DATA 624 - Project 2
Kory Martin
2023-11-26
Introduction
For this project, the focus is on looking to develop a model that allows us to predict the pH for our products and understand which factors are used in helping us to predict the output. In order to do this we will be looking at exploring different modeling techniques to try and predict the pH level. Given that we are looking to predict a continuous variable, we will use cnoider the following models:
- Linear Regression
- Decision Tree - Single Tree
- Random Forest
- Support Vector Machine
- Neural Network
Each of these have there own stengths and weaknesses, but given that one of our main objectives is to be able to explain the process associated with the manufacturing process, we anticipate one of our decision criteria will be based on our ability to explain the model and the relationship between our explanatory variables and the response variable.
Setup
We begin by importing the libraries that will be used for exploring the data as well as to build our machine learning models. The caret library will be the primary model that we will use in building our machine learning models.
Import and clean data
Next, we important the data that will be used to train and evaluate our models. One of the first pre-processing steps we are taking is to standardize the variable names for the variables in our dataframe.
sample_data_raw <- read_excel('StudentData.xlsx')
sample_data <- clean_names(sample_data_raw)
sample_data <- as_tibble(sample_data)Data Exploration
We will being by looking at the data and understanding the variables that are present and exploring the shape of the variables. Some of the main things that we are focused on exploring will be:
- Missing variables
- Shape and distribution of the response variable
- Shape and distribution of the explanatory variables
- Any outliers within the data
- Correlation between explanatory variables and the response variable
- Correlation between explanatory variables
sample_data %>% summary()## brand_code carb_volume fill_ounces pc_volume
## Length:2571 Min. :5.040 Min. :23.63 Min. :0.07933
## Class :character 1st Qu.:5.293 1st Qu.:23.92 1st Qu.:0.23917
## Mode :character Median :5.347 Median :23.97 Median :0.27133
## Mean :5.370 Mean :23.97 Mean :0.27712
## 3rd Qu.:5.453 3rd Qu.:24.03 3rd Qu.:0.31200
## Max. :5.700 Max. :24.32 Max. :0.47800
## NA's :10 NA's :38 NA's :39
## carb_pressure carb_temp psc psc_fill
## Min. :57.00 Min. :128.6 Min. :0.00200 Min. :0.0000
## 1st Qu.:65.60 1st Qu.:138.4 1st Qu.:0.04800 1st Qu.:0.1000
## Median :68.20 Median :140.8 Median :0.07600 Median :0.1800
## Mean :68.19 Mean :141.1 Mean :0.08457 Mean :0.1954
## 3rd Qu.:70.60 3rd Qu.:143.8 3rd Qu.:0.11200 3rd Qu.:0.2600
## Max. :79.40 Max. :154.0 Max. :0.27000 Max. :0.6200
## NA's :27 NA's :26 NA's :33 NA's :23
## psc_co2 mnf_flow carb_pressure1 fill_pressure
## Min. :0.00000 Min. :-100.20 Min. :105.6 Min. :34.60
## 1st Qu.:0.02000 1st Qu.:-100.00 1st Qu.:119.0 1st Qu.:46.00
## Median :0.04000 Median : 65.20 Median :123.2 Median :46.40
## Mean :0.05641 Mean : 24.57 Mean :122.6 Mean :47.92
## 3rd Qu.:0.08000 3rd Qu.: 140.80 3rd Qu.:125.4 3rd Qu.:50.00
## Max. :0.24000 Max. : 229.40 Max. :140.2 Max. :60.40
## NA's :39 NA's :2 NA's :32 NA's :22
## hyd_pressure1 hyd_pressure2 hyd_pressure3 hyd_pressure4
## Min. :-0.80 Min. : 0.00 Min. :-1.20 Min. : 52.00
## 1st Qu.: 0.00 1st Qu.: 0.00 1st Qu.: 0.00 1st Qu.: 86.00
## Median :11.40 Median :28.60 Median :27.60 Median : 96.00
## Mean :12.44 Mean :20.96 Mean :20.46 Mean : 96.29
## 3rd Qu.:20.20 3rd Qu.:34.60 3rd Qu.:33.40 3rd Qu.:102.00
## Max. :58.00 Max. :59.40 Max. :50.00 Max. :142.00
## NA's :11 NA's :15 NA's :15 NA's :30
## filler_level filler_speed temperature usage_cont carb_flow
## Min. : 55.8 Min. : 998 Min. :63.60 Min. :12.08 Min. : 26
## 1st Qu.: 98.3 1st Qu.:3888 1st Qu.:65.20 1st Qu.:18.36 1st Qu.:1144
## Median :118.4 Median :3982 Median :65.60 Median :21.79 Median :3028
## Mean :109.3 Mean :3687 Mean :65.97 Mean :20.99 Mean :2468
## 3rd Qu.:120.0 3rd Qu.:3998 3rd Qu.:66.40 3rd Qu.:23.75 3rd Qu.:3186
## Max. :161.2 Max. :4030 Max. :76.20 Max. :25.90 Max. :5104
## NA's :20 NA's :57 NA's :14 NA's :5 NA's :2
## density mfr balling pressure_vacuum
## Min. :0.240 Min. : 31.4 Min. :-0.170 Min. :-6.600
## 1st Qu.:0.900 1st Qu.:706.3 1st Qu.: 1.496 1st Qu.:-5.600
## Median :0.980 Median :724.0 Median : 1.648 Median :-5.400
## Mean :1.174 Mean :704.0 Mean : 2.198 Mean :-5.216
## 3rd Qu.:1.620 3rd Qu.:731.0 3rd Qu.: 3.292 3rd Qu.:-5.000
## Max. :1.920 Max. :868.6 Max. : 4.012 Max. :-3.600
## NA's :1 NA's :212 NA's :1
## ph oxygen_filler bowl_setpoint pressure_setpoint
## Min. :7.880 Min. :0.00240 Min. : 70.0 Min. :44.00
## 1st Qu.:8.440 1st Qu.:0.02200 1st Qu.:100.0 1st Qu.:46.00
## Median :8.540 Median :0.03340 Median :120.0 Median :46.00
## Mean :8.546 Mean :0.04684 Mean :109.3 Mean :47.62
## 3rd Qu.:8.680 3rd Qu.:0.06000 3rd Qu.:120.0 3rd Qu.:50.00
## Max. :9.360 Max. :0.40000 Max. :140.0 Max. :52.00
## NA's :4 NA's :12 NA's :2 NA's :12
## air_pressurer alch_rel carb_rel balling_lvl
## Min. :140.8 Min. :5.280 Min. :4.960 Min. :0.00
## 1st Qu.:142.2 1st Qu.:6.540 1st Qu.:5.340 1st Qu.:1.38
## Median :142.6 Median :6.560 Median :5.400 Median :1.48
## Mean :142.8 Mean :6.897 Mean :5.437 Mean :2.05
## 3rd Qu.:143.0 3rd Qu.:7.240 3rd Qu.:5.540 3rd Qu.:3.14
## Max. :148.2 Max. :8.620 Max. :6.060 Max. :3.66
## NA's :9 NA's :10 NA's :1sample_data %>% str()## tibble [2,571 × 33] (S3: tbl_df/tbl/data.frame)
## $ brand_code : chr [1:2571] "B" "A" "B" "A" ...
## $ carb_volume : num [1:2571] 5.34 5.43 5.29 5.44 5.49 ...
## $ fill_ounces : num [1:2571] 24 24 24.1 24 24.3 ...
## $ pc_volume : num [1:2571] 0.263 0.239 0.263 0.293 0.111 ...
## $ carb_pressure : num [1:2571] 68.2 68.4 70.8 63 67.2 66.6 64.2 67.6 64.2 72 ...
## $ carb_temp : num [1:2571] 141 140 145 133 137 ...
## $ psc : num [1:2571] 0.104 0.124 0.09 NA 0.026 0.09 0.128 0.154 0.132 0.014 ...
## $ psc_fill : num [1:2571] 0.26 0.22 0.34 0.42 0.16 ...
## $ psc_co2 : num [1:2571] 0.04 0.04 0.16 0.04 0.12 ...
## $ mnf_flow : num [1:2571] -100 -100 -100 -100 -100 -100 -100 -100 -100 -100 ...
## $ carb_pressure1 : num [1:2571] 119 122 120 115 118 ...
## $ fill_pressure : num [1:2571] 46 46 46 46.4 45.8 45.6 51.8 46.8 46 45.2 ...
## $ hyd_pressure1 : num [1:2571] 0 0 0 0 0 0 0 0 0 0 ...
## $ hyd_pressure2 : num [1:2571] NA NA NA 0 0 0 0 0 0 0 ...
## $ hyd_pressure3 : num [1:2571] NA NA NA 0 0 0 0 0 0 0 ...
## $ hyd_pressure4 : num [1:2571] 118 106 82 92 92 116 124 132 90 108 ...
## $ filler_level : num [1:2571] 121 119 120 118 119 ...
## $ filler_speed : num [1:2571] 4002 3986 4020 4012 4010 ...
## $ temperature : num [1:2571] 66 67.6 67 65.6 65.6 66.2 65.8 65.2 65.4 66.6 ...
## $ usage_cont : num [1:2571] 16.2 19.9 17.8 17.4 17.7 ...
## $ carb_flow : num [1:2571] 2932 3144 2914 3062 3054 ...
## $ density : num [1:2571] 0.88 0.92 1.58 1.54 1.54 1.52 0.84 0.84 0.9 0.9 ...
## $ mfr : num [1:2571] 725 727 735 731 723 ...
## $ balling : num [1:2571] 1.4 1.5 3.14 3.04 3.04 ...
## $ pressure_vacuum : num [1:2571] -4 -4 -3.8 -4.4 -4.4 -4.4 -4.4 -4.4 -4.4 -4.4 ...
## $ ph : num [1:2571] 8.36 8.26 8.94 8.24 8.26 8.32 8.4 8.38 8.38 8.5 ...
## $ oxygen_filler : num [1:2571] 0.022 0.026 0.024 0.03 0.03 0.024 0.066 0.046 0.064 0.022 ...
## $ bowl_setpoint : num [1:2571] 120 120 120 120 120 120 120 120 120 120 ...
## $ pressure_setpoint: num [1:2571] 46.4 46.8 46.6 46 46 46 46 46 46 46 ...
## $ air_pressurer : num [1:2571] 143 143 142 146 146 ...
## $ alch_rel : num [1:2571] 6.58 6.56 7.66 7.14 7.14 7.16 6.54 6.52 6.52 6.54 ...
## $ carb_rel : num [1:2571] 5.32 5.3 5.84 5.42 5.44 5.44 5.38 5.34 5.34 5.34 ...
## $ balling_lvl : num [1:2571] 1.48 1.56 3.28 3.04 3.04 3.02 1.44 1.44 1.44 1.38 ...sample_data %>% count(brand_code)## # A tibble: 5 × 2
## brand_code n
## <chr> <int>
## 1 A 293
## 2 B 1239
## 3 C 304
## 4 D 615
## 5 <NA> 120Our initial view of the data shows that the all of our explanatory variables are numerical - with the exception of the Brand. There are 2571 records in our dataset and 33 variables (including our resonse variable). Across our numerical variables, only the mfr variable has a significant number of missing values. The mfr variable has 212 missing values - representing 8.2% of the records. Outside of that variable, the largest number of missing values is 57, which represents 2.2% of the records included in our dataset. Finally, for the brand_code variable there are 120 missing values - which represents 4.6% of the total records. We’ll have to make a decision on how to evaluate this later.
In the following lines of code, we are focused on looking at the distribution of the numerical variables to see if there are any variables that have a problematic distribution
#Show histogram of variables
numeric_columns <- sample_data %>% select_if(., is.numeric) %>% colnames()
all_columns <- sample_data %>% colnames()
numeric_columns[[1]]## [1] "carb_volume"for(i in seq(numeric_columns)) {
#print(numeric_columns[[i]])
col_index = which(all_columns == numeric_columns[[i]])
plt <- sample_data %>%
ggplot(aes_string(x=numeric_columns[[i]])) +
geom_histogram(color="black", fill="white")
print(plt)
}## Warning: `aes_string()` was deprecated in ggplot2 3.0.0.
## ℹ Please use tidy evaluation idioms with `aes()`.
## ℹ See also `vignette("ggplot2-in-packages")` for more information.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 10 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 38 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 39 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 27 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 26 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 33 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 23 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 39 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 2 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 32 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 22 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 11 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 15 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 15 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 30 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 20 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 57 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 14 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 5 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 2 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 1 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 212 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 1 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 4 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 12 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 2 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 12 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 9 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 10 rows containing non-finite values (`stat_bin()`).
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## Warning: Removed 1 rows containing non-finite values (`stat_bin()`).
One of the first - and important - observations we make is that the ph
variable appears to follow a normal distribution. Based on this, we can
anticipate that we will not need to make an special adjustments or
transformations on that variable.However, there are several other
variables that we may want to consider making adjustments to. Some
notable variables to consider adjusting include:
- psc
- psc_fill
- psc_co2
- mnf_flow
- hyd_pressure1
- hyd_pressure2
- hyd_pressure3
- filler_level
- filler_speed
- temperature
- usage_cont
- carb_flow
- density
- mfr
- balling
- pressure_vacuum
- oxygen_filler
- bowl_setpoint
- pressure_setpoint
- air_pressurer
- alch_rel
- carb_rel
- balling_lvl
Next we will look at the correlation between the numerical variables
sample_data %>%
na.omit() %>%
select_if(is.numeric) %>%
cor()## carb_volume fill_ounces pc_volume carb_pressure
## carb_volume 1.000000000 0.022648037 -0.243234517 0.4278586099
## fill_ounces 0.022648037 1.000000000 -0.172701967 0.0044669914
## pc_volume -0.243234517 -0.172701967 1.000000000 -0.1300941805
## carb_pressure 0.427858610 0.004466991 -0.130094181 1.0000000000
## carb_temp -0.135080260 -0.020073098 0.008599427 0.8243350333
## psc -0.020625187 0.043056041 0.190961265 -0.0520086907
## psc_fill -0.012364141 0.069748652 -0.055914847 -0.0179275381
## psc_co2 -0.051977547 0.014705610 -0.055356932 -0.0174011136
## mnf_flow 0.100525364 -0.005787251 -0.203032882 0.0189865815
## carb_pressure1 0.081146212 -0.004186327 -0.238496181 0.0221844474
## fill_pressure -0.096123314 0.068377098 -0.095744421 -0.0850458606
## hyd_pressure1 -0.051894268 -0.146359721 0.271171741 -0.0704043564
## hyd_pressure2 0.033567314 -0.143047986 0.028768135 -0.0267152675
## hyd_pressure3 0.057610071 -0.107816453 -0.024741466 -0.0082140513
## hyd_pressure4 -0.567908627 0.056086556 0.132413786 -0.3229128556
## filler_level -0.018377226 0.010856354 0.220434988 0.0097134297
## filler_speed 0.003233452 0.044277046 -0.073766021 0.0320230919
## temperature -0.189760851 -0.011208233 0.142065435 -0.0490025241
## usage_cont 0.073297402 0.085289046 -0.269204704 0.0004415267
## carb_flow -0.088782919 -0.116991410 0.254535534 -0.0111938869
## density 0.801808803 -0.080414770 -0.167310345 0.4591743002
## mfr 0.010144634 0.043446000 -0.066678296 0.0292986178
## balling 0.818778791 -0.068439337 -0.204785622 0.4539307252
## pressure_vacuum -0.079190180 0.052165822 -0.051605143 0.0051279147
## ph 0.073251595 -0.118207504 0.097746859 0.0788617427
## oxygen_filler -0.112682435 -0.032064259 0.212285069 -0.0506609705
## bowl_setpoint 0.001993607 0.011188521 0.210972310 0.0187581570
## pressure_setpoint -0.160640146 0.065078231 -0.022210124 -0.1171228371
## air_pressurer -0.024854172 0.058101102 -0.036605483 0.0219237613
## alch_rel 0.824787575 -0.110432476 -0.177865067 0.4590380468
## carb_rel 0.841072250 -0.118666585 -0.167433257 0.4736224720
## balling_lvl 0.817993347 -0.063820071 -0.210081156 0.4589466196
## carb_temp psc psc_fill psc_co2
## carb_volume -0.135080260 -0.020625187 -0.012364141 -0.051977547
## fill_ounces -0.020073098 0.043056041 0.069748652 0.014705610
## pc_volume 0.008599427 0.190961265 -0.055914847 -0.055356932
## carb_pressure 0.824335033 -0.052008691 -0.017927538 -0.017401114
## carb_temp 1.000000000 -0.044380293 -0.010968792 0.023660514
## psc -0.044380293 1.000000000 0.193242059 0.053922116
## psc_fill -0.010968792 0.193242059 1.000000000 0.201967429
## psc_co2 0.023660514 0.053922116 0.201967429 1.000000000
## mnf_flow -0.030110321 0.042982180 -0.033801564 0.049931986
## carb_pressure1 -0.014312694 -0.011071066 -0.029772433 0.038316144
## fill_pressure -0.028433189 0.030908100 -0.010528126 0.078519840
## hyd_pressure1 -0.045431217 -0.019758828 -0.053220514 -0.049037869
## hyd_pressure2 -0.039543090 -0.033813839 -0.080467321 -0.021838338
## hyd_pressure3 -0.035120030 -0.020448188 -0.068958062 0.005757630
## hyd_pressure4 -0.002573081 0.036485173 0.019810466 0.054940092
## filler_level 0.011038984 -0.016061814 0.048227359 -0.042306839
## filler_speed 0.029355933 -0.002395360 -0.017295907 -0.008421406
## temperature 0.061914927 0.014217909 0.030080929 0.037007308
## usage_cont -0.043584359 0.049860720 -0.025727952 0.032502663
## carb_flow 0.045050619 -0.037361416 0.017579095 -0.013058433
## density 0.021593836 -0.067168725 -0.024708871 -0.045382139
## mfr 0.023442657 0.007659111 -0.008663604 -0.004263878
## balling 0.006005105 -0.064450826 -0.019108545 -0.043036807
## pressure_vacuum 0.037327836 0.055246194 0.033093793 -0.018438819
## ph 0.034594375 -0.069906559 -0.031374041 -0.088759922
## oxygen_filler 0.012010201 0.002765929 -0.005769535 -0.016971816
## bowl_setpoint 0.007550616 -0.020808347 0.049191096 -0.044820225
## pressure_setpoint -0.027029545 0.042979144 -0.005800041 0.082772777
## air_pressurer 0.038177076 0.061433424 -0.017995701 -0.006984004
## alch_rel 0.004966509 -0.060809035 -0.017045886 -0.050859393
## carb_rel 0.009020802 -0.065354969 -0.015365172 -0.066475871
## balling_lvl 0.010458674 -0.061576860 -0.016064731 -0.047296997
## mnf_flow carb_pressure1 fill_pressure hyd_pressure1
## carb_volume 0.100525364 0.081146212 -0.09612331 -0.051894268
## fill_ounces -0.005787251 -0.004186327 0.06837710 -0.146359721
## pc_volume -0.203032882 -0.238496181 -0.09574442 0.271171741
## carb_pressure 0.018986582 0.022184447 -0.08504586 -0.070404356
## carb_temp -0.030110321 -0.014312694 -0.02843319 -0.045431217
## psc 0.042982180 -0.011071066 0.03090810 -0.019758828
## psc_fill -0.033801564 -0.029772433 -0.01052813 -0.053220514
## psc_co2 0.049931986 0.038316144 0.07851984 -0.049037869
## mnf_flow 1.000000000 0.465355315 0.49111329 0.298997690
## carb_pressure1 0.465355315 1.000000000 0.22002705 -0.012711716
## fill_pressure 0.491113286 0.220027054 1.00000000 0.137594249
## hyd_pressure1 0.298997690 -0.012711716 0.13759425 1.000000000
## hyd_pressure2 0.648337926 0.284710255 0.31836940 0.711851345
## hyd_pressure3 0.764142833 0.375658047 0.43637028 0.605589019
## hyd_pressure4 0.073124200 0.038936494 0.29144885 0.051658157
## filler_level -0.608380668 -0.425646963 -0.45892889 -0.029684260
## filler_speed 0.024363769 0.012389362 -0.21347499 -0.022850088
## temperature -0.063477350 -0.070067643 0.06397894 -0.073341745
## usage_cont 0.560192617 0.354875344 0.30257079 0.118691986
## carb_flow -0.497211539 -0.273632871 -0.18715299 -0.169136801
## density 0.021343082 0.040888366 -0.20887500 -0.003601856
## mfr 0.019593355 0.013931715 -0.21290076 -0.035646540
## balling 0.112393982 0.073508154 -0.15676227 0.025863408
## pressure_vacuum -0.559943315 -0.284881843 -0.30810832 -0.324133005
## ph -0.457218313 -0.112108310 -0.31311751 -0.048631533
## oxygen_filler -0.550066399 -0.278795209 -0.24769592 -0.118317983
## bowl_setpoint -0.611580420 -0.432671459 -0.42609290 -0.029844050
## pressure_setpoint 0.482530739 0.234201877 0.81528556 0.190079720
## air_pressurer -0.041525175 0.061506659 0.03280945 -0.209227461
## alch_rel 0.027999719 0.009047055 -0.20467008 0.006667887
## carb_rel -0.027880871 0.004804776 -0.20272201 0.034387945
## balling_lvl 0.042267589 0.033601804 -0.17885789 -0.007754220
## hyd_pressure2 hyd_pressure3 hyd_pressure4 filler_level
## carb_volume 0.03356731 0.057610071 -0.567908627 -0.01837723
## fill_ounces -0.14304799 -0.107816453 0.056086556 0.01085635
## pc_volume 0.02876814 -0.024741466 0.132413786 0.22043499
## carb_pressure -0.02671527 -0.008214051 -0.322912856 0.00971343
## carb_temp -0.03954309 -0.035120030 -0.002573081 0.01103898
## psc -0.03381384 -0.020448188 0.036485173 -0.01606181
## psc_fill -0.08046732 -0.068958062 0.019810466 0.04822736
## psc_co2 -0.02183834 0.005757630 0.054940092 -0.04230684
## mnf_flow 0.64833793 0.764142833 0.073124200 -0.60838067
## carb_pressure1 0.28471026 0.375658047 0.038936494 -0.42564696
## fill_pressure 0.31836940 0.436370279 0.291448848 -0.45892889
## hyd_pressure1 0.71185135 0.605589019 0.051658157 -0.02968426
## hyd_pressure2 1.00000000 0.917661875 0.022980025 -0.41103521
## hyd_pressure3 0.91766187 1.000000000 0.046649319 -0.52796590
## hyd_pressure4 0.02298003 0.046649319 1.000000000 -0.10010344
## filler_level -0.41103521 -0.527965898 -0.100103442 1.00000000
## filler_speed 0.07028157 0.031692930 -0.259685780 0.06319357
## temperature -0.13725053 -0.120288087 0.283933984 0.05040323
## usage_cont 0.37224053 0.419761856 0.070916924 -0.35740823
## carb_flow -0.32187768 -0.363691033 -0.009527157 0.01743320
## density 0.05842429 0.040354370 -0.584381995 0.00772950
## mfr 0.04937645 0.012841039 -0.280014344 0.06935680
## balling 0.10480326 0.125869294 -0.594996647 0.01188558
## pressure_vacuum -0.62816754 -0.672258326 -0.054652623 0.35866815
## ph -0.21571713 -0.260535183 -0.184518087 0.34879540
## oxygen_filler -0.29438355 -0.363660021 0.009692254 0.22540536
## bowl_setpoint -0.41457787 -0.529215855 -0.117019292 0.97724166
## pressure_setpoint 0.34183180 0.459918732 0.288770453 -0.44595292
## air_pressurer -0.17237591 -0.067700529 0.021731381 -0.12573419
## alch_rel 0.03916606 0.049948952 -0.685576013 0.04355856
## carb_rel 0.03150549 0.013543248 -0.581827500 0.13107856
## balling_lvl 0.02885940 0.041595480 -0.575044299 0.05317923
## filler_speed temperature usage_cont carb_flow
## carb_volume 0.003233452 -0.18976085 0.0732974016 -0.088782919
## fill_ounces 0.044277046 -0.01120823 0.0852890459 -0.116991410
## pc_volume -0.073766021 0.14206543 -0.2692047044 0.254535534
## carb_pressure 0.032023092 -0.04900252 0.0004415267 -0.011193887
## carb_temp 0.029355933 0.06191493 -0.0435843590 0.045050619
## psc -0.002395360 0.01421791 0.0498607203 -0.037361416
## psc_fill -0.017295907 0.03008093 -0.0257279520 0.017579095
## psc_co2 -0.008421406 0.03700731 0.0325026634 -0.013058433
## mnf_flow 0.024363769 -0.06347735 0.5601926171 -0.497211539
## carb_pressure1 0.012389362 -0.07006764 0.3548753438 -0.273632871
## fill_pressure -0.213474995 0.06397894 0.3025707885 -0.187152994
## hyd_pressure1 -0.022850088 -0.07334175 0.1186919862 -0.169136801
## hyd_pressure2 0.070281568 -0.13725053 0.3722405332 -0.321877677
## hyd_pressure3 0.031692930 -0.12028809 0.4197618556 -0.363691033
## hyd_pressure4 -0.259685780 0.28393398 0.0709169237 -0.009527157
## filler_level 0.063193573 0.05040323 -0.3574082260 0.017433196
## filler_speed 1.000000000 -0.06794451 0.0466749382 -0.062641899
## temperature -0.067944506 1.00000000 -0.0950428265 0.140119431
## usage_cont 0.046674938 -0.09504283 1.0000000000 -0.488854072
## carb_flow -0.062641899 0.14011943 -0.4888540721 1.000000000
## density 0.025454377 -0.19369890 -0.0045531798 0.023058872
## mfr 0.951264445 -0.08156825 0.0397443802 -0.074148067
## balling 0.037938064 -0.23844781 0.0692694347 -0.102299824
## pressure_vacuum 0.045187804 0.04313148 -0.3196915871 0.290026894
## ph -0.047596639 -0.18152910 -0.3577042377 0.240757744
## oxygen_filler -0.044126285 0.09338308 -0.3164959408 0.385692082
## bowl_setpoint 0.027688288 0.05010937 -0.3584563273 0.012221618
## pressure_setpoint -0.068525700 0.05904194 0.2626733751 -0.166362628
## air_pressurer -0.006199674 0.06692883 -0.1025056258 0.137849013
## alch_rel -0.005598672 -0.24776232 -0.0237408078 0.007527013
## carb_rel 0.005300760 -0.13595194 -0.0390798757 -0.006940815
## balling_lvl 0.003823258 -0.22497504 0.0232069975 -0.055365859
## density mfr balling pressure_vacuum
## carb_volume 0.801808803 0.010144634 0.818778791 -0.079190180
## fill_ounces -0.080414770 0.043446000 -0.068439337 0.052165822
## pc_volume -0.167310345 -0.066678296 -0.204785622 -0.051605143
## carb_pressure 0.459174300 0.029298618 0.453930725 0.005127915
## carb_temp 0.021593836 0.023442657 0.006005105 0.037327836
## psc -0.067168725 0.007659111 -0.064450826 0.055246194
## psc_fill -0.024708871 -0.008663604 -0.019108545 0.033093793
## psc_co2 -0.045382139 -0.004263878 -0.043036807 -0.018438819
## mnf_flow 0.021343082 0.019593355 0.112393982 -0.559943315
## carb_pressure1 0.040888366 0.013931715 0.073508154 -0.284881843
## fill_pressure -0.208875002 -0.212900763 -0.156762274 -0.308108315
## hyd_pressure1 -0.003601856 -0.035646540 0.025863408 -0.324133005
## hyd_pressure2 0.058424287 0.049376448 0.104803262 -0.628167539
## hyd_pressure3 0.040354370 0.012841039 0.125869294 -0.672258326
## hyd_pressure4 -0.584381995 -0.280014344 -0.594996647 -0.054652623
## filler_level 0.007729500 0.069356797 0.011885578 0.358668150
## filler_speed 0.025454377 0.951264445 0.037938064 0.045187804
## temperature -0.193698895 -0.081568252 -0.238447810 0.043131483
## usage_cont -0.004553180 0.039744380 0.069269435 -0.319691587
## carb_flow 0.023058872 -0.074148067 -0.102299824 0.290026894
## density 1.000000000 0.039383665 0.953127632 -0.083762802
## mfr 0.039383665 1.000000000 0.051171065 0.043987724
## balling 0.953127632 0.051171065 1.000000000 -0.167052546
## pressure_vacuum -0.083762802 0.043987724 -0.167052546 1.000000000
## ph 0.095340535 -0.052377217 0.072297357 0.213173584
## oxygen_filler -0.046621720 -0.048876320 -0.113757056 0.278478283
## bowl_setpoint 0.019792536 0.042312091 0.027415802 0.358696671
## pressure_setpoint -0.250884909 -0.064390559 -0.210098742 -0.298520301
## air_pressurer -0.082976446 -0.008182360 -0.102307854 0.175090738
## alch_rel 0.922993173 -0.001003949 0.945725964 -0.054788103
## carb_rel 0.860322233 0.010884909 0.858990647 -0.008434181
## balling_lvl 0.956417297 0.016039036 0.988412082 -0.051626926
## ph oxygen_filler bowl_setpoint pressure_setpoint
## carb_volume 0.073251595 -0.112682435 0.001993607 -0.160640146
## fill_ounces -0.118207504 -0.032064259 0.011188521 0.065078231
## pc_volume 0.097746859 0.212285069 0.210972310 -0.022210124
## carb_pressure 0.078861743 -0.050660970 0.018758157 -0.117122837
## carb_temp 0.034594375 0.012010201 0.007550616 -0.027029545
## psc -0.069906559 0.002765929 -0.020808347 0.042979144
## psc_fill -0.031374041 -0.005769535 0.049191096 -0.005800041
## psc_co2 -0.088759922 -0.016971816 -0.044820225 0.082772777
## mnf_flow -0.457218313 -0.550066399 -0.611580420 0.482530739
## carb_pressure1 -0.112108310 -0.278795209 -0.432671459 0.234201877
## fill_pressure -0.313117513 -0.247695921 -0.426092896 0.815285564
## hyd_pressure1 -0.048631533 -0.118317983 -0.029844050 0.190079720
## hyd_pressure2 -0.215717133 -0.294383550 -0.414577866 0.341831797
## hyd_pressure3 -0.260535183 -0.363660021 -0.529215855 0.459918732
## hyd_pressure4 -0.184518087 0.009692254 -0.117019292 0.288770453
## filler_level 0.348795399 0.225405359 0.977241656 -0.445952917
## filler_speed -0.047596639 -0.044126285 0.027688288 -0.068525700
## temperature -0.181529097 0.093383081 0.050109366 0.059041937
## usage_cont -0.357704238 -0.316495941 -0.358456327 0.262673375
## carb_flow 0.240757744 0.385692082 0.012221618 -0.166362628
## density 0.095340535 -0.046621720 0.019792536 -0.250884909
## mfr -0.052377217 -0.048876320 0.042312091 -0.064390559
## balling 0.072297357 -0.113757056 0.027415802 -0.210098742
## pressure_vacuum 0.213173584 0.278478283 0.358696671 -0.298520301
## ph 1.000000000 0.162016479 0.358504933 -0.316950863
## oxygen_filler 0.162016479 1.000000000 0.224734144 -0.241221158
## bowl_setpoint 0.358504933 0.224734144 1.000000000 -0.440555036
## pressure_setpoint -0.316950863 -0.241221158 -0.440555036 1.000000000
## air_pressurer -0.008276544 0.116495267 -0.130939728 0.073805550
## alch_rel 0.156053375 -0.050111016 0.063332637 -0.268800293
## carb_rel 0.189575611 -0.039206163 0.152117160 -0.261257572
## balling_lvl 0.103279163 -0.074544984 0.070442468 -0.242343282
## air_pressurer alch_rel carb_rel balling_lvl
## carb_volume -0.024854172 0.824787575 0.841072250 0.817993347
## fill_ounces 0.058101102 -0.110432476 -0.118666585 -0.063820071
## pc_volume -0.036605483 -0.177865067 -0.167433257 -0.210081156
## carb_pressure 0.021923761 0.459038047 0.473622472 0.458946620
## carb_temp 0.038177076 0.004966509 0.009020802 0.010458674
## psc 0.061433424 -0.060809035 -0.065354969 -0.061576860
## psc_fill -0.017995701 -0.017045886 -0.015365172 -0.016064731
## psc_co2 -0.006984004 -0.050859393 -0.066475871 -0.047296997
## mnf_flow -0.041525175 0.027999719 -0.027880871 0.042267589
## carb_pressure1 0.061506659 0.009047055 0.004804776 0.033601804
## fill_pressure 0.032809452 -0.204670077 -0.202722006 -0.178857893
## hyd_pressure1 -0.209227461 0.006667887 0.034387945 -0.007754220
## hyd_pressure2 -0.172375910 0.039166055 0.031505491 0.028859401
## hyd_pressure3 -0.067700529 0.049948952 0.013543248 0.041595480
## hyd_pressure4 0.021731381 -0.685576013 -0.581827500 -0.575044299
## filler_level -0.125734192 0.043558564 0.131078561 0.053179225
## filler_speed -0.006199674 -0.005598672 0.005300760 0.003823258
## temperature 0.066928829 -0.247762315 -0.135951942 -0.224975043
## usage_cont -0.102505626 -0.023740808 -0.039079876 0.023206997
## carb_flow 0.137849013 0.007527013 -0.006940815 -0.055365859
## density -0.082976446 0.922993173 0.860322233 0.956417297
## mfr -0.008182360 -0.001003949 0.010884909 0.016039036
## balling -0.102307854 0.945725964 0.858990647 0.988412082
## pressure_vacuum 0.175090738 -0.054788103 -0.008434181 -0.051626926
## ph -0.008276544 0.156053375 0.189575611 0.103279163
## oxygen_filler 0.116495267 -0.050111016 -0.039206163 -0.074544984
## bowl_setpoint -0.130939728 0.063332637 0.152117160 0.070442468
## pressure_setpoint 0.073805550 -0.268800293 -0.261257572 -0.242343282
## air_pressurer 1.000000000 -0.082451451 -0.107386102 -0.085434755
## alch_rel -0.082451451 1.000000000 0.882791450 0.947343341
## carb_rel -0.107386102 0.882791450 1.000000000 0.872148303
## balling_lvl -0.085434755 0.947343341 0.872148303 1.000000000When we look at the correlation between the target variable (ph) and the different explanatory variables, we don’t find any significant correlation between any of the explanatory variables and the respone variable. However, there is higher levels of correlation for the following variables:
- mnf_flow
- fill_pressure
- filler_level
- usage_count
- bowl_setpoint
- pressure_setpoint
Next we will do some explorary analysis with the categorical variable brand. We will look at the count of the variable, the distribution of the variable. Additionally, we
sample_data %>% count(brand_code)## # A tibble: 5 × 2
## brand_code n
## <chr> <int>
## 1 A 293
## 2 B 1239
## 3 C 304
## 4 D 615
## 5 <NA> 120sample_data %>% select(brand_code) %>% table() %>% prop.table()## brand_code
## A B C D
## 0.119543 0.505508 0.124031 0.250918sample_data %>%
group_by(brand_code) %>%
summarize(median_ph = median(ph, na.rm=TRUE),
mean_ph = mean(ph, na.rm=TRUE),
sd_ph = sd(ph, na.rm=TRUE))## # A tibble: 5 × 4
## brand_code median_ph mean_ph sd_ph
## <chr> <dbl> <dbl> <dbl>
## 1 A 8.52 8.50 0.163
## 2 B 8.56 8.57 0.169
## 3 C 8.42 8.41 0.177
## 4 D 8.62 8.60 0.135
## 5 <NA> 8.51 8.49 0.177sample_data %>%
na.omit() %>%
ggplot(aes(x=brand_code, y=ph)) +
geom_boxplot()
sample_data %>%
mutate(brand_code = ifelse(is.na(brand_code), "A",brand_code)) %>%
na.omit() %>%
ggplot(aes(x=brand_code, y=ph)) +
geom_boxplot()
Some of the observations that we have when looking at the brand code is
that - as mentioned earlier, about 4.8% of the records have no brand
code. Amongst, those that do have a brand code, 50% are of brand code B,
25% have brand code D, with the remaining records being split relatively
evenly between brand code A and C.
Finally, when we look at the boxplot of the data - along with the summary metrics grouped by brand code - we see some evidence to suggest that there are meaningful differences in the ph levels amongst the groups, with the median ph being 8.52 for Group A, 8.56 for Group B, 8.42 for Group C and 8.62 for Group D.
Data Preparation
Now that we’ve done our initial exploratory analysis of the data, we will focus on making the following adjustments to our data:
Change records with NA brand code to brand code “A” - After reviewing the median and mean ph for the data based on brand code, we find the the values for the NA data are closest to Brand A. Thus we explored the impact on the Brand A metrics if we were to assign the NA records to a Brand Code of A and discovered that there would be minimal change to the data. Thus, we are going to convert the NA brand code records to having a Brand code of A
Change the Brand code to a factor variable
Since there were only a limited number of NA records in the data (since we’ve accouted for the blanks in the Brand Code variable), we will go ahead and just exclude the NA values from our overall dataset
#Set NA brand codes to Brand code A
sample_data <- sample_data %>%
mutate(brand_code = ifelse(is.na(brand_code), "A",brand_code))
#Change brand code variable to factor
sample_data_mod <- sample_data %>% mutate(brand_code = factor(brand_code, levels = sort(unique(brand_code))))
#Remove NA values from the dataset
sample_data_mod <- sample_data_mod %>% na.omit()Model Building & Evaluation
Setup training data
We will split the data between test and training data for use in testing our models. Additionally since our data exploration showed evidence that the brand of product appears to be a source of distinction in the composition of the ph, we will also upsample the test data to make sure that we train the model on a balanced set of data that includes examples from each of the different brand codes.
set.seed(1234)
index <- createDataPartition(sample_data_mod$brand_code, p=0.75, list=FALSE)
train <- sample_data_mod[index,]
test <- sample_data_mod[-index,]
trainX <- train %>% select(-ph)
trainY <- train %>% select(ph)
testX <- test %>% select(-ph)
testY <- test %>% select(ph)
set.seed(111)
train_up <- upSample(x=train[,-ncol(train)],y=train$brand_code)
train_up <- train_up %>% select(-Class)Linear Regression Model
We will use a multiple Linear regression model as the first model that we train and evaluate. After creating and reviewing the performance summary of our first model, we will then use backward elimination to remove parameters that are not statistically significant to our model in order to generate a simpler model and also, hopefully see improvements in our adjusted R-Squared along the way.
model_lm1 <- lm(ph ~ ., data=train_up)
summary(model_lm1)##
## Call:
## lm(formula = ph ~ ., data = train_up)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.53025 -0.07647 0.00549 0.09279 0.40321
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.203e+01 1.106e+00 10.873 < 2e-16 ***
## brand_codeB 7.980e-02 1.147e-02 6.957 4.23e-12 ***
## brand_codeC -8.392e-02 1.054e-02 -7.965 2.30e-15 ***
## brand_codeD 2.981e-03 1.682e-02 0.177 0.859330
## carb_volume -1.788e-01 9.225e-02 -1.938 0.052705 .
## fill_ounces -1.713e-01 3.029e-02 -5.655 1.70e-08 ***
## pc_volume -8.337e-02 5.433e-02 -1.535 0.125003
## carb_pressure 2.390e-03 4.575e-03 0.522 0.601381
## carb_temp -1.579e-03 3.552e-03 -0.445 0.656560
## psc 8.043e-02 5.273e-02 1.525 0.127274
## psc_fill -5.894e-02 2.032e-02 -2.901 0.003751 **
## psc_co2 -7.428e-02 5.831e-02 -1.274 0.202787
## mnf_flow -7.173e-04 4.414e-05 -16.250 < 2e-16 ***
## carb_pressure1 5.254e-03 6.721e-04 7.818 7.30e-15 ***
## fill_pressure -2.508e-03 1.477e-03 -1.698 0.089519 .
## hyd_pressure1 1.233e-03 3.345e-04 3.685 0.000233 ***
## hyd_pressure2 -2.611e-03 5.225e-04 -4.998 6.12e-07 ***
## hyd_pressure3 3.049e-03 5.679e-04 5.369 8.49e-08 ***
## hyd_pressure4 -2.460e-04 2.940e-04 -0.837 0.402843
## filler_level -1.067e-03 7.884e-04 -1.353 0.176164
## filler_speed 7.044e-05 2.196e-05 3.207 0.001354 **
## temperature -1.116e-02 2.489e-03 -4.482 7.65e-06 ***
## usage_cont -8.315e-03 1.102e-03 -7.544 5.93e-14 ***
## carb_flow -1.910e-05 4.001e-06 -4.775 1.88e-06 ***
## density -3.120e-03 2.502e-02 -0.125 0.900796
## mfr -4.438e-04 1.195e-04 -3.715 0.000207 ***
## balling -2.835e-02 1.475e-02 -1.922 0.054691 .
## pressure_vacuum -2.771e-02 6.625e-03 -4.183 2.96e-05 ***
## oxygen_filler -3.107e-01 7.550e-02 -4.115 3.96e-05 ***
## bowl_setpoint 2.111e-03 8.112e-04 2.602 0.009302 **
## pressure_setpoint -1.213e-03 2.013e-03 -0.602 0.546939
## air_pressurer 5.547e-04 2.489e-03 0.223 0.823657
## alch_rel 1.048e-01 2.783e-02 3.765 0.000170 ***
## carb_rel 2.259e-01 4.848e-02 4.660 3.30e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1326 on 3118 degrees of freedom
## Multiple R-squared: 0.4351, Adjusted R-squared: 0.4291
## F-statistic: 72.76 on 33 and 3118 DF, p-value: < 2.2e-16model_lm2 <- lm(ph ~ . -alch_rel -air_pressurer -pressure_vacuum -density
-filler_speed -filler_level -psc_co2 -pressure_setpoint -psc, data=train_up)
summary(model_lm2)##
## Call:
## lm(formula = ph ~ . - alch_rel - air_pressurer - pressure_vacuum -
## density - filler_speed - filler_level - psc_co2 - pressure_setpoint -
## psc, data = train_up)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.53439 -0.07831 0.00479 0.09196 0.41078
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.247e+01 1.040e+00 11.992 < 2e-16 ***
## brand_codeB 7.466e-02 1.128e-02 6.619 4.24e-11 ***
## brand_codeC -8.729e-02 1.023e-02 -8.533 < 2e-16 ***
## brand_codeD 5.461e-02 1.049e-02 5.205 2.07e-07 ***
## carb_volume -1.665e-01 9.176e-02 -1.815 0.069627 .
## fill_ounces -1.598e-01 3.032e-02 -5.269 1.47e-07 ***
## pc_volume -3.696e-02 4.972e-02 -0.743 0.457262
## carb_pressure 2.580e-03 4.566e-03 0.565 0.572068
## carb_temp -1.706e-03 3.548e-03 -0.481 0.630742
## psc_fill -5.657e-02 1.953e-02 -2.896 0.003806 **
## mnf_flow -6.878e-04 4.369e-05 -15.742 < 2e-16 ***
## carb_pressure1 4.993e-03 6.665e-04 7.492 8.76e-14 ***
## fill_pressure -2.706e-03 1.035e-03 -2.614 0.008979 **
## hyd_pressure1 8.971e-04 3.240e-04 2.769 0.005654 **
## hyd_pressure2 -2.291e-03 4.826e-04 -4.748 2.15e-06 ***
## hyd_pressure3 3.393e-03 5.216e-04 6.504 9.04e-11 ***
## hyd_pressure4 -2.865e-04 2.935e-04 -0.976 0.328980
## temperature -1.111e-02 2.463e-03 -4.510 6.74e-06 ***
## usage_cont -8.328e-03 1.091e-03 -7.635 2.98e-14 ***
## carb_flow -1.442e-05 3.726e-06 -3.871 0.000111 ***
## mfr -1.243e-04 4.279e-05 -2.905 0.003693 **
## balling 1.539e-03 7.768e-03 0.198 0.842977
## oxygen_filler -2.976e-01 7.509e-02 -3.963 7.57e-05 ***
## bowl_setpoint 1.133e-03 2.624e-04 4.319 1.62e-05 ***
## carb_rel 2.395e-01 4.579e-02 5.230 1.81e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1333 on 3127 degrees of freedom
## Multiple R-squared: 0.4275, Adjusted R-squared: 0.4231
## F-statistic: 97.27 on 24 and 3127 DF, p-value: < 2.2e-16Throught our backward elimination process, we were able to improve the adjusted R-squared from 0.4313 to 0.4323. Next we will check the reliability of the model using our test dataset.
predict(model_lm1, test)## 1 2 3 4 5 6 7 8
## 8.535312 8.552172 8.617410 8.599141 8.592518 8.487506 8.522059 8.537330
## 9 10 11 12 13 14 15 16
## 8.563412 8.513961 8.572605 8.559585 8.579524 8.594673 8.522230 8.534854
## 17 18 19 20 21 22 23 24
## 8.596204 8.633726 8.661215 8.591473 8.630996 8.618544 8.567257 8.599976
## 25 26 27 28 29 30 31 32
## 8.617948 8.638176 8.550960 8.583514 8.615816 8.618693 8.638351 8.599334
## 33 34 35 36 37 38 39 40
## 8.568214 8.686447 8.646237 8.595908 8.696560 8.536267 8.383022 8.410642
## 41 42 43 44 45 46 47 48
## 8.444845 8.609029 8.608778 8.545063 8.627627 8.608474 8.626407 8.634280
## 49 50 51 52 53 54 55 56
## 8.667126 8.556596 8.613031 8.650503 8.638972 8.553280 8.600839 8.644670
## 57 58 59 60 61 62 63 64
## 8.597825 8.536441 8.522494 8.528203 8.572763 8.598653 8.673867 8.622717
## 65 66 67 68 69 70 71 72
## 8.554984 8.540230 8.565499 8.701430 8.586814 8.584006 8.647969 8.476226
## 73 74 75 76 77 78 79 80
## 8.656510 8.652321 8.584494 8.662530 8.643258 8.698844 8.543026 8.704525
## 81 82 83 84 85 86 87 88
## 8.777679 8.825176 8.659717 8.737716 8.661774 8.645128 8.701232 8.485669
## 89 90 91 92 93 94 95 96
## 8.497086 8.472881 8.575344 8.719753 8.741559 8.682135 8.629311 8.596895
## 97 98 99 100 101 102 103 104
## 8.668197 8.615002 8.657667 8.611367 8.620442 8.713801 8.632544 8.607552
## 105 106 107 108 109 110 111 112
## 8.638635 8.688853 8.605818 8.634368 8.627923 8.519610 8.593470 8.622751
## 113 114 115 116 117 118 119 120
## 8.578370 8.486476 8.474408 8.703535 8.678214 8.732956 8.609192 8.714570
## 121 122 123 124 125 126 127 128
## 8.643319 8.583682 8.666468 8.558109 8.621107 8.729932 8.873657 8.653398
## 129 130 131 132 133 134 135 136
## 8.629262 8.715678 8.788885 8.674881 8.665523 8.439865 8.469357 8.500706
## 137 138 139 140 141 142 143 144
## 8.437576 8.458453 8.482004 8.498254 8.611552 8.689605 8.698472 8.682046
## 145 146 147 148 149 150 151 152
## 8.694364 8.739120 8.609007 8.649823 8.598848 8.610661 8.694824 8.726789
## 153 154 155 156 157 158 159 160
## 8.681785 8.652567 8.598238 8.617016 8.562521 8.663504 8.642907 8.643729
## 161 162 163 164 165 166 167 168
## 8.477988 8.689018 8.617325 8.585246 8.677114 8.512572 8.680095 8.803345
## 169 170 171 172 173 174 175 176
## 8.692787 8.666025 8.756548 8.770646 8.682911 8.734524 8.715625 8.725685
## 177 178 179 180 181 182 183 184
## 8.667368 8.554460 8.743824 8.761286 8.758101 8.722627 8.656748 8.482051
## 185 186 187 188 189 190 191 192
## 8.477618 8.457926 8.501116 8.478556 8.651998 8.654843 8.634054 8.610014
## 193 194 195 196 197 198 199 200
## 8.692841 8.634047 8.734718 8.722285 8.694298 8.788174 8.588965 8.560117
## 201 202 203 204 205 206 207 208
## 8.612198 8.675494 8.625220 8.629646 8.577714 8.701874 8.699637 8.560705
## 209 210 211 212 213 214 215 216
## 8.644735 8.667605 8.684891 8.639527 8.636594 8.688191 8.675120 8.747888
## 217 218 219 220 221 222 223 224
## 8.727718 8.565172 8.650489 8.631932 8.675710 8.618460 8.662707 8.737362
## 225 226 227 228 229 230 231 232
## 8.757956 8.611127 8.628337 8.631014 8.584984 8.623373 8.597118 8.637044
## 233 234 235 236 237 238 239 240
## 8.650168 8.690843 8.508010 8.505688 8.472799 8.547967 8.657178 8.601530
## 241 242 243 244 245 246 247 248
## 8.706025 8.772766 8.644880 8.692180 8.727837 8.706923 8.698254 8.673956
## 249 250 251 252 253 254 255 256
## 8.713024 8.651606 8.473119 8.493855 8.622884 8.482721 8.484194 8.539246
## 257 258 259 260 261 262 263 264
## 8.551515 8.554118 8.504529 8.418718 8.512044 8.453088 8.501460 8.394030
## 265 266 267 268 269 270 271 272
## 8.475523 8.524289 8.503736 8.612974 8.281501 8.656562 8.642244 8.537584
## 273 274 275 276 277 278 279 280
## 8.489459 8.441599 8.521856 8.408867 8.536206 8.533114 8.550946 8.509091
## 281 282 283 284 285 286 287 288
## 8.560228 8.565240 8.567228 8.588729 8.596048 8.532336 8.603991 8.587775
## 289 290 291 292 293 294 295 296
## 8.557889 8.567364 8.589820 8.638057 8.560941 8.604563 8.583663 8.328241
## 297 298 299 300 301 302 303 304
## 8.341659 8.370631 8.344864 8.268771 8.281730 8.546698 8.542892 8.529807
## 305 306 307 308 309 310 311 312
## 8.491231 8.544168 8.476425 8.470104 8.476578 8.566667 8.588488 8.566843
## 313 314 315 316 317 318 319 320
## 8.464548 8.490479 8.463749 8.337031 8.410468 8.451988 8.458341 8.454712
## 321 322 323 324 325 326 327 328
## 8.556154 8.569134 8.631975 8.588798 8.527219 8.556726 8.499378 8.497597
## 329 330 331 332 333 334 335 336
## 8.447940 8.505432 8.479001 8.391495 8.356680 8.272870 8.441314 8.416109
## 337 338 339 340 341 342 343 344
## 8.424972 8.416890 8.525539 8.498188 8.464682 8.466235 8.516341 8.505952
## 345 346 347 348 349 350 351 352
## 8.480390 8.472402 8.510878 8.271139 8.336505 8.323018 8.328623 8.277577
## 353 354 355 356 357 358 359 360
## 8.572634 8.430445 8.441532 8.417792 8.466240 8.438397 8.464246 8.455339
## 361 362 363 364 365 366 367 368
## 8.463211 8.436249 8.422607 8.457396 8.443884 8.452611 8.469901 8.440857
## 369 370 371 372 373 374 375 376
## 8.424638 8.556547 8.519020 8.500247 8.478301 8.464870 8.418684 8.434603
## 377 378 379 380 381 382 383 384
## 8.294607 8.412229 8.373675 8.446003 8.463647 8.509818 8.494136 8.536547
## 385 386 387 388 389 390 391 392
## 8.321739 8.457268 8.491027 8.418307 8.272420 8.282787 8.435603 8.498043
## 393 394 395 396 397 398 399 400
## 8.476977 8.507094 8.512056 8.349803 8.459772 8.394004 8.499125 8.540402
## 401 402 403 404 405 406 407 408
## 8.506349 8.496515 8.498837 8.476264 8.544528 8.529859 8.510080 8.432343
## 409 410 411 412 413 414 415 416
## 8.483909 8.570198 8.404957 8.348317 8.323134 8.347512 8.349837 8.328327
## 417 418 419 420 421 422 423 424
## 8.546584 8.527267 8.553337 8.537409 8.393011 8.524774 8.454039 8.306176
## 425 426 427 428 429 430 431 432
## 8.507428 8.496851 8.480885 8.470022 8.453689 8.492330 8.508521 8.391348
## 433 434 435 436 437 438 439 440
## 8.528258 8.540623 8.555924 8.563143 8.587529 8.579418 8.582648 8.536633
## 441 442 443 444 445 446 447 448
## 8.495592 8.511122 8.665363 8.506894 8.313142 8.336063 8.347746 8.327580
## 449 450 451 452 453 454 455 456
## 8.376291 8.561202 8.570062 8.491047 8.508612 8.545889 8.533885 8.561067
## 457 458 459 460 461 462 463 464
## 8.534692 8.591785 8.584092 8.625438 8.477412 8.600057 8.534042 8.482883
## 465 466 467 468 469 470 471 472
## 8.495659 8.496797 8.308670 8.472488 8.531140 8.531340 8.450722 8.522283
## 473 474 475 476 477 478 479 480
## 8.544303 8.490571 8.502829 8.540052 8.613147 8.538882 8.594406 8.615205
## 481 482 483 484 485 486 487 488
## 8.346103 8.314281 8.514047 8.458406 8.475322 8.465305 8.508352 8.581415
## 489 490 491 492 493 494 495 496
## 8.499972 8.520424 8.501832 8.593729 8.630822 8.416822 8.389844 8.429733
## 497 498 499 500 501 502 503 504
## 8.446820 8.583815 8.540383 8.550342 8.560813 8.509433 8.510931 8.533974
## 505 506 507 508 509 510 511 512
## 8.471296 8.506307 8.556053 8.520599 8.514418 8.577183 8.642422 8.538299
## 513 514 515 516 517 518 519 520
## 8.528060 8.566496 8.527929 8.553100 8.490989 8.540184 8.503175 8.649171
## 521 522 523 524 525 526 527 528
## 8.583863 8.544450 8.600710 8.547345 8.640192 8.344598 8.370644 8.323353
## 529 530 531
## 8.363332 8.317567 8.506639RMSE(predict(model_lm1, test), test$ph)## [1] 0.1408222The RMSE score for our Linear Regression model is 0.1358
Decision Tree Model (Single Tree)
The next model that we will try is a single tree Decision tree model. We will evaluate a model that uses all of our parameters and then a second one that uses only the parameters that we selected in our better performing Linear Regression model.
set.seed(1234)
model_tree1 <- rpart(ph ~., data = train)
#rpart.plot(tree)
predict(model_tree1, test)## 1 2 3 4 5 6 7 8
## 8.561120 8.561120 8.561120 8.561120 8.672667 8.492941 8.561120 8.561120
## 9 10 11 12 13 14 15 16
## 8.561120 8.561120 8.561120 8.561120 8.561120 8.561120 8.561120 8.561120
## 17 18 19 20 21 22 23 24
## 8.561120 8.672667 8.561120 8.561120 8.561120 8.561120 8.561120 8.561120
## 25 26 27 28 29 30 31 32
## 8.561120 8.561120 8.561120 8.719864 8.561120 8.672667 8.561120 8.561120
## 33 34 35 36 37 38 39 40
## 8.672667 8.561120 8.719864 8.719864 8.719864 8.719864 8.494231 8.251818
## 41 42 43 44 45 46 47 48
## 8.251818 8.719864 8.561120 8.561120 8.719864 8.719864 8.719864 8.561120
## 49 50 51 52 53 54 55 56
## 8.561120 8.561120 8.561120 8.561120 8.561120 8.561120 8.561120 8.561120
## 57 58 59 60 61 62 63 64
## 8.561120 8.561120 8.561120 8.561120 8.561120 8.561120 8.561120 8.719864
## 65 66 67 68 69 70 71 72
## 8.672667 8.561120 8.561120 8.672667 8.561120 8.561120 8.672667 8.251818
## 73 74 75 76 77 78 79 80
## 8.672667 8.561120 8.561120 8.561120 8.672667 8.672667 8.561120 8.561120
## 81 82 83 84 85 86 87 88
## 8.672667 8.672667 8.719864 8.719864 8.672667 8.719864 8.719864 8.251818
## 89 90 91 92 93 94 95 96
## 8.251818 8.494231 8.645595 8.719864 8.719864 8.645595 8.719864 8.719864
## 97 98 99 100 101 102 103 104
## 8.719864 8.719864 8.719864 8.719864 8.719864 8.719864 8.719864 8.719864
## 105 106 107 108 109 110 111 112
## 8.719864 8.719864 8.719864 8.645595 8.719864 8.719864 8.719864 8.645595
## 113 114 115 116 117 118 119 120
## 8.719864 8.494231 8.251818 8.719864 8.645595 8.645595 8.719864 8.719864
## 121 122 123 124 125 126 127 128
## 8.719864 8.645595 8.719864 8.719864 8.719864 8.719864 8.719864 8.719864
## 129 130 131 132 133 134 135 136
## 8.719864 8.719864 8.719864 8.719864 8.719864 8.494231 8.494231 8.494231
## 137 138 139 140 141 142 143 144
## 8.494231 8.494231 8.251818 8.494231 8.719864 8.719864 8.719864 8.719864
## 145 146 147 148 149 150 151 152
## 8.719864 8.719864 8.719864 8.719864 8.719864 8.719864 8.719864 8.719864
## 153 154 155 156 157 158 159 160
## 8.719864 8.672667 8.672667 8.672667 8.719864 8.719864 8.645595 8.719864
## 161 162 163 164 165 166 167 168
## 8.494231 8.645595 8.645595 8.645595 8.645595 8.645595 8.645595 8.645595
## 169 170 171 172 173 174 175 176
## 8.645595 8.645595 8.645595 8.719864 8.645595 8.645595 8.719864 8.719864
## 177 178 179 180 181 182 183 184
## 8.719864 8.719864 8.719864 8.719864 8.645595 8.645595 8.719864 8.494231
## 185 186 187 188 189 190 191 192
## 8.494231 8.494231 8.494231 8.494231 8.719864 8.719864 8.672667 8.561120
## 193 194 195 196 197 198 199 200
## 8.561120 8.645595 8.645595 8.645595 8.645595 8.719864 8.645595 8.645595
## 201 202 203 204 205 206 207 208
## 8.645595 8.719864 8.719864 8.645595 8.645595 8.645595 8.645595 8.494231
## 209 210 211 212 213 214 215 216
## 8.645595 8.645595 8.645595 8.719864 8.719864 8.719864 8.645595 8.645595
## 217 218 219 220 221 222 223 224
## 8.645595 8.645595 8.719864 8.719864 8.719864 8.719864 8.719864 8.719864
## 225 226 227 228 229 230 231 232
## 8.719864 8.719864 8.719864 8.719864 8.719864 8.719864 8.645595 8.645595
## 233 234 235 236 237 238 239 240
## 8.645595 8.719864 8.494231 8.494231 8.494231 8.494231 8.645595 8.719864
## 241 242 243 244 245 246 247 248
## 8.719864 8.719864 8.719864 8.719864 8.645595 8.645595 8.719864 8.645595
## 249 250 251 252 253 254 255 256
## 8.645595 8.645595 8.497333 8.530728 8.481053 8.481053 8.481053 8.530728
## 257 258 259 260 261 262 263 264
## 8.530728 8.530728 8.530728 8.530728 8.530728 8.530728 8.530728 8.481053
## 265 266 267 268 269 270 271 272
## 8.530728 8.530728 8.530728 8.608429 8.398605 8.481053 8.608429 8.530728
## 273 274 275 276 277 278 279 280
## 8.530728 8.530728 8.530728 8.530728 8.530728 8.530728 8.608429 8.530728
## 281 282 283 284 285 286 287 288
## 8.530728 8.530728 8.608429 8.608429 8.608429 8.530728 8.530728 8.530728
## 289 290 291 292 293 294 295 296
## 8.530728 8.530728 8.530728 8.530728 8.530728 8.530728 8.530728 8.530728
## 297 298 299 300 301 302 303 304
## 8.445138 8.530728 8.530728 8.445138 8.530728 8.530728 8.530728 8.530728
## 305 306 307 308 309 310 311 312
## 8.530728 8.530728 8.445138 8.445138 8.530728 8.608429 8.608429 8.608429
## 313 314 315 316 317 318 319 320
## 8.608429 8.530728 8.327917 8.530728 8.530728 8.530728 8.608429 8.608429
## 321 322 323 324 325 326 327 328
## 8.608429 8.608429 8.608429 8.608429 8.608429 8.608429 8.481053 8.530728
## 329 330 331 332 333 334 335 336
## 8.530728 8.530728 8.530728 8.327917 8.327917 8.327917 8.445138 8.445138
## 337 338 339 340 341 342 343 344
## 8.445138 8.445138 8.608429 8.608429 8.327917 8.327917 8.445138 8.445138
## 345 346 347 348 349 350 351 352
## 8.445138 8.445138 8.445138 8.327917 8.327917 8.327917 8.327917 8.327917
## 353 354 355 356 357 358 359 360
## 8.481053 8.481053 8.497333 8.445138 8.445138 8.445138 8.445138 8.445138
## 361 362 363 364 365 366 367 368
## 8.445138 8.445138 8.327917 8.445138 8.327917 8.327917 8.327917 8.445138
## 369 370 371 372 373 374 375 376
## 8.445138 8.481053 8.481053 8.608429 8.481053 8.481053 8.327917 8.327917
## 377 378 379 380 381 382 383 384
## 8.327917 8.327917 8.327917 8.327917 8.327917 8.481053 8.481053 8.327917
## 385 386 387 388 389 390 391 392
## 8.398605 8.327917 8.327917 8.327917 8.445138 8.445138 8.327917 8.445138
## 393 394 395 396 397 398 399 400
## 8.327917 8.608429 8.481053 8.327917 8.497333 8.327917 8.327917 8.445138
## 401 402 403 404 405 406 407 408
## 8.445138 8.445138 8.445138 8.445138 8.445138 8.445138 8.445138 8.497333
## 409 410 411 412 413 414 415 416
## 8.327917 8.608429 8.327917 8.327917 8.327917 8.327917 8.445138 8.327917
## 417 418 419 420 421 422 423 424
## 8.327917 8.327917 8.608429 8.608429 8.327917 8.327917 8.445138 8.445138
## 425 426 427 428 429 430 431 432
## 8.445138 8.327917 8.327917 8.327917 8.327917 8.327917 8.327917 8.327917
## 433 434 435 436 437 438 439 440
## 8.481053 8.481053 8.481053 8.481053 8.608429 8.608429 8.608429 8.481053
## 441 442 443 444 445 446 447 448
## 8.445138 8.497333 8.497333 8.497333 8.398605 8.398605 8.398605 8.398605
## 449 450 451 452 453 454 455 456
## 8.497333 8.608429 8.608429 8.497333 8.497333 8.497333 8.497333 8.497333
## 457 458 459 460 461 462 463 464
## 8.497333 8.608429 8.608429 8.608429 8.497333 8.497333 8.497333 8.497333
## 465 466 467 468 469 470 471 472
## 8.497333 8.497333 8.398605 8.481053 8.481053 8.481053 8.497333 8.497333
## 473 474 475 476 477 478 479 480
## 8.497333 8.497333 8.497333 8.497333 8.481053 8.497333 8.608429 8.608429
## 481 482 483 484 485 486 487 488
## 8.398605 8.398605 8.497333 8.497333 8.497333 8.497333 8.497333 8.608429
## 489 490 491 492 493 494 495 496
## 8.497333 8.497333 8.497333 8.481053 8.608429 8.497333 8.497333 8.497333
## 497 498 499 500 501 502 503 504
## 8.497333 8.608429 8.497333 8.497333 8.497333 8.608429 8.608429 8.608429
## 505 506 507 508 509 510 511 512
## 8.497333 8.497333 8.497333 8.497333 8.497333 8.497333 8.497333 8.497333
## 513 514 515 516 517 518 519 520
## 8.497333 8.497333 8.497333 8.608429 8.497333 8.497333 8.497333 8.608429
## 521 522 523 524 525 526 527 528
## 8.608429 8.608429 8.608429 8.608429 8.608429 8.171667 8.171667 8.171667
## 529 530 531
## 8.171667 8.398605 8.497333RMSE(predict(model_tree1, test), test$ph)## [1] 0.1344733#Model with selected variables based on Linear Regression model
set.seed(1234)
model_tree2 <- rpart(ph ~ . -alch_rel -air_pressurer -pressure_vacuum -density
-filler_speed -filler_level -psc_co2 -pressure_setpoint, data=train)
predict(model_tree2, test)## 1 2 3 4 5 6 7 8
## 8.605993 8.605993 8.605993 8.605993 8.605993 8.492941 8.605993 8.605993
## 9 10 11 12 13 14 15 16
## 8.605993 8.605993 8.605993 8.605993 8.605993 8.605993 8.605993 8.605993
## 17 18 19 20 21 22 23 24
## 8.605993 8.605993 8.605993 8.605993 8.605993 8.605993 8.605993 8.605993
## 25 26 27 28 29 30 31 32
## 8.605993 8.605993 8.605993 8.605993 8.605993 8.605993 8.605993 8.605993
## 33 34 35 36 37 38 39 40
## 8.605993 8.605993 8.724681 8.605993 8.605993 8.605993 8.494231 8.251818
## 41 42 43 44 45 46 47 48
## 8.251818 8.605993 8.605993 8.605993 8.605993 8.605993 8.605993 8.605993
## 49 50 51 52 53 54 55 56
## 8.605993 8.605993 8.605993 8.605993 8.605993 8.605993 8.605993 8.605993
## 57 58 59 60 61 62 63 64
## 8.605993 8.605993 8.605993 8.605993 8.605993 8.605993 8.605993 8.605993
## 65 66 67 68 69 70 71 72
## 8.605993 8.605993 8.605993 8.724681 8.605993 8.605993 8.724681 8.251818
## 73 74 75 76 77 78 79 80
## 8.605993 8.605993 8.605993 8.605993 8.605993 8.605993 8.605993 8.605993
## 81 82 83 84 85 86 87 88
## 8.724681 8.724681 8.605993 8.724681 8.724681 8.724681 8.724681 8.251818
## 89 90 91 92 93 94 95 96
## 8.251818 8.494231 8.704970 8.724681 8.704970 8.704970 8.704970 8.704970
## 97 98 99 100 101 102 103 104
## 8.704970 8.605993 8.704970 8.704970 8.704970 8.704970 8.704970 8.704970
## 105 106 107 108 109 110 111 112
## 8.605993 8.704970 8.704970 8.724681 8.724681 8.704970 8.704970 8.704970
## 113 114 115 116 117 118 119 120
## 8.704970 8.494231 8.251818 8.605993 8.605993 8.605993 8.704970 8.704970
## 121 122 123 124 125 126 127 128
## 8.605993 8.605993 8.605993 8.704970 8.704970 8.704970 8.704970 8.704970
## 129 130 131 132 133 134 135 136
## 8.605993 8.704970 8.704970 8.605993 8.704970 8.494231 8.494231 8.494231
## 137 138 139 140 141 142 143 144
## 8.494231 8.494231 8.251818 8.494231 8.605993 8.704970 8.704970 8.605993
## 145 146 147 148 149 150 151 152
## 8.704970 8.724681 8.704970 8.704970 8.704970 8.704970 8.704970 8.704970
## 153 154 155 156 157 158 159 160
## 8.704970 8.704970 8.704970 8.704970 8.704970 8.704970 8.704970 8.605993
## 161 162 163 164 165 166 167 168
## 8.494231 8.605993 8.605993 8.704970 8.704970 8.704970 8.704970 8.704970
## 169 170 171 172 173 174 175 176
## 8.704970 8.605993 8.605993 8.605993 8.605993 8.704970 8.704970 8.704970
## 177 178 179 180 181 182 183 184
## 8.704970 8.704970 8.724681 8.704970 8.704970 8.724681 8.724681 8.494231
## 185 186 187 188 189 190 191 192
## 8.494231 8.494231 8.494231 8.494231 8.704970 8.704970 8.704970 8.704970
## 193 194 195 196 197 198 199 200
## 8.704970 8.605993 8.704970 8.704970 8.704970 8.605993 8.704970 8.724681
## 201 202 203 204 205 206 207 208
## 8.704970 8.704970 8.605993 8.605993 8.704970 8.605993 8.605993 8.494231
## 209 210 211 212 213 214 215 216
## 8.704970 8.704970 8.605993 8.704970 8.704970 8.704970 8.605993 8.704970
## 217 218 219 220 221 222 223 224
## 8.704970 8.704970 8.704970 8.704970 8.704970 8.605993 8.704970 8.704970
## 225 226 227 228 229 230 231 232
## 8.605993 8.704970 8.605993 8.605993 8.605993 8.704970 8.605993 8.605993
## 233 234 235 236 237 238 239 240
## 8.704970 8.704970 8.494231 8.494231 8.494231 8.494231 8.704970 8.605993
## 241 242 243 244 245 246 247 248
## 8.704970 8.605993 8.605993 8.605993 8.704970 8.704970 8.704970 8.704970
## 249 250 251 252 253 254 255 256
## 8.704970 8.605993 8.493377 8.527018 8.498202 8.527018 8.498202 8.527018
## 257 258 259 260 261 262 263 264
## 8.527018 8.527018 8.527018 8.527018 8.527018 8.527018 8.527018 8.527018
## 265 266 267 268 269 270 271 272
## 8.527018 8.527018 8.527018 8.498202 8.301750 8.527018 8.527018 8.527018
## 273 274 275 276 277 278 279 280
## 8.527018 8.527018 8.527018 8.527018 8.527018 8.527018 8.527018 8.527018
## 281 282 283 284 285 286 287 288
## 8.527018 8.527018 8.560000 8.689756 8.689756 8.527018 8.527018 8.527018
## 289 290 291 292 293 294 295 296
## 8.527018 8.527018 8.527018 8.527018 8.527018 8.527018 8.527018 8.527018
## 297 298 299 300 301 302 303 304
## 8.301750 8.527018 8.527018 8.301750 8.527018 8.527018 8.527018 8.527018
## 305 306 307 308 309 310 311 312
## 8.527018 8.527018 8.424459 8.424459 8.527018 8.498202 8.689756 8.689756
## 313 314 315 316 317 318 319 320
## 8.301750 8.527018 8.424459 8.527018 8.527018 8.527018 8.527018 8.527018
## 321 322 323 324 325 326 327 328
## 8.689756 8.689756 8.689756 8.689756 8.527018 8.689756 8.527018 8.527018
## 329 330 331 332 333 334 335 336
## 8.527018 8.527018 8.527018 8.301750 8.301750 8.301750 8.424459 8.424459
## 337 338 339 340 341 342 343 344
## 8.424459 8.424459 8.560000 8.560000 8.560000 8.301750 8.424459 8.424459
## 345 346 347 348 349 350 351 352
## 8.424459 8.424459 8.424459 8.301750 8.301750 8.301750 8.301750 8.301750
## 353 354 355 356 357 358 359 360
## 8.498202 8.424459 8.301750 8.424459 8.424459 8.424459 8.424459 8.424459
## 361 362 363 364 365 366 367 368
## 8.424459 8.424459 8.301750 8.424459 8.424459 8.424459 8.424459 8.424459
## 369 370 371 372 373 374 375 376
## 8.424459 8.424459 8.301750 8.424459 8.493377 8.493377 8.424459 8.424459
## 377 378 379 380 381 382 383 384
## 8.498202 8.301750 8.301750 8.424459 8.424459 8.498202 8.424459 8.424459
## 385 386 387 388 389 390 391 392
## 8.301750 8.424459 8.424459 8.424459 8.301750 8.301750 8.424459 8.424459
## 393 394 395 396 397 398 399 400
## 8.493377 8.498202 8.498202 8.301750 8.498202 8.498202 8.560000 8.424459
## 401 402 403 404 405 406 407 408
## 8.424459 8.424459 8.424459 8.424459 8.424459 8.424459 8.424459 8.425373
## 409 410 411 412 413 414 415 416
## 8.425373 8.560000 8.425373 8.425373 8.425373 8.425373 8.425373 8.425373
## 417 418 419 420 421 422 423 424
## 8.493377 8.493377 8.560000 8.560000 8.425373 8.217143 8.217143 8.425373
## 425 426 427 428 429 430 431 432
## 8.217143 8.217143 8.217143 8.217143 8.217143 8.217143 8.217143 8.425373
## 433 434 435 436 437 438 439 440
## 8.498202 8.498202 8.498202 8.498202 8.560000 8.560000 8.560000 8.498202
## 441 442 443 444 445 446 447 448
## 8.217143 8.493377 8.493377 8.493377 8.425373 8.425373 8.425373 8.425373
## 449 450 451 452 453 454 455 456
## 8.425373 8.560000 8.560000 8.560000 8.493377 8.493377 8.493377 8.493377
## 457 458 459 460 461 462 463 464
## 8.493377 8.560000 8.560000 8.560000 8.425373 8.493377 8.493377 8.493377
## 465 466 467 468 469 470 471 472
## 8.493377 8.493377 8.425373 8.498202 8.498202 8.498202 8.560000 8.493377
## 473 474 475 476 477 478 479 480
## 8.493377 8.493377 8.493377 8.493377 8.498202 8.493377 8.560000 8.560000
## 481 482 483 484 485 486 487 488
## 8.425373 8.425373 8.493377 8.493377 8.493377 8.493377 8.493377 8.560000
## 489 490 491 492 493 494 495 496
## 8.560000 8.493377 8.493377 8.498202 8.560000 8.425373 8.425373 8.425373
## 497 498 499 500 501 502 503 504
## 8.425373 8.560000 8.493377 8.493377 8.493377 8.560000 8.560000 8.493377
## 505 506 507 508 509 510 511 512
## 8.560000 8.493377 8.493377 8.493377 8.493377 8.493377 8.493377 8.493377
## 513 514 515 516 517 518 519 520
## 8.493377 8.493377 8.493377 8.560000 8.560000 8.560000 8.560000 8.560000
## 521 522 523 524 525 526 527 528
## 8.560000 8.560000 8.560000 8.560000 8.560000 8.301750 8.301750 8.301750
## 529 530 531
## 8.301750 8.301750 8.493377RMSE(predict(model_tree2, test), test$ph)## [1] 0.1348702plot(model_tree1, uniform=TRUE, compress=FALSE, margin=.015)
text(model_tree1, all=TRUE, cex=.5)
Random Forest
#Model 1
set.seed(1234)
model_rf1 <- randomForest(ph ~ ., data=train_up,
ntree = 50, nodesize=15,)
model_rf1$rsq## [1] 0.6976031 0.7143194 0.7518379 0.7541110 0.7795894 0.7974019 0.8154279
## [8] 0.8247588 0.8372702 0.8470830 0.8562543 0.8600366 0.8646292 0.8654197
## [15] 0.8692691 0.8694372 0.8730366 0.8735679 0.8764539 0.8776930 0.8788295
## [22] 0.8793116 0.8804491 0.8810166 0.8811327 0.8806551 0.8812119 0.8821570
## [29] 0.8838412 0.8832307 0.8848415 0.8856629 0.8855656 0.8867830 0.8868331
## [36] 0.8876840 0.8882950 0.8888927 0.8895892 0.8891586 0.8891269 0.8893793
## [43] 0.8895869 0.8896685 0.8898406 0.8903273 0.8905896 0.8902392 0.8902180
## [50] 0.8904352RMSE(predict(model_rf1,test), test$ph)## [1] 0.1073807#Model 2
set.seed(1234)
model_rf2 <- randomForest(ph ~ . -alch_rel -air_pressurer -pressure_vacuum -density
-filler_speed -filler_level -psc_co2 -pressure_setpoint, data=train_up,
ntree = 50, nodesize=15,)
model_rf2$rsq## [1] 0.7484210 0.7413447 0.7350172 0.7615498 0.7885962 0.7956339 0.8041510
## [8] 0.8113253 0.8235833 0.8276234 0.8314446 0.8389428 0.8439495 0.8462311
## [15] 0.8458698 0.8481294 0.8500746 0.8524150 0.8546256 0.8570357 0.8589027
## [22] 0.8603767 0.8616162 0.8628927 0.8643926 0.8661009 0.8675507 0.8687223
## [29] 0.8697785 0.8697520 0.8710171 0.8715354 0.8727304 0.8725161 0.8728927
## [36] 0.8737550 0.8739372 0.8742044 0.8749446 0.8745692 0.8748572 0.8745765
## [43] 0.8748385 0.8758151 0.8758966 0.8767572 0.8777655 0.8784806 0.8791126
## [50] 0.8795768RMSE(predict(model_rf2,test), test$ph)## [1] 0.1102083Support Vector Machine
#Model 1
train_control <- trainControl(method = "cv", number = 5)
set.seed(1234)
model_svm <- svm(ph ~., data=train,
kernal = 'linear', cost=0.1)
summary(model_svm)##
## Call:
## svm(formula = ph ~ ., data = train, kernal = "linear", cost = 0.1)
##
##
## Parameters:
## SVM-Type: eps-regression
## SVM-Kernel: radial
## cost: 0.1
## gamma: 0.02857143
## epsilon: 0.1
##
##
## Number of Support Vectors: 1397ph_prediction <- predict(model_svm, test)
RMSE(test$ph,ph_prediction)## [1] 0.1318081#Model 2
train_control <- trainControl(method = "cv", number = 5)
set.seed(1234)
model_svm <- svm(ph ~. -alch_rel -air_pressurer -pressure_vacuum -density
-filler_speed -filler_level -psc_co2 -pressure_setpoint, data=train,
kernal = 'linear', cost=0.1)
summary(model_svm)##
## Call:
## svm(formula = ph ~ . - alch_rel - air_pressurer - pressure_vacuum -
## density - filler_speed - filler_level - psc_co2 - pressure_setpoint,
## data = train, kernal = "linear", cost = 0.1)
##
##
## Parameters:
## SVM-Type: eps-regression
## SVM-Kernel: radial
## cost: 0.1
## gamma: 0.03703704
## epsilon: 0.1
##
##
## Number of Support Vectors: 1413ph_prediction <- predict(model_svm, test)
RMSE(test$ph,ph_prediction)## [1] 0.1351914Neural Network
Finally, we will train a neural net model
library(nnet)
my.grid <- expand.grid(.decay = c(0.5,0,1), .size=c(5,6,7))
model_nnet <- train(ph ~ carb_volume, data=train, method="nnet",
maxit=1000, tuneGrid = my.grid, trace =F, linout = 1)## Warning in nominalTrainWorkflow(x = x, y = y, wts = weights, info = trainInfo,
## : There were missing values in resampled performance measures.predict(model_nnet, test)## 1 2 3 4 5 6 7 8
## 8.565224 8.547035 8.543868 8.537313 8.545383 8.536075 8.546192 8.547035
## 9 10 11 12 13 14 15 16
## 8.548825 8.545383 8.545383 8.546192 8.559843 8.571166 8.566657 8.552817
## 17 18 19 20 21 22 23 24
## 8.596397 8.582917 8.571166 8.558585 8.543161 8.537313 8.565936 8.543868
## 25 26 27 28 29 30 31 32
## 8.538726 8.541850 8.542488 8.545383 8.543161 8.541850 8.543868 8.545383
## 33 34 35 36 37 38 39 40
## 8.540136 8.536188 8.541245 8.543868 8.539632 8.536431 8.540136 8.539632
## 41 42 43 44 45 46 47 48
## 8.540136 8.574347 8.551767 8.539632 8.545383 8.539632 8.547035 8.566657
## 49 50 51 52 53 54 55 56
## 8.546192 8.540136 8.541850 8.540674 8.542488 8.538726 8.539162 8.543161
## 57 58 59 60 61 62 63 64
## 8.540136 8.536115 8.539632 8.540136 8.584737 8.574347 8.565224 8.581133
## 65 66 67 68 69 70 71 72
## 8.569628 8.547913 8.547913 8.542488 8.546192 8.539632 8.538726 8.536115
## 73 74 75 76 77 78 79 80
## 8.566657 8.563827 8.563827 8.536431 8.537953 8.539162 8.541850 8.566657
## 81 82 83 84 85 86 87 88
## 8.537616 8.543161 8.545383 8.569628 8.566657 8.545383 8.546192 8.543161
## 89 90 91 92 93 94 95 96
## 8.537616 8.537043 8.539162 8.558585 8.555020 8.572739 8.559843 8.550752
## 97 98 99 100 101 102 103 104
## 8.545383 8.540674 8.542488 8.584737 8.569628 8.577669 8.538726 8.541245
## 105 106 107 108 109 110 111 112
## 8.545383 8.568125 8.536150 8.552817 8.546192 8.545383 8.562464 8.539632
## 113 114 115 116 117 118 119 120
## 8.539632 8.543868 8.539632 8.581133 8.569628 8.561136 8.550752 8.537953
## 121 122 123 124 125 126 127 128
## 8.539632 8.547035 8.545383 8.550752 8.539162 8.539162 8.538726 8.559843
## 129 130 131 132 133 134 135 136
## 8.544608 8.561136 8.568125 8.543161 8.541850 8.543161 8.545383 8.536431
## 137 138 139 140 141 142 143 144
## 8.536115 8.542488 8.541245 8.542488 8.537953 8.545383 8.539632 8.541245
## 145 146 147 148 149 150 151 152
## 8.540674 8.572739 8.536075 8.538726 8.538323 8.541850 8.536602 8.569628
## 153 154 155 156 157 158 159 160
## 8.575991 8.543868 8.546192 8.543161 8.537313 8.537953 8.540136 8.538323
## 161 162 163 164 165 166 167 168
## 8.536293 8.557362 8.565224 8.569628 8.540136 8.543161 8.541245 8.574347
## 169 170 171 172 173 174 175 176
## 8.566657 8.563827 8.566657 8.562464 8.536431 8.537616 8.537043 8.569628
## 177 178 179 180 181 182 183 184
## 8.566657 8.563827 8.537953 8.536806 8.536068 8.538726 8.539632 8.537953
## 185 186 187 188 189 190 191 192
## 8.536293 8.536075 8.538726 8.536516 8.537313 8.543161 8.536115 8.536075
## 193 194 195 196 197 198 199 200
## 8.563827 8.562464 8.541850 8.540136 8.541245 8.539632 8.537616 8.537313
## 201 202 203 204 205 206 207 208
## 8.537313 8.540674 8.545383 8.559843 8.556174 8.551767 8.561136 8.536115
## 209 210 211 212 213 214 215 216
## 8.544608 8.540674 8.547913 8.541245 8.539162 8.541850 8.540674 8.566657
## 217 218 219 220 221 222 223 224
## 8.563827 8.552817 8.540674 8.537313 8.540674 8.537313 8.562464 8.562464
## 225 226 227 228 229 230 231 232
## 8.571166 8.540674 8.538726 8.536700 8.540136 8.538726 8.536188 8.538726
## 233 234 235 236 237 238 239 240
## 8.544608 8.572739 8.536093 8.539162 8.539632 8.541245 8.541245 8.537313
## 241 242 243 244 245 246 247 248
## 8.558585 8.575991 8.537616 8.539162 8.558585 8.558585 8.537953 8.537616
## 249 250 251 252 253 254 255 256
## 8.537953 8.536602 8.538323 8.538323 8.551767 8.546192 8.549771 8.536115
## 257 258 259 260 261 262 263 264
## 8.537313 8.537313 8.537313 8.536068 8.536115 8.545383 8.536806 8.545383
## 265 266 267 268 269 270 271 272
## 8.538323 8.540674 8.539162 8.582917 8.543161 8.551767 8.553901 8.536293
## 273 274 275 276 277 278 279 280
## 8.540674 8.537616 8.537313 8.577669 8.540136 8.537616 8.566657 8.537313
## 281 282 283 284 285 286 287 288
## 8.538726 8.542488 8.598464 8.596397 8.590408 8.537313 8.539632 8.540136
## 289 290 291 292 293 294 295 296
## 8.538726 8.536068 8.538323 8.538323 8.537953 8.538323 8.540136 8.540674
## 297 298 299 300 301 302 303 304
## 8.536115 8.538726 8.562464 8.537313 8.543868 8.541850 8.538726 8.539162
## 305 306 307 308 309 310 311 312
## 8.541850 8.538726 8.536602 8.539162 8.536150 8.574347 8.575991 8.582917
## 313 314 315 316 317 318 319 320
## 8.551767 8.536431 8.542488 8.536431 8.550752 8.552817 8.540674 8.557362
## 321 322 323 324 325 326 327 328
## 8.581133 8.581133 8.575991 8.565224 8.566657 8.561136 8.566657 8.543868
## 329 330 331 332 333 334 335 336
## 8.540674 8.536075 8.540674 8.562464 8.566657 8.542488 8.540674 8.541245
## 337 338 339 340 341 342 343 344
## 8.538323 8.537313 8.575991 8.594365 8.557362 8.555020 8.536293 8.538323
## 345 346 347 348 349 350 351 352
## 8.537616 8.536075 8.536602 8.542488 8.536806 8.541850 8.539632 8.543868
## 353 354 355 356 357 358 359 360
## 8.579383 8.586592 8.566657 8.542488 8.540136 8.550752 8.540674 8.536431
## 361 362 363 364 365 366 367 368
## 8.537313 8.537953 8.540674 8.542488 8.542488 8.547913 8.547035 8.537043
## 369 370 371 372 373 374 375 376
## 8.542488 8.582917 8.553901 8.577669 8.596397 8.596397 8.547913 8.537313
## 377 378 379 380 381 382 383 384
## 8.571166 8.548825 8.552817 8.545383 8.536240 8.548825 8.568125 8.539162
## 385 386 387 388 389 390 391 392
## 8.556174 8.557362 8.543868 8.547035 8.541245 8.543161 8.548825 8.540136
## 393 394 395 396 397 398 399 400
## 8.547913 8.582917 8.590408 8.555020 8.579383 8.594365 8.592369 8.540136
## 401 402 403 404 405 406 407 408
## 8.548825 8.540136 8.545383 8.543161 8.538726 8.536602 8.536115 8.537170
## 409 410 411 412 413 414 415 416
## 8.542488 8.600566 8.547913 8.542488 8.547913 8.546192 8.537616 8.552817
## 417 418 419 420 421 422 423 424
## 8.545383 8.547035 8.574347 8.584737 8.557362 8.543868 8.543868 8.536093
## 425 426 427 428 429 430 431 432
## 8.543868 8.547035 8.542488 8.566657 8.548825 8.543161 8.545383 8.544608
## 433 434 435 436 437 438 439 440
## 8.582917 8.577669 8.571166 8.577669 8.572739 8.568125 8.577669 8.588482
## 441 442 443 444 445 446 447 448
## 8.539632 8.542488 8.540136 8.548825 8.545383 8.538323 8.540136 8.539162
## 449 450 451 452 453 454 455 456
## 8.562464 8.569628 8.563827 8.562464 8.543161 8.541245 8.538323 8.536431
## 457 458 459 460 461 462 463 464
## 8.536188 8.594365 8.594365 8.584737 8.539162 8.538323 8.552817 8.543868
## 465 466 467 468 469 470 471 472
## 8.537313 8.548825 8.537043 8.577669 8.581133 8.607086 8.558585 8.536806
## 473 474 475 476 477 478 479 480
## 8.539632 8.541850 8.541245 8.539162 8.571166 8.540136 8.574347 8.572739
## 481 482 483 484 485 486 487 488
## 8.537313 8.542488 8.555020 8.541245 8.545383 8.550752 8.540136 8.556174
## 489 490 491 492 493 494 495 496
## 8.584737 8.536293 8.536806 8.577669 8.575991 8.541850 8.541850 8.544608
## 497 498 499 500 501 502 503 504
## 8.543868 8.581133 8.536068 8.536602 8.536188 8.577669 8.575991 8.562464
## 505 506 507 508 509 510 511 512
## 8.575991 8.536431 8.536115 8.539632 8.537616 8.541850 8.541245 8.537616
## 513 514 515 516 517 518 519 520
## 8.537616 8.536806 8.537953 8.565224 8.574347 8.561136 8.569628 8.553901
## 521 522 523 524 525 526 527 528
## 8.553901 8.571166 8.562464 8.571166 8.557362 8.536093 8.536093 8.537616
## 529 530 531
## 8.537313 8.540674 8.543161RMSE(predict(model_nnet, test), test$ph)## [1] 0.1715755Out neural net model has an RMSE of 0.167.
Model Selection
Across our three models, we had the following RMSE when we applied the models to our test data set
- Linear Regression - 0.1358189
- Decision Tree (Single Tree) - 0.1266664
- Random Forest - 0.1060108
- Suppoer Vector Machine - 0.1240094
- Neural Network - 0.1673001
Based on the RMSE values, we find that the Random Forest tree - with all data included, has performed the best amongst our models and thus will be using that as the model to predict our ph.
Final Prediction on Eval Data
eval_data_raw <- read_excel('StudentEvaluation.xlsx')
eval_data <- clean_names(eval_data_raw)
eval_data <- as_tibble(eval_data)
eval_data <- eval_data %>%
mutate(brand_code = ifelse(is.na(brand_code), "A",brand_code))
#Change brand code variable to factor
eval_data <- eval_data %>% mutate(brand_code = factor(brand_code, levels = sort(unique(brand_code))))
eval_data <- eval_data %>% select(-ph)
eval_data <- eval_data %>% mutate(across(where(is.numeric), ~replace_na(., median(., na.rm=TRUE))))
ph_prediction <- predict(model_rf1,eval_data)
eval_data$ph <- ph_prediction
write_xlsx(eval_data, 'StudentEvaluation_final.xlsx')For making our final predictions, we used the median value of our numeric columns when imputed the missing NA values. From there we used our Random Forest model to predict the ph for the the different combinations in our dataset.
Conclusion
While the usage of the Random Forest model is the best model in terms of predicting the ph of the product based on the root mean square error, one of the main shortcoming is that it generates a model that is not interpretable. As a result, we are able to feel slightly confident in our prediction for the ph, but we have a limited ability to interpret the model. If indeed we are primarily interested in understanding the model - and are willing to sacrifice some of the predictive ability, than we should use the Single decision tree - which has a higher RMSE but we are able to generate a visual illustration that helps us understand the factors that are used to determine the variables that impact the ph.