these models should be considered.
6.3 A chemical manufacturing process for a pharmaceutical product was discussed in Sect. 1.4. In this problem, the objective is to understand the relationship between biological measurements of the raw materials (predictors), measurements of the manufacturing process (predictors), and the response of product yield. Biological predictors cannot be changed but can be used to assess the quality of the raw material before processing. On the other hand, manufacturing process predictors can be changed in the manufacturing process. Improving product yield by 1% will boost revenue by approximately one hundred thousand dollars per batch:
A. Start R and use these commands to load the data.
library(AppliedPredictiveModeling)
data("ChemicalManufacturingProcess")The matrix processPredictors contains the 57 predictors (12 describing the input biological material and 45 decribing the process predictors) for the 176 manufaccturing runs. yield contains the percent yield for each run.
According to ?ChemicalManufacturingProcess, this data set contains information about a chemical manufacturing process, in which the goal is to understand the relationship between the process and the resuling final product yield. Raw material in this process is put through a sequence of 27 steps to generate the final pharmaceutical product. The starting material is generated from a biological unit and has a range of quality and characteristics. The objective in this project was to develop a model to predict percent yield of the manufacturing process.
Of the 57 characteristics, there were 12 measurements of the biological starting material, and 45 measurements of the manufacturing process. The process variables included measurements such as temperature, drying time, washing time, and concentrations of by–products at various steps. Some of the process measurements can be controlled, while others are observed. Predictors are continuous, count, categorical; some are correlated, and some contain missing values. Samples are not independent because sets of samples come from the same batch of biological starting material.
Let’s take a look at the head of the dataset.
colnames(ChemicalManufacturingProcess)## [1] "Yield" "BiologicalMaterial01"
## [3] "BiologicalMaterial02" "BiologicalMaterial03"
## [5] "BiologicalMaterial04" "BiologicalMaterial05"
## [7] "BiologicalMaterial06" "BiologicalMaterial07"
## [9] "BiologicalMaterial08" "BiologicalMaterial09"
## [11] "BiologicalMaterial10" "BiologicalMaterial11"
## [13] "BiologicalMaterial12" "ManufacturingProcess01"
## [15] "ManufacturingProcess02" "ManufacturingProcess03"
## [17] "ManufacturingProcess04" "ManufacturingProcess05"
## [19] "ManufacturingProcess06" "ManufacturingProcess07"
## [21] "ManufacturingProcess08" "ManufacturingProcess09"
## [23] "ManufacturingProcess10" "ManufacturingProcess11"
## [25] "ManufacturingProcess12" "ManufacturingProcess13"
## [27] "ManufacturingProcess14" "ManufacturingProcess15"
## [29] "ManufacturingProcess16" "ManufacturingProcess17"
## [31] "ManufacturingProcess18" "ManufacturingProcess19"
## [33] "ManufacturingProcess20" "ManufacturingProcess21"
## [35] "ManufacturingProcess22" "ManufacturingProcess23"
## [37] "ManufacturingProcess24" "ManufacturingProcess25"
## [39] "ManufacturingProcess26" "ManufacturingProcess27"
## [41] "ManufacturingProcess28" "ManufacturingProcess29"
## [43] "ManufacturingProcess30" "ManufacturingProcess31"
## [45] "ManufacturingProcess32" "ManufacturingProcess33"
## [47] "ManufacturingProcess34" "ManufacturingProcess35"
## [49] "ManufacturingProcess36" "ManufacturingProcess37"
## [51] "ManufacturingProcess38" "ManufacturingProcess39"
## [53] "ManufacturingProcess40" "ManufacturingProcess41"
## [55] "ManufacturingProcess42" "ManufacturingProcess43"
## [57] "ManufacturingProcess44" "ManufacturingProcess45"str(ChemicalManufacturingProcess)## 'data.frame': 176 obs. of 58 variables:
## $ Yield : num 38 42.4 42 41.4 42.5 ...
## $ BiologicalMaterial01 : num 6.25 8.01 8.01 8.01 7.47 6.12 7.48 6.94 6.94 6.94 ...
## $ BiologicalMaterial02 : num 49.6 61 61 61 63.3 ...
## $ BiologicalMaterial03 : num 57 67.5 67.5 67.5 72.2 ...
## $ BiologicalMaterial04 : num 12.7 14.7 14.7 14.7 14 ...
## $ BiologicalMaterial05 : num 19.5 19.4 19.4 19.4 17.9 ...
## $ BiologicalMaterial06 : num 43.7 53.1 53.1 53.1 54.7 ...
## $ BiologicalMaterial07 : num 100 100 100 100 100 100 100 100 100 100 ...
## $ BiologicalMaterial08 : num 16.7 19 19 19 18.2 ...
## $ BiologicalMaterial09 : num 11.4 12.6 12.6 12.6 12.8 ...
## $ BiologicalMaterial10 : num 3.46 3.46 3.46 3.46 3.05 3.78 3.04 3.85 3.85 3.85 ...
## $ BiologicalMaterial11 : num 138 154 154 154 148 ...
## $ BiologicalMaterial12 : num 18.8 21.1 21.1 21.1 21.1 ...
## $ ManufacturingProcess01: num NA 0 0 0 10.7 12 11.5 12 12 12 ...
## $ ManufacturingProcess02: num NA 0 0 0 0 0 0 0 0 0 ...
## $ ManufacturingProcess03: num NA NA NA NA NA NA 1.56 1.55 1.56 1.55 ...
## $ ManufacturingProcess04: num NA 917 912 911 918 924 933 929 928 938 ...
## $ ManufacturingProcess05: num NA 1032 1004 1015 1028 ...
## $ ManufacturingProcess06: num NA 210 207 213 206 ...
## $ ManufacturingProcess07: num NA 177 178 177 178 178 177 178 177 177 ...
## $ ManufacturingProcess08: num NA 178 178 177 178 178 178 178 177 177 ...
## $ ManufacturingProcess09: num 43 46.6 45.1 44.9 45 ...
## $ ManufacturingProcess10: num NA NA NA NA NA NA 11.6 10.2 9.7 10.1 ...
## $ ManufacturingProcess11: num NA NA NA NA NA NA 11.5 11.3 11.1 10.2 ...
## $ ManufacturingProcess12: num NA 0 0 0 0 0 0 0 0 0 ...
## $ ManufacturingProcess13: num 35.5 34 34.8 34.8 34.6 34 32.4 33.6 33.9 34.3 ...
## $ ManufacturingProcess14: num 4898 4869 4878 4897 4992 ...
## $ ManufacturingProcess15: num 6108 6095 6087 6102 6233 ...
## $ ManufacturingProcess16: num 4682 4617 4617 4635 4733 ...
## $ ManufacturingProcess17: num 35.5 34 34.8 34.8 33.9 33.4 33.8 33.6 33.9 35.3 ...
## $ ManufacturingProcess18: num 4865 4867 4877 4872 4886 ...
## $ ManufacturingProcess19: num 6049 6097 6078 6073 6102 ...
## $ ManufacturingProcess20: num 4665 4621 4621 4611 4659 ...
## $ ManufacturingProcess21: num 0 0 0 0 -0.7 -0.6 1.4 0 0 1 ...
## $ ManufacturingProcess22: num NA 3 4 5 8 9 1 2 3 4 ...
## $ ManufacturingProcess23: num NA 0 1 2 4 1 1 2 3 1 ...
## $ ManufacturingProcess24: num NA 3 4 5 18 1 1 2 3 4 ...
## $ ManufacturingProcess25: num 4873 4869 4897 4892 4930 ...
## $ ManufacturingProcess26: num 6074 6107 6116 6111 6151 ...
## $ ManufacturingProcess27: num 4685 4630 4637 4630 4684 ...
## $ ManufacturingProcess28: num 10.7 11.2 11.1 11.1 11.3 11.4 11.2 11.1 11.3 11.4 ...
## $ ManufacturingProcess29: num 21 21.4 21.3 21.3 21.6 21.7 21.2 21.2 21.5 21.7 ...
## $ ManufacturingProcess30: num 9.9 9.9 9.4 9.4 9 10.1 11.2 10.9 10.5 9.8 ...
## $ ManufacturingProcess31: num 69.1 68.7 69.3 69.3 69.4 68.2 67.6 67.9 68 68.5 ...
## $ ManufacturingProcess32: num 156 169 173 171 171 173 159 161 160 164 ...
## $ ManufacturingProcess33: num 66 66 66 68 70 70 65 65 65 66 ...
## $ ManufacturingProcess34: num 2.4 2.6 2.6 2.5 2.5 2.5 2.5 2.5 2.5 2.5 ...
## $ ManufacturingProcess35: num 486 508 509 496 468 490 475 478 491 488 ...
## $ ManufacturingProcess36: num 0.019 0.019 0.018 0.018 0.017 0.018 0.019 0.019 0.019 0.019 ...
## $ ManufacturingProcess37: num 0.5 2 0.7 1.2 0.2 0.4 0.8 1 1.2 1.8 ...
## $ ManufacturingProcess38: num 3 2 2 2 2 2 2 2 3 3 ...
## $ ManufacturingProcess39: num 7.2 7.2 7.2 7.2 7.3 7.2 7.3 7.3 7.4 7.1 ...
## $ ManufacturingProcess40: num NA 0.1 0 0 0 0 0 0 0 0 ...
## $ ManufacturingProcess41: num NA 0.15 0 0 0 0 0 0 0 0 ...
## $ ManufacturingProcess42: num 11.6 11.1 12 10.6 11 11.5 11.7 11.4 11.4 11.3 ...
## $ ManufacturingProcess43: num 3 0.9 1 1.1 1.1 2.2 0.7 0.8 0.9 0.8 ...
## $ ManufacturingProcess44: num 1.8 1.9 1.8 1.8 1.7 1.8 2 2 1.9 1.9 ...
## $ ManufacturingProcess45: num 2.4 2.2 2.3 2.1 2.1 2 2.2 2.2 2.1 2.4 ...head(ChemicalManufacturingProcess)## Yield BiologicalMaterial01 BiologicalMaterial02 BiologicalMaterial03
## 1 38.00 6.25 49.58 56.97
## 2 42.44 8.01 60.97 67.48
## 3 42.03 8.01 60.97 67.48
## 4 41.42 8.01 60.97 67.48
## 5 42.49 7.47 63.33 72.25
## 6 43.57 6.12 58.36 65.31
## BiologicalMaterial04 BiologicalMaterial05 BiologicalMaterial06
## 1 12.74 19.51 43.73
## 2 14.65 19.36 53.14
## 3 14.65 19.36 53.14
## 4 14.65 19.36 53.14
## 5 14.02 17.91 54.66
## 6 15.17 21.79 51.23
## BiologicalMaterial07 BiologicalMaterial08 BiologicalMaterial09
## 1 100 16.66 11.44
## 2 100 19.04 12.55
## 3 100 19.04 12.55
## 4 100 19.04 12.55
## 5 100 18.22 12.80
## 6 100 18.30 12.13
## BiologicalMaterial10 BiologicalMaterial11 BiologicalMaterial12
## 1 3.46 138.09 18.83
## 2 3.46 153.67 21.05
## 3 3.46 153.67 21.05
## 4 3.46 153.67 21.05
## 5 3.05 147.61 21.05
## 6 3.78 151.88 20.76
## ManufacturingProcess01 ManufacturingProcess02 ManufacturingProcess03
## 1 NA NA NA
## 2 0.0 0 NA
## 3 0.0 0 NA
## 4 0.0 0 NA
## 5 10.7 0 NA
## 6 12.0 0 NA
## ManufacturingProcess04 ManufacturingProcess05 ManufacturingProcess06
## 1 NA NA NA
## 2 917 1032.2 210.0
## 3 912 1003.6 207.1
## 4 911 1014.6 213.3
## 5 918 1027.5 205.7
## 6 924 1016.8 208.9
## ManufacturingProcess07 ManufacturingProcess08 ManufacturingProcess09
## 1 NA NA 43.00
## 2 177 178 46.57
## 3 178 178 45.07
## 4 177 177 44.92
## 5 178 178 44.96
## 6 178 178 45.32
## ManufacturingProcess10 ManufacturingProcess11 ManufacturingProcess12
## 1 NA NA NA
## 2 NA NA 0
## 3 NA NA 0
## 4 NA NA 0
## 5 NA NA 0
## 6 NA NA 0
## ManufacturingProcess13 ManufacturingProcess14 ManufacturingProcess15
## 1 35.5 4898 6108
## 2 34.0 4869 6095
## 3 34.8 4878 6087
## 4 34.8 4897 6102
## 5 34.6 4992 6233
## 6 34.0 4985 6222
## ManufacturingProcess16 ManufacturingProcess17 ManufacturingProcess18
## 1 4682 35.5 4865
## 2 4617 34.0 4867
## 3 4617 34.8 4877
## 4 4635 34.8 4872
## 5 4733 33.9 4886
## 6 4786 33.4 4862
## ManufacturingProcess19 ManufacturingProcess20 ManufacturingProcess21
## 1 6049 4665 0.0
## 2 6097 4621 0.0
## 3 6078 4621 0.0
## 4 6073 4611 0.0
## 5 6102 4659 -0.7
## 6 6115 4696 -0.6
## ManufacturingProcess22 ManufacturingProcess23 ManufacturingProcess24
## 1 NA NA NA
## 2 3 0 3
## 3 4 1 4
## 4 5 2 5
## 5 8 4 18
## 6 9 1 1
## ManufacturingProcess25 ManufacturingProcess26 ManufacturingProcess27
## 1 4873 6074 4685
## 2 4869 6107 4630
## 3 4897 6116 4637
## 4 4892 6111 4630
## 5 4930 6151 4684
## 6 4871 6128 4687
## ManufacturingProcess28 ManufacturingProcess29 ManufacturingProcess30
## 1 10.7 21.0 9.9
## 2 11.2 21.4 9.9
## 3 11.1 21.3 9.4
## 4 11.1 21.3 9.4
## 5 11.3 21.6 9.0
## 6 11.4 21.7 10.1
## ManufacturingProcess31 ManufacturingProcess32 ManufacturingProcess33
## 1 69.1 156 66
## 2 68.7 169 66
## 3 69.3 173 66
## 4 69.3 171 68
## 5 69.4 171 70
## 6 68.2 173 70
## ManufacturingProcess34 ManufacturingProcess35 ManufacturingProcess36
## 1 2.4 486 0.019
## 2 2.6 508 0.019
## 3 2.6 509 0.018
## 4 2.5 496 0.018
## 5 2.5 468 0.017
## 6 2.5 490 0.018
## ManufacturingProcess37 ManufacturingProcess38 ManufacturingProcess39
## 1 0.5 3 7.2
## 2 2.0 2 7.2
## 3 0.7 2 7.2
## 4 1.2 2 7.2
## 5 0.2 2 7.3
## 6 0.4 2 7.2
## ManufacturingProcess40 ManufacturingProcess41 ManufacturingProcess42
## 1 NA NA 11.6
## 2 0.1 0.15 11.1
## 3 0.0 0.00 12.0
## 4 0.0 0.00 10.6
## 5 0.0 0.00 11.0
## 6 0.0 0.00 11.5
## ManufacturingProcess43 ManufacturingProcess44 ManufacturingProcess45
## 1 3.0 1.8 2.4
## 2 0.9 1.9 2.2
## 3 1.0 1.8 2.3
## 4 1.1 1.8 2.1
## 5 1.1 1.7 2.1
## 6 2.2 1.8 2.0B. A small percentage of cells in the predictor set contain missing values. Use an imputation function to fill in these missing values (e.g., see Sec 3.8).
What is missing from this dataset?
is_na <- sort(colSums(is.na(ChemicalManufacturingProcess)))
is_na[is_na > 0]## ManufacturingProcess01 ManufacturingProcess04 ManufacturingProcess05
## 1 1 1
## ManufacturingProcess07 ManufacturingProcess08 ManufacturingProcess12
## 1 1 1
## ManufacturingProcess14 ManufacturingProcess22 ManufacturingProcess23
## 1 1 1
## ManufacturingProcess24 ManufacturingProcess40 ManufacturingProcess41
## 1 1 1
## ManufacturingProcess06 ManufacturingProcess02 ManufacturingProcess25
## 2 3 5
## ManufacturingProcess26 ManufacturingProcess27 ManufacturingProcess28
## 5 5 5
## ManufacturingProcess29 ManufacturingProcess30 ManufacturingProcess31
## 5 5 5
## ManufacturingProcess33 ManufacturingProcess34 ManufacturingProcess35
## 5 5 5
## ManufacturingProcess36 ManufacturingProcess10 ManufacturingProcess11
## 5 9 10
## ManufacturingProcess03
## 15Given the numerous different columns that exist in this dataframe, we will impute the missing data using the KNN algorithm (with 3 neighbors).
# Reference: https://www.r-bloggers.com/missing-value-treatment/
library(DMwR)## Loading required package: gridknn_output_df <- knnImputation(ChemicalManufacturingProcess[, 1:57], k = 3, meth = "weighAvg")
anyNA(knn_output_df)## [1] FALSEWe have successfully imputed the missing data points with KNN with 3 neighbors. Let us take a look at the head of this dataset.
head(knn_output_df)## Yield BiologicalMaterial01 BiologicalMaterial02 BiologicalMaterial03
## 1 38.00 6.25 49.58 56.97
## 2 42.44 8.01 60.97 67.48
## 3 42.03 8.01 60.97 67.48
## 4 41.42 8.01 60.97 67.48
## 5 42.49 7.47 63.33 72.25
## 6 43.57 6.12 58.36 65.31
## BiologicalMaterial04 BiologicalMaterial05 BiologicalMaterial06
## 1 12.74 19.51 43.73
## 2 14.65 19.36 53.14
## 3 14.65 19.36 53.14
## 4 14.65 19.36 53.14
## 5 14.02 17.91 54.66
## 6 15.17 21.79 51.23
## BiologicalMaterial07 BiologicalMaterial08 BiologicalMaterial09
## 1 100 16.66 11.44
## 2 100 19.04 12.55
## 3 100 19.04 12.55
## 4 100 19.04 12.55
## 5 100 18.22 12.80
## 6 100 18.30 12.13
## BiologicalMaterial10 BiologicalMaterial11 BiologicalMaterial12
## 1 3.46 138.09 18.83
## 2 3.46 153.67 21.05
## 3 3.46 153.67 21.05
## 4 3.46 153.67 21.05
## 5 3.05 147.61 21.05
## 6 3.78 151.88 20.76
## ManufacturingProcess01 ManufacturingProcess02 ManufacturingProcess03
## 1 11.57415 21.43705 1.547340
## 2 0.00000 0.00000 1.553157
## 3 0.00000 0.00000 1.544921
## 4 0.00000 0.00000 1.552545
## 5 10.70000 0.00000 1.550000
## 6 12.00000 0.00000 1.550000
## ManufacturingProcess04 ManufacturingProcess05 ManufacturingProcess06
## 1 933.3301 988.3955 206.1187
## 2 917.0000 1032.2000 210.0000
## 3 912.0000 1003.6000 207.1000
## 4 911.0000 1014.6000 213.3000
## 5 918.0000 1027.5000 205.7000
## 6 924.0000 1016.8000 208.9000
## ManufacturingProcess07 ManufacturingProcess08 ManufacturingProcess09
## 1 177.266 177.266 43.00
## 2 177.000 178.000 46.57
## 3 178.000 178.000 45.07
## 4 177.000 177.000 44.92
## 5 178.000 178.000 44.96
## 6 178.000 178.000 45.32
## ManufacturingProcess10 ManufacturingProcess11 ManufacturingProcess12
## 1 8.963704 9.175013 0
## 2 9.561514 10.221835 0
## 3 9.337431 9.594266 0
## 4 9.261935 10.158557 0
## 5 8.905721 9.786672 0
## 6 8.944334 9.817275 0
## ManufacturingProcess13 ManufacturingProcess14 ManufacturingProcess15
## 1 35.5 4898 6108
## 2 34.0 4869 6095
## 3 34.8 4878 6087
## 4 34.8 4897 6102
## 5 34.6 4992 6233
## 6 34.0 4985 6222
## ManufacturingProcess16 ManufacturingProcess17 ManufacturingProcess18
## 1 4682 35.5 4865
## 2 4617 34.0 4867
## 3 4617 34.8 4877
## 4 4635 34.8 4872
## 5 4733 33.9 4886
## 6 4786 33.4 4862
## ManufacturingProcess19 ManufacturingProcess20 ManufacturingProcess21
## 1 6049 4665 0.0
## 2 6097 4621 0.0
## 3 6078 4621 0.0
## 4 6073 4611 0.0
## 5 6102 4659 -0.7
## 6 6115 4696 -0.6
## ManufacturingProcess22 ManufacturingProcess23 ManufacturingProcess24
## 1 6.694152 4.322566 11.57939
## 2 3.000000 0.000000 3.00000
## 3 4.000000 1.000000 4.00000
## 4 5.000000 2.000000 5.00000
## 5 8.000000 4.000000 18.00000
## 6 9.000000 1.000000 1.00000
## ManufacturingProcess25 ManufacturingProcess26 ManufacturingProcess27
## 1 4873 6074 4685
## 2 4869 6107 4630
## 3 4897 6116 4637
## 4 4892 6111 4630
## 5 4930 6151 4684
## 6 4871 6128 4687
## ManufacturingProcess28 ManufacturingProcess29 ManufacturingProcess30
## 1 10.7 21.0 9.9
## 2 11.2 21.4 9.9
## 3 11.1 21.3 9.4
## 4 11.1 21.3 9.4
## 5 11.3 21.6 9.0
## 6 11.4 21.7 10.1
## ManufacturingProcess31 ManufacturingProcess32 ManufacturingProcess33
## 1 69.1 156 66
## 2 68.7 169 66
## 3 69.3 173 66
## 4 69.3 171 68
## 5 69.4 171 70
## 6 68.2 173 70
## ManufacturingProcess34 ManufacturingProcess35 ManufacturingProcess36
## 1 2.4 486 0.019
## 2 2.6 508 0.019
## 3 2.6 509 0.018
## 4 2.5 496 0.018
## 5 2.5 468 0.017
## 6 2.5 490 0.018
## ManufacturingProcess37 ManufacturingProcess38 ManufacturingProcess39
## 1 0.5 3 7.2
## 2 2.0 2 7.2
## 3 0.7 2 7.2
## 4 1.2 2 7.2
## 5 0.2 2 7.3
## 6 0.4 2 7.2
## ManufacturingProcess40 ManufacturingProcess41 ManufacturingProcess42
## 1 0.0 0.00 11.6
## 2 0.1 0.15 11.1
## 3 0.0 0.00 12.0
## 4 0.0 0.00 10.6
## 5 0.0 0.00 11.0
## 6 0.0 0.00 11.5
## ManufacturingProcess43 ManufacturingProcess44
## 1 3.0 1.8
## 2 0.9 1.9
## 3 1.0 1.8
## 4 1.1 1.8
## 5 1.1 1.7
## 6 2.2 1.8C. Split the data into a training and a test set, pre-process the data, and tune a model of your choice from this chapter. What is the optimal value of the performance metric?
Let us evaluate multiple different models.
First, remove zero or near zero variables.
near_zero <- nearZeroVar(knn_output_df)
knn_output_df <- knn_output_df[,-near_zero]Split the data into training and testing data sets.
# Reference: https://topepo.github.io/caret/model-training-and-tuning.html
library(caret)
inTraining <- createDataPartition(knn_output_df$Yield, p = 0.80, list=FALSE)
training <- knn_output_df[ inTraining,]
testing <- knn_output_df[-inTraining,]
X <- training[,2:(length(training))]
Y <- training$Yield
X_test <- testing[,2:(length(testing))]
Y_test <- testing$YieldLet us create a repeated K-fold validation using the training data.
fitControl <- trainControl(## 10-fold CV
method = "repeatedcv",
number = 10,
## repeated ten times
repeats = 10)Now let us try to fit multiple different models. (We also center and scale the data.)
Linear Regression:
lmFit1 <- train(Yield ~ ., data = training,
method = "lm",
trControl = fitControl,
preProcess = c("center", "scale"))
lmFit1## Linear Regression
##
## 144 samples
## 55 predictor
##
## Pre-processing: centered (55), scaled (55)
## Resampling: Cross-Validated (10 fold, repeated 10 times)
## Summary of sample sizes: 131, 128, 131, 129, 131, 130, ...
## Resampling results:
##
## RMSE Rsquared MAE
## 3.755443 0.3699803 1.884209
##
## Tuning parameter 'intercept' was held constant at a value of TRUElm_predict <- predict(lmFit1, X_test)
pls.eval = data.frame(obs = Y_test, pred=lm_predict)
defaultSummary(pls.eval)## RMSE Rsquared MAE
## 35.501311024 0.004407862 7.060969737L1 Regularization (Lasso):
enetGrid <- expand.grid(.lambda = c(0, 0.01, .1),
.fraction = seq(.05, 1, length = 20))
enetTune <- train(X, Y,
method = "enet",
tuneGrid = enetGrid,
trControl = ctrl,
preProc = c("center", "scale"))
enetTune## Elasticnet
##
## 144 samples
## 55 predictor
##
## Pre-processing: centered (55), scaled (55)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 129, 129, 132, 130, 129, 128, ...
## Resampling results across tuning parameters:
##
## lambda fraction RMSE Rsquared MAE
## 0.00 0.05 1.293704 0.5727579 1.0568796
## 0.00 0.10 1.184856 0.5856747 0.9585989
## 0.00 0.15 1.454452 0.5259946 1.0348906
## 0.00 0.20 1.642978 0.5069678 1.1012191
## 0.00 0.25 1.694782 0.4968470 1.1265830
## 0.00 0.30 1.699699 0.4912833 1.1438488
## 0.00 0.35 1.637276 0.4904836 1.1415942
## 0.00 0.40 1.479154 0.5005173 1.1359263
## 0.00 0.45 1.726742 0.4401298 1.2208945
## 0.00 0.50 2.049722 0.3917739 1.3190214
## 0.00 0.55 2.287806 0.3737031 1.3881701
## 0.00 0.60 2.513011 0.3601518 1.4553294
## 0.00 0.65 2.727934 0.3499126 1.5191724
## 0.00 0.70 2.895844 0.3417457 1.5696663
## 0.00 0.75 3.037842 0.3346265 1.6125323
## 0.00 0.80 3.134344 0.3300129 1.6417944
## 0.00 0.85 3.224111 0.3255748 1.6692207
## 0.00 0.90 3.314813 0.3211657 1.6974550
## 0.00 0.95 3.399163 0.3164010 1.7241917
## 0.00 1.00 3.472128 0.3118088 1.7470118
## 0.01 0.05 1.556894 0.5300258 1.2520307
## 0.01 0.10 1.323959 0.5761841 1.0772309
## 0.01 0.15 1.215661 0.5774209 1.0039198
## 0.01 0.20 1.182875 0.5858000 0.9584362
## 0.01 0.25 1.163978 0.5947054 0.9316350
## 0.01 0.30 1.320091 0.5428456 0.9909924
## 0.01 0.35 1.451525 0.5229400 1.0416603
## 0.01 0.40 1.543579 0.5164543 1.0712703
## 0.01 0.45 1.618062 0.5104602 1.0964253
## 0.01 0.50 1.655560 0.5050705 1.1131580
## 0.01 0.55 1.671184 0.5010500 1.1242356
## 0.01 0.60 1.646733 0.4999308 1.1237128
## 0.01 0.65 1.603948 0.4998720 1.1178206
## 0.01 0.70 1.550242 0.4999949 1.1111020
## 0.01 0.75 1.524891 0.4975729 1.1122286
## 0.01 0.80 1.456444 0.5009853 1.0956618
## 0.01 0.85 1.390057 0.5113694 1.0824917
## 0.01 0.90 1.340287 0.5289814 1.0669862
## 0.01 0.95 1.341712 0.5266201 1.0661031
## 0.01 1.00 1.397548 0.4991660 1.1038129
## 0.10 0.05 1.668319 0.4620199 1.3414137
## 0.10 0.10 1.506849 0.5447006 1.2124366
## 0.10 0.15 1.365241 0.5728497 1.1032467
## 0.10 0.20 1.262713 0.5784281 1.0344739
## 0.10 0.25 1.215298 0.5753146 1.0025234
## 0.10 0.30 1.197250 0.5766440 0.9784317
## 0.10 0.35 1.180563 0.5844490 0.9531289
## 0.10 0.40 1.169930 0.5900891 0.9396729
## 0.10 0.45 1.265215 0.5562936 0.9732962
## 0.10 0.50 1.382741 0.5392798 1.0094634
## 0.10 0.55 1.449038 0.5323189 1.0321538
## 0.10 0.60 1.501292 0.5289396 1.0508984
## 0.10 0.65 1.556295 0.5251250 1.0720837
## 0.10 0.70 1.569835 0.5191121 1.0869812
## 0.10 0.75 1.552184 0.5126267 1.0923810
## 0.10 0.80 1.559334 0.5032859 1.1023833
## 0.10 0.85 1.570190 0.4953804 1.1115091
## 0.10 0.90 1.571994 0.4905710 1.1167554
## 0.10 0.95 1.566373 0.4883040 1.1188133
## 0.10 1.00 1.562787 0.4863931 1.1217230
##
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were fraction = 0.25 and lambda = 0.01.lasso_predict <- predict(enetTune, X_test)
pls.eval = data.frame(obs = Y_test, pred=lasso_predict)
defaultSummary(pls.eval)## RMSE Rsquared MAE
## 1.1671136 0.5665374 0.8481957L2 Ridge Regression:
ridgeGrid <- data.frame(.lambda = seq(0, .1, length=15))
ridgeFit <- train(X, Y,
method = "ridge",
trControl = ctrl,
tuneGrid = ridgeGrid,
preProc = c("center", "scale"))
ridgeFit## Ridge Regression
##
## 144 samples
## 55 predictor
##
## Pre-processing: centered (55), scaled (55)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 129, 129, 130, 130, 131, 129, ...
## Resampling results across tuning parameters:
##
## lambda RMSE Rsquared MAE
## 0.000000000 3.462524 0.4042364 1.728815
## 0.007142857 1.685784 0.5059081 1.194027
## 0.014285714 1.373601 0.5486381 1.074836
## 0.021428571 1.290372 0.5675075 1.048995
## 0.028571429 1.296655 0.5742562 1.043411
## 0.035714286 1.337157 0.5634232 1.065428
## 0.042857143 1.379675 0.5532834 1.081110
## 0.050000000 1.416236 0.5463701 1.092619
## 0.057142857 1.446525 0.5416899 1.101289
## 0.064285714 1.471596 0.5384086 1.108356
## 0.071428571 1.492538 0.5360243 1.113998
## 0.078571429 1.510242 0.5342394 1.118955
## 0.085714286 1.525399 0.5328709 1.123079
## 0.092857143 1.538535 0.5318016 1.126555
## 0.100000000 1.550056 0.5309537 1.129562
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was lambda = 0.02142857.ridge_predict <- predict(ridgeFit, X_test, s = 1, mode = "fraction")
pls.eval = data.frame(obs = Y_test, pred=ridge_predict)
defaultSummary(pls.eval)## RMSE Rsquared MAE
## 9.38631653 0.03346543 2.37322610Partial Least Squares:
plsTune <- train(X, Y,
method = "pls",
tuneLength = 20,
trControl = ctrl,
preProc = c("center", "scale"))
summary(plsTune)## Data: X dimension: 144 55
## Y dimension: 144 1
## Fit method: oscorespls
## Number of components considered: 3
## TRAINING: % variance explained
## 1 comps 2 comps 3 comps
## X 19.03 29.47 36.01
## .outcome 46.07 60.68 67.60plot(plsTune)
We will be using ncomp=3.
pls_predict <- predict(plsTune, X_test , ncomp = 3)
pls.eval = data.frame(obs = Y_test, pred=pls_predict)
defaultSummary(pls.eval)## RMSE Rsquared MAE
## 1.2345155 0.5448403 0.9492541Comparing the RMSE and R-squared values, the L1 Lasso Regression is the best model to use here.
D. Predict the response for the test set. What is the value of the performance metric and how does this compare with the resampled perfomance metric on the training set?
The training set RMSE is 1.192336. The testing set RMSE is 1.0321298. The testing dataset overall performed well in the testing data set. (In other words, they are comparable.)
The predicted responses are below.
lasso_predict## 5 9 11 15 16 27 52 53
## 42.77002 41.75710 41.52341 40.82298 41.24813 36.27978 41.47797 42.21578
## 64 65 67 71 77 81 86 91
## 39.40874 41.85053 41.65124 39.51028 40.31666 39.89095 40.45945 39.50926
## 97 99 100 102 108 125 132 144
## 39.47321 38.55110 38.31076 38.15294 35.83006 39.79271 40.25121 40.00585
## 151 158 163 166 167 170 174 175
## 38.74265 39.12953 40.13682 38.46058 38.03143 39.67253 42.06144 40.49185E. Which predictors are most important in the model you have trained? Do either the biological or process predictors dominate the list?
From the model we have developed from cross-validation, the lasso model demonstrates an optimal model of fraction = 0.1 and lambda = 0.0. Let us rebuild the model using the enet function such that we can see the most important coefficients or predictors that have contributed to the model.
# Rebuild the model
lassoModel <- enet(x = as.matrix(X), y = Y,
lambda = 0.0, normalize = TRUE)
lassoCoef <- predict(lassoModel, newx = as.matrix(X_test),
s=.1, mode = "fraction", type = "coefficients")
sort(lassoCoef$coefficients)## ManufacturingProcess36 ManufacturingProcess17 ManufacturingProcess13
## -2.455557e+02 -1.993376e-01 -1.370443e-01
## ManufacturingProcess37 BiologicalMaterial01 BiologicalMaterial02
## -1.024329e-01 0.000000e+00 0.000000e+00
## BiologicalMaterial04 BiologicalMaterial06 BiologicalMaterial08
## 0.000000e+00 0.000000e+00 0.000000e+00
## BiologicalMaterial09 BiologicalMaterial10 BiologicalMaterial11
## 0.000000e+00 0.000000e+00 0.000000e+00
## BiologicalMaterial12 ManufacturingProcess01 ManufacturingProcess02
## 0.000000e+00 0.000000e+00 0.000000e+00
## ManufacturingProcess03 ManufacturingProcess04 ManufacturingProcess05
## 0.000000e+00 0.000000e+00 0.000000e+00
## ManufacturingProcess07 ManufacturingProcess08 ManufacturingProcess10
## 0.000000e+00 0.000000e+00 0.000000e+00
## ManufacturingProcess11 ManufacturingProcess12 ManufacturingProcess14
## 0.000000e+00 0.000000e+00 0.000000e+00
## ManufacturingProcess15 ManufacturingProcess16 ManufacturingProcess18
## 0.000000e+00 0.000000e+00 0.000000e+00
## ManufacturingProcess19 ManufacturingProcess20 ManufacturingProcess22
## 0.000000e+00 0.000000e+00 0.000000e+00
## ManufacturingProcess23 ManufacturingProcess24 ManufacturingProcess25
## 0.000000e+00 0.000000e+00 0.000000e+00
## ManufacturingProcess26 ManufacturingProcess27 ManufacturingProcess28
## 0.000000e+00 0.000000e+00 0.000000e+00
## ManufacturingProcess31 ManufacturingProcess33 ManufacturingProcess35
## 0.000000e+00 0.000000e+00 0.000000e+00
## ManufacturingProcess38 ManufacturingProcess40 ManufacturingProcess41
## 0.000000e+00 0.000000e+00 0.000000e+00
## ManufacturingProcess42 ManufacturingProcess43 ManufacturingProcess30
## 0.000000e+00 0.000000e+00 7.597564e-04
## BiologicalMaterial03 ManufacturingProcess06 BiologicalMaterial05
## 1.401715e-02 2.461960e-02 2.964591e-02
## ManufacturingProcess39 ManufacturingProcess44 ManufacturingProcess29
## 3.568471e-02 5.826039e-02 1.105585e-01
## ManufacturingProcess32 ManufacturingProcess09 ManufacturingProcess34
## 1.384560e-01 2.793302e-01 1.673640e+00There are a few variables with 0 coefficients. In other words, in a lasso model, these variables are selected out. Let us remove these variables.
list_coef <- lassoCoef$coefficients
sort(list_coef[list_coef != 0])## ManufacturingProcess36 ManufacturingProcess17 ManufacturingProcess13
## -2.455557e+02 -1.993376e-01 -1.370443e-01
## ManufacturingProcess37 ManufacturingProcess30 BiologicalMaterial03
## -1.024329e-01 7.597564e-04 1.401715e-02
## ManufacturingProcess06 BiologicalMaterial05 ManufacturingProcess39
## 2.461960e-02 2.964591e-02 3.568471e-02
## ManufacturingProcess44 ManufacturingProcess29 ManufacturingProcess32
## 5.826039e-02 1.105585e-01 1.384560e-01
## ManufacturingProcess09 ManufacturingProcess34
## 2.793302e-01 1.673640e+00There are 9 Manufacturing Processing variables vs. 1 Biological Material variables. In this particular model, it appears that the manufacturing process dominates over the biological material variables.
F. Explore the relationships between each of the top predictors and the response. How could this information be helpful in improving yield in future runs of the manufacturing process?
The above coefficients can help guide us in regards to the manufacturing process. The top responses appear to be Manufacturing Process 35 and Manufacturing Process 36. Because yield is what we are interested in, if we desire to increase the yield, a data scientist’s recommendations would be to increase the Manufacturing Process 34 and decrease the Manufacturing Process 36. The former appears to have aimprovement when increased, whereas the latter appears to decrease the yield every time process 36 is increased.