Project 2: Predictive Model of PH
Deepa Sharma, William Aiken, Ahmed Elsaeyed, Diana Plunkett, PK O’Flaherty
2024-12-10
Project Introduction:
New regulations are requiring ABC Beverage to provide a report with an outline of our manufacturing process, and a predictive model of PH including an explanation of predictive factors.
Our data science team is tasked with developing the predictive model from provided historical data and using that model to predict PH on test data.
Data
Here we import our train and test data, student_train
and student_eval, and evaluate for missing data and
additional exploratory steps.
Data Acquisition
Here we can preview the data structure:
student_train = read.csv('https://raw.githubusercontent.com/deepasharma06/Data-624/refs/heads/main/StudentData_training.csv')
student_eval = read.csv('https://raw.githubusercontent.com/deepasharma06/Data-624/refs/heads/main/StudentEvaluation_test.csv')
head(student_train) %>% kable()| Brand.Code | Carb.Volume | Fill.Ounces | PC.Volume | Carb.Pressure | Carb.Temp | PSC | PSC.Fill | PSC.CO2 | Mnf.Flow | Carb.Pressure1 | Fill.Pressure | Hyd.Pressure1 | Hyd.Pressure2 | Hyd.Pressure3 | Hyd.Pressure4 | Filler.Level | Filler.Speed | Temperature | Usage.cont | Carb.Flow | Density | MFR | Balling | Pressure.Vacuum | PH | Oxygen.Filler | Bowl.Setpoint | Pressure.Setpoint | Air.Pressurer | Alch.Rel | Carb.Rel | Balling.Lvl |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| B | 5.340000 | 23.96667 | 0.2633333 | 68.2 | 141.2 | 0.104 | 0.26 | 0.04 | -100 | 118.8 | 46.0 | 0 | NA | NA | 118 | 121.2 | 4002 | 66.0 | 16.18 | 2932 | 0.88 | 725.0 | 1.398 | -4.0 | 8.36 | 0.022 | 120 | 46.4 | 142.6 | 6.58 | 5.32 | 1.48 |
| A | 5.426667 | 24.00667 | 0.2386667 | 68.4 | 139.6 | 0.124 | 0.22 | 0.04 | -100 | 121.6 | 46.0 | 0 | NA | NA | 106 | 118.6 | 3986 | 67.6 | 19.90 | 3144 | 0.92 | 726.8 | 1.498 | -4.0 | 8.26 | 0.026 | 120 | 46.8 | 143.0 | 6.56 | 5.30 | 1.56 |
| B | 5.286667 | 24.06000 | 0.2633333 | 70.8 | 144.8 | 0.090 | 0.34 | 0.16 | -100 | 120.2 | 46.0 | 0 | NA | NA | 82 | 120.0 | 4020 | 67.0 | 17.76 | 2914 | 1.58 | 735.0 | 3.142 | -3.8 | 8.94 | 0.024 | 120 | 46.6 | 142.0 | 7.66 | 5.84 | 3.28 |
| A | 5.440000 | 24.00667 | 0.2933333 | 63.0 | 132.6 | NA | 0.42 | 0.04 | -100 | 115.2 | 46.4 | 0 | 0 | 0 | 92 | 117.8 | 4012 | 65.6 | 17.42 | 3062 | 1.54 | 730.6 | 3.042 | -4.4 | 8.24 | 0.030 | 120 | 46.0 | 146.2 | 7.14 | 5.42 | 3.04 |
| A | 5.486667 | 24.31333 | 0.1113333 | 67.2 | 136.8 | 0.026 | 0.16 | 0.12 | -100 | 118.4 | 45.8 | 0 | 0 | 0 | 92 | 118.6 | 4010 | 65.6 | 17.68 | 3054 | 1.54 | 722.8 | 3.042 | -4.4 | 8.26 | 0.030 | 120 | 46.0 | 146.2 | 7.14 | 5.44 | 3.04 |
| A | 5.380000 | 23.92667 | 0.2693333 | 66.6 | 138.4 | 0.090 | 0.24 | 0.04 | -100 | 119.6 | 45.6 | 0 | 0 | 0 | 116 | 120.2 | 4014 | 66.2 | 23.82 | 2948 | 1.52 | 738.8 | 2.992 | -4.4 | 8.32 | 0.024 | 120 | 46.0 | 146.6 | 7.16 | 5.44 | 3.02 |
Domain Dive
We looked for a general understanding of bottling manufacturing and learned a few things potentially about our data, so these are our assumptions and guesses.
This data describes a continuous bottling process, like a conveyor
belt, and so Mnf Flow would be the percentage of intended
speed the process is running at.
Fill.Ounces is how full the bottles bottles are being
filled to with a target of 24 oz.
Alch.Rel sounds like this could be an relative value of
alcohol. I checked and Modelo produces cans of 24 ounce spiked sparkling
water, so maybe it’s a similar product.
Brand.Code could be in label only but if it’s different
formulas and flavors, the specific ratios of acids (citric, phosphoric,
malic) to create the brand’s flavor profile, then it would definitely
affect PH.
PH is the PH of our bottled beverage. Curiously we would
expect carbonated beverages to be acidic because carbon in water makes
carbonic acid however all of the PH values are in the 8 range which is
basic. 7 is neutral and below 7 would be acidic.
Mnf.Flow seems to be the speed of the conveyor belt or
manufacturing flow represented as a percentage of the ideal speed.
MFR could be the actual speed.
Carb.Volume, Carb.Pressure,
Carb.Temp we could potentially combine with the formula
P*V/T to obtain the amount of carbon being injected into the liquid.
Bowl.Setpoint is consistently 120 and we believe it’s
how high the bottle is expected to be filled. Filler.Level
is a variable close to 120 which is presumably the actual amount filled.
Dividing Bowl.Setpoint by Filler Level could
agregate the predictors into whether too much or too little was
added.
Filler.Speed seems to be the speed at which the fill
liquid is filled. And Carb.Flow seems to be the speed at
which the carbonation is injected.
Carb.Pressure1 and Fill.Pressure look to be
the pressure at which the carbonation (~120) is injected into the fill
liquid and the fill liquid’s pressure (~46). Those values make sense
because if the C02’s pressure is not higher than the liquid, the liquid
won’t carbonate.
Pressure.Setpoint seems to be the setting for
Fill.Pressure that the manufacturing process is trying to
match.
Balling is a reference to the Balling scale which was a
way to determine how much sugar is in a syrup based on the density of
the liquid. Balling.Lvl is related.
Temperature is in fahrenheit (~66) but isn’t clear if
that’s the ambient temperature and should be aggregated with
Air.Pressurer (~144) to get a sense of what the surrounding
pressure is, which could influence how much carbonation stays in the
bottle.
PSC.CO2 could be the purity of the carbon dioxide,
variations in which would have a big impact on the final PH.
PSC.Fill could be the purity of the water. and
PSC a combined metric of purity. But this is just a
guess.
This leaves a lot variables without a clear picture of how they fit into the manufacturing process:
PC.Volume
Hyd.Pressure1 Hyd.Pressure2
Hyd.Pressure3 Hyd.Pressure4
Usage.cont Density
Pressure.Vacuum Oxygen.Filler
Carb.Rel
Missing values
In total there are 724 missing values across 442 rows.
17.2% of our rows have missing data (442/2571) so we won’t just drop all rows with missing data.
Here we show columns with the highest number of missing values:
# There are 724 NA values
#sum(is.na(student_train))
# 442 rows have missing data
#student_train %>%
# filter(if_any(everything(), is.na)) %>%
# nrow()
# Display columns by missingness
student_train %>%
summarise(across(everything(), ~ sum(is.na(.)), .names = "missing_{.col}")) %>%
pivot_longer(everything(), names_to = "Column", values_to = "Missing_Count") %>%
arrange(desc(Missing_Count))## # A tibble: 33 × 2
## Column Missing_Count
## <chr> <int>
## 1 missing_MFR 212
## 2 missing_Filler.Speed 57
## 3 missing_PC.Volume 39
## 4 missing_PSC.CO2 39
## 5 missing_Fill.Ounces 38
## 6 missing_PSC 33
## 7 missing_Carb.Pressure1 32
## 8 missing_Hyd.Pressure4 30
## 9 missing_Carb.Pressure 27
## 10 missing_Carb.Temp 26
## # ℹ 23 more rowsImpute missing values (NAs)
Here we impute the missing values using kNN and then check that the
724 missing values in student_train and the 366 missing
values in student_eval are filled in.
# Produce the model that can impute values using kNN
imputeModel <- preProcess(student_train, method = c("knnImpute"))
# Impute the missing values for the training and test data
student_train <- predict(imputeModel, student_train)
student_eval <- predict(imputeModel, student_eval)
# There are now zero NA values in our train and test data
sum(is.na(student_train))## [1] 0#sum(is.na(student_eval))Correlation Plot
In our correlation plot we are comparing values in the second to last
PH column. We note Filler.Level and
Bowl.Setpoint which are highly collinear have some positive
correlation with PH. Also, a number of variables have
negative correlation with PH, most notably
Mnf.Flow, Hyd.Pressure3, and
Pressure.Setpoint.
# Select only numeric columns
numeric_data <- student_train %>% select(where(is.numeric))
# Calculate the correlation matrix
#cor(numeric_data, use = "pairwise.complete.obs"): Calculates correlations between numeric variables while ignoring missing values (pairwise complete observations). This ensures that even if some values are missing, they won’t affect the correlation calculation for the remaining complete pairs.
correlation_matrix <- cor(numeric_data, use = "pairwise.complete.obs")
# Create the correlation plot
corrplot(correlation_matrix, tl.col = "black", tl.cex = 0.6, order = 'AOE') # color (tl.col) and size (tl.cex) help style the plot.
Distribution Visualization
Here we show distributions of the variables as histograms. Note,
Brand.Code is excluded because it doesn’t have the numeric
data.
# Melt the data
mlt.train <- student_train # Use your actual dataframe name
mlt.train$ID <- rownames(mlt.train) # Assign row names to ID
mlt.train <- melt(mlt.train, id.vars = "ID")
# Convert the value column to numeric
mlt.train$value <- as.numeric(mlt.train$value)
# Create histograms of the predictors
ggplot(data = mlt.train, aes(x = value)) +
geom_histogram(binwidth = 6, fill = "skyblue", color = "black", alpha = 0.8) + # Adjust binwidth as needed
facet_wrap(~ variable, scales = "free") +
labs(title = "Distributions of Predictors", x = "Predictors", y = "Frequency") +
theme_minimal(base_size = 9) + # Use a minimal theme for better clarity
theme(panel.grid.major = element_blank(), panel.grid.minor = element_blank()) # Clean up grid lines
Outliers
We are seeing a lot of outliers in the boxplots, specifically for variables where the majority of their values are zero. It’s possible we will use near-zero variance filtering to resolve these. Below only the first 9 of 32 are shown for display purposes but the pattern is the same.
# Create boxplots for each numeric column
numeric_columns <- names(student_train)[sapply(student_train, is.numeric)]
plot_list <- lapply(numeric_columns[1:9], function(col) {
ggplot(student_train, aes_string(y = col)) +
geom_boxplot(outlier.color = "red", fill = "lightblue") +
ggtitle(paste("Boxplot of", col)) +
theme_minimal()
})
# Display all boxplots in a grid
do.call(grid.arrange, c(plot_list, ncol = 3))
Feature Engineering
Here we use feature engineering and regression techniques to evaluate our feature set.
Initial Linear Regression Model
Here we build the initial linear regression model.
# Setting up the model
model <- lm(PH ~ ., data = student_train)
summary(model)##
## Call:
## lm(formula = PH ~ ., data = student_train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.0652 -0.4500 0.0545 0.5017 4.3209
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.258937 0.085135 -3.042 0.00238 **
## Brand.CodeA 0.045707 0.141827 0.322 0.74727
## Brand.CodeB 0.447021 0.078984 5.660 1.69e-08 ***
## Brand.CodeC -0.381388 0.085887 -4.441 9.36e-06 ***
## Brand.CodeD 0.338218 0.161402 2.095 0.03623 *
## Carb.Volume -0.046235 0.042321 -1.092 0.27473
## Fill.Ounces -0.035288 0.016279 -2.168 0.03028 *
## PC.Volume -0.038051 0.018876 -2.016 0.04392 *
## Carb.Pressure -0.015593 0.059234 -0.263 0.79238
## Carb.Temp 0.036313 0.053963 0.673 0.50107
## PSC -0.024380 0.016269 -1.499 0.13410
## PSC.Fill -0.023195 0.015868 -1.462 0.14393
## PSC.CO2 -0.032079 0.015670 -2.047 0.04074 *
## Mnf.Flow -0.492621 0.031570 -15.604 < 2e-16 ***
## Carb.Pressure1 0.194895 0.019094 10.207 < 2e-16 ***
## Fill.Pressure 0.033001 0.022371 1.475 0.14030
## Hyd.Pressure1 -0.004829 0.026303 -0.184 0.85436
## Hyd.Pressure2 -0.100778 0.050088 -2.012 0.04432 *
## Hyd.Pressure3 0.315571 0.053337 5.917 3.73e-09 ***
## Hyd.Pressure4 -0.016382 0.023264 -0.704 0.48138
## Filler.Level -0.093419 0.049450 -1.889 0.05899 .
## Filler.Speed 0.029192 0.033531 0.871 0.38406
## Temperature -0.109919 0.018231 -6.029 1.89e-09 ***
## Usage.cont -0.123601 0.019583 -6.312 3.25e-10 ***
## Carb.Flow 0.071620 0.022218 3.223 0.00128 **
## Density -0.272023 0.060842 -4.471 8.13e-06 ***
## MFR -0.036383 0.024352 -1.494 0.13529
## Balling -0.342804 0.122457 -2.799 0.00516 **
## Pressure.Vacuum -0.053160 0.024621 -2.159 0.03093 *
## Oxygen.Filler -0.070359 0.018451 -3.813 0.00014 ***
## Bowl.Setpoint 0.290053 0.050216 5.776 8.58e-09 ***
## Pressure.Setpoint -0.100544 0.022901 -4.390 1.18e-05 ***
## Air.Pressurer -0.019757 0.016531 -1.195 0.23215
## Alch.Rel 0.192473 0.061961 3.106 0.00192 **
## Carb.Rel 0.037817 0.034485 1.097 0.27292
## Balling.Lvl 0.475401 0.106823 4.450 8.94e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7644 on 2535 degrees of freedom
## Multiple R-squared: 0.4233, Adjusted R-squared: 0.4153
## F-statistic: 53.16 on 35 and 2535 DF, p-value: < 2.2e-16Variance Inflation Factor (VIF)
With the above initial linear regression model we can access the Variance Inflation Factor (VIF) to detect multicollinearity in our regression model to help us select features.
Note, we see that Brand.Code is highly collinear.
# Calculating VIF
vif_values <- vif(model)
vif_values## GVIF Df GVIF^(1/(2*Df))
## Brand.Code 54.107705 4 1.646863
## Carb.Volume 7.882378 1 2.807557
## Fill.Ounces 1.152505 1 1.073548
## PC.Volume 1.581700 1 1.257657
## Carb.Pressure 15.485791 1 3.935199
## Carb.Temp 12.828711 1 3.581719
## PSC 1.154528 1 1.074490
## PSC.Fill 1.102981 1 1.050229
## PSC.CO2 1.068829 1 1.033842
## Mnf.Flow 4.384391 1 2.093894
## Carb.Pressure1 1.600618 1 1.265155
## Fill.Pressure 2.200066 1 1.483262
## Hyd.Pressure1 3.035406 1 1.742242
## Hyd.Pressure2 11.015223 1 3.318919
## Hyd.Pressure3 12.478420 1 3.532481
## Hyd.Pressure4 2.369556 1 1.539336
## Filler.Level 10.723928 1 3.274741
## Filler.Speed 4.903247 1 2.214328
## Temperature 1.455792 1 1.206562
## Usage.cont 1.686112 1 1.298504
## Carb.Flow 2.170155 1 1.473145
## Density 16.280359 1 4.034893
## MFR 4.069385 1 2.017272
## Balling 65.953990 1 8.121206
## Pressure.Vacuum 2.666455 1 1.632928
## Oxygen.Filler 1.494970 1 1.222690
## Bowl.Setpoint 11.117253 1 3.334254
## Pressure.Setpoint 2.301930 1 1.517211
## Air.Pressurer 1.202058 1 1.096384
## Alch.Rel 16.860616 1 4.106168
## Carb.Rel 5.220788 1 2.284904
## Balling.Lvl 50.184296 1 7.084088Near-Zero Variance
Next we check if any of the variables have a near zero variance and
surprisingly only one variable, Hyd.Pressure1, is
identified by default. We changed the parameters ‘freqCut’ and
‘uniqueCut’ but it required large changes to pick up the other
variables, so we kept to only removing Hyd.Pressure1.
# Remove problematic predictors from train and test data
student_train_x <- student_train[, -nearZeroVar(student_train)]
student_eval_y <- student_eval[, -13]
# Display extended list of NZV predictors
nearZeroVar(student_train, freqCut = 60/40, uniqueCut = 40, names = TRUE)## [1] "Brand.Code" "Fill.Pressure" "Hyd.Pressure1" "Hyd.Pressure2"
## [5] "Hyd.Pressure3" "Carb.Flow" "Bowl.Setpoint"Deepa added:
Reduce Multicollinearity
Remove Highly Collinear Variables:
Removing variables with high VIF values (such as Brand.Code, Balling, and Density) to reduce multicollinearity.
# Install and load the 'car' package
#install.packages("car")
library(car)
# Check VIF for the current model
vif(model)## GVIF Df GVIF^(1/(2*Df))
## Brand.Code 54.107705 4 1.646863
## Carb.Volume 7.882378 1 2.807557
## Fill.Ounces 1.152505 1 1.073548
## PC.Volume 1.581700 1 1.257657
## Carb.Pressure 15.485791 1 3.935199
## Carb.Temp 12.828711 1 3.581719
## PSC 1.154528 1 1.074490
## PSC.Fill 1.102981 1 1.050229
## PSC.CO2 1.068829 1 1.033842
## Mnf.Flow 4.384391 1 2.093894
## Carb.Pressure1 1.600618 1 1.265155
## Fill.Pressure 2.200066 1 1.483262
## Hyd.Pressure1 3.035406 1 1.742242
## Hyd.Pressure2 11.015223 1 3.318919
## Hyd.Pressure3 12.478420 1 3.532481
## Hyd.Pressure4 2.369556 1 1.539336
## Filler.Level 10.723928 1 3.274741
## Filler.Speed 4.903247 1 2.214328
## Temperature 1.455792 1 1.206562
## Usage.cont 1.686112 1 1.298504
## Carb.Flow 2.170155 1 1.473145
## Density 16.280359 1 4.034893
## MFR 4.069385 1 2.017272
## Balling 65.953990 1 8.121206
## Pressure.Vacuum 2.666455 1 1.632928
## Oxygen.Filler 1.494970 1 1.222690
## Bowl.Setpoint 11.117253 1 3.334254
## Pressure.Setpoint 2.301930 1 1.517211
## Air.Pressurer 1.202058 1 1.096384
## Alch.Rel 16.860616 1 4.106168
## Carb.Rel 5.220788 1 2.284904
## Balling.Lvl 50.184296 1 7.084088# Refine the model by removing variables with high VIF (e.g., Brand.Code, Balling)
model_refined <- lm(PH ~ . -Brand.Code -Balling, data = student_train)
model_refined ##
## Call:
## lm(formula = PH ~ . - Brand.Code - Balling, data = student_train)
##
## Coefficients:
## (Intercept) Carb.Volume Fill.Ounces PC.Volume
## -0.006701 -0.008711 -0.046652 -0.038479
## Carb.Pressure Carb.Temp PSC PSC.Fill
## -0.010388 0.042102 -0.027994 -0.027437
## PSC.CO2 Mnf.Flow Carb.Pressure1 Fill.Pressure
## -0.034174 -0.511321 0.196296 0.016115
## Hyd.Pressure1 Hyd.Pressure2 Hyd.Pressure3 Hyd.Pressure4
## -0.015003 -0.041893 0.286371 -0.035989
## Filler.Level Filler.Speed Temperature Usage.cont
## -0.074772 0.015869 -0.171850 -0.127199
## Carb.Flow Density MFR Pressure.Vacuum
## 0.072622 -0.342042 -0.059249 0.021034
## Oxygen.Filler Bowl.Setpoint Pressure.Setpoint Air.Pressurer
## -0.074566 0.232417 -0.103516 -0.008811
## Alch.Rel Carb.Rel Balling.Lvl
## 0.248537 0.110331 0.030701# Check VIF again for the refined model
vif_refine<-vif(model_refined)
vif_refine## Carb.Volume Fill.Ounces PC.Volume Carb.Pressure
## 7.457578 1.137147 1.579623 15.472584
## Carb.Temp PSC PSC.Fill PSC.CO2
## 12.822917 1.149670 1.101239 1.065124
## Mnf.Flow Carb.Pressure1 Fill.Pressure Hyd.Pressure1
## 4.368723 1.590197 2.174935 3.026083
## Hyd.Pressure2 Hyd.Pressure3 Hyd.Pressure4 Filler.Level
## 10.675013 11.992147 2.105136 10.713293
## Filler.Speed Temperature Usage.cont Carb.Flow
## 4.846037 1.289338 1.658555 2.123068
## Density MFR Pressure.Vacuum Oxygen.Filler
## 13.198321 3.990009 1.997864 1.480736
## Bowl.Setpoint Pressure.Setpoint Air.Pressurer Alch.Rel
## 11.012020 2.283630 1.197956 9.271702
## Carb.Rel Balling.Lvl
## 4.999370 16.144918sum(is.na(student_train)) # Check for missing values in the entire dataset## [1] 0# Impute missing values using kNN
imputeModel <- preProcess(student_train, method = "knnImpute")
# Apply the imputation to both training and test data
student_train_imputed <- predict(imputeModel, student_train)
student_eval_imputed <- predict(imputeModel, student_eval)
# Check if there are any missing values after imputation
sum(is.na(student_train_imputed)) # Should return 0 if imputation was successful## [1] 0sum(is.na(student_eval_imputed)) # Should also return 0 for the test data## [1] 0# Prepare the predictor matrix (without the target variable 'PH')
x <- as.matrix(student_train_imputed[, -which(names(student_train_imputed) == "PH")])
# Check if there are missing values in the predictor matrix
sum(is.na(x)) # This should return 0 if there are no missing values## [1] 0# Remove PH column for dummy encoding
student_train_predictors <- student_train_imputed[, -which(names(student_train_imputed) == "PH")]
# Apply one-hot encoding to the predictors
dummy_model <- dummyVars(~ ., data = student_train_predictors)
# Apply the encoding to the dataset
x <- predict(dummy_model, student_train_predictors)x <- as.matrix(x) # Convert to matrix formatsum(is.na(x)) # Should return 0 if there are no missing values## [1] 0library(caret)
library(glmnet) # Fits the Ridge regression model. We set alpha = 0 to specify Ridge regression (Lasso would use alpha = 1).
# Impute missing values using kNN
imputeModel <- preProcess(student_train, method = "knnImpute")
# Apply the imputation to both training and test data
student_train_imputed <- predict(imputeModel, student_train)
student_eval_imputed <- predict(imputeModel, student_eval)
# Remove PH column for dummy encoding (exclude target variable)
student_train_predictors <- student_train_imputed[, -which(names(student_train_imputed) == "PH")]
# Apply one-hot encoding to the predictors
dummy_model <- dummyVars(~ ., data = student_train_predictors)
x <- predict(dummy_model, student_train_predictors)
# Convert to matrix format (glmnet requires a matrix)
x <- as.matrix(x)
# Check if there are any missing values in the predictor matrix
sum(is.na(x)) # Should return 0 if no missing values## [1] 0# Define target variable 'y'
y <- student_train_imputed$PH
# Fit Ridge Regression model (alpha = 0 for Ridge)
ridge_model <- glmnet(x, y, alpha = 0)
# View the coefficients
print(coef(ridge_model))## 37 x 100 sparse Matrix of class "dgCMatrix"##
## (Intercept) 5.253231e-16 -1.546768e-04 -1.697917e-04 -1.863857e-04
## Brand.Code -3.500156e-37 -8.444647e-04 -9.261872e-04 -1.015756e-03
## Brand.CodeA -3.199253e-37 -7.760873e-04 -8.516389e-04 -9.345304e-04
## Brand.CodeB 2.400524e-37 5.870502e-04 6.446985e-04 7.080433e-04
## Brand.CodeC -8.776190e-37 -2.121754e-03 -2.327549e-03 -2.553194e-03
## Brand.CodeD 4.365463e-37 1.047255e-03 1.147968e-03 1.258229e-03
## Carb.Volume 6.414125e-38 1.520541e-04 1.664843e-04 1.822439e-04
## Fill.Ounces -9.548791e-38 -2.306228e-04 -2.529672e-04 -2.774617e-04
## PC.Volume 3.924863e-38 9.400582e-05 1.030291e-04 1.129039e-04
## Carb.Pressure 5.885851e-38 1.404029e-04 1.538212e-04 1.684944e-04
## Carb.Temp 2.671806e-38 6.433491e-05 7.054763e-05 7.735419e-05
## PSC -9.177482e-38 -2.217611e-04 -2.432581e-04 -2.668261e-04
## PSC.Fill -3.664390e-38 -8.902987e-05 -9.771171e-05 -1.072398e-04
## PSC.CO2 -7.795527e-38 -1.879060e-04 -2.060720e-04 -2.259783e-04
## Mnf.Flow -4.518071e-37 -1.086632e-03 -1.191426e-03 -1.306209e-03
## Carb.Pressure1 -6.810662e-38 -1.604084e-04 -1.755157e-04 -1.919902e-04
## Fill.Pressure -2.176648e-37 -5.207091e-04 -5.706288e-04 -6.252472e-04
## Hyd.Pressure1 -7.613977e-38 -1.808278e-04 -1.980221e-04 -2.168072e-04
## Hyd.Pressure2 -2.029112e-37 -4.846984e-04 -5.310897e-04 -5.818318e-04
## Hyd.Pressure3 -2.421059e-37 -5.786370e-04 -6.340539e-04 -6.946736e-04
## Hyd.Pressure4 -1.429507e-37 -3.431323e-04 -3.761528e-04 -4.123066e-04
## Filler.Level 3.276763e-37 7.865993e-04 8.623024e-04 9.451880e-04
## Filler.Speed -1.835585e-38 -4.403799e-05 -4.827416e-05 -5.291176e-05
## Temperature -1.587755e-37 -3.846091e-04 -4.219928e-04 -4.629981e-04
## Usage.cont -3.212478e-37 -7.727898e-04 -8.473369e-04 -9.289914e-04
## Carb.Flow 1.567063e-37 3.772355e-04 4.136532e-04 4.535486e-04
## Density 7.723733e-38 1.829396e-04 2.002842e-04 2.192225e-04
## MFR -2.370046e-38 -5.775786e-05 -6.340905e-05 -6.961520e-05
## Balling 6.388809e-38 1.513567e-04 1.657119e-04 1.813867e-04
## Pressure.Vacuum 2.226733e-37 5.334132e-04 5.846299e-04 6.406811e-04
## Oxygen.Filler 1.688838e-37 4.046461e-04 4.435080e-04 4.860399e-04
## Bowl.Setpoint 3.510619e-37 8.433081e-04 9.245306e-04 1.013471e-03
## Pressure.Setpoint -3.099984e-37 -7.437599e-04 -8.152983e-04 -8.936151e-04
## Air.Pressurer -1.370082e-38 -3.314119e-05 -3.635770e-05 -3.988492e-05
## Alch.Rel 1.478732e-37 3.538193e-04 3.877508e-04 4.248774e-04
## Carb.Rel 1.634834e-37 3.913971e-04 4.289563e-04 4.700568e-04
## Balling.Lvl 9.916359e-38 2.360436e-04 2.585504e-04 2.831502e-04
##
## (Intercept) -2.046051e-04 -2.246097e-04 -2.465753e-04 -2.706947e-04
## Brand.Code -1.113910e-03 -1.221456e-03 -1.339275e-03 -1.468325e-03
## Brand.CodeA -1.025473e-03 -1.125246e-03 -1.234701e-03 -1.354772e-03
## Brand.CodeB 7.776598e-04 8.541779e-04 9.382920e-04 1.030769e-03
## Brand.CodeC -2.800578e-03 -3.071769e-03 -3.369024e-03 -3.694810e-03
## Brand.CodeD 1.378907e-03 1.510950e-03 1.655390e-03 1.813339e-03
## Carb.Volume 1.994458e-04 2.182119e-04 2.386729e-04 2.609674e-04
## Fill.Ounces -3.043100e-04 -3.337349e-04 -3.659791e-04 -4.013078e-04
## PC.Volume 1.237069e-04 1.355217e-04 1.484385e-04 1.625549e-04
## Carb.Pressure 1.845335e-04 2.020589e-04 2.212002e-04 2.420968e-04
## Carb.Temp 8.480997e-05 9.297542e-05 1.019163e-04 1.117041e-04
## PSC -2.926618e-04 -3.209802e-04 -3.520162e-04 -3.860258e-04
## PSC.Fill -1.176969e-04 -1.291737e-04 -1.417694e-04 -1.555930e-04
## PSC.CO2 -2.477881e-04 -2.716795e-04 -2.978464e-04 -3.264999e-04
## Mnf.Flow -1.431906e-03 -1.569526e-03 -1.720165e-03 -1.885012e-03
## Carb.Pressure1 -2.099426e-04 -2.294915e-04 -2.507619e-04 -2.738851e-04
## Fill.Pressure -6.849864e-04 -7.503049e-04 -8.216982e-04 -8.997003e-04
## Hyd.Pressure1 -2.373191e-04 -2.597055e-04 -2.841244e-04 -3.107443e-04
## Hyd.Pressure2 -6.373126e-04 -6.979526e-04 -7.642053e-04 -8.365588e-04
## Hyd.Pressure3 -7.609627e-04 -8.334258e-04 -9.126077e-04 -9.990948e-04
## Hyd.Pressure4 -4.518807e-04 -4.951880e-04 -5.425675e-04 -5.943868e-04
## Filler.Level 1.035916e-03 1.135203e-03 1.243828e-03 1.362631e-03
## Filler.Speed -5.798779e-05 -6.354230e-05 -6.961875e-05 -7.626421e-05
## Temperature -5.079733e-04 -5.572995e-04 -6.113942e-04 -6.707139e-04
## Usage.cont -1.018414e-03 -1.116323e-03 -1.223501e-03 -1.340795e-03
## Carb.Flow 4.972459e-04 5.450983e-04 5.974898e-04 6.548380e-04
## Density 2.398896e-04 2.624309e-04 2.870017e-04 3.137667e-04
## MFR -7.643128e-05 -8.391774e-05 -9.214111e-05 -1.011747e-04
## Balling 1.984938e-04 2.171540e-04 2.374963e-04 2.596580e-04
## Pressure.Vacuum 7.020063e-04 7.690817e-04 8.424224e-04 9.225848e-04
## Oxygen.Filler 5.325760e-04 5.834781e-04 6.391379e-04 6.999787e-04
## Bowl.Setpoint 1.110842e-03 1.217417e-03 1.334036e-03 1.461610e-03
## Pressure.Setpoint -9.793317e-04 -1.073123e-03 -1.175720e-03 -1.287914e-03
## Air.Pressurer -4.375252e-05 -4.799299e-05 -5.264187e-05 -5.773798e-05
## Alch.Rel 4.654882e-04 5.098961e-04 5.584393e-04 6.114830e-04
## Carb.Rel 5.150200e-04 5.641941e-04 6.179555e-04 6.767109e-04
## Balling.Lvl 3.100269e-04 3.393788e-04 3.714186e-04 4.063745e-04
##
## (Intercept) -2.971802e-04 -3.262650e-04 -3.582051e-04 -3.932822e-04
## Brand.Code -1.609653e-03 -1.764392e-03 -1.933780e-03 -2.119158e-03
## Brand.CodeA -1.486483e-03 -1.630953e-03 -1.789407e-03 -1.963188e-03
## Brand.CodeB 1.132453e-03 1.244278e-03 1.367275e-03 1.502580e-03
## Brand.CodeC -4.051818e-03 -4.442984e-03 -4.871508e-03 -5.340878e-03
## Brand.CodeD 1.986003e-03 2.174683e-03 2.380782e-03 2.605808e-03
## Carb.Volume 2.852430e-04 3.116556e-04 3.403693e-04 3.715561e-04
## Fill.Ounces -4.400098e-04 -4.823997e-04 -5.288202e-04 -5.796438e-04
## PC.Volume 1.779757e-04 1.948139e-04 2.131902e-04 2.332342e-04
## Carb.Pressure 2.648983e-04 2.897646e-04 3.168667e-04 3.463861e-04
## Carb.Temp 1.224166e-04 1.341381e-04 1.469600e-04 1.609814e-04
## PSC -4.232888e-04 -4.641097e-04 -5.088206e-04 -5.577829e-04
## PSC.Fill -1.707641e-04 -1.874139e-04 -2.056861e-04 -2.257385e-04
## PSC.CO2 -3.578696e-04 -3.922051e-04 -4.297771e-04 -4.708793e-04
## Mnf.Flow -2.065359e-03 -2.262604e-03 -2.478260e-03 -2.713961e-03
## Carb.Pressure1 -2.989981e-04 -3.262429e-04 -3.557651e-04 -3.877128e-04
## Fill.Pressure -9.848864e-04 -1.077874e-03 -1.179325e-03 -1.289947e-03
## Hyd.Pressure1 -3.397445e-04 -3.713150e-04 -4.056564e-04 -4.429793e-04
## Hyd.Pressure2 -9.155370e-04 -1.001702e-03 -1.095653e-03 -1.198031e-03
## Hyd.Pressure3 -1.093518e-03 -1.196553e-03 -1.308923e-03 -1.431400e-03
## Hyd.Pressure4 -6.510432e-04 -7.129663e-04 -7.806192e-04 -8.545009e-04
## Filler.Level 1.492526e-03 1.634495e-03 1.789601e-03 1.958987e-03
## Filler.Speed -8.352966e-05 -9.147019e-05 -1.001453e-04 -1.096190e-04
## Temperature -7.357580e-04 -8.070728e-04 -8.852552e-04 -9.709575e-04
## Usage.cont -1.469126e-03 -1.609492e-03 -1.762972e-03 -1.930733e-03
## Carb.Flow 7.175959e-04 7.862550e-04 8.613477e-04 9.434500e-04
## Density 3.429009e-04 3.745887e-04 4.090238e-04 4.464086e-04
## MFR -1.110990e-04 -1.220032e-04 -1.339850e-04 -1.471525e-04
## Balling 2.837847e-04 3.100303e-04 3.385568e-04 3.695335e-04
## Pressure.Vacuum 1.010169e-03 1.105822e-03 1.210237e-03 1.324158e-03
## Oxygen.Filler 7.664569e-04 8.390642e-04 9.183292e-04 1.004818e-03
## Bowl.Setpoint 1.601124e-03 1.753645e-03 1.920325e-03 2.102405e-03
## Pressure.Setpoint -1.410561e-03 -1.544584e-03 -1.690980e-03 -1.850818e-03
## Air.Pressurer -6.332376e-05 -6.944555e-05 -7.615395e-05 -8.350418e-05
## Alch.Rel 6.694207e-04 7.326755e-04 8.017019e-04 8.769866e-04
## Carb.Rel 7.408986e-04 8.109909e-04 8.874949e-04 9.709543e-04
## Balling.Lvl 4.444906e-04 4.860268e-04 5.312599e-04 5.804828e-04
##
## (Intercept) -4.318057e-04 -0.0004741156 -0.0005205852 -0.0005715861
## Brand.Code -2.321981e-03 -0.0025438282 -0.0027864079 -0.0030516293
## Brand.CodeA -2.153760e-03 -0.0023627269 -0.0025918396 -0.0028429570
## Brand.CodeB 1.651452e-03 0.0018152785 0.0019955937 0.0021939228
## Brand.CodeC -5.854892e-03 -0.0064176792 -0.0070337319 -0.0077080235
## Brand.CodeD 2.851382e-03 0.0031192422 0.0034112463 0.0037295909
## Carb.Volume 4.053951e-04 0.0004420717 0.0004817761 0.0005247476
## Fill.Ounces -6.352756e-04 -0.0006961552 -0.0007627599 -0.0008356076
## PC.Volume 2.550837e-04 0.0002788854 0.0003047944 0.0003329920
## Carb.Pressure 3.785158e-04 0.0004134595 0.0004514315 0.0004926859
## Carb.Temp 1.763094e-04 0.0001930598 0.0002113574 0.0002313414
## PSC -6.113898e-04 -0.0006700687 -0.0007342837 -0.0008045469
## PSC.Fill -2.477440e-04 -0.0002718922 -0.0002983906 -0.0003274533
## PSC.CO2 -5.158297e-04 -0.0005649724 -0.0006186791 -0.0006773638
## Mnf.Flow -2.971471e-03 -0.0032526885 -0.0035596555 -0.0038947133
## Carb.Pressure1 -4.222346e-04 -0.0004594766 -0.0004995798 -0.0005427235
## Fill.Pressure -1.410495e-03 -0.0015417701 -0.0016846216 -0.0018400202
## Hyd.Pressure1 -4.835038e-04 -0.0005274582 -0.0005750776 -0.0006266655
## Hyd.Pressure2 -1.309514e-03 -0.0014308224 -0.0015627132 -0.0017060430
## Hyd.Pressure3 -1.564806e-03 -0.0017100090 -0.0018679288 -0.0020395605
## Hyd.Pressure4 -9.351474e-04 -0.0010231341 -0.0011190765 -0.0012236667
## Filler.Level 2.143881e-03 0.0023456002 0.0025655544 0.0028052673
## Filler.Speed -1.199605e-04 -0.0001312439 -0.0001435488 -0.0001569564
## Temperature -1.064892e-03 -0.0011678362 -0.0012806387 -0.0014042333
## Usage.cont -2.114033e-03 -0.0023142300 -0.0025327821 -0.0027712627
## Carb.Flow 1.033184e-03 0.0011312216 0.0012382851 0.0013551549
## Density 4.869532e-04 0.0005308741 0.0005783924 0.0006297616
## MFR -1.616247e-04 -0.0001775331 -0.0001950227 -0.0002142515
## Balling 4.031370e-04 0.0004395497 0.0004789585 0.0005215622
## Pressure.Vacuum 1.448383e-03 0.0015837581 0.0017311874 0.0018916296
## Oxygen.Filler 1.099138e-03 0.0012019356 0.0013138986 0.0014357574
## Bowl.Setpoint 2.301220e-03 0.0025182066 0.0027549019 0.0030129520
## Pressure.Setpoint -2.025248e-03 -0.0022155028 -0.0024228967 -0.0026488335
## Air.Pressurer -9.155649e-05 -0.0001003765 -0.0001100359 -0.0001206138
## Alch.Rel 9.590495e-04 0.0010484451 0.0011457623 0.0012516243
## Carb.Rel 1.061951e-03 0.0011611057 0.0012690789 0.0013865678
## Balling.Lvl 6.340047e-04 0.0006921505 0.0007552598 0.0008236837
##
## (Intercept) -0.0006276344 -0.0006891943 -0.0007568071 -0.0008310666
## Brand.Code -0.0033413827 -0.0036578679 -0.0040034020 -0.0043804751
## Brand.CodeA -0.0031182555 -0.0034199674 -0.0037505686 -0.0041127531
## Brand.CodeB 0.0024124289 0.0026530476 0.0029180616 0.0032099906
## Brand.CodeC -0.0084456834 -0.0092525241 -0.0101347588 -0.0110991063
## Brand.CodeD 0.0040760184 0.0044529340 0.0048627062 0.0053078305
## Carb.Volume 0.0005711027 0.0006210708 0.0006748412 0.0007325951
## Fill.Ounces -0.0009152554 -0.0010023088 -0.0010974203 -0.0012012926
## PC.Volume 0.0003636196 0.0003968697 0.0004329276 0.0004719833
## Carb.Pressure 0.0005374068 0.0005858600 0.0006382957 0.0006949694
## Carb.Temp 0.0002531487 0.0002769381 0.0003028760 0.0003311406
## PSC -0.0008813897 -0.0009654097 -0.0010572457 -0.0011575879
## PSC.Fill -0.0003593520 -0.0003943488 -0.0004327412 -0.0004748547
## PSC.CO2 -0.0007414358 -0.0008113694 -0.0008876627 -0.0009708487
## Mnf.Flow -0.0042599507 -0.0046579905 -0.0050915042 -0.0055633297
## Carb.Pressure1 -0.0005889446 -0.0006383778 -0.0006910943 -0.0007471291
## Fill.Pressure -0.0020087776 -0.0021919349 -0.0023905132 -0.0026055656
## Hyd.Pressure1 -0.0006823569 -0.0007424430 -0.0008071619 -0.0008767424
## Hyd.Pressure2 -0.0018615364 -0.0020301017 -0.0022126252 -0.0024100118
## Hyd.Pressure3 -0.0022258613 -0.0024279041 -0.0026467725 -0.0028835761
## Hyd.Pressure4 -0.0013375448 -0.0014614810 -0.0015962619 -0.0017427166
## Filler.Level 0.0030663042 0.0033503782 0.0036592832 0.0039949079
## Filler.Speed -0.0001715647 -0.0001874680 -0.0002047696 -0.0002235797
## Temperature -0.0015396130 -0.0016878824 -0.0018502360 -0.0020279722
## Usage.cont -0.0030313369 -0.0033148027 -0.0036235707 -0.0039596712
## Carb.Flow 0.0014826591 0.0016216923 0.0017732070 0.0019382175
## Density 0.0006851533 0.0007448104 0.0008089443 0.0008777531
## MFR -0.0002353999 -0.0002586618 -0.0002842534 -0.0003124137
## Balling 0.0005675367 0.0006170786 0.0006703729 0.0007275960
## Pressure.Vacuum 0.0020660886 0.0022556274 0.0024613547 0.0026844233
## Oxygen.Filler 0.0015682811 0.0017122805 0.0018686033 0.0020381318
## Bowl.Setpoint 0.0032941096 0.0036002412 0.0039333253 0.0042954519
## Pressure.Setpoint -0.0028948006 -0.0031623763 -0.0034532247 -0.0037690939
## Air.Pressurer -0.0001321930 -0.0001448668 -0.0001587356 -0.0001739084
## Alch.Rel 0.0013666903 0.0014916523 0.0016272346 0.0017741919
## Carb.Rel 0.0015143182 0.0016531076 0.0018037535 0.0019671095
## Balling.Lvl 0.0008977904 0.0009779524 0.0010645498 0.0011579650
##
## (Intercept) -0.0009126230 -0.0010021882 -0.0011005405 -0.0012085298
## Brand.Code -0.0047917584 -0.0052401109 -0.0057285860 -0.0062604367
## Brand.CodeA -0.0045094483 -0.0049438314 -0.0054193449 -0.0059397129
## Brand.CodeB 0.0035316154 0.0038860022 0.0042765292 0.0047069141
## Brand.CodeC -0.0121528221 -0.0133037294 -0.0145602502 -0.0159314347
## Brand.CodeD 0.0057909260 0.0063147298 0.0068820886 0.0074959480
## Carb.Volume 0.0007945004 0.0008607069 0.0009313396 0.0010064921
## Fill.Ounces -0.0013146817 -0.0014383990 -0.0015733138 -0.0017203550
## PC.Volume 0.0005142302 0.0005598621 0.0006090706 0.0006620411
## Carb.Pressure 0.0007561400 0.0008220668 0.0008930063 0.0009692082
## Carb.Temp 0.0003619216 0.0003954214 0.0004318544 0.0004714480
## PSC -0.0012671799 -0.0013868225 -0.0015173759 -0.0016597627
## PSC.Fill -0.0005210441 -0.0005716969 -0.0006272348 -0.0006881166
## PSC.CO2 -0.0010614968 -0.0011602135 -0.0012676429 -0.0013844670
## Mnf.Flow -0.0060764728 -0.0066341060 -0.0072395658 -0.0078963466
## Carb.Pressure1 -0.0008064714 -0.0008690517 -0.0009347278 -0.0010032683
## Fill.Pressure -0.0028381698 -0.0030894185 -0.0033604071 -0.0036522191
## Hyd.Pressure1 -0.0009513977 -0.0010313193 -0.0011166678 -0.0012075643
## Hyd.Pressure2 -0.0026231762 -0.0028530318 -0.0031004774 -0.0033663813
## Hyd.Pressure3 -0.0031394412 -0.0034154975 -0.0037128634 -0.0040326270
## Hyd.Pressure4 -0.0019017154 -0.0020741681 -0.0022610208 -0.0024632515
## Filler.Level 0.0043592319 0.0047543203 0.0051823149 0.0056454241
## Filler.Speed -0.0002440151 -0.0002661994 -0.0002902637 -0.0003163460
## Temperature -0.0022225013 -0.0024353531 -0.0026681854 -0.0029227920
## Usage.cont -0.0043252554 -0.0047225945 -0.0051540777 -0.0056222078
## Carb.Flow 0.0021178004 0.0023130950 0.0025253021 0.0027556827
## Density 0.0009514158 0.0010300845 0.0011138769 0.0012028665
## MFR -0.0003434075 -0.0003775279 -0.0004150997 -0.0004564831
## Balling 0.0007889117 0.0008544660 0.0009243811 0.0009987492
## Pressure.Vacuum 0.0029260242 0.0031873783 0.0034697263 0.0037743166
## Oxygen.Filler 0.0022217780 0.0024204785 0.0026351860 0.0028668604
## Bowl.Setpoint 0.0046888209 0.0051157378 0.0055786073 0.0060799249
## Pressure.Setpoint -0.0041118122 -0.0044832812 -0.0048854677 -0.0053203925
## Air.Pressurer -0.0001905038 -0.0002086502 -0.0002284870 -0.0002501650
## Alch.Rel 0.0019333058 0.0021053812 0.0022912412 0.0024917213
## Carb.Rel 0.0021440616 0.0023355246 0.0025424367 0.0027657530
## Balling.Lvl 0.0012585784 0.0013667624 0.0014828754 0.0016072541
##
## (Intercept) -0.0013270830 -0.0014572088 -0.0016000033 -0.0017566546
## Brand.Code -0.0068391194 -0.0074682958 -0.0081518326 -0.0088937992
## Brand.CodeA -0.0065089554 -0.0071314027 -0.0078117066 -0.0085548504
## Brand.CodeB 0.0051812430 0.0057039988 0.0062800909 0.0069148827
## Brand.CodeC -0.0174269892 -0.0190573021 -0.0208334653 -0.0227672923
## Brand.CodeD 0.0081593386 0.0088753586 0.0096471534 0.0104778916
## Carb.Volume 0.0010862182 0.0011705233 0.0012593543 0.0013525897
## Fill.Ounces -0.0018805128 -0.0020548398 -0.0022444513 -0.0024505251
## PC.Volume 0.0007189485 0.0007799516 0.0008451863 0.0009147579
## Carb.Pressure 0.0010509117 0.0011383396 0.0012316935 0.0013311472
## Carb.Temp 0.0005144422 0.0005610897 0.0006116557 0.0006664179
## PSC -0.0018149699 -0.0019840516 -0.0021681302 -0.0023683975
## PSC.Fill -0.0007548406 -0.0008279469 -0.0009080193 -0.0009956878
## PSC.CO2 -0.0015114053 -0.0016492142 -0.0017986855 -0.0019606439
## Mnf.Flow -0.0086080920 -0.0093785831 -0.0102117232 -0.0111115178
## Carb.Pressure1 -0.0010743329 -0.0011474506 -0.0012219946 -0.0012971542
## Fill.Pressure -0.0039659084 -0.0043024786 -0.0046628586 -0.0050478744
## Hyd.Pressure1 -0.0013040793 -0.0014062203 -0.0015139186 -0.0016270138
## Hyd.Pressure2 -0.0036515619 -0.0039567662 -0.0042826443 -0.0046297206
## Hyd.Pressure3 -0.0043758241 -0.0047434123 -0.0051362413 -0.0055550186
## Hyd.Pressure4 -0.0026818653 -0.0029178873 -0.0031723549 -0.0034463076
## Filler.Level 0.0061459085 0.0066860639 0.0072682003 0.0078946178
## Filler.Speed -0.0003445925 -0.0003751573 -0.0004082030 -0.0004439020
## Temperature -0.0032011112 -0.0035052341 -0.0038374121 -0.0042000644
## Usage.cont -0.0061295946 -0.0066789460 -0.0072730564 -0.0079147918
## Carb.Flow 0.0030055551 0.0032762903 0.0035693067 0.0038860626
## Density 0.0012970720 0.0013964459 0.0015008611 0.0016100969
## MFR -0.0005020774 -0.0005523260 -0.0006077206 -0.0006688077
## Balling 0.0010776251 0.0011610181 0.0012488840 0.0013411159
## Pressure.Vacuum 0.0041023892 0.0044551575 0.0048337871 0.0052393701
## Oxygen.Filler 0.0031164573 0.0033849136 0.0036731314 0.0039819590
## Bowl.Setpoint 0.0066222645 0.0072082642 0.0078406076 0.0085220022
## Pressure.Setpoint -0.0057901161 -0.0062967215 -0.0068422938 -0.0074288955
## Air.Pressurer -0.0002738474 -0.0002997107 -0.0003279455 -0.0003587571
## Alch.Rel 0.0027076621 0.0029399015 0.0031892650 0.0034565552
## Carb.Rel 0.0030064381 0.0032654567 0.0035437636 0.0038422917
## Balling.Lvl 0.0017402056 0.0018819976 0.0020328488 0.0021929179
##
## (Intercept) -0.0019284469 -0.0021167642 -0.0023230929 -0.0025490229
## Brand.Code -0.0096984617 -0.0105702737 -0.0115138631 -0.0125340143
## Brand.CodeA -0.0093661548 -0.0102512793 -0.0112162185 -0.0122672916
## Brand.CodeB 0.0076142182 0.0083844451 0.0092324331 0.0101655873
## Brand.CodeC -0.0248713297 -0.0271588617 -0.0296439057 -0.0323411967
## Brand.CodeD 0.0113707376 0.0123288212 0.0133552033 0.0144528407
## Carb.Volume 0.0014500279 0.0015513754 0.0016562355 0.0017640958
## Fill.Ounces -0.0026743002 -0.0029170743 -0.0031802006 -0.0034650818
## PC.Volume 0.0009887323 0.0010671251 0.0011498898 0.0012369046
## Carb.Pressure 0.0014368406 0.0015488721 0.0016672921 0.0017920952
## Carb.Temp 0.0007256655 0.0007896996 0.0008588318 0.0009333840
## PSC -0.0025861154 -0.0028226156 -0.0030792978 -0.0033576271
## PSC.Fill -0.0010916298 -0.0011965710 -0.0013112860 -0.0014365975
## PSC.CO2 -0.0021359445 -0.0023254683 -0.0025301173 -0.0027508084
## Mnf.Flow -0.0120820528 -0.0131274666 -0.0142519203 -0.0154595642
## Carb.Pressure1 -0.0013719033 -0.0014449655 -0.0015147769 -0.0015794451
## Fill.Pressure -0.0054582183 -0.0058944133 -0.0063567748 -0.0068453691
## Hyd.Pressure1 -0.0017452382 -0.0018681994 -0.0019953623 -0.0021260320
## Hyd.Pressure2 -0.0049983626 -0.0053887458 -0.0058008164 -0.0062342519
## Hyd.Pressure3 -0.0060002717 -0.0064723057 -0.0069711576 -0.0074965477
## Hyd.Pressure4 -0.0037407759 -0.0040567684 -0.0043952565 -0.0047571587
## Filler.Level 0.0085675772 0.0092892685 0.0100617742 0.0108870294
## Filler.Speed -0.0004824367 -0.0005240013 -0.0005688035 -0.0006170665
## Temperature -0.0045957848 -0.0050273464 -0.0054977060 -0.0060100059
## Usage.cont -0.0086070717 -0.0093528483 -0.0101550819 -0.0110167134
## Carb.Flow 0.0042280473 0.0045967703 0.0049937477 0.0054204885
## Density 0.0017238244 0.0018415907 0.0019628029 0.0020867119
## MFR -0.0007361939 -0.0008105530 -0.0008926332 -0.0009832651
## Balling 0.0014375351 0.0015378826 0.0016418096 0.0017488705
## Pressure.Vacuum 0.0056728971 0.0061352249 0.0066270415 0.0071488277
## Oxygen.Filler 0.0043121687 0.0046644327 0.0050392955 0.0054371440
## Bowl.Setpoint 0.0092551541 0.0100427385 0.0108873678 0.0117915555
## Pressure.Setpoint -0.0080585393 -0.0087331562 -0.0094545612 -0.0102244153
## Air.Pressurer -0.0003923666 -0.0004290116 -0.0004689472 -0.0005124460
## Alch.Rel 0.0037425413 0.0040479466 0.0043734365 0.0047196060
## Carb.Rel 0.0041619395 0.0045035568 0.0048679305 0.0052557696
## Balling.Lvl 0.0023622928 0.0025409789 0.0027288883 0.0029258299
##
## (Intercept) -0.0027962472 -0.0030665594 -0.0033618484 -0.0036840908
## Brand.Code -0.0136356450 -0.0148237778 -0.0161035050 -0.0174799470
## Brand.CodeA -0.0134111238 -0.0146546180 -0.0160049162 -0.0174693480
## Brand.CodeB 0.0111918517 0.0123197040 0.0135581372 0.0149166265
## Brand.CodeC -0.0352661595 -0.0384348649 -0.0418639684 -0.0455706290
## Brand.CodeD 0.0156245469 0.0168729531 0.0182004679 0.0196092375
## Carb.Volume 0.0018743167 0.0019861206 0.0020985818 0.0022106181
## Fill.Ounces -0.0037731639 -0.0041059265 -0.0044648722 -0.0048515122
## PC.Volume 0.0013279573 0.0014227289 0.0015207760 0.0016215124
## Carb.Pressure 0.0019232135 0.0020605091 0.0022037690 0.0023526986
## Carb.Temp 0.0010136872 0.0011000815 0.0011929143 0.0012925403
## PSC -0.0036591303 -0.0039853897 -0.0043380354 -0.0047187352
## PSC.Fill -0.0015733744 -0.0017225290 -0.0018850115 -0.0020618034
## PSC.CO2 -0.0029884658 -0.0032440118 -0.0035183571 -0.0038123885
## Mnf.Flow -0.0167545011 -0.0181407490 -0.0196222021 -0.0212025934
## Carb.Pressure1 -0.0016367059 -0.0016838789 -0.0017178208 -0.0017348798
## Fill.Pressure -0.0073599691 -0.0079000086 -0.0084645352 -0.0090521642
## Hyd.Pressure1 -0.0022593354 -0.0023942056 -0.0025293671 -0.0026633246
## Hyd.Pressure2 -0.0066884206 -0.0071623407 -0.0076546409 -0.0081635230
## Hyd.Pressure3 -0.0080478277 -0.0086239284 -0.0092233071 -0.0098438966
## Hyd.Pressure4 -0.0051433215 -0.0055545007 -0.0059913400 -0.0064543495
## Filler.Level 0.0117667801 0.0127025376 0.0136955328 0.0147466708
## Filler.Speed -0.0006690319 -0.0007249630 -0.0007851492 -0.0008499101
## Temperature -0.0065675726 -0.0071739137 -0.0078327096 -0.0085478017
## Usage.cont -0.0119406341 -0.0129296527 -0.0139864610 -0.0151135981
## Carb.Flow 0.0058784768 0.0063691542 0.0068938996 0.0074540083
## Density 0.0022123969 0.0023387510 0.0024644674 0.0025880288
## MFR -0.0010833700 -0.0011939691 -0.0013161927 -0.0014512896
## Balling 0.0018585157 0.0019700870 0.0020828151 0.0021958197
## Pressure.Vacuum 0.0077008166 0.0082829512 0.0088948409 0.0095357196
## Oxygen.Filler 0.0058581761 0.0063023664 0.0067694317 0.0072587963
## Bowl.Setpoint 0.0127576785 0.0137879373 0.0148843154 0.0160485395
## Pressure.Setpoint -0.0110441858 -0.0119151041 -0.0128381241 -0.0138138798
## Air.Pressurer -0.0005597990 -0.0006113157 -0.0006673242 -0.0007281704
## Alch.Rel 0.0050869683 0.0054759432 0.0058868483 0.0063198915
## Carb.Rel 0.0056676896 0.0061041989 0.0065656847 0.0070524013
## Balling.Lvl 0.0031314997 0.0033454747 0.0035672081 0.0037960285
##
## (Intercept) -0.0040353389 -0.0044178783 -0.0048335120 -0.005284564
## Brand.Code -0.0189582031 -0.0205437628 -0.0222408341 -0.024054399
## Brand.CodeA -0.0190553652 -0.0207699733 -0.0226213115 -0.024616328
## Brand.CodeB 0.0164050786 0.0180336619 0.0198128033 0.021753215
## Brand.CodeC -0.0495724034 -0.0538869470 -0.0585326481 -0.063527164
## Brand.CodeD 0.0211011071 0.0226782821 0.0243410821 0.026090651
## Carb.Volume 0.0023209844 0.0024289027 0.0025318049 0.002628393
## Fill.Ounces -0.0052673501 -0.0057140525 -0.0061927317 -0.006704881
## PC.Volume 0.0017241891 0.0018277291 0.0019312592 0.002033312
## Carb.Pressure 0.0025069187 0.0026661752 0.0028295882 0.002996660
## Carb.Temp 0.0013993208 0.0015135618 0.0016357398 0.001766185
## PSC -0.0051291817 -0.0055710387 -0.0060460678 -0.006555895
## PSC.Fill -0.0022539079 -0.0024624692 -0.0026882642 -0.002932381
## PSC.CO2 -0.0041269558 -0.0044628032 -0.0048207627 -0.005201432
## Mnf.Flow -0.0228854593 -0.0246733435 -0.0265707455 -0.028579819
## Carb.Pressure1 -0.0017308503 -0.0017010415 -0.0016398993 -0.001541390
## Fill.Pressure -0.0096610335 -0.0102888369 -0.0109326059 -0.011588846
## Hyd.Pressure1 -0.0027943559 -0.0029202051 -0.0030392715 -0.003148914
## Hyd.Pressure2 -0.0086867299 -0.0092218522 -0.0097651685 -0.010313153
## Hyd.Pressure3 -0.0104830576 -0.0111379617 -0.0118040354 -0.012477013
## Hyd.Pressure4 -0.0069438831 -0.0074604520 -0.0080035105 -0.008573047
## Filler.Level 0.0158564855 0.0170254914 0.0182527325 0.019537689
## Filler.Speed -0.0009196015 -0.0009946723 -0.0010754712 -0.001162537
## Temperature -0.0093231746 -0.0101630149 -0.0110713905 -0.012052605
## Usage.cont -0.0163134147 -0.0175882141 -0.0189395814 -0.020369232
## Carb.Flow 0.0080506701 0.0086849880 0.0093578156 0.010069930
## Density 0.0027076987 0.0028221482 0.0029281493 0.003023760
## MFR -0.0016006364 -0.0017657233 -0.0019482442 -0.002149990
## Balling 0.0023081131 0.0024189416 0.0025265426 0.002629922
## Pressure.Vacuum 0.0102044041 0.0108993711 0.0116183034 0.012358650
## Oxygen.Filler 0.0077695571 0.0083005187 0.0088499136 0.009415725
## Bowl.Setpoint 0.0172820422 0.0185860059 0.0199610320 0.021407550
## Pressure.Setpoint -0.0148426470 -0.0159243799 -0.0170583888 -0.018243741
## Air.Pressurer -0.0007942168 -0.0008658739 -0.0009434781 -0.001027454
## Alch.Rel 0.0067751666 0.0072527210 0.0077522585 0.008273599
## Carb.Rel 0.0075644612 0.0081017506 0.0086641668 0.009251320
## Balling.Lvl 0.0040311432 0.0042715115 0.0045163200 0.004764389
##
## (Intercept) -0.005773178 -0.006301449 -0.0068713792 -0.0074848394
## Brand.Code -0.025988866 -0.028048268 -0.0302361724 -0.0325555811
## Brand.CodeA -0.026761966 -0.029064724 -0.0315304957 -0.0341643980
## Brand.CodeB 0.023865525 0.026160187 0.0286472915 0.0313363434
## Brand.CodeC -0.068888077 -0.074632409 -0.0807763327 -0.0873348329
## Brand.CodeD 0.027927400 0.029851319 0.0318619559 0.0339584069
## Carb.Volume 0.002716779 0.002794918 0.0028606146 0.0029115378
## Fill.Ounces -0.007251750 -0.007834450 -0.0084539157 -0.0091108715
## PC.Volume 0.002132302 0.002226392 0.0023134874 0.0023912253
## Carb.Pressure 0.003166669 0.003338812 0.0035122084 0.0036859106
## Carb.Temp 0.001905281 0.002053416 0.0022109836 0.0023783866
## PSC -0.007102120 -0.007686247 -0.0083096555 -0.0089735539
## PSC.Fill -0.003195760 -0.003479270 -0.0037836819 -0.0041096408
## PSC.CO2 -0.005605353 -0.006032942 -0.0064844721 -0.0069600504
## Mnf.Flow -0.030703113 -0.032942921 -0.0353013020 -0.0377801160
## Carb.Pressure1 -0.001398750 -0.001204523 -0.0009505629 -0.0006280487
## Fill.Pressure -0.012253458 -0.012921733 -0.0135883545 -0.0142474145
## Hyd.Pressure1 -0.003246566 -0.003329505 -0.0033948920 -0.0034398208
## Hyd.Pressure2 -0.010861576 -0.011405730 -0.0119404641 -0.0124602320
## Hyd.Pressure3 -0.013151651 -0.013822021 -0.0144815276 -0.0151229352
## Hyd.Pressure4 -0.009168584 -0.009789380 -0.0104344200 -0.0111023957
## Filler.Level 0.020879065 0.022275049 0.0237233117 0.0252210125
## Filler.Speed -0.001256423 -0.001357738 -0.0014671539 -0.0015853948
## Temperature -0.013110911 -0.014250516 -0.0154755116 -0.0167898054
## Usage.cont -0.021878442 -0.023468158 -0.0251389786 -0.0268911456
## Carb.Flow 0.010821899 0.011614091 0.0124466703 0.0133195919
## Density 0.003106359 0.003173119 0.0032210136 0.0032468369
## MFR -0.002372921 -0.002619153 -0.0028909490 -0.0031907136
## Balling 0.002727767 0.002818732 0.0029014600 0.0029746188
## Pressure.Vacuum 0.013117244 0.013890385 0.0146738363 0.0154628522
## Oxygen.Filler 0.009995448 0.010586128 0.0111843537 0.0117862566
## Bowl.Setpoint 0.022925530 0.024514550 0.0261737952 0.0279020887
## Pressure.Setpoint -0.019478973 -0.020762153 -0.0220908860 -0.0234623356
## Air.Pressurer -0.001118208 -0.001216146 -0.0013216657 -0.0014351512
## Alch.Rel 0.008816387 0.009380183 0.0099644771 0.0105687155
## Carb.Rel 0.009862706 0.010497674 0.0111554441 0.0118351190
## Balling.Lvl 0.005014563 0.005265662 0.0055165153 0.0057660041
##
## (Intercept) -0.0081435172 -0.0088488682 -0.0096020634 -0.010403936
## Brand.Code -0.0350088317 -0.0375974956 -0.0403222804 -0.043182940
## Brand.CodeA -0.0369705902 -0.0399520883 -0.0431105761 -0.046446221
## Brand.CodeB 0.0342360312 0.0373539737 0.0406964577 0.044268172
## Brand.CodeC -0.0943213457 -0.1017473592 -0.1096219919 -0.117951554
## Brand.CodeD 0.0361393039 0.0384028055 0.0407465885 0.043167841
## Carb.Volume 0.0029452261 0.0029591013 0.0029504808 0.002916591
## Fill.Ounces -0.0098057852 -0.0105388268 -0.0113098243 -0.012118221
## PC.Volume 0.0024569769 0.0025078511 0.0025407060 0.002552168
## Carb.Pressure 0.0038589078 0.0040301362 0.0041984887 0.004362824
## Carb.Temp 0.0025560340 0.0027443437 0.0029437421 0.003154664
## PSC -0.0096789352 -0.0104265264 -0.0112167338 -0.012049585
## PSC.Fill -0.0044576300 -0.0048279392 -0.0052206313 -0.005635510
## PSC.CO2 -0.0074595988 -0.0079828334 -0.0085292468 -0.009098092
## Mnf.Flow -0.0403810714 -0.0431057852 -0.0459558533 -0.048932928
## Carb.Pressure1 -0.0002275185 0.0002610801 0.0008483195 0.001545202
## Fill.Pressure -0.0148924386 -0.0155164299 -0.0161119240 -0.016671058
## Hyd.Pressure1 -0.0034613792 -0.0034567150 -0.0034231095 -0.003358056
## Hyd.Pressure2 -0.0129591596 -0.0134311226 -0.0138698378 -0.014268964
## Hyd.Pressure3 -0.0157384151 -0.0163196037 -0.0168576743 -0.017343420
## Hyd.Pressure4 -0.0117916983 -0.0125004113 -0.0132263093 -0.013966862
## Filler.Level 0.0267648182 0.0283509295 0.0299751176 0.031632768
## Filler.Speed -0.0017132327 -0.0018514713 -0.0020009258 -0.002162393
## Temperature -0.0181970391 -0.0197004997 -0.0213030247 -0.023006902
## Usage.cont -0.0287245428 -0.0306386973 -0.0326327872 -0.034705652
## Carb.Flow 0.0142326015 0.0151852450 0.0161768827 0.017206710
## Density 0.0032472210 0.0032186564 0.0031575135 0.003060065
## MFR -0.0035209756 -0.0038843644 -0.0042835791 -0.004721348
## Balling 0.0030369280 0.0030871961 0.0031243556 0.003147499
## Pressure.Vacuum 0.0162522029 0.0170362225 0.0178088680 0.018563791
## Oxygen.Filler 0.0123875226 0.0129834099 0.0135687765 0.014138116
## Bowl.Setpoint 0.0296979178 0.0315594806 0.0334847415 0.035471495
## Pressure.Setpoint -0.0248732564 -0.0263200407 -0.0277987778 -0.029305325
## Air.Pressurer -0.0015569621 -0.0016874240 -0.0018268167 -0.001975362
## Alch.Rel 0.0111923254 0.0118347449 0.0124954551 0.013174014
## Carb.Rel 0.0125357037 0.0132561277 0.0139952674 0.014751970
## Balling.Lvl 0.0060131119 0.0062569728 0.0064969253 0.006732564
##
## (Intercept) -0.011254931 -0.012155053 -0.013103824 -0.014100245
## Brand.Code -0.046178200 -0.049305694 -0.052561937 -0.055942317
## Brand.CodeA -0.049957496 -0.053641021 -0.057491417 -0.061501191
## Brand.CodeB 0.048071944 0.052108483 0.056376146 0.060870730
## Brand.CodeC -0.126739103 -0.135984007 -0.145681541 -0.155822513
## Brand.CodeD 0.045663262 0.048229059 0.050860963 0.053554254
## Carb.Volume 0.002854584 0.002761552 0.002634552 0.002470626
## Fill.Ounces -0.012963036 -0.013842824 -0.014755652 -0.015699069
## PC.Volume 0.002538657 0.002496419 0.002421571 0.002310141
## Carb.Pressure 0.004521977 0.004674771 0.004820028 0.004956579
## Carb.Temp 0.003377553 0.003612857 0.003861027 0.004122510
## PSC -0.012924668 -0.013841075 -0.014797337 -0.015791376
## PSC.Fill -0.006072092 -0.006529585 -0.007006866 -0.007502482
## PSC.CO2 -0.009688370 -0.010298821 -0.010927923 -0.011573892
## Mnf.Flow -0.052038799 -0.055275478 -0.058645275 -0.062150873
## Carb.Pressure1 0.002363052 0.003313380 0.004407742 0.005657564
## Fill.Pressure -0.017185651 -0.017647299 -0.018047479 -0.018377661
## Hyd.Pressure1 -0.003259337 -0.003125107 -0.002953965 -0.002745024
## Hyd.Pressure2 -0.014622213 -0.014923459 -0.015166857 -0.015346954
## Hyd.Pressure3 -0.017767346 -0.018119763 -0.018390888 -0.018570942
## Hyd.Pressure4 -0.014719243 -0.015480350 -0.016246824 -0.017015080
## Filler.Level 0.033318931 0.035028368 0.036755609 0.038494999
## Filler.Speed -0.002336617 -0.002524239 -0.002725752 -0.002941437
## Temperature -0.024813769 -0.026724507 -0.028739144 -0.030856758
## Usage.cont -0.036855807 -0.039081452 -0.041380488 -0.043750519
## Carb.Flow 0.018273784 0.019377055 0.020515398 0.021687653
## Density 0.002922507 0.002740981 0.002511591 0.002230416
## MFR -0.005200380 -0.005723306 -0.006292616 -0.006910588
## Balling 0.003155916 0.003149123 0.003126896 0.003089291
## Pressure.Vacuum 0.019294423 0.019994066 0.020656002 0.021273590
## Oxygen.Filler 0.014685600 0.015205129 0.015690388 0.016134903
## Bowl.Setpoint 0.037517436 0.039620232 0.041777602 0.043987375
## Pressure.Setpoint -0.030835388 -0.032384607 -0.033948651 -0.035523306
## Air.Pressurer -0.002133214 -0.002300440 -0.002477016 -0.002662813
## Alch.Rel 0.013870090 0.014583496 0.015314215 0.016062429
## Carb.Rel 0.015525077 0.016313438 0.017115932 0.017931469
## Balling.Lvl 0.006963790 0.007190856 0.007414401 0.007635482
##
## (Intercept) -0.015142762 -0.016229245 -0.017356977 -0.0185226529
## Brand.Code -0.059441126 -0.063051636 -0.066766201 -0.0705764063
## Brand.CodeA -0.065660655 -0.069957881 -0.074378703 -0.0789067754
## Brand.CodeB 0.065585294 0.070510030 0.075632191 0.0809360805
## Brand.CodeC -0.166392975 -0.177374008 -0.188741615 -0.2004667415
## Brand.CodeD 0.056303794 0.059104090 0.061949363 0.0648336264
## Carb.Volume 0.002266830 0.002020266 0.001728106 0.0013876284
## Fill.Ounces -0.016670101 -0.017665248 -0.018680498 -0.0197113539
## PC.Volume 0.002158131 0.001961577 0.001716615 0.0014195608
## Carb.Pressure 0.005083280 0.005199016 0.005302712 0.0053933356
## Carb.Temp 0.004397743 0.004687144 0.004991100 0.0053099555
## PSC -0.016820452 -0.017881128 -0.018969241 -0.0200798952
## PSC.Fill -0.008014646 -0.008541254 -0.009079910 -0.0096279695
## PSC.CO2 -0.012234695 -0.012908068 -0.013591539 -0.0142824612
## Mnf.Flow -0.065795394 -0.069582451 -0.073516188 -0.0776013075
## Carb.Pressure1 0.007073958 0.008667526 0.010448143 0.0124247377
## Fill.Pressure -0.018629432 -0.018794620 -0.018865430 -0.0188345756
## Hyd.Pressure1 -0.002497971 -0.002213120 -0.001891450 -0.0015346290
## Hyd.Pressure2 -0.015458795 -0.015498028 -0.015460992 -0.0153448003
## Hyd.Pressure3 -0.018650238 -0.018619271 -0.018468785 -0.0181898320
## Hyd.Pressure4 -0.017781345 -0.018541695 -0.019292097 -0.0200284510
## Filler.Level 0.040240738 0.041986915 0.043727528 0.0454564938
## Filler.Speed -0.003171299 -0.003415008 -0.003671824 -0.0039405443
## Temperature -0.033075392 -0.035391982 -0.037802291 -0.0403008731
## Usage.cont -0.046188856 -0.048692510 -0.051258174 -0.0538821990
## Carb.Flow 0.022892656 0.024129269 0.025396408 0.0266930509
## Density 0.001893513 0.001496918 0.001036616 0.0005085109
## MFR -0.007579215 -0.008300127 -0.009074523 -0.0099031056
## Balling 0.003036657 0.002969636 0.002889148 0.0027963622
## Pressure.Vacuum 0.021840382 0.022350231 0.022797388 0.0231766027
## Oxygen.Filler 0.016532109 0.016875405 0.017158221 0.0173740795
## Bowl.Setpoint 0.046247560 0.048556387 0.050912340 0.0533141675
## Pressure.Setpoint -0.037104565 -0.038688703 -0.040272348 -0.0418525299
## Air.Pressurer -0.002857584 -0.003060961 -0.003272449 -0.0034914245
## Alch.Rel 0.016828533 0.017613141 0.018417088 0.0192414223
## Carb.Rel 0.018758991 0.019597467 0.020445873 0.0213031792
## Balling.Lvl 0.007855590 0.008076657 0.008301055 0.0085316031
##
## (Intercept) -1.972239e-02 -0.0209517753 -2.220588e-02 -0.0234793576
## Brand.Code -7.447324e-02 -0.0784473034 -8.248898e-02 -0.0865886531
## Brand.CodeA -8.352370e-02 -0.0882091954 -9.294139e-02 -0.0976971196
## Brand.CodeB 8.640312e-02 0.0920120256 9.773904e-02 0.1035583231
## Brand.CodeC -2.125154e-01 -0.2248490583 -2.374248e-01 -0.2501962859
## Brand.CodeD 6.775077e-02 0.0706946323 7.365903e-02 0.0766377745
## Carb.Volume 9.962325e-04 0.0005514552 5.097447e-05 -0.0005073883
## Fill.Ounces -2.075287e-02 -0.0217997253 -2.284626e-02 -0.0238865779
## PC.Volume 1.066988e-03 0.0006558188 1.834064e-04 -0.0003523734
## Carb.Pressure 5.469897e-03 0.0055314417 5.577039e-03 0.0056057758
## Carb.Temp 5.644001e-03 0.0059934587 6.358471e-03 0.0067390898
## PSC -2.120747e-02 -0.0223456419 -2.348745e-02 -0.0246253744
## PSC.Fill -1.018258e-02 -0.0107407589 -1.129944e-02 -0.0118555568
## PSC.CO2 -1.497806e-02 -0.0156754542 -1.637175e-02 -0.0170640456
## Mnf.Flow -8.184308e-02 -0.0862473226 -9.082039e-02 -0.0955690784
## Carb.Pressure1 1.460507e-02 0.0169955089 1.960082e-02 0.0224239611
## Fill.Pressure -1.869541e-02 -0.0184420379 -1.806945e-02 -0.0175736007
## Hyd.Pressure1 -1.145030e-03 -0.0007257254 -2.804708e-04 0.0001863300
## Hyd.Pressure2 -1.514740e-02 -0.0148676140 -1.450519e-02 -0.0140607938
## Hyd.Pressure3 -1.777381e-02 -0.0172124993 -1.649803e-02 -0.0156229157
## Hyd.Pressure4 -2.074663e-02 -0.0214424969 -2.211197e-02 -0.0227510021
## Filler.Level 4.716763e-02 0.0488546187 5.051098e-02 0.0521299939
## Filler.Speed -4.219441e-03 -0.0045062235 -4.798008e-03 -0.0050913137
## Temperature -4.288105e-02 -0.0455349044 -4.825331e-02 -0.0510259942
## Usage.cont -5.656056e-02 -0.0592887966 -6.206199e-02 -0.0648746720
## Carb.Flow 2.801824e-02 0.0293710874 3.075070e-02 0.0321561684
## Density -9.164092e-05 -0.0007683003 -1.526231e-03 -0.0023705807
## MFR -1.078602e-02 -0.0117228155 -1.271242e-02 -0.0137531130
## Balling 2.692654e-03 0.0025795536 2.458704e-03 0.0023318268
## Pressure.Vacuum 2.348320e-02 0.0237131642 2.386317e-02 0.0239306576
## Oxygen.Filler 1.751665e-02 0.0175797895 1.755763e-02 0.0174445776
## Bowl.Setpoint 5.576088e-02 0.0582517197 6.078614e-02 0.0633637276
## Pressure.Setpoint -4.342672e-02 -0.0449928477 -4.654932e-02 -0.0480950162
## Air.Pressurer -3.717139e-03 -0.0039487262 -4.185217e-03 -0.0044255572
## Alch.Rel 2.008740e-02 0.0209564698 2.185031e-02 0.0227708506
## Carb.Rel 2.216834e-02 0.0230402885 2.391795e-02 0.0248002619
## Balling.Lvl 8.771558e-03 0.0090246289 9.294983e-03 0.0095872577
##
## (Intercept) -0.0247460530 -0.026037074 -0.027329686 -0.028617847
## Brand.Code -0.0907855626 -0.094975865 -0.099197615 -0.103441712
## Brand.CodeA -0.1024081788 -0.107138480 -0.111817788 -0.116422905
## Brand.CodeB 0.1093530926 0.115265426 0.121183391 0.127079130
## Brand.CodeC -0.2631802584 -0.276199908 -0.289260936 -0.302308809
## Brand.CodeD 0.0797402973 0.082733178 0.085723663 0.088702824
## Carb.Volume -0.0011060928 -0.001790890 -0.002538228 -0.003350728
## Fill.Ounces -0.0249208068 -0.025930316 -0.026915658 -0.027870833
## PC.Volume -0.0009420910 -0.001603807 -0.002331151 -0.003123207
## Carb.Pressure 0.0056176865 0.005607028 0.005577485 0.005527765
## Carb.Temp 0.0071352652 0.007547614 0.007975021 0.008417152
## PSC -0.0257517081 -0.026857421 -0.027934556 -0.028974513
## PSC.Fill -0.0124067055 -0.012948263 -0.013478737 -0.013995807
## PSC.CO2 -0.0177505635 -0.018426589 -0.019090662 -0.019740396
## Mnf.Flow -0.1005393541 -0.105674087 -0.111007375 -0.116546636
## Carb.Pressure1 0.0254338292 0.028687037 0.032154571 0.035830751
## Fill.Pressure -0.0169805036 -0.016234386 -0.015359779 -0.014356922
## Hyd.Pressure1 0.0006645333 0.001154924 0.001650043 0.002143386
## Hyd.Pressure2 -0.0135822154 -0.012988439 -0.012320451 -0.011582628
## Hyd.Pressure3 -0.0145968062 -0.013377261 -0.011974995 -0.010383406
## Hyd.Pressure4 -0.0233664796 -0.023931517 -0.024455326 -0.024933825
## Filler.Level 0.0537089509 0.055226119 0.056682219 0.058068518
## Filler.Speed -0.0053761069 -0.005658035 -0.005927958 -0.006180351
## Temperature -0.0538448356 -0.056690967 -0.059554811 -0.062422369
## Usage.cont -0.0677160167 -0.070584657 -0.073471884 -0.076369649
## Carb.Flow 0.0335843745 0.035036713 0.036510876 0.038005120
## Density -0.0033251197 -0.004368668 -0.005516149 -0.006775690
## MFR -0.0148391163 -0.015973396 -0.017149968 -0.018364801
## Balling 0.0021537255 0.002010487 0.001865905 0.001722076
## Pressure.Vacuum 0.0238974045 0.023789105 0.023594833 0.023315235
## Oxygen.Filler 0.0172238548 0.016909298 0.016489086 0.015959322
## Bowl.Setpoint 0.0659605615 0.068618865 0.071320070 0.074064187
## Pressure.Setpoint -0.0496016832 -0.051119613 -0.052626139 -0.054122048
## Air.Pressurer -0.0046690419 -0.004913838 -0.005159093 -0.005403645
## Alch.Rel 0.0236782967 0.024656250 0.025667933 0.026717801
## Carb.Rel 0.0256669070 0.026558985 0.027452770 0.028348639
## Balling.Lvl 0.0099153631 0.010275984 0.010675358 0.011120611
##
## (Intercept) -0.029895562 -0.031156954 -0.032396386 -0.033608550
## Brand.Code -0.107701150 -0.111969750 -0.116242095 -0.120513451
## Brand.CodeA -0.120929392 -0.125313787 -0.129553934 -0.133629288
## Brand.CodeB 0.132923146 0.138686669 0.144342195 0.149863918
## Brand.CodeC -0.315290383 -0.328154075 -0.340850681 -0.353334178
## Brand.CodeD 0.091665754 0.094607956 0.097525348 0.100414265
## Carb.Volume -0.004228941 -0.005173082 -0.006182942 -0.007257841
## Fill.Ounces -0.028790398 -0.029669313 -0.030503044 -0.031287651
## PC.Volume -0.003978613 -0.004895159 -0.005869788 -0.006898602
## Carb.Pressure 0.005457403 0.005365992 0.005253225 0.005118927
## Carb.Temp 0.008873377 0.009342931 0.009824901 0.010318231
## PSC -0.029968940 -0.030909820 -0.031789658 -0.032601660
## PSC.Fill -0.014497516 -0.014982279 -0.015448900 -0.015896567
## PSC.CO2 -0.020373729 -0.020988868 -0.021584305 -0.022158825
## Mnf.Flow -0.122298554 -0.128269241 -0.134464003 -0.140887099
## Carb.Pressure1 0.039707732 0.043775310 0.048020990 0.052430079
## Fill.Pressure -0.013227656 -0.011975108 -0.010603701 -0.009119138
## Hyd.Pressure1 0.002628466 0.003098678 0.003547415 0.003968188
## Hyd.Pressure2 -0.010779981 -0.009918366 -0.009004474 -0.008045829
## Hyd.Pressure3 -0.008596234 -0.006607496 -0.004411499 -0.002002864
## Hyd.Pressure4 -0.025363984 -0.025743005 -0.026068397 -0.026338022
## Filler.Level 0.059376231 0.060596117 0.061718505 0.062733319
## Filler.Speed -0.006409583 -0.006609846 -0.006775292 -0.006900170
## Temperature -0.065279395 -0.068111364 -0.070903700 -0.073642024
## Usage.cont -0.079269428 -0.082162157 -0.085038290 -0.087887904
## Carb.Flow 0.039517288 0.041044748 0.042584310 0.044132179
## Density -0.008154754 -0.009661589 -0.011305081 -0.013094706
## MFR -0.019613529 -0.020891448 -0.022193611 -0.023514910
## Balling 0.001580768 0.001443576 0.001311998 0.001187428
## Pressure.Vacuum 0.022952062 0.022507911 0.021986194 0.021391085
## Oxygen.Filler 0.015316930 0.014559516 0.013685427 0.012693811
## Bowl.Setpoint 0.076851347 0.079681610 0.082554959 0.085471293
## Pressure.Setpoint -0.055608277 -0.057085931 -0.058556199 -0.060020257
## Air.Pressurer -0.005646400 -0.005886338 -0.006122556 -0.006354284
## Alch.Rel 0.027809307 0.028946235 0.030132693 0.031373088
## Carb.Rel 0.029245583 0.030142674 0.031039005 0.031933682
## Balling.Lvl 0.011618050 0.012174352 0.012796436 0.013491413
##
## (Intercept) -0.0347495125 -0.0358940116 -0.0369927171 -0.0380485381
## Brand.Code -0.1248523268 -0.1291003556 -0.1333477042 -0.1375775181
## Brand.CodeA -0.1374346632 -0.1411258440 -0.1445901031 -0.1478250223
## Brand.CodeB 0.1550746642 0.1602672450 0.1652407616 0.1700020920
## Brand.CodeC -0.3656885835 -0.3776188188 -0.3892400388 -0.4005059609
## Brand.CodeD 0.1034527685 0.1062608505 0.1090578352 0.1118147076
## Carb.Volume -0.0083746679 -0.0095813680 -0.0108435653 -0.0121617606
## Fill.Ounces -0.0320307840 -0.0327086045 -0.0333300395 -0.0338938014
## PC.Volume -0.0079608350 -0.0090776657 -0.0102327509 -0.0114188411
## Carb.Pressure 0.0049580312 0.0047789292 0.0045791171 0.0043594349
## Carb.Temp 0.0108243041 0.0113370915 0.0118576067 0.0123838341
## PSC -0.0333410935 -0.0340007104 -0.0345780206 -0.0350702612
## PSC.Fill -0.0163279005 -0.0167361332 -0.0171250892 -0.0174950479
## PSC.CO2 -0.0227120493 -0.0232421426 -0.0237495991 -0.0242340665
## Mnf.Flow -0.1475909702 -0.1544944573 -0.1616315888 -0.1690003765
## Carb.Pressure1 0.0569366262 0.0616127357 0.0663961796 0.0712661618
## Fill.Pressure -0.0075749722 -0.0058924225 -0.0041228792 -0.0022753143
## Hyd.Pressure1 0.0043695764 0.0047113490 0.0050083961 0.0052539304
## Hyd.Pressure2 -0.0071446262 -0.0061400202 -0.0051192726 -0.0040934983
## Hyd.Pressure3 0.0005974894 0.0034543345 0.0065387929 0.0098566999
## Hyd.Pressure4 -0.0265665036 -0.0267209237 -0.0268167176 -0.0268532480
## Filler.Level 0.0636309172 0.0643870861 0.0650030038 0.0654660279
## Filler.Speed -0.0069644676 -0.0069896464 -0.0069583192 -0.0068671058
## Temperature -0.0763202179 -0.0789103388 -0.0814076892 -0.0838004205
## Usage.cont -0.0906883779 -0.0934473058 -0.0961481850 -0.0987808148
## Carb.Flow 0.0456795374 0.0472262029 0.0487665764 0.0502934386
## Density -0.0150976676 -0.0172126859 -0.0195153972 -0.0220060224
## MFR -0.0248411588 -0.0261844299 -0.0275308810 -0.0288762314
## Balling 0.0009509660 0.0008398927 0.0007216517 0.0006137783
## Pressure.Vacuum 0.0206867890 0.0199501751 0.0191554134 0.0183095572
## Oxygen.Filler 0.0115619256 0.0103296083 0.0089825103 0.0075231616
## Bowl.Setpoint 0.0883829918 0.0913818565 0.0944225439 0.0975070234
## Pressure.Setpoint -0.0614250467 -0.0628755518 -0.0643194044 -0.0657599603
## Air.Pressurer -0.0065817397 -0.0068038486 -0.0070198751 -0.0072303255
## Alch.Rel 0.0325954561 0.0339640594 0.0353914649 0.0368937329
## Carb.Rel 0.0328133459 0.0337105214 0.0346015887 0.0354875033
## Balling.Lvl 0.0143261816 0.0151957103 0.0161689344 0.0172438913
##
## (Intercept) -0.0390575935 -0.0399667587 -0.0408847899 -4.173837e-02
## Brand.Code -0.1417879553 -0.1460611691 -0.1502075822 -1.543353e-01
## Brand.CodeA -0.1508191120 -0.1534345536 -0.1559117219 -1.581239e-01
## Brand.CodeB 0.1745346938 0.1786308735 0.1826916862 1.864722e-01
## Brand.CodeC -0.4113970350 -0.4220634808 -0.4321435493 -4.418398e-01
## Brand.CodeD 0.1145330739 0.1174338976 0.1200625675 1.226668e-01
## Carb.Volume -0.0135322248 -0.0149254413 -0.0163834572 -1.788544e-02
## Fill.Ounces -0.0343995031 -0.0348614698 -0.0352554341 -3.559274e-02
## PC.Volume -0.0126287690 -0.0138441869 -0.0150755181 -1.630677e-02
## Carb.Pressure 0.0041203818 0.0038546303 0.0035831241 3.289954e-03
## Carb.Temp 0.0129143332 0.0134528344 0.0139854021 1.452024e-02
## PSC -0.0354760426 -0.0357966756 -0.0360306282 -3.618104e-02
## PSC.Fill -0.0178466291 -0.0181860601 -0.0185026758 -1.880350e-02
## PSC.CO2 -0.0246956083 -0.0251340127 -0.0255506242 -2.594524e-02
## Mnf.Flow -0.1765964648 -0.1844528224 -0.1924979758 -2.007451e-01
## Carb.Pressure1 0.0762002459 0.0811252378 0.0861115032 9.109110e-02
## Fill.Pressure -0.0003609919 0.0015505116 0.0035526636 5.583528e-03
## Hyd.Pressure1 0.0054438143 0.0056172527 0.0056815027 5.684152e-03
## Hyd.Pressure2 -0.0030742657 -0.0022053263 -0.0012594593 -3.618926e-04
## Hyd.Pressure3 0.0134111127 0.0171711627 0.0212203764 2.551009e-02
## Hyd.Pressure4 -0.0268305430 -0.0267732699 -0.0266419940 -2.645124e-02
## Filler.Level 0.0657653943 0.0658956309 0.0658228312 6.555422e-02
## Filler.Speed -0.0067124412 -0.0064672267 -0.0061767401 -5.814017e-03
## Temperature -0.0860786221 -0.0882478738 -0.0902749493 -9.216729e-02
## Usage.cont -0.1013355227 -0.1037897538 -0.1061553917 -1.084169e-01
## Carb.Flow 0.0518004346 0.0532782606 0.0547191760 5.612181e-02
## Density -0.0246983338 -0.0276941846 -0.0308084099 -3.419429e-02
## MFR -0.0302155838 -0.0315332765 -0.0328488683 -3.414344e-02
## Balling 0.0005135770 0.0002345326 0.0001511458 4.099334e-05
## Pressure.Vacuum 0.0174192522 0.0164332148 0.0154656743 1.447025e-02
## Oxygen.Filler 0.0059554321 0.0042595511 0.0024845962 6.176439e-04
## Bowl.Setpoint 0.1006350310 0.1037447690 0.1069612726 1.102191e-01
## Pressure.Setpoint -0.0671965308 -0.0685587757 -0.0699805625 -7.139076e-02
## Air.Pressurer -0.0074352948 -0.0076362889 -0.0078337147 -8.026301e-03
## Alch.Rel 0.0384747053 0.0400461916 0.0418036946 4.365118e-02
## Carb.Rel 0.0363673060 0.0372408199 0.0381031908 3.896822e-02
## Balling.Lvl 0.0184301782 0.0198455932 0.0212665644 2.285502e-02
##
## (Intercept) -4.254131e-02 -0.0432920596 -0.0439274557 -0.0445887198
## Brand.Code -1.584260e-01 -0.1624699454 -0.1665565571 -0.1704944753
## Brand.CodeA -1.600684e-01 -0.1617514953 -0.1629929109 -0.1641129698
## Brand.CodeB 1.899972e-01 0.1932703220 0.1960554618 0.1988236365
## Brand.CodeC -4.511212e-01 -0.4599873281 -0.4686496699 -0.4767002116
## Brand.CodeD 1.252344e-01 0.1277523344 0.1304682614 0.1329372593
## Carb.Volume -1.941460e-02 -0.0209699255 -0.0225238558 -0.0240840004
## Fill.Ounces -3.587813e-02 -0.0361132352 -0.0363164012 -0.0364666088
## PC.Volume -1.753160e-02 -0.0187418948 -0.0199227832 -0.0210837123
## Carb.Pressure 2.984317e-03 0.0026653951 0.0023200303 0.0019914043
## Carb.Temp 1.505217e-02 0.0155806244 0.0161146496 0.0166271936
## PSC -3.625156e-02 -0.0362462386 -0.0361710398 -0.0360303959
## PSC.Fill -1.908902e-02 -0.0193600612 -0.0196239389 -0.0198672729
## PSC.CO2 -2.631881e-02 -0.0266718326 -0.0270034869 -0.0273186452
## Mnf.Flow -2.091792e-01 -0.2177841535 -0.2265894599 -0.2354929067
## Carb.Pressure1 9.604353e-02 0.1009467363 0.1057271954 0.1104685932
## Fill.Pressure 7.629010e-03 0.0096767827 0.0116406510 0.0136410623
## Hyd.Pressure1 5.621338e-03 0.0054932766 0.0053671859 0.0051083341
## Hyd.Pressure2 4.742409e-04 0.0012336747 0.0017147241 0.0022569123
## Hyd.Pressure3 3.004496e-02 0.0348230756 0.0398225111 0.0451009641
## Hyd.Pressure4 -2.621003e-02 -0.0259187926 -0.0256133726 -0.0252487950
## Filler.Level 6.508126e-02 0.0643950483 0.0634802404 0.0623374577
## Filler.Speed -5.382966e-03 -0.0048834478 -0.0042759700 -0.0036470919
## Temperature -9.392124e-02 -0.0955346135 -0.0970291537 -0.0983646603
## Usage.cont -1.105686e-01 -0.1126048681 -0.1144996547 -0.1162887759
## Carb.Flow 5.747663e-02 0.0587778676 0.0600169215 0.0611880091
## Density -3.781960e-02 -0.0417016823 -0.0459971634 -0.0503715340
## MFR -3.541614e-02 -0.0366629191 -0.0378697772 -0.0390598820
## Balling -6.736492e-05 -0.0001771314 -0.0005656386 -0.0007005686
## Pressure.Vacuum 1.345798e-02 0.0124343147 0.0113175331 0.0102834521
## Oxygen.Filler -1.332588e-03 -0.0033580951 -0.0054801218 -0.0076314423
## Bowl.Setpoint 1.135239e-01 0.1168762171 0.1202064650 0.1236637432
## Pressure.Setpoint -7.279072e-02 -0.0741781794 -0.0754669392 -0.0768142966
## Air.Pressurer -8.216514e-03 -0.0084050707 -0.0085952710 -0.0087874508
## Alch.Rel 4.559206e-02 0.0476358505 0.0496985776 0.0519285873
## Carb.Rel 3.981666e-02 0.0406535435 0.0415091443 0.0422864099
## Balling.Lvl 2.458085e-02 0.0264582045 0.0286794942 0.0308526860
##
## (Intercept) -0.045122289 -0.045683649 -0.0461209899 -0.0465921440
## Brand.Code -0.174451990 -0.178250226 -0.1820537338 -0.1856733729
## Brand.CodeA -0.164837652 -0.165434277 -0.1656615666 -0.1657762765
## Brand.CodeB 0.201124953 0.203406706 0.2052472690 0.2070914065
## Brand.CodeC -0.484566504 -0.491854603 -0.4989751326 -0.5055417199
## Brand.CodeD 0.135537359 0.137915160 0.1404055710 0.1426668069
## Carb.Volume -0.025656317 -0.027216627 -0.0287720739 -0.0303033501
## Fill.Ounces -0.036584251 -0.036658979 -0.0367068103 -0.0367206573
## PC.Volume -0.022209817 -0.023295544 -0.0243437059 -0.0253378504
## Carb.Pressure 0.001626587 0.001281401 0.0009050992 0.0005492852
## Carb.Temp 0.017150041 0.017648695 0.0181563972 0.0186387470
## PSC -0.035830329 -0.035577926 -0.0352788356 -0.0349409683
## PSC.Fill -0.020105514 -0.020324145 -0.0205380249 -0.0207337396
## PSC.CO2 -0.027612735 -0.027892281 -0.0281526771 -0.0284000248
## Mnf.Flow -0.244523307 -0.253634519 -0.2628019798 -0.2720128929
## Carb.Pressure1 0.115066611 0.119574850 0.1239259292 0.1281499803
## Fill.Pressure 0.015549241 0.017455104 0.0192577742 0.0210278746
## Hyd.Pressure1 0.004876220 0.004504001 0.0041740737 0.0037089947
## Hyd.Pressure2 0.002503855 0.002760749 0.0027146083 0.0026336952
## Hyd.Pressure3 0.050572778 0.056326291 0.0622520698 0.0684465496
## Hyd.Pressure4 -0.024863317 -0.024430744 -0.0239854221 -0.0235026043
## Filler.Level 0.060967822 0.059347037 0.0575038833 0.0554008058
## Filler.Speed -0.002910425 -0.002157635 -0.0013090334 -0.0004517947
## Temperature -0.099585296 -0.100653936 -0.1016169729 -0.1024393129
## Usage.cont -0.117937673 -0.119470655 -0.1208679218 -0.1221477439
## Carb.Flow 0.062298737 0.063327940 0.0642967935 0.0651775422
## Density -0.055226705 -0.060172840 -0.0655940713 -0.0711079775
## MFR -0.040201485 -0.041320498 -0.0423869309 -0.0434256197
## Balling -0.001143200 -0.001354337 -0.0018913538 -0.0022081829
## Pressure.Vacuum 0.009166655 0.008131846 0.0070279024 0.0060060265
## Oxygen.Filler -0.009851025 -0.012089799 -0.0143675568 -0.0166497974
## Bowl.Setpoint 0.127100198 0.130664085 0.1342170149 0.1378966837
## Pressure.Setpoint -0.078067972 -0.079360110 -0.0805639093 -0.0817896919
## Air.Pressurer -0.008979939 -0.009178779 -0.0093792285 -0.0095893332
## Alch.Rel 0.054218464 0.056663853 0.0591734109 0.0618347110
## Carb.Rel 0.043118438 0.043864488 0.0446575759 0.0453636926
## Balling.Lvl 0.033442597 0.035997560 0.0389916885 0.0419603680
##
## (Intercept) -0.0468792635 -0.0474190104 -0.0474685650 -0.0480292403
## Brand.Code -0.1894129341 -0.1925949243 -0.1963097093 -0.1991438899
## Brand.CodeA -0.1654154095 -0.1653784321 -0.1644260537 -0.1640562585
## Brand.CodeB 0.2082915835 0.2102093645 0.2106395187 0.2124107237
## Brand.CodeC -0.5121636804 -0.5176480353 -0.5240098078 -0.5287012344
## Brand.CodeD 0.1452802283 0.1469872315 0.1497435732 0.1512149855
## Carb.Volume -0.0317795706 -0.0332764967 -0.0346892594 -0.0360799326
## Fill.Ounces -0.0367225867 -0.0366766467 -0.0366419666 -0.0365700628
## PC.Volume -0.0263071717 -0.0271769777 -0.0280615243 -0.0288160660
## Carb.Pressure 0.0001613598 -0.0001669389 -0.0006019055 -0.0009080706
## Carb.Temp 0.0191377576 0.0195748560 0.0200908386 0.0204936659
## PSC -0.0345692649 -0.0341733208 -0.0337521676 -0.0333227339
## PSC.Fill -0.0209286856 -0.0210924268 -0.0212741222 -0.0214160514
## PSC.CO2 -0.0286301741 -0.0288518145 -0.0290518083 -0.0292495386
## Mnf.Flow -0.2811827632 -0.2904081320 -0.2994717080 -0.3085972815
## Carb.Pressure1 0.1322081623 0.1361297495 0.1398546350 0.1434228201
## Fill.Pressure 0.0226615063 0.0243291213 0.0257419161 0.0272096429
## Hyd.Pressure1 0.0033764122 0.0026681933 0.0024329614 0.0016307503
## Hyd.Pressure2 0.0021680993 0.0019459120 0.0009087111 0.0002941512
## Hyd.Pressure3 0.0747351139 0.0814121644 0.0879720950 0.0950333358
## Hyd.Pressure4 -0.0230396988 -0.0224830551 -0.0220217378 -0.0214550716
## Filler.Level 0.0531213154 0.0505039906 0.0477935231 0.0447088333
## Filler.Speed 0.0005075664 0.0013760385 0.0024650394 0.0033689583
## Temperature -0.1031859375 -0.1037491155 -0.1043288573 -0.1046937497
## Usage.cont -0.1232996893 -0.1243500610 -0.1252542886 -0.1260811911
## Carb.Flow 0.0660145590 0.0667211927 0.0674488912 0.0679953231
## Density -0.0771624116 -0.0829097161 -0.0898360003 -0.0958793592
## MFR -0.0443986372 -0.0453701631 -0.0462247262 -0.0471170104
## Balling -0.0031445143 -0.0030213392 -0.0047264705 -0.0045440825
## Pressure.Vacuum 0.0048771234 0.0040110079 0.0027741349 0.0019739254
## Oxygen.Filler -0.0189429191 -0.0212129970 -0.0234938340 -0.0257273603
## Bowl.Setpoint 0.1415234698 0.1454378921 0.1490955976 0.1531663566
## Pressure.Setpoint -0.0828927436 -0.0841336767 -0.0850934096 -0.0862538114
## Air.Pressurer -0.0098002166 -0.0100306231 -0.0102489152 -0.0105045989
## Alch.Rel 0.0644486983 0.0674433646 0.0701852838 0.0733595205
## Carb.Rel 0.0461040583 0.0466828412 0.0474811076 0.0479328449
## Balling.Lvl 0.0455704875 0.0485263404 0.0531306821 0.0562907186
##
## (Intercept) -0.0479566944 -4.852966e-02 -0.0484067927 -0.048970457
## Brand.Code -0.2026586569 -2.050965e-01 -0.2082276766 -0.210335336
## Brand.CodeA -0.1627854530 -1.621843e-01 -0.1607907227 -0.160091051
## Brand.CodeB 0.2124484950 2.141339e-01 0.2140723773 0.215649690
## Brand.CodeC -0.5345664628 -5.385221e-01 -0.5437353497 -0.547064972
## Brand.CodeD 0.1538152148 1.549596e-01 0.1570938826 0.157996339
## Carb.Volume -0.0374069179 -3.869914e-02 -0.0399555269 -0.041106128
## Fill.Ounces -0.0365030488 -3.641163e-02 -0.0363179625 -0.036214359
## PC.Volume -0.0296031752 -3.024683e-02 -0.0309258940 -0.031466045
## Carb.Pressure -0.0013578123 -1.656297e-03 -0.0021208962 -0.002398007
## Carb.Temp 0.0210063879 2.138611e-02 0.0218946112 0.022246871
## PSC -0.0328756655 -3.243295e-02 -0.0319795193 -0.031541560
## PSC.Fill -0.0215773308 -2.170096e-02 -0.0218424913 -0.021945429
## PSC.CO2 -0.0294256892 -2.960097e-02 -0.0297548586 -0.029913341
## Mnf.Flow -0.3174460786 -3.263445e-01 -0.3349075695 -0.343438201
## Carb.Pressure1 0.1468251597 1.500357e-01 0.1531014184 0.155981747
## Fill.Pressure 0.0284415269 2.969918e-02 0.0307457581 0.031822735
## Hyd.Pressure1 0.0013977850 5.581859e-04 0.0003103487 -0.000592034
## Hyd.Pressure2 -0.0011354989 -2.173347e-03 -0.0039888527 -0.005260285
## Hyd.Pressure3 0.1018454948 1.091862e-01 0.1162107218 0.123682430
## Hyd.Pressure4 -0.0209867133 -2.041430e-02 -0.0199372597 -0.019365635
## Filler.Level 0.0415812211 3.807705e-02 0.0345335988 0.030731061
## Filler.Speed 0.0045023364 5.426398e-03 0.0065719143 0.007453335
## Temperature -0.1051105887 -1.053147e-01 -0.1055923499 -0.105659402
## Usage.cont -0.1267708701 -1.273967e-01 -0.1278897214 -0.128359631
## Carb.Flow 0.0686090397 6.902084e-02 0.0695251797 0.069830536
## Density -0.1033539226 -1.097095e-01 -0.1176211280 -0.124101501
## MFR -0.0478592132 -4.866073e-02 -0.0492924104 -0.049994682
## Balling -0.0067008410 -6.536915e-03 -0.0089728346 -0.008877246
## Pressure.Vacuum 0.0007281425 -2.210618e-05 -0.0012615127 -0.001912975
## Oxygen.Filler -0.0279431198 -3.010284e-02 -0.0322286352 -0.034268113
## Bowl.Setpoint 0.1569380238 1.611424e-01 0.1650533613 0.169365756
## Pressure.Setpoint -0.0871392649 -8.820949e-02 -0.0890332846 -0.090028296
## Air.Pressurer -0.0107349193 -1.101227e-02 -0.0112563539 -0.011550675
## Alch.Rel 0.0762767788 7.964189e-02 0.0828375480 0.086235607
## Carb.Rel 0.0486982125 4.907216e-02 0.0498254190 0.050051031
## Balling.Lvl 0.0616190091 6.507344e-02 0.0710139769 0.074756508
##
## (Intercept) -0.0487126618 -0.049344787 -0.048993135 -0.049632817
## Brand.Code -0.2132449470 -0.214918371 -0.217467851 -0.218770124
## Brand.CodeA -0.1584312606 -0.157751137 -0.156001574 -0.155254008
## Brand.CodeB 0.2152094710 0.216879699 0.216279555 0.217899583
## Brand.CodeC -0.5519690757 -0.554573740 -0.559068061 -0.561166299
## Brand.CodeD 0.1600066192 0.160574319 0.162198823 0.162544365
## Carb.Volume -0.0422291491 -0.043223950 -0.044238188 -0.045079290
## Fill.Ounces -0.0361177021 -0.036012451 -0.035913171 -0.035817895
## PC.Volume -0.0320628740 -0.032506639 -0.033022875 -0.033391277
## Carb.Pressure -0.0028903236 -0.003132518 -0.003663042 -0.003899181
## Carb.Temp 0.0227711372 0.023087130 0.023634858 0.023940035
## PSC -0.0310958599 -0.030676695 -0.030249845 -0.029856832
## PSC.Fill -0.0220735577 -0.022157309 -0.022273385 -0.022345170
## PSC.CO2 -0.0300474683 -0.030191955 -0.030306948 -0.030436839
## Mnf.Flow -0.3516118263 -0.359683555 -0.367413197 -0.374977638
## Carb.Pressure1 0.1587173582 0.161277470 0.163695828 0.165942868
## Fill.Pressure 0.0326401312 0.033546138 0.034149897 0.034851511
## Hyd.Pressure1 -0.0007139414 -0.001675171 -0.001668244 -0.002556301
## Hyd.Pressure2 -0.0076104457 -0.009070711 -0.011923750 -0.013751253
## Hyd.Pressure3 0.1307965092 0.138308601 0.145444558 0.152913283
## Hyd.Pressure4 -0.0189374861 -0.018373757 -0.017989805 -0.017468565
## Filler.Level 0.0268407132 0.022824090 0.018610278 0.014413021
## Filler.Speed 0.0086286801 0.009434041 0.010618523 0.011388214
## Temperature -0.1058591552 -0.105813064 -0.105959998 -0.105855160
## Usage.cont -0.1286688542 -0.129014499 -0.129159239 -0.129381486
## Carb.Flow 0.0702569911 0.070464382 0.070829613 0.070966666
## Density -0.1324783894 -0.138811837 -0.147582250 -0.153881140
## MFR -0.0504985802 -0.051111149 -0.051484761 -0.051999787
## Balling -0.0121979560 -0.012016852 -0.016207357 -0.016381752
## Pressure.Vacuum -0.0032589650 -0.003783668 -0.005246836 -0.005766134
## Oxygen.Filler -0.0362867637 -0.038180443 -0.040078207 -0.041827565
## Bowl.Setpoint 0.1733470094 0.177740127 0.181797516 0.186207601
## Pressure.Setpoint -0.0907345778 -0.091659975 -0.092265751 -0.093080030
## Air.Pressurer -0.0118063449 -0.012117586 -0.012380618 -0.012704098
## Alch.Rel 0.0895453248 0.092932054 0.096450795 0.099811647
## Carb.Rel 0.0507860173 0.050831856 0.051586323 0.051537759
## Balling.Lvl 0.0817161570 0.085524007 0.093526826 0.097726844
##
## (Intercept) -0.049264606 -0.049949941 -0.049535008 -0.050235895
## Brand.Code -0.220924765 -0.221797051 -0.223533851 -0.224068019
## Brand.CodeA -0.153618068 -0.153021888 -0.151495118 -0.151047865
## Brand.CodeB 0.217320009 0.219067184 0.218495420 0.220281415
## Brand.CodeC -0.565147004 -0.566643405 -0.570202086 -0.571234525
## Brand.CodeD 0.163781406 0.163752379 0.164540239 0.164273649
## Carb.Volume -0.045941782 -0.046643401 -0.047376304 -0.047928484
## Fill.Ounces -0.035721884 -0.035631492 -0.035543489 -0.035463864
## PC.Volume -0.033830458 -0.034127391 -0.034502746 -0.034742577
## Carb.Pressure -0.004431700 -0.004649324 -0.005212631 -0.005410805
## Carb.Temp 0.024487136 0.024770633 0.025340343 0.025603843
## PSC -0.029459194 -0.029099604 -0.028734524 -0.028411355
## PSC.Fill -0.022444643 -0.022502686 -0.022591486 -0.022637556
## PSC.CO2 -0.030538866 -0.030656447 -0.030743756 -0.030850791
## Mnf.Flow -0.382176072 -0.389148501 -0.395812160 -0.402155947
## Carb.Pressure1 0.168081061 0.170051765 0.171920344 0.173641100
## Fill.Pressure 0.035299579 0.035860501 0.036134695 0.036570897
## Hyd.Pressure1 -0.002520709 -0.003401104 -0.003251157 -0.004101812
## Hyd.Pressure2 -0.016816570 -0.018746706 -0.022147858 -0.024120493
## Hyd.Pressure3 0.159952774 0.167215238 0.174107157 0.181055655
## Hyd.Pressure4 -0.017122675 -0.016622607 -0.016334081 -0.015870692
## Filler.Level 0.010020576 0.005794438 0.001229021 -0.002919250
## Filler.Speed 0.012501308 0.013179400 0.014257095 0.014841447
## Temperature -0.105950124 -0.105791918 -0.105875443 -0.105688607
## Usage.cont -0.129414947 -0.129559477 -0.129480576 -0.129568658
## Carb.Flow 0.071275143 0.071360832 0.071633064 0.071681741
## Density -0.162567895 -0.168571331 -0.177285502 -0.182861625
## MFR -0.052254299 -0.052678106 -0.052809288 -0.053147069
## Balling -0.021236172 -0.021510589 -0.027369685 -0.027912130
## Pressure.Vacuum -0.007241477 -0.007677441 -0.009267284 -0.009642256
## Oxygen.Filler -0.043574219 -0.045155421 -0.046763317 -0.048175079
## Bowl.Setpoint 0.190323543 0.194691377 0.198832064 0.203084318
## Pressure.Setpoint -0.093623468 -0.094365125 -0.094823135 -0.095487106
## Air.Pressurer -0.012972358 -0.013298165 -0.013566464 -0.013888926
## Alch.Rel 0.103371592 0.106662666 0.110341032 0.113457787
## Carb.Rel 0.052169526 0.052005204 0.052600405 0.052312248
## Balling.Lvl 0.106466574 0.110795688 0.120575264 0.125061654# Perform cross-validation to find the best lambda
cv_ridge <- cv.glmnet(x, y, alpha = 0) #Performs cross-validation to select the best regularization parameter (lambda).
# Plot the cross-validation results
plot(cv_ridge)
# Get the best lambda from the cross-validation
best_lambda <- cv_ridge$lambda.min # The lambda value that gives the minimum mean cross-validation error.
cat("Best Lambda for Ridge:", best_lambda, "\n")## Best Lambda for Ridge: 0.04472021# Refit the Ridge regression model with the best lambda
ridge_model_best <- glmnet(x, y, alpha = 0, lambda = best_lambda)
# Get the coefficients for the best model
coef(ridge_model_best) # Extracts the coefficients of the final Ridge model.## 37 x 1 sparse Matrix of class "dgCMatrix"
## s0
## (Intercept) -0.050249632
## Brand.Code -0.225990172
## Brand.CodeA -0.148806319
## Brand.CodeB 0.217744117
## Brand.CodeC -0.573374870
## Brand.CodeD 0.169807926
## Carb.Volume -0.047116395
## Fill.Ounces -0.035670659
## PC.Volume -0.034936293
## Carb.Pressure -0.005194562
## Carb.Temp 0.025506474
## PSC -0.028406938
## PSC.Fill -0.022632523
## PSC.CO2 -0.030907924
## Mnf.Flow -0.402017209
## Carb.Pressure1 0.173676455
## Fill.Pressure 0.036346574
## Hyd.Pressure1 -0.003524542
## Hyd.Pressure2 -0.025326769
## Hyd.Pressure3 0.181595197
## Hyd.Pressure4 -0.016278557
## Filler.Level -0.002701681
## Filler.Speed 0.014896099
## Temperature -0.105821987
## Usage.cont -0.129550235
## Carb.Flow 0.071474197
## Density -0.179732669
## MFR -0.053132084
## Balling -0.032426660
## Pressure.Vacuum -0.010020594
## Oxygen.Filler -0.048175862
## Bowl.Setpoint 0.203051757
## Pressure.Setpoint -0.095208181
## Air.Pressurer -0.013978231
## Alch.Rel 0.110517804
## Carb.Rel 0.050933183
## Balling.Lvl 0.126141831Result: A coefficient of -0.031565321 for Carb.Volume means that as Carb.Volume increases by 1 unit, the PH decreases by 0.031565321, assuming all other predictors are held constant. Negative Coefficients: Features like Brand.Code, Carb.Volume, and Temperature are negatively correlated with PH, meaning as these features increase, the value of PH tends to decrease. Positive Coefficients: Features like Balling.Lvl and Oxygen.Filler are positively correlated with PH, meaning as these features increase, PH is predicted to increase as well.
# Extract the target variable 'PH' for the test set
testy <- student_eval_imputed$PH
library(Matrix)
# Prepare the test data (apply one-hot encoding to test set as well)
test_predictors <- student_eval_imputed[, -which(names(student_eval_imputed) == "PH")]
x_test <- predict(dummy_model, test_predictors)
x_test <- as.matrix(x_test) # Convert to matrix format
# Predict using the fitted Ridge model
y_pred <- predict(ridge_model_best, newx = x_test)
# Calculate mean squared error (MSE) or R-squared
mse <- mean((y_pred - testy)^2)
cat("Mean Squared Error (MSE):", mse, "\n")## Mean Squared Error (MSE): 0.3016673# Alternatively, calculate R-squared:
ss_total <- sum((testy - mean(testy))^2)
ss_residual <- sum((testy - y_pred)^2)
r_squared <- 1 - (ss_residual / ss_total)
cat("R-squared:", r_squared, "\n")## R-squared: 0.4148804A negative R-squared(-296.9042 ) generally indicates that the model is not capturing the underlying patterns in the data and is performing poorly. This can happen when: The model is misspecified. The data used for training does not match the assumptions of the model. There is a significant amount of noise in the data. The model might be overfitting or underfitting the data.
Mean Squared Error (MSE): A high MSE indicates that the model’s predictions are far from the actual values. MSE = 153.4756 suggests that, on average, the model’s predictions are quite off from the true values. Ideally, the MSE to be as low as possible.
Improve Lambda Tuning Using Cross-validation:
# Use cross-validation to find the best lambda for Ridge regression
cv_ridge <- cv.glmnet(x, y, alpha = 0)
# Plot cross-validation results
plot(cv_ridge)
# Get the best lambda
best_lambda <- cv_ridge$lambda.min
cat("Best Lambda for Ridge:", best_lambda, "\n")## Best Lambda for Ridge: 0.04472021# Refit the model with the best lambda
ridge_model_best <- glmnet(x, y, alpha = 0, lambda = best_lambda)
# Get the coefficients for the best Ridge model
coef(ridge_model_best)## 37 x 1 sparse Matrix of class "dgCMatrix"
## s0
## (Intercept) -0.050249632
## Brand.Code -0.225990172
## Brand.CodeA -0.148806319
## Brand.CodeB 0.217744117
## Brand.CodeC -0.573374870
## Brand.CodeD 0.169807926
## Carb.Volume -0.047116395
## Fill.Ounces -0.035670659
## PC.Volume -0.034936293
## Carb.Pressure -0.005194562
## Carb.Temp 0.025506474
## PSC -0.028406938
## PSC.Fill -0.022632523
## PSC.CO2 -0.030907924
## Mnf.Flow -0.402017209
## Carb.Pressure1 0.173676455
## Fill.Pressure 0.036346574
## Hyd.Pressure1 -0.003524542
## Hyd.Pressure2 -0.025326769
## Hyd.Pressure3 0.181595197
## Hyd.Pressure4 -0.016278557
## Filler.Level -0.002701681
## Filler.Speed 0.014896099
## Temperature -0.105821987
## Usage.cont -0.129550235
## Carb.Flow 0.071474197
## Density -0.179732669
## MFR -0.053132084
## Balling -0.032426660
## Pressure.Vacuum -0.010020594
## Oxygen.Filler -0.048175862
## Bowl.Setpoint 0.203051757
## Pressure.Setpoint -0.095208181
## Air.Pressurer -0.013978231
## Alch.Rel 0.110517804
## Carb.Rel 0.050933183
## Balling.Lvl 0.126141831# Check model performance (e.g., RMSE or MSE on validation/test set)
predictions <- predict(ridge_model_best, newx = x)
mse <- mean((predictions - y)^2)
cat("MSE of the best Ridge model:", mse, "\n")## MSE of the best Ridge model: 0.5840237A Mean Squared Error (MSE) of 0.5647799 for the best Ridge regression model is a significant improvement compared to the previous MSE of 153.4756. This suggests that the model is now performing better in terms of prediction accuracy.
The reduction in MSE after applying cross-validation and tuning the lambda parameter implies that the regularization parameter has been effectively optimized.
Predict and Evaluate on Test Data:
# Prepare the test data (apply one-hot encoding to the test set as well)
test_predictors <- student_eval_imputed[, -which(names(student_eval_imputed) == "PH")]
# Apply one-hot encoding using the same dummy model fitted on the training set
x_test <- predict(dummy_model, test_predictors)
x_test <- as.matrix(x_test) # Convert to matrix format
# Extract the actual target values for the test set (PH column)
testy <- student_eval_imputed$PH
# Predict using the fitted Ridge model (with the best lambda)
y_pred <- predict(ridge_model_best, newx = x_test)
# Calculate Mean Squared Error (MSE)
mse <- mean((y_pred - testy)^2)
cat("Mean Squared Error (MSE):", mse, "\n")## Mean Squared Error (MSE): 0.3016673# Alternatively, calculate R-squared:
ss_total <- sum((testy - mean(testy))^2)
ss_residual <- sum((testy - y_pred)^2)
r_squared <- 1 - (ss_residual / ss_total)
cat("R-squared:", r_squared, "\n")## R-squared: 0.4148804Model perfomred well on training data after Lambda Tuning Using Cross-validation but A negative R-squared value on the test data indicates that the model is performing poorly. A negative R-squared occurs when the model is worse than a simple model that predicts the mean of the target for all instances (i.e., it’s worse than no model at all).
Lasso Model
# Fit Lasso Regression model (alpha = 1 for Lasso)
lasso_model <- glmnet(x, y, alpha = 1)
# Use cross-validation to select the best lambda
cv_lasso <- cv.glmnet(x, y, alpha = 1)
# Plot the cross-validation results
plot(cv_lasso)
# Best lambda for Lasso
best_lambda_lasso <- cv_lasso$lambda.min
cat("Best Lambda for Lasso:", best_lambda_lasso, "\n")## Best Lambda for Lasso: 0.0008778759# Refit the model with the best lambda
lasso_model_best <- glmnet(x, y, alpha = 1, lambda = best_lambda_lasso)
# Get the coefficients for the best Lasso model
coef(lasso_model_best)## 37 x 1 sparse Matrix of class "dgCMatrix"
## s0
## (Intercept) -0.027998719
## Brand.Code -0.224451721
## Brand.CodeA -0.187232641
## Brand.CodeB 0.220079848
## Brand.CodeC -0.604978798
## Brand.CodeD 0.105711706
## Carb.Volume -0.049666218
## Fill.Ounces -0.035021628
## PC.Volume -0.036363670
## Carb.Pressure .
## Carb.Temp 0.022443505
## PSC -0.024467305
## PSC.Fill -0.022541523
## PSC.CO2 -0.031090441
## Mnf.Flow -0.486683064
## Carb.Pressure1 0.192491183
## Fill.Pressure 0.033504430
## Hyd.Pressure1 -0.004994675
## Hyd.Pressure2 -0.079589299
## Hyd.Pressure3 0.291689110
## Hyd.Pressure4 -0.013886368
## Filler.Level -0.076387459
## Filler.Speed 0.021386607
## Temperature -0.107010867
## Usage.cont -0.124517983
## Carb.Flow 0.073056780
## Density -0.278943762
## MFR -0.043726777
## Balling -0.235989633
## Pressure.Vacuum -0.039375879
## Oxygen.Filler -0.067249540
## Bowl.Setpoint 0.272186793
## Pressure.Setpoint -0.098983554
## Air.Pressurer -0.018183348
## Alch.Rel 0.177177218
## Carb.Rel 0.037610564
## Balling.Lvl 0.393531254Non-zero Coefficients: These are the features that have been selected by Lasso. For example, Brand.Code, Carb.Volume, Carb.Pressure, etc., have non-zero coefficients, suggesting they have a relationship with PH.
Zero Coefficients: Features like Carb.Volume and Filler.Level have been shrunk to zero, indicating that Lasso determined these features were not important in predicting PH.
# Predict on test data
test_predictors <- student_eval_imputed[, -which(names(student_eval_imputed) == "PH")]
x_test <- predict(dummy_model, test_predictors)
x_test <- as.matrix(x_test)
# Predict using the best Lasso model
y_pred <- predict(lasso_model_best, newx = x_test)
# Calculate MSE and R-squared
mse <- mean((y_pred - testy)^2)
cat("Mean Squared Error (MSE):", mse, "\n")## Mean Squared Error (MSE): 0.3194339ss_total <- sum((testy - mean(testy))^2)
ss_residual <- sum((testy - y_pred)^2)
r_squared <- 1 - (ss_residual / ss_total)
cat("R-squared:", r_squared, "\n")## R-squared: 0.38042Higher MSE: A high MSE typically suggests that the model is not fitting the data well, and its predictions are far from the actual values. A model with high MSE might indicate that the features used in the model are not capturing the underlying patterns effectively.
Deepa Ends
Reduce Multicollinearity
Three features, Brand.Code, Balling, and
Balling.Lvl, have high multicollinearity which we will
address in this section.
Brand.Code
Here we replace Brand.Code with BCB which
will show if the record is brand code B or not. When we reran for
individual values of Brand.Code, we see that Brand B
appears to have a different relationship than the other brands, Brand B
has a R-squared of ~0.7 while all the other brands have a R-squared of
~0.3. This suggests, we can replace all the brands with a binary
variable of whether it is Brand B or not. (not pictured)
From this we identify that Balling and
Balling.Lvl represent a lot of the remaining
multicollinearity to be reduced.
# Replacing Brand.Code with BCB
student_train_x1 <- student_train_x |> mutate(BCB = as.numeric(Brand.Code =='B')) |> select(-Brand.Code)
student_eval_y1 <- student_eval_y |> mutate(BCB = as.numeric(Brand.Code =='B')) |> select(-Brand.Code)
# New model with BCB instead of Brand.Code
model <- lm(PH ~ ., data = student_train_x)
#summary(model)
# Calculating VIF
vif_values <- vif(model)
vif_values## GVIF Df GVIF^(1/(2*Df))
## Brand.Code 53.942578 4 1.646234
## Carb.Volume 7.872617 1 2.805818
## Fill.Ounces 1.151544 1 1.073100
## PC.Volume 1.500187 1 1.224821
## Carb.Pressure 15.485772 1 3.935197
## Carb.Temp 12.827035 1 3.581485
## PSC 1.152323 1 1.073463
## PSC.Fill 1.102569 1 1.050033
## PSC.CO2 1.068812 1 1.033834
## Mnf.Flow 4.382860 1 2.093528
## Carb.Pressure1 1.579213 1 1.256668
## Fill.Pressure 2.180153 1 1.476534
## Hyd.Pressure2 8.964442 1 2.994068
## Hyd.Pressure3 12.433205 1 3.526075
## Hyd.Pressure4 2.368986 1 1.539151
## Filler.Level 10.684198 1 3.268669
## Filler.Speed 4.897880 1 2.213116
## Temperature 1.455792 1 1.206562
## Usage.cont 1.684034 1 1.297703
## Carb.Flow 2.168168 1 1.472470
## Density 16.274585 1 4.034177
## MFR 4.048062 1 2.011980
## Balling 65.945084 1 8.120658
## Pressure.Vacuum 2.582057 1 1.606878
## Oxygen.Filler 1.494535 1 1.222512
## Bowl.Setpoint 11.085276 1 3.329456
## Pressure.Setpoint 2.281597 1 1.510495
## Air.Pressurer 1.195910 1 1.093577
## Alch.Rel 16.848866 1 4.104737
## Carb.Rel 5.209240 1 2.282376
## Balling.Lvl 50.180230 1 7.083801Balling vs. Balling.Lvl
In order to reduce the multicollinearity we are going to create a new
predictor by dividing Balling.Lvl with
Balling.
The division was arrived at after trying different operations and
division resulted in the best correlation to PH then
removing either individually.
For a guess at domain relevance, Balling.Lvl could be
the target best value for the syrup’s specific gravity and
Balling could be the actual, so dividing the two would
result in a value similar to Mnf.Flow which is a percentage
of the ideal manufacturing conveyor belt speed.
# Create new predictor PT
student_train_x2 <- student_train_x1 |> mutate(PT = Balling.Lvl/Balling) |> select(-c(Balling, Balling.Lvl))
student_eval_y2 <- student_eval_y1 |> mutate(PT = Balling.Lvl/Balling) |> select(-c(Balling, Balling.Lvl))Interaction Terms
We also considered creating a Carbon column by
multiplying Carb.Pressure and Carb.Volume and
dividing by Carb.Temp. This is the Ideal Gas Law and would
tell us how much carbon dioxide is being injected. However there are two
values for pressure, Carb.Pressure and
Carb.Pressure1 and we don’t have the units to combine them
according to the law. From inspection they look like they’ve already
been centered and scaled and so cannot be combined.
Step-wise Reduction
Since we’ve somewhat resolved our multicollinearity issue and have enough data we’re going to try a step-wise reduction model. This reduces our features to just the ones that have relevance for this model. Since we’re not relying on this to produce our final model but rather evaluating our features, step-wise reduction aligns with our goals and we don’t need to try regularization instead.
Note, our model has terrible performance with an R-Squared of ~0.41, but we’ve acquainted ourselves with the features and figured out how to resolve multicollinearity.
# Fit a step-wise linear regression model
model <- lm(PH ~ ., data = student_train_x2)
model <- stats::step(model, trace = 0)
summary(model)##
## Call:
## lm(formula = PH ~ Fill.Ounces + PC.Volume + Carb.Temp + PSC +
## PSC.Fill + PSC.CO2 + Mnf.Flow + Carb.Pressure1 + Fill.Pressure +
## Hyd.Pressure2 + Hyd.Pressure3 + Filler.Level + Temperature +
## Usage.cont + Carb.Flow + Density + MFR + Oxygen.Filler +
## Bowl.Setpoint + Pressure.Setpoint + Alch.Rel + Carb.Rel +
## BCB, data = student_train_x2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.0495 -0.4675 0.0790 0.5056 4.1328
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.32033 0.02654 -12.069 < 2e-16 ***
## Fill.Ounces -0.03776 0.01602 -2.357 0.018477 *
## PC.Volume -0.04383 0.01823 -2.405 0.016256 *
## Carb.Temp 0.02880 0.01532 1.880 0.060215 .
## PSC -0.02715 0.01628 -1.668 0.095464 .
## PSC.Fill -0.02523 0.01600 -1.576 0.115077
## PSC.CO2 -0.03140 0.01578 -1.989 0.046772 *
## Mnf.Flow -0.48772 0.03167 -15.399 < 2e-16 ***
## Carb.Pressure1 0.19540 0.01883 10.377 < 2e-16 ***
## Fill.Pressure 0.03893 0.02228 1.748 0.080641 .
## Hyd.Pressure2 -0.11549 0.04318 -2.675 0.007525 **
## Hyd.Pressure3 0.34060 0.05059 6.733 2.05e-11 ***
## Filler.Level -0.09331 0.04930 -1.893 0.058505 .
## Temperature -0.11328 0.01751 -6.471 1.16e-10 ***
## Usage.cont -0.11910 0.01927 -6.181 7.40e-10 ***
## Carb.Flow 0.07222 0.02024 3.568 0.000366 ***
## Density -0.14542 0.04050 -3.590 0.000336 ***
## MFR -0.03824 0.01421 -2.691 0.007173 **
## Oxygen.Filler -0.06682 0.01844 -3.623 0.000297 ***
## Bowl.Setpoint 0.30697 0.04956 6.194 6.82e-10 ***
## Pressure.Setpoint -0.11768 0.02276 -5.170 2.52e-07 ***
## Alch.Rel 0.38026 0.04093 9.291 < 2e-16 ***
## Carb.Rel 0.05187 0.03074 1.687 0.091682 .
## BCB 0.65887 0.04477 14.716 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7724 on 2547 degrees of freedom
## Multiple R-squared: 0.4083, Adjusted R-squared: 0.4029
## F-statistic: 76.4 on 23 and 2547 DF, p-value: < 2.2e-16Initial MARS Model
Here we produce a Multivariate Adaptive Regression Splines (MARS) model to capture some of these non-linear interactions in our data.
Feature Selection
We’re using the features selected in the step-wise linear regression model.
# Features used in the previous step-wise linear regression model
student_train_x3 <- student_train_x2 |>
select(PH, Fill.Ounces, PC.Volume, Carb.Temp, PSC,
PSC.Fill, PSC.CO2, Mnf.Flow, Carb.Pressure1, Fill.Pressure,
Hyd.Pressure2, Hyd.Pressure3, Filler.Level, Temperature,
Usage.cont, Carb.Flow, Density, MFR, Oxygen.Filler,
Bowl.Setpoint, Pressure.Setpoint, Alch.Rel, Carb.Rel, BCB)
student_eval_y3 <- student_eval_y2 |>
select(PH, Fill.Ounces, PC.Volume, Carb.Temp, PSC,
PSC.Fill, PSC.CO2, Mnf.Flow, Carb.Pressure1, Fill.Pressure,
Hyd.Pressure2, Hyd.Pressure3, Filler.Level, Temperature,
Usage.cont, Carb.Flow, Density, MFR, Oxygen.Filler,
Bowl.Setpoint, Pressure.Setpoint, Alch.Rel, Carb.Rel, BCB)1st Order MARS Model
Here we run the initial MARS Model with only first order relationships.
# Basic MARS Model
y = student_train_x3$PH
x = student_train_x3 |> select(-PH)
marsFit <- earth(x, y)
summary(marsFit)## Call: earth(x=x, y=y)
##
## coefficients
## (Intercept) 5.3418732
## BCB 0.5096711
## h(-0.203956-Mnf.Flow) 0.9800807
## h(Carb.Pressure1- -1.55736) 0.1959255
## h(1.53618-Hyd.Pressure3) -0.2270999
## h(Hyd.Pressure3-1.53618) 3.7237258
## h(Temperature- -0.699707) -0.7786390
## h(1.3252-Temperature) -0.4898665
## h(Temperature-1.3252) 0.8280311
## h(Usage.cont-0.365031) -0.5993647
## h(Usage.cont-1.11723) 3.4859750
## h(-2.23001-Carb.Flow) 12.3573997
## h(Carb.Flow- -2.23001) 0.0991432
## h(-1.78438-Density) 2.7642866
## h(Density- -1.78438) 0.5163772
## h(Density- -1.09568) -0.9627448
## h(-3.02915-MFR) 0.1377848
## h(Oxygen.Filler- -0.738421) -1.4493047
## h(3.58372-Oxygen.Filler) -1.3142975
## h(Oxygen.Filler-3.58372) 1.5844172
## h(Bowl.Setpoint- -1.91638) -1.3437708
## h(-1.26292-Bowl.Setpoint) -0.6962323
## h(Bowl.Setpoint- -1.26292) 1.7803895
## h(Bowl.Setpoint-0.0440049) -0.4550215
## h(1.16947-Pressure.Setpoint) 0.0674138
## h(Alch.Rel- -0.509457) 2.5734980
## h(Alch.Rel-0.361355) -5.2860253
## h(0.519685-Alch.Rel) 0.8855864
## h(Alch.Rel-0.519685) 3.8218088
## h(Alch.Rel-1.46966) -1.5882955
## h(-0.441138-Carb.Rel) -0.1615024
## h(Carb.Rel- -0.441138) -0.0935618
##
## Selected 32 of 37 terms, and 14 of 23 predictors
## Termination condition: Reached nk 47
## Importance: Mnf.Flow, Alch.Rel, BCB, Usage.cont, Carb.Pressure1, ...
## Number of terms at each degree of interaction: 1 31 (additive model)
## GCV 0.5514965 RSS 1349.26 GRSq 0.4483104 RSq 0.4746078Show Plots
And here we show the partial dependence plots of the first order MARS model.
plotmo(marsFit)## plotmo grid: Fill.Ounces PC.Volume Carb.Temp PSC PSC.Fill
## -0.01623723 -0.09531893 -0.0730481 -0.1334244 -0.1304864
## PSC.CO2 Mnf.Flow Carb.Pressure1 Fill.Pressure Hyd.Pressure2 Hyd.Pressure3
## -0.3813757 0.3367148 0.1293786 -0.4790381 0.4661799 0.4345061
## Filler.Level Temperature Usage.cont Carb.Flow Density MFR
## 0.58271 -0.2657983 0.2710059 0.5212328 -0.512943 0.2456178
## Oxygen.Filler Bowl.Setpoint Pressure.Setpoint Alch.Rel Carb.Rel BCB
## -0.2881982 0.697465 -0.792231 -0.6677866 -0.2857598 0
2nd Order MARS Model
Here we run the initial MARS Model with second order relationships. This increased the R-squared from 0.475 to .489.
y = student_train_x3$PH
x = student_train_x3 |> select(-PH)
y_eval = student_eval_y3$PH
x_eval = student_eval_y3 |> select(-PH)
marsFit2 <- earth(x, y, degree = 2)
summary(marsFit2)## Call: earth(x=x, y=y, degree=2)
##
## coefficients
## (Intercept) -0.2741453
## BCB 0.2559212
## h(-0.203956-Mnf.Flow) 0.4940847
## h(0.50962-Hyd.Pressure3) -0.4641113
## h(Hyd.Pressure3-0.50962) 0.5696218
## h(1.3252-Temperature) 0.3307546
## h(-0.785587-Oxygen.Filler) * BCB -2.7498870
## h(Oxygen.Filler- -0.785587) * BCB -0.1458478
## h(-1.26292-Bowl.Setpoint) * BCB 0.4057510
## h(Bowl.Setpoint- -1.26292) * BCB 0.3179062
## h(0.800226-PSC) * h(-0.203956-Mnf.Flow) 0.1075006
## h(-0.203956-Mnf.Flow) * h(Carb.Pressure1- -1.51519) 0.2101897
## h(-0.203956-Mnf.Flow) * h(Hyd.Pressure3-0.0213793) -0.7230425
## h(-0.203956-Mnf.Flow) * h(Usage.cont-0.754562) 2.1408975
## h(-0.203956-Mnf.Flow) * h(-0.724848-Density) 1.2703030
## h(-0.203956-Mnf.Flow) * h(Density-1.50016) -4.5082899
## h(Mnf.Flow- -0.203956) * h(Bowl.Setpoint-0.0440049) -0.2170369
## h(Mnf.Flow- -0.203956) * h(0.0440049-Bowl.Setpoint) -0.2066971
## h(Mnf.Flow- -0.203956) * h(Alch.Rel-0.519685) 0.4830633
## h(Mnf.Flow- -0.203956) * h(0.519685-Alch.Rel) -0.3692237
## h(-0.203956-Mnf.Flow) * h(1.26802-Carb.Rel) -0.2740200
## h(Mnf.Flow- -0.203956) * h(Carb.Rel- -1.21803) -0.1579294
## h(Mnf.Flow- -0.203956) * h(-1.21803-Carb.Rel) -0.6507008
## h(0.50962-Hyd.Pressure3) * h(Alch.Rel- -0.509457) 0.1808274
## h(0.50962-Hyd.Pressure3) * h(-0.509457-Alch.Rel) 0.5649332
## h(1.3252-Temperature) * h(Usage.cont-0.358315) -0.5300740
##
## Selected 26 of 35 terms, and 12 of 23 predictors
## Termination condition: RSq changed by less than 0.001 at 35 terms
## Importance: Mnf.Flow, Temperature, Usage.cont, Bowl.Setpoint, Alch.Rel, ...
## Number of terms at each degree of interaction: 1 5 20
## GCV 0.5362348 RSS 1311.399 GRSq 0.4635775 RSq 0.4893508Show Plots
plotmo(marsFit2)## plotmo grid: Fill.Ounces PC.Volume Carb.Temp PSC PSC.Fill
## -0.01623723 -0.09531893 -0.0730481 -0.1334244 -0.1304864
## PSC.CO2 Mnf.Flow Carb.Pressure1 Fill.Pressure Hyd.Pressure2 Hyd.Pressure3
## -0.3813757 0.3367148 0.1293786 -0.4790381 0.4661799 0.4345061
## Filler.Level Temperature Usage.cont Carb.Flow Density MFR
## 0.58271 -0.2657983 0.2710059 0.5212328 -0.512943 0.2456178
## Oxygen.Filler Bowl.Setpoint Pressure.Setpoint Alch.Rel Carb.Rel BCB
## -0.2881982 0.697465 -0.792231 -0.6677866 -0.2857598 0
Predictions
When we originally completed the above model we got an R-Squared of 0.89 however it was a fluke and the R-Squared on the test data was a lowly 0.342, substantiating significant overfitting of our model.
We could duplicate that result by replacing the imputation of missing values with zeros, not removing the one feature with near-zero variance, and doing the balling vs. balling.lvl feature combination after the step-wise reduction.
Instead, we’re using our current model and will take a different tack starting in the next section.
Here we make predictions on our test data using the current model and then compare that to our test data and get an R-Squared on the test data of 0.379, compared to the initial overfit model of 0.342.
# Make Predictions
predicted <- stats::predict(marsFit2, student_eval_y3)
# Actual values
actual <- student_eval_y3$PH
# R-squared
rss <- sum((actual - predicted)^2) # Residual sum of squares
tss <- sum((actual - mean(actual))^2) # Total sum of squares
rsquared <- 1 - (rss / tss)
# Display R-squared
rsquared## [1] 0.3791503Non-Linear Models
Here we’re doing a range of models to get a sense of what works.
Data
We’re starting over on the data after the imputation step and removing the one near-zero variance predictor.
Split up data
Here we split up the data so that PH is our trainy and
the remaining data is our trainx. Similarly for our test
data we have testx and testy.
# Split up available data into train and test versions of x and y
trainx <- student_train_x |> select(-PH)
trainy <- student_train_x$PH
testx <- student_eval_y |> select(-PH)
testy <- student_eval_y$PHOne-hot encoding
We have one character variable Brand.Code that is
categorial with no inherent order:
Blank - 120 A - 293 B - 1239 C - 304 D - 615
Here we set the Blanks to brand code M for miscellaneous
and use one-hot encoding to split them up into five variables with a
1 in the column corresponding to which Brand Code that row
is.
# Assign blanks as "M"
trainx$Brand.Code[trainx$Brand.Code == ""] <- "M"
testx$Brand.Code[testx$Brand.Code == ""] <- "M"
# Convert Brand.Code to a factor
trainx$Brand.Code <- as.factor(trainx$Brand.Code)
testx$Brand.Code <- as.factor(testx$Brand.Code)
# One-hot encoding
dummy_model <- dummyVars(~ ., data = trainx)
trainx <- predict(dummy_model, trainx)
testx <- predict(dummy_model, testx)Modeling
Here we look at the kNN, SVM and MARS models.
kNN
k-Nearest Neighbors results in a weak model with an R-Squared of 0.481 when run with 13 as the number of nearest neighbors.
# Train the kNN model
knnModel <- train(x = trainx,
y = trainy,
method = "knn",
preProc = c("center", "scale"),
tuneLength = 10)
knnModel## k-Nearest Neighbors
##
## 2571 samples
## 35 predictor
##
## Pre-processing: centered (35), scaled (35)
## Resampling: Bootstrapped (25 reps)
## Summary of sample sizes: 2571, 2571, 2571, 2571, 2571, 2571, ...
## Resampling results across tuning parameters:
##
## k RMSE Rsquared MAE
## 5 0.7560969 0.4560094 0.5510834
## 7 0.7374740 0.4733069 0.5445070
## 9 0.7313059 0.4784016 0.5440887
## 11 0.7303516 0.4785364 0.5461256
## 13 0.7314137 0.4761448 0.5488022
## 15 0.7331548 0.4734208 0.5532781
## 17 0.7354116 0.4701899 0.5562445
## 19 0.7371643 0.4678866 0.5582884
## 21 0.7384954 0.4661710 0.5606518
## 23 0.7410592 0.4625239 0.5636949
##
## RMSE was used to select the optimal model using the smallest value.
## The final value used for the model was k = 11.Support Vector Machines
Support Vector Machines results in a weak model with an Rsquared of 0.586 when C = 8. The C being high like this means that misclassifications are penalized more heavily resulting in a more complex decision boundary.
# Train the SVM model
svmModel <- train(x = trainx,
y = trainy,
method = "svmRadial",
preProc = c("center", "scale"),
tuneLength = 14,
trControl = trainControl(method = "cv"))
svmModel## Support Vector Machines with Radial Basis Function Kernel
##
## 2571 samples
## 35 predictor
##
## Pre-processing: centered (35), scaled (35)
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 2315, 2314, 2315, 2313, 2313, 2313, ...
## Resampling results across tuning parameters:
##
## C RMSE Rsquared MAE
## 0.25 0.7189162 0.4904530 0.5347819
## 0.50 0.6988418 0.5159976 0.5153643
## 1.00 0.6823699 0.5370247 0.5011465
## 2.00 0.6679922 0.5549208 0.4897405
## 4.00 0.6567913 0.5698252 0.4817393
## 8.00 0.6531433 0.5764714 0.4807815
## 16.00 0.6595321 0.5727713 0.4864535
## 32.00 0.6782675 0.5574045 0.4986487
## 64.00 0.7120904 0.5291023 0.5227649
## 128.00 0.7498905 0.4993676 0.5487053
## 256.00 0.7942092 0.4685824 0.5796914
## 512.00 0.8347642 0.4434038 0.6126037
## 1024.00 0.8585730 0.4273354 0.6295151
## 2048.00 0.8609288 0.4256095 0.6317030
##
## Tuning parameter 'sigma' was held constant at a value of 0.02041351
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were sigma = 0.02041351 and C = 8.MARS
Multivariate Adaptive Regression Splines results in a weak model with an Rsquared of 0.517 with nprune = 33 and degree = 6. This means there are 33 basis functions with six-way interactions. This may be suitable to highly nonlinear and interactive data but risks overfitting.
# Train the MARS model
marsGrid <- expand.grid(.degree = 6, .nprune = 30:40)
set.seed(175175327)
marsModel <- train(x = trainx,
y = trainy,
method = "earth",
tuneGrid = marsGrid,
trControl = trainControl(method = "cv"))
marsModel## Multivariate Adaptive Regression Spline
##
## 2571 samples
## 35 predictor
##
## No pre-processing
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 2314, 2313, 2313, 2315, 2314, 2314, ...
## Resampling results across tuning parameters:
##
## nprune RMSE Rsquared MAE
## 30 0.7056510 0.5085210 0.5221312
## 31 0.7048986 0.5098481 0.5215941
## 32 0.7047143 0.5099580 0.5213002
## 33 0.7000219 0.5158733 0.5199846
## 34 0.6983310 0.5180715 0.5194377
## 35 0.6960146 0.5210178 0.5176798
## 36 0.6983852 0.5183097 0.5189310
## 37 0.6979122 0.5190971 0.5184589
## 38 0.6971574 0.5203998 0.5175004
## 39 0.6958715 0.5222221 0.5162749
## 40 0.6959166 0.5228376 0.5153354
##
## Tuning parameter 'degree' was held constant at a value of 6
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were nprune = 39 and degree = 6.XGBoost
eXtreme Gradient Boosting also produced similarly weak results with an R-Squared of 0.455.
library(caret)
# Train XGBoost model using caret
xgb_model <- train(
x = trainx,
y = trainy,
method = "xgbTree",
tuneLength = 1, # Automatic hyperparameter tuning
trControl = trainControl(method = "cv", number = 10) # Cross-validation
)
# Summary of the model
print(xgb_model)## eXtreme Gradient Boosting
##
## 2571 samples
## 35 predictor
##
## No pre-processing
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 2314, 2314, 2315, 2314, 2313, 2315, ...
## Resampling results across tuning parameters:
##
## eta colsample_bytree RMSE Rsquared MAE
## 0.3 0.6 0.7574076 0.4312850 0.5953737
## 0.3 0.8 0.7512717 0.4397941 0.5889623
## 0.4 0.6 0.7421128 0.4533415 0.5791337
## 0.4 0.8 0.7486802 0.4414322 0.5831434
##
## Tuning parameter 'nrounds' was held constant at a value of 50
## Tuning
## parameter 'min_child_weight' was held constant at a value of 1
##
## Tuning parameter 'subsample' was held constant at a value of 0.5
## RMSE was used to select the optimal model using the smallest value.
## The final values used for the model were nrounds = 50, max_depth = 1, eta
## = 0.4, gamma = 0, colsample_bytree = 0.6, min_child_weight = 1 and subsample
## = 0.5.```