setwd("C:/Users/obunadic8159/OneDrive - ARCADIS/Desktop/Data_Analysis_11DCE_14DOX/Recent_New_Data/WW")
#Load necessary libraries
#install.packages("car")
#install.packages("corrplot")
#install.packages("MASS")
#install.packages("lmtest")
library(MASS)
library(dplyr)
library(ggplot2)
library(caret)
library(car)
library(corrplot)
library(e1071)
library(forecast)
library(lubridate)
library(lmtest)Linear Regression and Time Series Model_WW for 1,1-Dichloroethene
Importing the Data from our directory
# Load the dataset
data <- read.csv("C:/Users/obunadic8159/OneDrive - ARCADIS/Desktop/Data_Analysis_11DCE_14DOX/Recent_New_Data/WW/11DCE.csv")Visualizing the Data Type and Structure
##################Checking the Data Type and Structure ###############
head(data) SiteName Date Quarter X11DCE
1 WW01 7/16/2015 Q3 0.004790
2 WW01 8/24/2015 Q3 0.001600
3 WW01 8/24/2015 Q3 0.002250
4 WW01 9/28/2015 Q3 0.000434
5 WW01 11/3/2015 Q4 0.001280
6 WW01 12/8/2015 Q4 0.001460str(data)'data.frame': 2680 obs. of 4 variables:
$ SiteName: chr "WW01" "WW01" "WW01" "WW01" ...
$ Date : chr "7/16/2015" "8/24/2015" "8/24/2015" "9/28/2015" ...
$ Quarter : chr "Q3" "Q3" "Q3" "Q3" ...
$ X11DCE : num 0.00479 0.0016 0.00225 0.000434 0.00128 0.00146 0.00189 0.00131 0.000216 0.00243 ...Convert Date to Date format and factorizing Site Name
# Convert Date to Date format and factorize SiteName
data$Date <- as.Date(data$Date, format="%m/%d/%Y")
data$SiteName <- as.factor(data$SiteName)Checking for Missing data using For-Loop
#### Handling Missing Data
vnames <- colnames(data)
n <- nrow(data)
out <- NULL
for (j in 1:ncol(data)){
vname <- colnames(data)[j]
x <- as.vector(data[,j])
n1 <- sum(is.na(x), na.rm=TRUE) # NA
n2 <- sum(x=="NA", na.rm=TRUE) # "NA"
n3 <- sum(x==" ", na.rm=TRUE) # missing
nmiss <- n1 + n2 + n3
nmiss <- sum(is.na(x))
ncomplete <- n-nmiss
out <- rbind(out, c(col.num=j, v.name=vname, mode=mode(x),
n.level=length(unique(x)),
ncom=ncomplete, nmiss= nmiss, miss.prop=nmiss/n))
}
out <- as.data.frame(out)
row.names(out) <- NULL
out col.num v.name mode n.level ncom nmiss miss.prop
1 1 SiteName character 131 2680 0 0
2 2 Date numeric 531 2680 0 0
3 3 Quarter character 4 2680 0 0
4 4 X11DCE numeric 834 2680 0 0for (j in 1:NCOL(data)){
print(head(colnames(data)[j]))
print(head(table(data[,j], useNA="ifany")))
}[1] "SiteName"
WW01 WW02 WW03 WW04 WW06 WW07
28 23 52 50 33 20
[1] "Date"
2015-07-16 2015-07-17 2015-08-18 2015-08-20 2015-08-21 2015-08-22
6 3 1 2 12 8
[1] "Quarter"
Q1 Q2 Q3 Q4
669 614 726 671
[1] "X11DCE"
0.000188 0.000192 0.000194 0.000197 2e-04 0.000201
36 1395 1 1 9 3 Checking for Outliers using Boxplot
# Check for outliers using boxplots for DCE
ggplot(data, aes(x = SiteName, y = X11DCE)) + geom_boxplot() + ggtitle("Boxplot for DCE by SiteName")
Calculating and Filtering Outliers in 11DCE
# Calculate IQR and detect outliers for X11DCE
Q1_DCE <- quantile(data$X11DCE, 0.25)
Q3_DCE <- quantile(data$X11DCE, 0.75)
IQR_DCE <- Q3_DCE - Q1_DCE
outliers_DCE <- data %>% filter(X11DCE < (Q1_DCE - 1.5 * IQR_DCE) | X11DCE > (Q3_DCE + 1.5 * IQR_DCE))There are 491 observations of 1,1Dichloroethene (DCE) with outliers. Therefore, it is important to remove them before building the linear regression model and time series forecast.
# Outliers in 11DCE
#print(outliers_DCE)
#print("Outliers in X11DCE:")
print(head(outliers_DCE)) SiteName Date Quarter X11DCE
1 WW03 2015-08-18 Q3 0.1570
2 WW03 2015-08-21 Q3 0.1570
3 WW03 2015-08-21 Q3 0.1400
4 WW03 2015-08-21 Q3 0.1700
5 WW03 2015-09-04 Q3 0.0573
6 WW03 2015-09-19 Q3 0.1500Checking for skewness in 1,1-Dichloroethene
[1] "Skewness for X11DCE: 3.16218481398556"Interpretation of Skewness Values:
Skewness for 11DCE (3.16):This is also a high positive skewness value. The distribution of DCE is positively skewed, with most data points clustered towards the lower end and a long tail to the right. This indicates the presence of some high values. The high skewness values suggest that the data for 11DCE are not normally distributed and are influenced by a few very large values. When working with these variables in analyses, it might be necessary to consider transformations (such as log or square root transformations) to reduce skewness and achieve a more normal distribution, which can be beneficial for certain statistical analyses and models.~
# Visualize distributions
par(mfrow=c(1, 2))
hist(data$X11DCE, main="11DCE", xlab="11DCE", col = 'red', breaks=20)
Transforming data due to skewness by using log transformation.
# Transform data if skewness is high (optional, example using log transformation)
if(abs(skewness_DCE) > 1){data$X11DCE <- log1p(data$X11DCE)}Removing the Outliers for 11DCE.
# Remove outliers (optional, depending on analysis)
data <- data %>% filter(X11DCE >= (Q1_DCE - 1.5 * IQR_DCE) & X11DCE <= (Q3_DCE + 1.5 * IQR_DCE))
##################Checking the Data Type and Structure ###############
str(data)'data.frame': 2194 obs. of 4 variables:
$ SiteName: Factor w/ 131 levels "WW01","WW02",..: 1 1 1 1 1 1 1 1 1 1 ...
$ Date : Date, format: "2015-07-16" "2015-08-24" ...
$ Quarter : chr "Q3" "Q3" "Q3" "Q3" ...
$ X11DCE : num 0.004779 0.001599 0.002247 0.000434 0.001279 ...head(data) SiteName Date Quarter X11DCE
1 WW01 2015-07-16 Q3 0.0047785645
2 WW01 2015-08-24 Q3 0.0015987214
3 WW01 2015-08-24 Q3 0.0022474725
4 WW01 2015-09-28 Q3 0.0004339058
5 WW01 2015-11-03 Q4 0.0012791815
6 WW01 2015-12-08 Q4 0.0014589352Convert Date to numeric for VIF calculation
data$Date_numeric <- as.numeric(data$Date)Check for multicollinearity using VIF
# Check for multicollinearity using VIF
vif_data <- lm(X11DCE ~ Date_numeric + SiteName, data = data)
vif(vif_data) GVIF Df GVIF^(1/(2*Df))
Date_numeric 1.101557 1 1.049551
SiteName 1.101557 129 1.000375In this case, both Date_numeric and SiteName have GVIF values close to 1, indicating that there is little to no multicollinearity between these predictors in the model. This suggests that the model is not adversely affected by multicollinearity, and the estimates of the regression coefficients should be reliable.
Check correlation matrix for predictor variable (11DCE) and Date
1: Perfect positive correlation (as one variable increases, the other variable also increases).
-1: Perfect negative correlation (as one variable increases, the other variable decreases).
0: No linear correlation (the variables do not have a linear relationship).
# Check correlation matrix for predictor and dependent variables
cor_matrix <- cor(data %>% select(X11DCE, Date_numeric))
print(cor_matrix) X11DCE Date_numeric
X11DCE 1.00000000 0.05653293
Date_numeric 0.05653293 1.00000000Implications of Correlation Matrix on the Model:
The correlation results suggest that there is almost no linear relationship between X11DCE and Date_numeric. This low correlation indicates that Date_numeric is not a strong predictor of X11DCE based on linear association alone.
# Visualize the correlation matrix
corrplot(cor_matrix, method = "circle")
Removing the 5th column (Date Numeric) for Linear Regression Model
# Convert Date back to as.Date data type and removing the 5th column
data$Date <- as.Date(data$Date, format="%m/%d/%Y")
head(data) SiteName Date Quarter X11DCE Date_numeric
1 WW01 2015-07-16 Q3 0.0047785645 16632
2 WW01 2015-08-24 Q3 0.0015987214 16671
3 WW01 2015-08-24 Q3 0.0022474725 16671
4 WW01 2015-09-28 Q3 0.0004339058 16706
5 WW01 2015-11-03 Q4 0.0012791815 16742
6 WW01 2015-12-08 Q4 0.0014589352 16777colnames(data)[1] "SiteName" "Date" "Quarter" "X11DCE" "Date_numeric"data= data[, -(5)]
head(data) SiteName Date Quarter X11DCE
1 WW01 2015-07-16 Q3 0.0047785645
2 WW01 2015-08-24 Q3 0.0015987214
3 WW01 2015-08-24 Q3 0.0022474725
4 WW01 2015-09-28 Q3 0.0004339058
5 WW01 2015-11-03 Q4 0.0012791815
6 WW01 2015-12-08 Q4 0.0014589352Splitting the data into training and testing sets for the Linear Regression Modelling of 11DCE
# Split the data into training and testing sets for X11DCE
set.seed(123)
trainIndex <- createDataPartition(data$X11DCE, p = .8, list = FALSE, times = 1)
trainData <- data[trainIndex,]
testData <- data[-trainIndex,]This line uses the createDataPartition function from the caret package to create an index for splitting the data.X11DCE: The target variable, which is 1,1 Dicholorethene, is used to ensure that the split maintains the same distribution of this variable in both the training and test sets.p = .8: This specifies that 80% of the data should be used for training. The remaining 20% will be used for testing.list = FALSE: By setting this to FALSE, the function returns the indices as a vector instead of a list.times = 1: This specifies that only one partition should be created.
Using set.seed(123) ensures that every time you run this code, the training and test splits will be the same, allowing for consistent and reproducible results.
Factoring SiteName in testData to Match with the trainData (Categorical Variable)
# Ensure factor levels in test set match training set
testData$SiteName <- factor(testData$SiteName, levels = levels(trainData$SiteName))The above code ensures that the SiteName factor levels in the testData dataset match those in the trainData dataset. This step is crucial when you want to make predictions on the test data using a model trained on the training data, especially when dealing with categorical variables.
Linear Regression Model and StepWise Backward Elimination Model Selection
# Model for X11DCE using stepwise backward elimination
full_model_DCE <- lm(X11DCE ~ Date + SiteName, data = trainData)
step_model_DCE <- step(full_model_DCE, direction = "backward", trace = 0)Purpose: The goal of step-wise model selection is to improve the model by removing predictors that do not contribute significantly to the prediction of the response variable. This can lead to a more parsimonious model that is easier to interpret and may perform better on new data.
Backward Elimination: In this specific procedure, predictors are removed one by one based on their statistical significance, starting with the least significant predictor. The process continues until only predictors that contribute meaningfully to the model remain.
Summary and Interpretation of the Linear Regression Model
# Summary of the final model
summary(step_model_DCE)
Call:
lm(formula = X11DCE ~ Date + SiteName, data = trainData)
Residuals:
Min 1Q Median 3Q Max
-0.0083848 -0.0001865 -0.0000063 0.0000994 0.0120007
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.926e-03 1.450e-03 -1.328 0.184318
Date 1.686e-07 7.846e-08 2.148 0.031837 *
SiteNameWW02 -8.046e-04 7.039e-04 -1.143 0.253184
SiteNameWW03 1.782e-04 7.559e-04 0.236 0.813617
SiteNameWW04 3.345e-03 9.516e-04 3.515 0.000452 ***
SiteNameWW06 6.286e-03 6.445e-04 9.754 < 2e-16 ***
SiteNameWW07 -8.383e-04 7.148e-04 -1.173 0.241034
SiteNameWW08 -6.857e-04 2.221e-03 -0.309 0.757569
SiteNameWW10 -6.938e-04 1.188e-03 -0.584 0.559303
SiteNameWW100 -8.317e-04 7.401e-04 -1.124 0.261233
SiteNameWW101 -8.076e-04 6.392e-04 -1.263 0.206614
SiteNameWW102 -8.222e-04 6.941e-04 -1.185 0.236363
SiteNameWW103 -8.381e-04 8.133e-04 -1.030 0.302961
SiteNameWW104 -8.153e-04 6.852e-04 -1.190 0.234290
SiteNameWW105 -7.972e-04 7.039e-04 -1.133 0.257590
SiteNameWW106 9.965e-04 6.769e-04 1.472 0.141187
SiteNameWW107 -7.306e-04 6.941e-04 -1.053 0.292672
SiteNameWW11 -7.594e-04 8.394e-04 -0.905 0.365748
SiteNameWW116 5.035e-03 6.853e-04 7.347 3.19e-13 ***
SiteNameWW117 -8.407e-04 6.561e-04 -1.281 0.200207
SiteNameWW12 2.549e-03 1.608e-03 1.585 0.113106
SiteNameWW13 -7.935e-04 8.392e-04 -0.946 0.344522
SiteNameWW14 1.125e-03 1.009e-03 1.115 0.265012
SiteNameWW156 -8.635e-04 6.564e-04 -1.316 0.188510
SiteNameWW156-2 -8.779e-04 8.701e-04 -1.009 0.313130
SiteNameWW16 4.814e-03 6.694e-04 7.191 9.73e-13 ***
SiteNameWW17 1.060e-03 6.624e-04 1.601 0.109641
SiteNameWW178 -8.522e-04 6.771e-04 -1.259 0.208383
SiteNameWW18 -2.700e-04 7.148e-04 -0.378 0.705664
SiteNameWW180 -7.990e-04 6.941e-04 -1.151 0.249865
SiteNameWW181 -8.400e-04 6.345e-04 -1.324 0.185695
SiteNameWW183 -7.835e-04 7.267e-04 -1.078 0.281109
SiteNameWW184 -6.958e-04 2.221e-03 -0.313 0.754095
SiteNameWW185 -6.960e-04 2.221e-03 -0.313 0.754037
SiteNameWW186 -6.960e-04 2.221e-03 -0.313 0.754037
SiteNameWW188 -8.123e-04 6.851e-04 -1.186 0.235948
SiteNameWW189 -8.115e-04 6.851e-04 -1.184 0.236445
SiteNameWW19 5.620e-03 6.852e-04 8.202 4.75e-16 ***
SiteNameWW195 -6.961e-04 2.221e-03 -0.313 0.753979
SiteNameWW196 -6.961e-04 2.221e-03 -0.313 0.753979
SiteNameWW197 -7.522e-04 7.272e-04 -1.034 0.301120
SiteNameWW200 -6.976e-04 2.221e-03 -0.314 0.753458
SiteNameWW201 -8.507e-04 7.551e-04 -1.127 0.260091
SiteNameWW21 -8.267e-04 7.040e-04 -1.174 0.240405
SiteNameWW210 -7.921e-04 1.009e-03 -0.785 0.432331
SiteNameWW210/211 -8.704e-04 8.137e-04 -1.070 0.284893
SiteNameWW211 -7.539e-04 1.187e-03 -0.635 0.525464
SiteNameWW215 -8.098e-04 1.341e-03 -0.604 0.546150
SiteNameWW217 -8.623e-04 9.517e-04 -0.906 0.365029
SiteNameWW22 -8.084e-04 7.147e-04 -1.131 0.258198
SiteNameWW221 -8.435e-04 9.064e-04 -0.931 0.352204
SiteNameWW222 -8.273e-04 1.083e-03 -0.764 0.445159
SiteNameWW223 -8.358e-04 6.942e-04 -1.204 0.228722
SiteNameWW23 -8.279e-04 7.040e-04 -1.176 0.239710
SiteNameWW230 -6.978e-04 2.221e-03 -0.314 0.753400
SiteNameWW24 -8.070e-04 7.400e-04 -1.091 0.275652
SiteNameWW25 -9.435e-04 7.424e-04 -1.271 0.203952
SiteNameWW26 -8.193e-04 7.039e-04 -1.164 0.244632
SiteNameWW28 -8.006e-04 6.941e-04 -1.153 0.248920
SiteNameWW29 -8.553e-04 7.150e-04 -1.196 0.231776
SiteNameWW30 -8.541e-04 7.721e-04 -1.106 0.268797
SiteNameWW31 -8.701e-04 8.137e-04 -1.069 0.285094
SiteNameWW32 -7.891e-04 6.695e-04 -1.179 0.238706
SiteNameWW34 -8.643e-04 7.553e-04 -1.144 0.252693
SiteNameWW35 -8.268e-04 1.607e-03 -0.515 0.606915
SiteNameWW36 4.937e-03 6.561e-04 7.524 8.74e-14 ***
SiteNameWW36X -9.334e-06 9.084e-04 -0.010 0.991803
SiteNameWW37 3.021e-03 6.305e-04 4.792 1.80e-06 ***
SiteNameWW38 2.683e-03 1.009e-03 2.659 0.007913 **
SiteNameWW39-1 -6.922e-04 2.221e-03 -0.312 0.755310
SiteNameWW41 -6.924e-04 2.221e-03 -0.312 0.755252
SiteNameWW42 4.533e-03 6.561e-04 6.909 6.99e-12 ***
SiteNameWW43 -2.181e-04 9.515e-04 -0.229 0.818731
SiteNameWW44 7.635e-03 8.701e-04 8.775 < 2e-16 ***
SiteNameWW45 2.699e-03 6.501e-04 4.151 3.47e-05 ***
SiteNameWW46 1.615e-03 8.394e-04 1.924 0.054514 .
SiteNameWW47 4.586e-04 9.075e-04 0.505 0.613424
SiteNameWW48 -7.962e-04 1.187e-03 -0.671 0.502396
SiteNameWW49 -4.503e-04 6.694e-04 -0.673 0.501276
SiteNameWW50 6.577e-03 7.039e-04 9.344 < 2e-16 ***
SiteNameWW51 3.547e-03 9.070e-04 3.911 9.59e-05 ***
SiteNameWW56 2.365e-04 7.720e-04 0.306 0.759396
SiteNameWW57 -6.922e-04 2.221e-03 -0.312 0.755310
SiteNameWW58 -6.922e-04 2.221e-03 -0.312 0.755310
SiteNameWW59 -8.176e-04 7.267e-04 -1.125 0.260718
SiteNameWW60 -6.922e-04 2.221e-03 -0.312 0.755310
SiteNameWW61 2.268e-04 1.607e-03 0.141 0.887785
SiteNameWW62 -8.143e-04 6.851e-04 -1.189 0.234801
SiteNameWW63 -8.258e-04 7.550e-04 -1.094 0.274223
SiteNameWW64 -8.164e-04 6.694e-04 -1.220 0.222792
SiteNameWW65 -6.924e-04 2.221e-03 -0.312 0.755252
SiteNameWW66 -7.960e-04 7.148e-04 -1.114 0.265624
SiteNameWW67 3.633e-03 6.625e-04 5.483 4.83e-08 ***
SiteNameWW68 -8.382e-04 7.551e-04 -1.110 0.267111
SiteNameWW69 -8.537e-04 6.502e-04 -1.313 0.189379
SiteNameWW70 -2.369e-04 6.624e-04 -0.358 0.720696
SiteNameWW71 -7.599e-04 8.137e-04 -0.934 0.350500
SiteNameWW74 -6.943e-04 2.221e-03 -0.313 0.754615
SiteNameWW76 -7.110e-04 6.945e-04 -1.024 0.306126
SiteNameWW77 -8.788e-04 6.398e-04 -1.374 0.169776
SiteNameWW77-2 -9.756e-04 8.168e-04 -1.195 0.232455
SiteNameWW78 1.128e-03 6.854e-04 1.646 0.099972 .
SiteNameWW78-2 -8.950e-05 6.695e-04 -0.134 0.893669
SiteNameWW79 -5.385e-04 6.852e-04 -0.786 0.432048
SiteNameWW80 1.950e-03 6.699e-04 2.912 0.003646 **
SiteNameWW81 -7.783e-06 8.728e-04 -0.009 0.992887
SiteNameWW82-1 4.630e-04 6.564e-04 0.705 0.480727
SiteNameWW82-2 7.073e-04 6.948e-04 1.018 0.308821
SiteNameWW83 6.662e-03 6.771e-04 9.840 < 2e-16 ***
SiteNameWW84 1.012e-03 6.771e-04 1.494 0.135421
SiteNameWW85 -3.937e-04 6.392e-04 -0.616 0.538007
SiteNameWW85-2 -8.870e-04 8.711e-04 -1.018 0.308707
SiteNameWW86 1.239e-03 6.444e-04 1.922 0.054766 .
SiteNameWW87 -6.709e-04 6.852e-04 -0.979 0.327690
SiteNameWW88 -8.309e-04 6.344e-04 -1.310 0.190464
SiteNameWW88-2 -9.170e-04 1.188e-03 -0.772 0.440155
SiteNameWW89 -8.223e-04 6.694e-04 -1.228 0.219500
SiteNameWW90 -7.445e-04 1.009e-03 -0.738 0.460541
SiteNameWW91 -8.322e-04 1.009e-03 -0.825 0.409481
SiteNameWW92 -8.287e-04 7.148e-04 -1.159 0.246449
SiteNameWW93 -5.551e-04 2.221e-03 -0.250 0.802652
SiteNameWW96 -4.315e-04 6.694e-04 -0.645 0.519241
SiteNameWW97 -7.665e-04 7.267e-04 -1.055 0.291676
SiteNameWW98 -8.137e-04 6.343e-04 -1.283 0.199761
SiteNameWW99 -8.331e-04 7.550e-04 -1.103 0.270029
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.002167 on 1632 degrees of freedom
Multiple R-squared: 0.5035, Adjusted R-squared: 0.4657
F-statistic: 13.34 on 124 and 1632 DF, p-value: < 2.2e-16Residuals: Min ( -0.0083349), 1Q(-0.0001863), Median(-0.0000063), 3Q(0.0000994) and Max(0.0119175) shows the distribution of the residuals (difference between observed and predicted values). The small values of the residuals indicate that the model’s prediction are close to the observed values.
Date: The coefficient for Date is significant at 5% level (p<0.05), suggesting that there is a relationship between Date and XIIDCE.
Significant SiteName Coefficients:
SiteNameWW04: 3.316e-03 (p = 0.000468)
SiteNameWW06: 6.250e-03 (p < 2e-16)
SiteNameWW116: 5.010e-03 (p = 2.99e-13)
SiteNameWW16: 4.788e-03 (p = 9.37e-13)
SiteNameWW19: 5.592e-03 (p = 4.40e-16)
SiteNameWW36: 4.905e-03 (p = 8.93e-14)
SiteNameWW37: 3.008e-03 (p = 1.73e-06)
SiteNameWW38: 2.662e-03 (p = 0.008027)
SiteNameWW42: 4.514e-03 (p = 6.37e-12)
SiteNameWW44: 7.586e-03 (p < 2e-16)
SiteNameWW45: 2.684e-03 (p = 3.44e-05)
SiteNameWW50: 6.536e-03 (p < 2e-16)
SiteNameWW51: 3.519e-03 (p = 9.85e-05)
SiteNameWW67: 3.615e-03 (p = 4.67e-08)
SiteNameWW83: 6.620e-03 (p < 2e-16)
SiteNameWW80: 1.938e-03 (p = 0.003651)
These significant coefficients indicate that the corresponding SiteName levels have a statistically significant effect on X11DCE.
Model Fit Statistics
Residual standard error: 0.002153 on 1632 degrees of freedom
Multiple R-squared (0.504): This indicates that approximately 50.4% of the variability in
X11DCEis explained by the model.Adjusted R-squared (0.4663): This adjusts the R-squared value for the number of predictors in the model, providing a more accurate measure of model fit.
F-statistic: 13.37 on 124 and 1632 DF (p-value: < 2.2e-16): The F-statistic tests the overall significance of the model. A very low p-value (< 2.2e-16) indicates that the model is statistically significant.
Identifying High Leverage Points:
Leverage Values: These values indicate how much influence each data point has on the fitted values of the model. High leverage points are those that can potentially have a large impact on the model.
Data Points with High Leverage Points:
# Print high leverage points
#print(high_leverage_points)
print(head(high_leverage_points))68 69 70 73 74 75
53 54 55 56 57 58 print(tail(high_leverage_points))2165 2167 2168 2169 2171 2194
1733 1734 1735 1736 1737 1757 Removing the Data with High Leverage Points for Improved Model:
# Remove high leverage points and refit the model if necessary
trainData_cleaned <- trainData[-high_leverage_points, ]Refitting the Model after removing high leverage points:
# Refit the model without high leverage points
full_model_DCE_cleaned <- lm(X11DCE ~ Date + SiteName, data = trainData_cleaned)
step_model_DCE_cleaned <- step(full_model_DCE_cleaned, direction = "backward", trace = 0)Summary and Interpretation of the Re-fitted Linear Regression Model
# Summary of the final model without high leverage points
summary(step_model_DCE_cleaned)
Call:
lm(formula = X11DCE ~ SiteName, data = trainData_cleaned)
Residuals:
Min 1Q Median 3Q Max
-0.0085112 -0.0000824 -0.0000014 0.0000000 0.0122920
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.010e-03 4.735e-04 2.133 0.03304 *
SiteNameWW02 -8.178e-04 6.880e-04 -1.189 0.23475
SiteNameWW03 9.881e-05 7.379e-04 0.134 0.89350
SiteNameWW06 6.269e-03 6.298e-04 9.953 < 2e-16 ***
SiteNameWW07 -8.178e-04 6.986e-04 -1.171 0.24192
SiteNameWW100 -8.182e-04 7.233e-04 -1.131 0.25812
SiteNameWW101 -8.173e-04 6.247e-04 -1.308 0.19096
SiteNameWW102 -8.182e-04 6.784e-04 -1.206 0.22795
SiteNameWW103 -8.182e-04 7.949e-04 -1.029 0.30347
SiteNameWW104 -7.957e-04 6.697e-04 -1.188 0.23492
SiteNameWW105 -8.106e-04 6.880e-04 -1.178 0.23891
SiteNameWW106 9.973e-04 6.616e-04 1.507 0.13192
SiteNameWW107 -7.359e-04 6.784e-04 -1.085 0.27822
SiteNameWW11 -8.058e-04 8.201e-04 -0.982 0.32604
SiteNameWW116 5.067e-03 6.697e-04 7.566 6.52e-14 ***
SiteNameWW117 -8.176e-04 6.411e-04 -1.275 0.20243
SiteNameWW13 -8.186e-04 8.201e-04 -0.998 0.31836
SiteNameWW156 -8.121e-04 6.411e-04 -1.267 0.20548
SiteNameWW156-2 -8.182e-04 8.500e-04 -0.963 0.33587
SiteNameWW16 4.821e-03 6.542e-04 7.369 2.77e-13 ***
SiteNameWW17 1.060e-03 6.475e-04 1.638 0.10164
SiteNameWW178 -8.173e-04 6.616e-04 -1.235 0.21692
SiteNameWW18 -2.475e-04 6.986e-04 -0.354 0.72314
SiteNameWW180 -7.963e-04 6.784e-04 -1.174 0.24066
SiteNameWW181 -8.117e-04 6.200e-04 -1.309 0.19064
SiteNameWW183 -7.806e-04 7.103e-04 -1.099 0.27195
SiteNameWW188 -8.182e-04 6.697e-04 -1.222 0.22193
SiteNameWW189 -8.182e-04 6.697e-04 -1.222 0.22193
SiteNameWW19 5.643e-03 6.697e-04 8.427 < 2e-16 ***
SiteNameWW197 -8.127e-04 7.103e-04 -1.144 0.25269
SiteNameWW201 -8.182e-04 7.379e-04 -1.109 0.26766
SiteNameWW21 -8.096e-04 6.880e-04 -1.177 0.23949
SiteNameWW210/211 -8.161e-04 7.949e-04 -1.027 0.30476
SiteNameWW22 -8.182e-04 6.986e-04 -1.171 0.24165
SiteNameWW221 -8.182e-04 8.859e-04 -0.924 0.35580
SiteNameWW223 -8.170e-04 6.784e-04 -1.204 0.22867
SiteNameWW23 -8.122e-04 6.880e-04 -1.181 0.23795
SiteNameWW24 -8.094e-04 7.233e-04 -1.119 0.26327
SiteNameWW25 -8.172e-04 7.233e-04 -1.130 0.25874
SiteNameWW26 -8.182e-04 6.880e-04 -1.189 0.23450
SiteNameWW28 -8.126e-04 6.784e-04 -1.198 0.23120
SiteNameWW29 -8.168e-04 6.986e-04 -1.169 0.24246
SiteNameWW30 -8.182e-04 7.544e-04 -1.085 0.27827
SiteNameWW31 -8.161e-04 7.949e-04 -1.027 0.30476
SiteNameWW32 -8.096e-04 6.542e-04 -1.237 0.21612
SiteNameWW34 -8.148e-04 7.379e-04 -1.104 0.26967
SiteNameWW36 4.968e-03 6.411e-04 7.749 1.66e-14 ***
SiteNameWW36X 1.231e-04 8.859e-04 0.139 0.88954
SiteNameWW37 3.086e-03 6.155e-04 5.014 5.93e-07 ***
SiteNameWW42 4.561e-03 6.411e-04 7.113 1.72e-12 ***
SiteNameWW44 7.693e-03 8.500e-04 9.051 < 2e-16 ***
SiteNameWW45 2.674e-03 6.353e-04 4.209 2.72e-05 ***
SiteNameWW46 1.657e-03 8.201e-04 2.021 0.04348 *
SiteNameWW47 5.585e-04 8.859e-04 0.630 0.52851
SiteNameWW49 -4.427e-04 6.542e-04 -0.677 0.49872
SiteNameWW50 6.587e-03 6.880e-04 9.574 < 2e-16 ***
SiteNameWW51 3.619e-03 8.859e-04 4.085 4.63e-05 ***
SiteNameWW56 2.061e-04 7.544e-04 0.273 0.78475
SiteNameWW59 -8.182e-04 7.103e-04 -1.152 0.24949
SiteNameWW62 -8.182e-04 6.697e-04 -1.222 0.22193
SiteNameWW63 -8.182e-04 7.379e-04 -1.109 0.26766
SiteNameWW64 -8.182e-04 6.542e-04 -1.251 0.21125
SiteNameWW66 -8.182e-04 6.986e-04 -1.171 0.24165
SiteNameWW67 3.654e-03 6.475e-04 5.644 1.97e-08 ***
SiteNameWW68 -8.165e-04 7.379e-04 -1.107 0.26867
SiteNameWW69 -8.165e-04 6.353e-04 -1.285 0.19891
SiteNameWW70 -2.360e-04 6.475e-04 -0.365 0.71550
SiteNameWW71 -8.148e-04 7.949e-04 -1.025 0.30552
SiteNameWW76 -6.574e-04 6.784e-04 -0.969 0.33264
SiteNameWW77 -8.169e-04 6.247e-04 -1.308 0.19120
SiteNameWW77-2 -8.142e-04 7.949e-04 -1.024 0.30584
SiteNameWW78 1.166e-03 6.697e-04 1.741 0.08184 .
SiteNameWW78-2 -6.077e-05 6.542e-04 -0.093 0.92601
SiteNameWW79 -5.226e-04 6.697e-04 -0.780 0.43530
SiteNameWW80 2.004e-03 6.542e-04 3.064 0.00222 **
SiteNameWW81 1.527e-04 8.500e-04 0.180 0.85747
SiteNameWW82-1 5.156e-04 6.411e-04 0.804 0.42140
SiteNameWW82-2 7.731e-04 6.784e-04 1.140 0.25464
SiteNameWW83 6.690e-03 6.616e-04 10.112 < 2e-16 ***
SiteNameWW84 1.048e-03 6.616e-04 1.584 0.11340
SiteNameWW85 -3.978e-04 6.247e-04 -0.637 0.52440
SiteNameWW85-2 -7.781e-04 8.500e-04 -0.915 0.36008
SiteNameWW86 1.238e-03 6.298e-04 1.965 0.04960 *
SiteNameWW87 -6.521e-04 6.697e-04 -0.974 0.33031
SiteNameWW88 -8.053e-04 6.200e-04 -1.299 0.19415
SiteNameWW89 -8.062e-04 6.542e-04 -1.232 0.21807
SiteNameWW92 -8.124e-04 6.986e-04 -1.163 0.24505
SiteNameWW96 -4.229e-04 6.542e-04 -0.646 0.51818
SiteNameWW97 -7.703e-04 7.103e-04 -1.084 0.27832
SiteNameWW98 -8.174e-04 6.200e-04 -1.318 0.18756
SiteNameWW99 -8.182e-04 7.379e-04 -1.109 0.26766
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.002118 on 1569 degrees of freedom
Multiple R-squared: 0.5172, Adjusted R-squared: 0.4895
F-statistic: 18.68 on 90 and 1569 DF, p-value: < 2.2e-16Residuals: Min ( -0.0084612), 1Q(-0.0000823), Median(-0.0000014), 3Q(0.0000000) and Max(0.0122086) shows the distribution of the residuals (difference between observed and predicted values). The small values of the residuals indicate that the model’s prediction are close to the observed values.
Intercept: The intercept is statistically significant at the 5% level, indicating a small but significant base level of
X11DCEwhen allSiteNamefactors are zero (though in practice,SiteNamefactors wouldn’t be zero).
Significant SiteName Coefficients:
- SiteNameWW06: 6.232e-03 (p < 2e-16)
- SiteNameWW116: 5.042e-03 (p = 6.09e-14)
- SiteNameWW16: 4.795e-03 (p = 2.67e-13)
- SiteNameWW19: 5.615e-03 (p < 2e-16)
- SiteNameWW36: 4.936e-03 (p = 1.70e-14)
- SiteNameWW37: 3.073e-03 (p = 5.68e-07)
- SiteNameWW42: 4.542e-03 (p = 1.56e-12)
- SiteNameWW44: 7.644e-03 (p < 2e-16)
- SiteNameWW45: 2.659e-03 (p = 2.69e-05)
- SiteNameWW50: 6.545e-03 (p < 2e-16)
- SiteNameWW51: 3.591e-03 (p = 4.76e-05)
- SiteNameWW67: 3.636e-03 (p = 1.90e-08)
- SiteNameWW83: 6.648e-03 (p < 2e-16)
- SiteNameWW80: 1.992e-03 (p = 0.00222)
These significant coefficients indicate that the corresponding SiteName levels have a statistically significant effect on X11DCE.
Model Fit Statistics
Residual standard error: 0.002105 on 1569 degrees of freedom
Multiple R-squared (0.5177): This indicates that approximately 51.77% of the variability in
X11DCEis explained by the model.Adjusted R-squared (0.49): This adjusts the R-squared value for the number of predictors in the model, providing a more accurate measure of model fit.
F-statistic: 18.71 on 90 and 1569 DF (p-value: < 2.2e-16): The F-statistic tests the overall significance of the model. A very low p-value (< 2.2e-16) indicates that the model is statistically significant.
Conclusion:
The model explains a significant portion of the variance in X11DCE (R-squared = 51.77%). Several SiteName levels are significant, indicating that these levels are important for predicting X11DCE. The model has a good fit, but there might be room for improvement or further refinement to increase the explained variance.
Checking for Model Assumption:
# Check for model assumptions
par(mfrow=c(2, 2))
plot(step_model_DCE_cleaned)
Durbin-Watson test:
The Durbin-Watson test is used to detect the presence of autocorrelation in the residuals of a regression analysis.
# Durbin-Watson test for autocorrelation
dwtest(step_model_DCE)
Durbin-Watson test
data: step_model_DCE
DW = 1.2875, p-value < 2.2e-16
alternative hypothesis: true autocorrelation is greater than 0DW = 1.2869: The Durbin-Watson statistic value is approximately. The DW statistic ranges from 0 to 4. A value around 2 suggests no autocorrelation. Values less than 2 indicate positive autocorrelation, while values greater than 2 indicate negative autocorrelation.
P-value = <2.2e-16: The p-value is <2.2e-16. This value is used to determine the significance of the test result. Typically, a p-value less than 0.05 indicates that the null hypothesis can be rejected with a very high level of confidence.
Alternative hypothesis: true autocorrelation is greater than 0: The alternative hypothesis in this test suggests that there is positive autocorrelation in the residuals (i.e., the residuals are positively correlated).
Conclusion
The Durbin-Watson test statistic of 1.2869 and the very low p-value suggest that there is significant positive autocorrelation in the residuals of the model step_model_DCE. Positive autocorrelation means that consecutive residuals are correlated with each other, which can indicate that the model might be missing some important variables or that there are patterns in the data that are not captured by the model.
Residual Plots Vs Time or Fitted Values to Visualize Autocorrelation
####Create Plots of Residuals Vs Time or Fitted Values to Visually Inspect the AutoCorrelation
plot(residuals(step_model_DCE_cleaned), type = "l", main = "Residuals over Time")
acf(residuals(step_model_DCE_cleaned), main = "ACF of Residuals")
Factoring SiteName in testData to Match with the Non-High Leverage trainData (Categorical Variable)
# Ensure factor levels in test set match training set
testData$SiteName <- factor(testData$SiteName , levels = levels(trainData_cleaned$SiteName))The above code ensures that the SiteName factor levels in the testData matches those in the Non-High Leverage trainData dataset. This step is crucial when you want to make predictions on the test data using a model trained on the training data, especially when dealing with categorical variables.
Removing Data Points with High Levels for Prediction of Actual Vs Predicted Values
# Filter out rows in testData that have levels not present in trainData_cleaned
testData <- testData[!testData$SiteName %in% c('WW04', 'WW10', 'WW12','WW14','WW177','WW186','WW193',
'WW198','WW206','WW210','WW214','WW217','WW222','WW35',
'WW39-2','WW43','WW48','WW88-2','WW91'), ]Predicting on test data
Having Trained our model to predict X11DCE based on the testdata. We shall now proceed to investigate how efficient our model is,
# Predict on test data
predictions_DCE_cleaned <- predict(step_model_DCE_cleaned, newdata = testData)Model Evaluation
# Evaluate the model
DCE_results_cleaned <- data.frame(WellName = testData$SiteName, Actual = testData$X11DCE,
Predicted = predictions_DCE_cleaned)Result of the Prediction
The correlation coefficient of 0.7145267 indicates a strong positive relationship between the actual and predicted values, suggesting that the model performs well. However, further analysis and refinement can be conducted to improve the model’s accuracy and reliability.
print(DCE_results_cleaned) WellName Actual Predicted
1 WW01 0.0047785645 0.0010102240
2 WW01 0.0015987214 0.0010102240
3 WW01 0.0022474725 0.0010102240
9 WW01 0.0002159767 0.0010102240
11 WW01 0.0004309071 0.0010102240
21 WW01 0.0001919816 0.0010102240
25 WW01 0.0004548965 0.0010102240
27 WW01 0.0002159767 0.0010102240
35 WW02 0.0002159767 0.0001924259
38 WW02 0.0001919816 0.0001924259
39 WW02 0.0001919816 0.0001924259
46 WW02 0.0001919816 0.0001924259
51 WW02 0.0001919816 0.0001924259
55 WW03 0.0002159767 0.0011090335
66 WW03 0.0002159767 0.0011090335
80 WW06 0.0108410232 0.0072788301
83 WW06 0.0092669291 0.0072788301
88 WW06 0.0100493359 0.0072788301
95 WW06 0.0001999800 0.0072788301
100 WW06 0.0081963182 0.0072788301
105 WW06 0.0063398605 0.0072788301
113 WW07 0.0001919816 0.0001924521
126 WW07 0.0001919816 0.0001924521
130 WW07 0.0001919816 0.0001924521
139 WW11 0.0001919816 0.0002044786
144 WW11 0.0001919816 0.0002044786
146 WW11 0.0001919816 0.0002044786
149 WW11 0.0001919816 0.0002044786
157 WW13 0.0001919816 0.0001915816
161 WW13 0.0001919816 0.0001915816
182 WW16 0.0001919816 0.0058314623
184 WW16 0.0001919816 0.0058314623
189 WW16 0.0105442139 0.0058314623
191 WW16 0.0091777553 0.0058314623
192 WW16 0.0075613409 0.0058314623
200 WW16 0.0128175037 0.0058314623
203 WW16 0.0028559180 0.0058314623
204 WW16 0.0035038543 0.0058314623
216 WW17 0.0001919816 0.0020706851
219 WW17 0.0012891687 0.0020706851
221 WW17 0.0025467543 0.0020706851
227 WW17 0.0020179625 0.0020706851
233 WW18 0.0006397953 0.0007626981
237 WW18 0.0010594386 0.0007626981
241 WW18 0.0022774047 0.0007626981
248 WW18 0.0008046762 0.0007626981
251 WW18 0.0001919816 0.0007626981
256 WW19 0.0089101861 0.0066533200
258 WW19 0.0086921139 0.0066533200
265 WW19 0.0087317669 0.0066533200
271 WW19 0.0047188487 0.0066533200
272 WW19 0.0039322585 0.0066533200
278 WW19 0.0016785904 0.0066533200
279 WW19 0.0041414125 0.0066533200
282 WW21 0.0001919816 0.0002006462
292 WW21 0.0001919816 0.0002006462
295 WW21 0.0001919816 0.0002006462
297 WW21 0.0001919816 0.0002006462
302 WW21 0.0001919816 0.0002006462
306 WW22 0.0001919816 0.0001919816
313 WW22 0.0001919816 0.0001919816
316 WW22 0.0001919816 0.0001919816
317 WW22 0.0001919816 0.0001919816
321 WW22 0.0001919816 0.0001919816
327 WW22 0.0001919816 0.0001919816
329 WW23 0.0001919816 0.0001979801
331 WW23 0.0001919816 0.0001979801
332 WW23 0.0001919816 0.0001979801
339 WW23 0.0001919816 0.0001979801
350 WW24 0.0002999550 0.0002007795
356 WW24 0.0001919816 0.0002007795
360 WW24 0.0001919816 0.0002007795
365 WW24 0.0001919816 0.0002007795
366 WW24 0.0001919816 0.0002007795
367 WW24 0.0001919816 0.0002007795
368 WW24 0.0001919816 0.0002007795
371 WW24 0.0001919816 0.0002007795
373 WW25 0.0001919816 0.0001930480
377 WW25 0.0001919816 0.0001930480
399 WW26 0.0001919816 0.0001919816
400 WW26 0.0001919816 0.0001919816
402 WW26 0.0001919816 0.0001919816
405 WW26 0.0001919816 0.0001919816
412 WW28 0.0001919816 0.0001976644
420 WW28 0.0001919816 0.0001976644
432 WW28 0.0001919816 0.0001976644
433 WW28 0.0001919816 0.0001976644
436 WW29 0.0001919816 0.0001933930
441 WW29 0.0001919816 0.0001933930
444 WW29 0.0001919816 0.0001933930
447 WW29 0.0001919816 0.0001933930
456 WW30 0.0001919816 0.0001919816
460 WW30 0.0001919816 0.0001919816
461 WW30 0.0001919816 0.0001919816
464 WW30 0.0001919816 0.0001919816
471 WW30 0.0001919816 0.0001919816
472 WW30 0.0001919816 0.0001919816
475 WW30 0.0002159767 0.0001919816
477 WW31 0.0001919816 0.0001941629
485 WW31 0.0001919816 0.0001941629
488 WW31 0.0001919816 0.0001941629
489 WW31 0.0001919816 0.0001941629
491 WW32 0.0002049790 0.0002006389
506 WW32 0.0001919816 0.0002006389
508 WW32 0.0001919816 0.0002006389
513 WW32 0.0001919816 0.0002006389
514 WW32 0.0001919816 0.0002006389
517 WW32 0.0001919816 0.0002006389
521 WW34 0.0002159767 0.0001954094
525 WW34 0.0001919816 0.0001954094
527 WW34 0.0001919816 0.0001954094
528 WW34 0.0001919816 0.0001954094
543 WW36 0.0081467251 0.0059781283
544 WW36 0.0076308112 0.0059781283
550 WW36 0.0002159767 0.0059781283
559 WW36 0.0118297518 0.0059781283
560 WW36 0.0106431601 0.0059781283
564 WW36 0.0018582723 0.0059781283
568 WW36 0.0020279423 0.0059781283
573 WW36X 0.0002069786 0.0011332756
582 WW37 0.0017684354 0.0040965377
583 WW37 0.0001919816 0.0040965377
586 WW37 0.0011892925 0.0040965377
587 WW37 0.0025467543 0.0040965377
588 WW37 0.0001919816 0.0040965377
589 WW37 0.0017484705 0.0040965377
599 WW37 0.0100493359 0.0040965377
605 WW37 0.0063299237 0.0040965377
633 WW42 0.0001919816 0.0055707646
637 WW42 0.0058926045 0.0055707646
640 WW42 0.0081467251 0.0055707646
643 WW42 0.0001919816 0.0055707646
652 WW42 0.0025068552 0.0055707646
669 WW44 0.0001919816 0.0087031425
673 WW44 0.0089696521 0.0087031425
677 WW44 0.0103462921 0.0087031425
680 WW44 0.0024370280 0.0087031425
681 WW44 0.0012691942 0.0087031425
687 WW45 0.0079880107 0.0036838519
691 WW45 0.0001919816 0.0036838519
714 WW46 0.0001919816 0.0026675315
720 WW47 0.0106431601 0.0015686958
725 WW47 0.0001919816 0.0015686958
748 WW49 0.0001919816 0.0005675184
754 WW49 0.0001919816 0.0005675184
756 WW49 0.0026365213 0.0005675184
762 WW50 0.0113355097 0.0075971507
763 WW50 0.0110388471 0.0075971507
768 WW50 0.0001919816 0.0075971507
773 WW50 0.0104452579 0.0075971507
778 WW50 0.0138042810 0.0075971507
783 WW50 0.0054948755 0.0075971507
784 WW51 0.0139029052 0.0046292026
790 WW51 0.0001919816 0.0046292026
800 WW56 0.0001919816 0.0012163176
803 WW56 0.0001919816 0.0012163176
804 WW56 0.0001919816 0.0012163176
805 WW56 0.0001919816 0.0012163176
807 WW56 0.0001919816 0.0012163176
819 WW59 0.0001919816 0.0001919816
820 WW59 0.0001919816 0.0001919816
828 WW59 0.0001919816 0.0001919816
839 WW62 0.0001919816 0.0001919816
843 WW62 0.0001919816 0.0001919816
858 WW62 0.0001919816 0.0001919816
859 WW63 0.0001919816 0.0001919816
864 WW63 0.0001919816 0.0001919816
865 WW63 0.0001919816 0.0001919816
873 WW63 0.0001919816 0.0001919816
877 WW63 0.0001919816 0.0001919816
885 WW64 0.0001919816 0.0001919816
888 WW64 0.0001919816 0.0001919816
894 WW64 0.0001919816 0.0001919816
913 WW66 0.0001919816 0.0001919816
917 WW66 0.0001919816 0.0001919816
920 WW66 0.0001919816 0.0001919816
923 WW66 0.0001919816 0.0001919816
925 WW66 0.0001919816 0.0001919816
927 WW67 0.0025567288 0.0046642967
934 WW67 0.0036034996 0.0046642967
939 WW67 0.0036134636 0.0046642967
950 WW67 0.0018981973 0.0046642967
957 WW68 0.0001919816 0.0001936955
958 WW68 0.0001919816 0.0001936955
959 WW68 0.0001919816 0.0001936955
961 WW68 0.0001919816 0.0001936955
964 WW68 0.0001919816 0.0001936955
965 WW68 0.0001919816 0.0001936955
970 WW68 0.0001919816 0.0001936955
974 WW68 0.0002159767 0.0001936955
975 WW69 0.0001919816 0.0001937412
977 WW69 0.0001919816 0.0001937412
1001 WW69 0.0001879823 0.0001937412
1003 WW70 0.0009745250 0.0007742007
1009 WW70 0.0001919816 0.0007742007
1018 WW70 0.0010194802 0.0007742007
1022 WW70 0.0001919816 0.0007742007
1036 WW71 0.0001919816 0.0001954354
1039 WW71 0.0001919816 0.0001954354
1045 WW76 0.0001919816 0.0003527732
1047 WW76 0.0001919816 0.0003527732
1052 WW76 0.0001919816 0.0003527732
1056 WW76 0.0001919816 0.0003527732
1057 WW76 0.0001919816 0.0003527732
1061 WW76 0.0001919816 0.0003527732
1066 WW76 0.0007297337 0.0003527732
1071 WW77 0.0001919816 0.0001933146
1080 WW77 0.0001919816 0.0001933146
1081 WW77 0.0001919816 0.0001933146
1088 WW77 0.0001919816 0.0001933146
1094 WW77 0.0001919816 0.0001933146
1097 WW77 0.0002159767 0.0001933146
1114 WW78 0.0004848824 0.0021762364
1123 WW78 0.0014888911 0.0021762364
1128 WW78 0.0031649861 0.0021762364
1130 WW78 0.0047984689 0.0021762364
1133 WW78 0.0010294699 0.0021762364
1135 WW78 0.0098612180 0.0021762364
1143 WW78-2 0.0001919816 0.0009494567
1147 WW78-2 0.0007727014 0.0009494567
1156 WW78-2 0.0001919816 0.0009494567
1159 WW78-2 0.0020578811 0.0009494567
1164 WW78-2 0.0002159767 0.0009494567
1169 WW79 0.0003089523 0.0004876607
1174 WW79 0.0001919816 0.0004876607
1182 WW79 0.0009755240 0.0004876607
1184 WW79 0.0006947586 0.0004876607
1186 WW79 0.0005708370 0.0004876607
1195 WW80 0.0002539677 0.0030147114
1204 WW80 0.0001919816 0.0030147114
1205 WW80 0.0039521798 0.0030147114
1217 WW81 0.0001919816 0.0011628994
1229 WW82-1 0.0001919816 0.0015258353
1232 WW82-1 0.0001919816 0.0015258353
1249 WW82-1 0.0007597113 0.0015258353
1253 WW82-1 0.0003079526 0.0015258353
1255 WW82-2 0.0001919816 0.0017833141
1258 WW82-2 0.0011193733 0.0017833141
1259 WW82-2 0.0001919816 0.0017833141
1266 WW82-2 0.0033444013 0.0017833141
1274 WW82-2 0.0058130713 0.0017833141
1276 WW82-2 0.0011193733 0.0017833141
1280 WW83 0.0067471865 0.0077004251
1286 WW83 0.0103462921 0.0077004251
1288 WW83 0.0001919816 0.0077004251
1289 WW83 0.0001919816 0.0077004251
1295 WW83 0.0139029052 0.0077004251
1299 WW83 0.0140015196 0.0077004251
1307 WW83 0.0003959216 0.0077004251
1313 WW84 0.0001919816 0.0020582470
1315 WW84 0.0001919816 0.0020582470
1318 WW84 0.0012192564 0.0020582470
1326 WW84 0.0022075615 0.0020582470
1332 WW84 0.0001879823 0.0020582470
1336 WW85 0.0003819271 0.0006124424
1337 WW85 0.0003529377 0.0006124424
1343 WW85 0.0004668910 0.0006124424
1347 WW85 0.0005428526 0.0006124424
1354 WW85 0.0007737006 0.0006124424
1361 WW85 0.0001919816 0.0006124424
1381 WW86 0.0020678605 0.0022477939
1383 WW86 0.0001919816 0.0022477939
1388 WW86 0.0001919816 0.0022477939
1399 WW86 0.0037031349 0.0022477939
1400 WW86 0.0031948909 0.0022477939
1411 WW87 0.0001919816 0.0003581187
1418 WW87 0.0001919816 0.0003581187
1419 WW87 0.0001919816 0.0003581187
1423 WW87 0.0001919816 0.0003581187
1432 WW88 0.0002069786 0.0002049072
1440 WW88 0.0002939568 0.0002049072
1457 WW88 0.0001919816 0.0002049072
1472 WW89 0.0001919816 0.0002040694
1473 WW89 0.0001919816 0.0002040694
1475 WW89 0.0001919816 0.0002040694
1476 WW89 0.0001919816 0.0002040694
1477 WW89 0.0001919816 0.0002040694
1490 WW89 0.0001919816 0.0002040694
1491 WW89 0.0001919816 0.0002040694
1495 WW89 0.0002159767 0.0002040694
1512 WW92 0.0003299456 0.0001978625
1514 WW92 0.0001919816 0.0001978625
1517 WW92 0.0001919816 0.0001978625
1518 WW92 0.0001919816 0.0001978625
1529 WW92 0.0001919816 0.0001978625
1532 WW92 0.0001919816 0.0001978625
1533 WW92 0.0001919816 0.0001978625
1534 WW92 0.0001919816 0.0001978625
1535 WW92 0.0002159767 0.0001978625
1536 WW92 0.0002159767 0.0001978625
1544 WW96 0.0001919816 0.0005873781
1554 WW96 0.0015088611 0.0005873781
1555 WW96 0.0008816113 0.0005873781
1557 WW96 0.0009505481 0.0005873781
1558 WW96 0.0001919816 0.0005873781
1567 WW97 0.0001919816 0.0002399539
1568 WW97 0.0001919816 0.0002399539
1570 WW97 0.0001919816 0.0002399539
1572 WW97 0.0001919816 0.0002399539
1574 WW97 0.0001919816 0.0002399539
1587 WW97 0.0001919816 0.0002399539
1589 WW97 0.0001919816 0.0002399539
1590 WW97 0.0001919816 0.0002399539
1595 WW98 0.0001919816 0.0001928385
1614 WW98 0.0001919816 0.0001928385
1615 WW98 0.0001919816 0.0001928385
1622 WW99 0.0001919816 0.0001919816
1623 WW99 0.0001919816 0.0001919816
1630 WW99 0.0001919816 0.0001919816
1631 WW99 0.0001919816 0.0001919816
1633 WW99 0.0001919816 0.0001919816
1636 WW99 0.0001919816 0.0001919816
1642 WW99 0.0002159767 0.0001919816
1644 WW100 0.0001919816 0.0001919816
1645 WW100 0.0001919816 0.0001919816
1646 WW100 0.0001919816 0.0001919816
1648 WW100 0.0001919816 0.0001919816
1650 WW100 0.0001919816 0.0001919816
1661 WW100 0.0001919816 0.0001919816
1686 WW101 0.0001919816 0.0001928703
1693 WW102 0.0001919816 0.0001919816
1699 WW102 0.0001919816 0.0001919816
1703 WW102 0.0001919816 0.0001919816
1704 WW102 0.0001919816 0.0001919816
1720 WW103 0.0001919816 0.0001919816
1724 WW103 0.0001919816 0.0001919816
1726 WW103 0.0001919816 0.0001919816
1731 WW104 0.0001919816 0.0002145260
1732 WW104 0.0001919816 0.0002145260
1737 WW104 0.0002559672 0.0002145260
1750 WW104 0.0002109777 0.0002145260
1754 WW105 0.0001919816 0.0001996462
1764 WW105 0.0001919816 0.0001996462
1766 WW105 0.0001919816 0.0001996462
1767 WW105 0.0001919816 0.0001996462
1770 WW105 0.0001919816 0.0001996462
1776 WW105 0.0001919816 0.0001996462
1780 WW106 0.0031649861 0.0020075373
1788 WW106 0.0001919816 0.0020075373
1791 WW106 0.0031550177 0.0020075373
1792 WW106 0.0032347625 0.0020075373
1795 WW106 0.0021277348 0.0020075373
1798 WW106 0.0021377135 0.0020075373
1806 WW107 0.0002669644 0.0002743624
1813 WW107 0.0004269089 0.0002743624
1825 WW107 0.0001919816 0.0002743624
1826 WW107 0.0002159767 0.0002743624
1829 WW116 0.0024769299 0.0060768788
1830 WW116 0.0027861152 0.0060768788
1832 WW116 0.0038127223 0.0060768788
1840 WW116 0.0075514162 0.0060768788
1857 WW117 0.0001919816 0.0001926481
1859 WW117 0.0001919816 0.0001926481
1861 WW117 0.0001919816 0.0001926481
1874 WW117 0.0001919816 0.0001926481
1875 WW117 0.0001919816 0.0001926481
1876 WW117 0.0001919816 0.0001926481
1877 WW117 0.0001919816 0.0001926481
1880 WW117 0.0002159767 0.0001926481
1884 WW156 0.0001919816 0.0001981468
1891 WW156 0.0001919816 0.0001981468
1895 WW156 0.0001919816 0.0001981468
1896 WW156 0.0001919816 0.0001981468
1918 WW156-2 0.0001919816 0.0001919816
1923 WW178 0.0001919816 0.0001929338
1925 WW178 0.0001919816 0.0001929338
1927 WW178 0.0001919816 0.0001929338
1928 WW178 0.0001919816 0.0001929338
1941 WW178 0.0001919816 0.0001929338
1947 WW178 0.0002159767 0.0001929338
1948 WW178 0.0001879823 0.0001929338
1950 WW180 0.0001919816 0.0002139223
1955 WW180 0.0001919816 0.0002139223
1956 WW180 0.0001919816 0.0002139223
1959 WW180 0.0001919816 0.0002139223
1963 WW180 0.0001919816 0.0002139223
1969 WW180 0.0001919816 0.0002139223
1989 WW181 0.0001919816 0.0001985157
2003 WW181 0.0001879823 0.0001985157
2006 WW183 0.0001919816 0.0002296534
2009 WW183 0.0001919816 0.0002296534
2012 WW183 0.0001919816 0.0002296534
2022 WW183 0.0001919816 0.0002296534
2024 WW183 0.0001919816 0.0002296534
2033 WW188 0.0001919816 0.0001919816
2037 WW188 0.0001919816 0.0001919816
2046 WW188 0.0001919816 0.0001919816
2050 WW188 0.0001919816 0.0001919816
2055 WW189 0.0001919816 0.0001919816
2072 WW189 0.0001919816 0.0001919816
2088 WW197 0.0001919816 0.0001974803
2100 WW201 0.0001919816 0.0001919816
2101 WW201 0.0001919816 0.0001919816
2106 WW201 0.0001919816 0.0001919816
2114 WW201 0.0001919816 0.0001919816
2129 WW210/211 0.0001919816 0.0001941629
2132 WW210/211 0.0001919816 0.0001941629
2135 WW210/211 0.0001919816 0.0001941629
2177 WW223 0.0001919816 0.0001932445
2185 WW223 0.0001919816 0.0001932445
2186 WW223 0.0001919816 0.0001932445print(cor(DCE_results_cleaned$Actual, DCE_results_cleaned$Predicted))[1] 0.7142108A correlation of 0.7145267 suggests a strong positive linear relationship between the actual and predicted values. This means that as the actual values increase, the predicted values tend to also increase, and vice versa. While the correlation provides a measure of the strength of the linear relationship, it does not directly indicate how well the model fits the data in terms of variance explained. For this, metrics such as R-squared, RMSE (Root Mean Square Error), and MAE (Mean Absolute Error) are also important.
Visual Inspection of Actual Vs Predicted Values
## Visual Inspection ##
plot(DCE_results_cleaned$Actual, DCE_results_cleaned$Predicted, main = "Actual vs Predicted Values", xlab = "Actual", ylab = "Predicted")
abline(lm(DCE_results_cleaned$Predicted ~ DCE_results_cleaned$Actual), col = "red")
Residual Analysis
Analyze the residuals (actual - predicted) to check for patterns that might indicate model deficiencies.
### Residual Analysis###
residuals <- DCE_results_cleaned$Actual - DCE_results_cleaned$Predicted
plot(residuals, main = "Residuals", ylab = "Residuals")
Model Evaluation Metrics
### Model Evaluation Metrics ##
rmse <- sqrt(mean((DCE_results_cleaned$Actual - DCE_results_cleaned$Predicted)^2))
print("Root Mean Square Error:")[1] "Root Mean Square Error:"print(rmse)[1] 0.002004078mae <- mean(abs(DCE_results_cleaned$Actual - DCE_results_cleaned$Predicted))
print("Mean Absolute Error:")[1] "Mean Absolute Error:"print(mae)[1] 0.0009357635r_squared <- summary(lm(DCE_results_cleaned$Predicted ~ DCE_results_cleaned$Actual))$r.squared
print("Root Square Error:")[1] "Root Square Error:"print(r_squared)[1] 0.5100971Summary of Model Performance
RMSE (0.001992572): Indicates the average error magnitude in predicting the dependent variable. A very low value suggests that the model’s predictions are close to the actual values.
MAE (0.0009310584): Shows the average absolute error in predictions. The low value reinforces the indication from RMSE that the model performs well.
R^2 (0.5105484): Indicates that about 51.05% of the variance in the dependent variable is explained by the model. While this shows a decent fit, there’s still room for improvement to capture more variance.
Time series forecasting for 1,1-Dichloroethene
Aggregating data by month for a better time series analysis
For the Time Series Forecasting,
# Time series forecasting for X11DCE
# Aggregating data by month for a better time series analysis
data_ts <- data %>%
group_by(Date, SiteName) %>%
summarize(X11DCE = mean(X11DCE), .groups = "drop")Convert to time series object
# Convert to time series object
ts_DCE <- ts(data_ts$X11DCE, start = c(year(min(data_ts$Date)), month(min(data_ts$Date))), frequency = 12)# Decompose time series
decomp_DCE <- stl(ts_DCE, s.window="periodic")# Plot decompositions
plot(decomp_DCE)
Fit ARIMA model for DCE
# Fit ARIMA model
fit_DCE <- auto.arima(ts_DCE)
summary(fit_DCE)Series: ts_DCE
ARIMA(1,0,1)(2,0,0)[12] with non-zero mean
Coefficients:
ar1 ma1 sar1 sar2 mean
0.9213 -0.7837 -0.0012 -0.0125 0.0014
s.e. 0.0170 0.0266 0.0223 0.0221 0.0002
sigma^2 = 7.778e-06: log likelihood = 9584.73
AIC=-19157.47 AICc=-19157.43 BIC=-19123.44
Training set error measures:
ME RMSE MAE MPE MAPE MASE
Training set -5.260923e-07 0.002785721 0.001646678 -363.7399 384.7188 0.8422742
ACF1
Training set 0.005685058Forecast for DCE in the next 12 months
# Forecast for the next 12 months
forecast_DCE <- forecast(fit_DCE, h = 12)
plot(forecast_DCE)
# Combine forecasts into a data frame
future_forecasts_DCE <- data.frame(
Date = seq.Date(from = max(data$Date) + 1, by = "month", length.out = 48),
DCE_Forecast = as.numeric(forecast_DCE$mean)
)# Print future forecasts
print(future_forecasts_DCE) Date DCE_Forecast
1 2024-02-02 0.0009098581
2 2024-03-02 0.0009637816
3 2024-04-02 0.0009272554
4 2024-05-02 0.0010462195
5 2024-06-02 0.0010768909
6 2024-07-02 0.0010912384
7 2024-08-02 0.0009701374
8 2024-09-02 0.0010015895
9 2024-10-02 0.0011526537
10 2024-11-02 0.0011926688
11 2024-12-02 0.0012104960
12 2025-01-02 0.0012269204
13 2025-02-02 0.0009098581
14 2025-03-02 0.0009637816
15 2025-04-02 0.0009272554
16 2025-05-02 0.0010462195
17 2025-06-02 0.0010768909
18 2025-07-02 0.0010912384
19 2025-08-02 0.0009701374
20 2025-09-02 0.0010015895
21 2025-10-02 0.0011526537
22 2025-11-02 0.0011926688
23 2025-12-02 0.0012104960
24 2026-01-02 0.0012269204
25 2026-02-02 0.0009098581
26 2026-03-02 0.0009637816
27 2026-04-02 0.0009272554
28 2026-05-02 0.0010462195
29 2026-06-02 0.0010768909
30 2026-07-02 0.0010912384
31 2026-08-02 0.0009701374
32 2026-09-02 0.0010015895
33 2026-10-02 0.0011526537
34 2026-11-02 0.0011926688
35 2026-12-02 0.0012104960
36 2027-01-02 0.0012269204
37 2027-02-02 0.0009098581
38 2027-03-02 0.0009637816
39 2027-04-02 0.0009272554
40 2027-05-02 0.0010462195
41 2027-06-02 0.0010768909
42 2027-07-02 0.0010912384
43 2027-08-02 0.0009701374
44 2027-09-02 0.0010015895
45 2027-10-02 0.0011526537
46 2027-11-02 0.0011926688
47 2027-12-02 0.0012104960
48 2028-01-02 0.0012269204Method2: Time Series Forecast for each Site w.r.t 11DCE
# Create an empty list to store forecasts for each site
forecasts_list <- list()# Iterate over each site
for(site in unique(data_ts$SiteName)) {site_data <- data_ts %>% filter(SiteName == site)
# Check if there is enough data for decomposition
if(nrow(site_data) < 24) { # Ensure at least two full years of data for monthly series
warning(paste("Not enough data for site", site, ". Skipping decomposition and ARIMA modeling."))
next
}
# Convert to time series object
ts_DCE <- ts(site_data$X11DCE, start = c(year(min(site_data$Date)), month(min(site_data$Date))), frequency = 12)
# Check if time series has sufficient length for decomposition
if (length(ts_DCE) < 24) { # Ensure at least two full periods of data for decomposition
warning(paste("Series is too short for STL decomposition for site", site, ". Skipping decomposition and ARIMA modeling."))
next
}
# Decompose time series
decomp_DCE <- tryCatch({
stl(ts_DCE, s.window = "periodic")
}, error = function(e) {
warning(paste("Error in STL decomposition for site", site, ":", e$message))
NULL
})
if (!is.null(decomp_DCE)) {
# Plot decompositions
plot(decomp_DCE, main = paste("Decomposition for Site", site))
}
# Fit ARIMA model
fit_DCE <- auto.arima(ts_DCE)
summary(fit_DCE)
# Forecast for the next 12 months
forecast_DCE <- forecast(fit_DCE, h = 12)
plot(forecast_DCE, main = paste("Forecast for Site", site))
# Store forecast in the list
forecasts_list[[site]] <- data.frame(
Date = seq.Date(from = max(site_data$Date) + 1, by = "month", length.out = 12),
SiteName = site,
DCE_Forecast = as.numeric(forecast_DCE$mean)
)
}
Warning: Not enough data for site WW02 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW03 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW04 . Skipping decomposition and ARIMA
modeling.
Warning: Not enough data for site WW10 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW08 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW28 . Skipping decomposition and ARIMA
modeling.
Warning: Not enough data for site WW21 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW22 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW29 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW30 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW34 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW07 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW11 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW23 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW24 . Skipping decomposition and ARIMA
modeling.
Warning: Not enough data for site WW39-1 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW39-2 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW57 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW58 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW59 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW60 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW41 . Skipping decomposition and ARIMA
modeling.
Warning: Not enough data for site WW61 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW62 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW63 . Skipping decomposition and ARIMA
modeling.
Warning: Not enough data for site WW65 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW66 . Skipping decomposition and ARIMA
modeling.
Warning: Not enough data for site WW79 . Skipping decomposition and ARIMA
modeling.
Warning in value[[3L]](cond): Error in STL decomposition for site WW82-2 :
series is not periodic or has less than two periods
Warning in value[[3L]](cond): Error in STL decomposition for site WW97 : series
is not periodic or has less than two periods
Warning: Not enough data for site WW99 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW100 . Skipping decomposition and ARIMA
modeling.
Warning: Not enough data for site WW102 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW103 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW116 . Skipping decomposition and ARIMA
modeling.
Warning in value[[3L]](cond): Error in STL decomposition for site WW49 : series
is not periodic or has less than two periods
Warning: Not enough data for site WW68 . Skipping decomposition and ARIMA
modeling.
Warning: Not enough data for site WW104 . Skipping decomposition and ARIMA
modeling.Warning in value[[3L]](cond): Error in STL decomposition for site WW105 :
series is not periodic or has less than two periods
Warning: Not enough data for site WW50 . Skipping decomposition and ARIMA
modeling.
Warning in value[[3L]](cond): Error in STL decomposition for site WW87 : series
is not periodic or has less than two periods
Warning: Not enough data for site WW90 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW91 . Skipping decomposition and ARIMA
modeling.
Warning: Not enough data for site WW93 . Skipping decomposition and ARIMA
modeling.
Warning: Not enough data for site WW107 . Skipping decomposition and ARIMA
modeling.Warning in value[[3L]](cond): Error in STL decomposition for site WW70 : series
is not periodic or has less than two periods
Warning: Not enough data for site WW71 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW47 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW74 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW12 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW51 . Skipping decomposition and ARIMA
modeling.
Warning: Not enough data for site WW177 . Skipping decomposition and ARIMA
modeling.
Warning: Not enough data for site WW183 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW184 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW185 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW186 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW193 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW195 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW196 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW197 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW198 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW18 . Skipping decomposition and ARIMA
modeling.Warning in value[[3L]](cond): Error in STL decomposition for site WW188 :
series is not periodic or has less than two periods
Warning: Not enough data for site WW189 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW200 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW201 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW214 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW217 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW230 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW223 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW26 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW206 . Skipping decomposition and ARIMA
modeling.
Warning: Not enough data for site WW210 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW211 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW46 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW13 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW43 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW81 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW14 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW38 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW44 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW56 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW36X . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW48 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW156-2 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW210/211 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW31 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW25 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW221 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW222 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW215 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW35 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW88-2 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW85-2 . Skipping decomposition and ARIMA
modeling.Warning: Not enough data for site WW77-2 . Skipping decomposition and ARIMA
modeling.
# Combine all forecasts into a single data frame
all_forecasts <- do.call(rbind, forecasts_list)
# Print future forecasts
print(all_forecasts) Date SiteName DCE_Forecast
WW01.1 2021-12-07 WW01 0.0003295223
WW01.2 2022-01-07 WW01 0.0003295223
WW01.3 2022-02-07 WW01 0.0003295223
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