Artificial Neural Network correlation

In zFactor: Calculate the Compressibility Factor 'z' for Hydrocarbon Gases

knitr::opts_chunk$set(echo=T, comment=NA, error=T, warning=F, message = F, fig.align = 'center', results="hold")

How do we find the limits of accuracy in the ANN10 correlation

[@Kamyab2010]

Get z at selected Ppr and Tpr

Use the the correlation to calculate z and from Standing-Katz chart obtain z a digitized point at the given Tpr and Ppr.

# get a z value
library(zFactor)
ppr <- 1.5
tpr <- 2.0

z.calc <- z.Ann10(pres.pr = ppr, temp.pr = tpr)
# get a z value from the SK chart at the same Ppr and Tpr
z.chart <- getStandingKatzMatrix(tpr_vector = tpr, 
                      pprRange = "lp")[1, as.character(ppr)]

# calculate the APE
ape <- abs((z.calc - z.chart) / z.chart) * 100
df <- as.data.frame(list(Ppr = ppr,  z.calc =z.calc, z.chart = z.chart, ape=ape))
rownames(df) <- tpr
df
# HY = 0.9580002; # DAK = 0.9551087

Get z at selected Ppr and Tpr=1.1

From the Standing-Katz chart we read z at a digitized point:

library(zFactor)
ppr <- 1.5
tpr <- 1.1

z.calc <- z.Ann10(pres.pr = ppr, temp.pr = tpr)

# From the Standing-Katz chart we obtain a digitized point:
z.chart <- getStandingKatzMatrix(tpr_vector = tpr, 
                      pprRange = "lp")[1, as.character(ppr)]

# calculate the APE (Average Percentage Error)
ape <- abs((z.calc - z.chart) / z.chart) * 100
df <- as.data.frame(list(Ppr = ppr,  z.calc =z.calc, z.chart = z.chart, ape=ape))
rownames(df) <- tpr
df

At lower Tpr there is some small error. We see a difference between the values of z from the ANN10 calculation and the value read from the Standing-Katz chart.

Get values of z for combinations of Ppr and Tpr

In this example we provide vectors instead of a single point. With the same ppr and tpr vectors that we use for the correlation, we do the same for the Standing-Katz chart. We want to compare both and find the absolute percentage error or APE.

# test with 1st-derivative using the values from paper 
ppr <- c(0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5) 
tpr <- c(1.05, 1.1, 1.7, 2) 

# calculate using the correlation
z.calc <- z.Ann10(ppr, tpr)

# With the same ppr and tpr vector, we do the same for the Standing-Katz chart
z.chart <- getStandingKatzMatrix(ppr_vector = ppr, tpr_vector = tpr)
ape <- abs((z.calc - z.chart) / z.chart) * 100

# calculate the APE
cat("z.correlation \n"); print(z.calc)
cat("\n z.chart \n"); print(z.chart)
cat("\n APE \n"); print(ape)

Analyze the error at the isotherms

Applying the function summary over the transpose of the matrix:

sum_t_ape <- summary(t(ape))
sum_t_ape

Analyze the error for greater values of Tpr

library(zFactor)
# enter vectors for Tpr and Ppr
tpr2 <- c(1.2, 1.3, 1.5, 2.0, 3.0) 
ppr2 <- c(0.5, 1.5, 2.5, 3.5, 4.5, 5.5) 

# get z values from the SK chart
z.chart <- getStandingKatzMatrix(ppr_vector = ppr2, tpr_vector = tpr2, pprRange = "lp")

# We do the same with the HY correlation:
# calculate z values at lower values of Tpr
z.calc <- z.Ann10(pres.pr = ppr2, temp.pr = tpr2) 
ape <- abs((z.calc - z.chart) / z.chart) * 100

# calculate the APE
cat("z.correlation \n"); print(z.calc)
cat("\n z.chart \n"); print(z.chart)
cat("\n APE \n"); print(ape)

Analyze the error at the isotherms

Applying the function summary over the transpose of the matrix to observe the error of the correlation at each isotherm.

sum_t_ape <- summary(t(ape))
sum_t_ape
 # Hall-Yarborough
 #      1.2               1.3              1.5               2         
 # Min.   :0.05224   Min.   :0.1105   Min.   :0.1021   Min.   :0.0809  
 # 1st Qu.:0.09039   1st Qu.:0.2080   1st Qu.:0.1623   1st Qu.:0.1814  
 # Median :0.28057   Median :0.3181   Median :0.1892   Median :0.1975  
 # Mean   :0.30122   Mean   :0.3899   Mean   :0.2597   Mean   :0.2284  
 # 3rd Qu.:0.51710   3rd Qu.:0.5355   3rd Qu.:0.3533   3rd Qu.:0.2627  
 # Max.   :0.57098   Max.   :0.8131   Max.   :0.5162   Max.   :0.4338  
 #       3          
 # Min.   :0.09128  
 # 1st Qu.:0.17466  
 # Median :0.35252  
 # Mean   :0.34820  
 # 3rd Qu.:0.52184  
 # Max.   :0.59923  

Prepare to plot SK vs N10 correlation

library(zFactor)
library(tibble)
library(ggplot2)

tpr2 <- c(1.05, 1.1, 1.2, 1.3) 
ppr2 <- c(0.5, 1.0, 1.5, 2, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0, 6.5) 

sk_dak_2 <- createTidyFromMatrix(ppr2, tpr2, correlation = "N10")
as_tible(sk_dak_2)

p <- ggplot(sk_dak_2, aes(x=Ppr, y=z.calc, group=Tpr, color=Tpr)) +
    geom_line() +
    geom_point() +
    geom_errorbar(aes(ymin=z.calc-dif, ymax=z.calc+dif), width=.4,
                  position=position_dodge(0.05))
print(p)

Analysis at the lowest Tpr

This is the isotherm where we usually see the greatest error.

library(zFactor)

sk_dak_3 <- sk_dak_2[sk_dak_2$Tpr==1.05,]
sk_dak_3

p <- ggplot(sk_dak_3, aes(x=Ppr, y=z.calc, group=Tpr, color=Tpr)) +
    geom_line() +
    geom_point() +
    geom_errorbar(aes(ymin=z.calc-dif, ymax=z.calc+dif), width=.2,
                  position=position_dodge(0.05))
print(p)

Analyzing performance of the N10 correlation for all the Tpr curves

In this last example, we compare the values of z at all the isotherms. We use the function getStandingKatzTpr to obtain all the isotherms or Tpr curves in the Standing-Katz chart that have been digitized. The next function createTidyFromMatrix calculates z using the correlation and prepares a tidy dataset ready to plot.

library(ggplot2)
library(tibble)

# get all `lp` Tpr curves
tpr_all <- getStandingKatzTpr(pprRange = "lp")
ppr <- c(0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5) 
sk_corr_all <- createTidyFromMatrix(ppr, tpr_all, correlation = "N10")
as_tible(sk_corr_all)

p <- ggplot(sk_corr_all, aes(x=Ppr, y=z.calc, group=Tpr, color=Tpr)) +
    geom_line() +
    geom_point() +
    geom_errorbar(aes(ymin=z.calc-dif, ymax=z.calc+dif), width=.4,
                  position=position_dodge(0.05))
print(p)

Range of applicability of the correlation

# MSE: Mean Squared Error
# RMSE: Root Mean Squared Error
# RSS: residual sum of square
# ARE:  Average Relative Error, %
# AARE: Average Absolute Relative Error, %
library(dplyr)
grouped <- group_by(sk_corr_all, Tpr, Ppr)

smry_tpr_ppr <- summarise(grouped, 
          RMSE= sqrt(mean((z.chart-z.calc)^2)), 
          MPE = sum((z.calc - z.chart) / z.chart) * 100 / n(),
          MAPE = sum(abs((z.calc - z.chart) / z.chart)) * 100 / n(), 
          MSE = sum((z.calc - z.chart)^2) / n(), 
          RSS = sum((z.calc - z.chart)^2),
          MAE = sum(abs(z.calc - z.chart)) / n(),
          RMLSE = sqrt(1/n()*sum((log(z.calc +1)-log(z.chart +1))^2))
          )

ggplot(smry_tpr_ppr, aes(Ppr, Tpr)) + 
    geom_tile(data=smry_tpr_ppr, aes(fill=MAPE), color="white") +
    scale_fill_gradient2(low="blue", high="red", mid="yellow", na.value = "pink",
                         midpoint=12.5, limit=c(0, 25), name="MAPE") + 
    theme(axis.text.x = element_text(angle=45, vjust=1, size=11, hjust=1)) + 
    coord_equal() +
    ggtitle("Artificial Neural Network", subtitle = "N10")

Plotting the Tpr and Ppr values that show more error

The MAPE (mean average percentage error) gradient bar indicates that the more red the square is, the more error there is.

library(dplyr)

sk_corr_all %>%
    filter(Tpr %in% c("1.05", "1.1")) %>%
    ggplot(aes(x = z.chart, y=z.calc, group = Tpr, color = Tpr)) +
    geom_point(size = 3) +
    geom_line(aes(x = z.chart, y = z.chart), color = "black") +
    facet_grid(. ~ Tpr, scales = "free") +
    geom_errorbar(aes(ymin=z.calc-abs(dif), ymax=z.calc+abs(dif)), 
                  position=position_dodge(0.5))

Looking numerically at the errors

Finally, the dataframe with the calculated errors between the z from the correlation and the z read from the chart:

as_tibble(smry_tpr_ppr)

References

zFactor documentation built on Aug. 1, 2019, 5:04 p.m.