Artificial Neural Network correlation

Get `z` at selected `Ppr` and `Tpr`

# get a z value using DPR correlation
library(zFactor)
# library(rJava)

z.Ann10(pres.pr = 1.5, temp.pr = 2.0)
# HY = 0.9580002

[1] 0.9572277

From the Standing-Katz chart we obtain a digitized point at the same Ppr and Tpr:

# get a z value from the SK chart at the same Ppr and Tpr
library(zFactor)

tpr_vec <- c(2.0)
getStandingKatzMatrix(tpr_vector = tpr_vec, 
                      pprRange = "lp")[1, "1.5"]

  1.5 
0.956

It looks pretty good.

Get `z` at selected `Ppr` and `Tpr`

library(zFactor)
z.Ann10(pres.pr = 1.5, temp.pr = 1.1)

[1] 0.4309125

From the Standing-Katz chart we obtain a digitized point:

library(zFactor)
tpr_vec <- c(1.1)
getStandingKatzMatrix(tpr_vector = tpr_vec, 
                      pprRange = "lp")[1, "1.5"]

  1.5 
0.426

At lower Tpr there is some 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 several `Ppr` and `Tpr`

# test HY 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(1.3, 1.5, 1.7, 2) 
 
dpr <- z.Ann10(ppr, tpr)
print(dpr)

# From Hall-Yarborough
#    0.5       1.5       2.5       3.5       4.5       5.5       6.5
# 1.3 0.9176300 0.7534433 0.6399020 0.6323003 0.6881127 0.7651710 0.8493794
# 1.5 0.9496855 0.8581232 0.7924067 0.7687902 0.7868071 0.8316848 0.8906351
# 1.7 0.9682547 0.9134862 0.8756412 0.8605668 0.8694525 0.8978885 0.9396353
# 2   0.9838234 0.9580002 0.9426939 0.9396286 0.9490995 0.9697839 0.9994317

# From Dranchuk-AbouKassem
#  0.5       1.5       2.5       3.5       4.5       5.5       6.5
# 1.3 0.9203019 0.7543694 0.6377871 0.6339357 0.6898314 0.7663247 0.8499523
# 1.5 0.9509373 0.8593144 0.7929993 0.7710525 0.7896224 0.8331893 0.8904317
# 1.7 0.9681353 0.9128087 0.8753784 0.8619509 0.8721085 0.9003962 0.9409634
# 2   0.9824731 0.9551087 0.9400752 0.9385273 0.9497137 0.9715388 1.0015560

          0.5       1.5       2.5       3.5       4.5       5.5       6.5
1.3 0.9196115 0.7567070 0.6394479 0.6341957 0.6857549 0.7611212 0.8471371
1.5 0.9508509 0.8607096 0.7940885 0.7685691 0.7867923 0.8323518 0.8918874
1.7 0.9682749 0.9146453 0.8767457 0.8581919 0.8672123 0.8978116 0.9413442
2   0.9839990 0.9572277 0.9414698 0.9352303 0.9453140 0.9693022 1.0014522

With the same ppr and tpr vectors, we do the same for the Standing-Katz chart:

library(zFactor)

sk <- getStandingKatzMatrix(ppr_vector = ppr, tpr_vector = tpr)
print(sk)

       0.5   1.5   2.5   3.5   4.5   5.5   6.5
1.30 0.916 0.756 0.638 0.633 0.684 0.759 0.844
1.50 0.948 0.859 0.794 0.770 0.790 0.836 0.892
1.70 0.968 0.914 0.876 0.857 0.864 0.897 0.942
2.00 0.982 0.956 0.941 0.937 0.945 0.969 1.003

Subtract the two matrices and find the difference:

err <- round((sk - dpr) / sk * 100, 2)
err

# DAK
# 0.5   1.5  2.5   3.5   4.5   5.5   6.5
# 1.30 -0.47  0.22 0.03 -0.15 -0.85 -0.97 -0.71
# 1.50 -0.31 -0.04 0.13 -0.14  0.05  0.34  0.18
# 1.70 -0.01  0.13 0.07 -0.58 -0.94 -0.38  0.11
# 2.00 -0.05  0.09 0.10 -0.16 -0.50 -0.26  0.14

       0.5   1.5   2.5   3.5   4.5   5.5   6.5
1.30 -0.39 -0.09 -0.23 -0.19 -0.26 -0.28 -0.37
1.50 -0.30 -0.20 -0.01  0.19  0.41  0.44  0.01
1.70 -0.03 -0.07 -0.09 -0.14 -0.37 -0.09  0.07
2.00 -0.20 -0.13 -0.05  0.19 -0.03 -0.03  0.15

Error by `Ppr` and by `PPr`

print(colSums(err))

  0.5   1.5   2.5   3.5   4.5   5.5   6.5 
-0.92 -0.49 -0.38  0.05 -0.25  0.04 -0.14

print(rowSums(err))

 1.30  1.50  1.70  2.00 
-1.81  0.54 -0.72 -0.10

Analyze the error for smaller values of `Tpr`

library(zFactor)

tpr2 <- c(1.05, 1.1) 
ppr2 <- c(0.5, 1.5, 2.5, 3.5, 4.5, 5.5) 

sk2 <- getStandingKatzMatrix(ppr_vector = ppr2, tpr_vector = tpr2, pprRange = "lp")
sk2

       0.5   1.5   2.5   3.5   4.5   5.5
1.05 0.829 0.253 0.343 0.471 0.598 0.727
1.10 0.854 0.426 0.393 0.500 0.615 0.729

We do the same with the DPR correlation:

# calculate z values at lower values of Tpr
library(zFactor)

dpr2 <- z.Ann10(ppr2, tpr2)
print(dpr2)

           0.5       1.5       2.5       3.5       4.5       5.5
1.05 0.8324799 0.2526076 0.3420322 0.4693520 0.5991874 0.7254470
1.1  0.8547310 0.4309125 0.3930420 0.4983162 0.6136523 0.7278621

Subtract the matrices and calculate the error in percentage:

err2 <- round((sk2 - dpr2) / sk2 * 100, 2)
err2

# DAK
# 0.5    1.5    2.5   3.5   4.5   5.5
# 1.05 -0.13 -12.15 -12.78 -7.49 -4.34 -1.68
# 1.10 -0.36  -4.79  -4.97 -3.56 -2.14 -1.21

       0.5   1.5   2.5  3.5   4.5  5.5
1.05 -0.42  0.16  0.28 0.35 -0.20 0.21
1.10 -0.09 -1.15 -0.01 0.34  0.22 0.16

Transposing the matrix with Tpr as columns and Ppr as rows:

t_err2 <- t(err2)
t_err2

     1.05  1.10
0.5 -0.42 -0.09
1.5  0.16 -1.15
2.5  0.28 -0.01
3.5  0.35  0.34
4.5 -0.20  0.22
5.5  0.21  0.16

A statistical summary by Tpr curve:

sum_t_err2 <- summary(t_err2)
sum_t_err2

      1.05               1.10         
 Min.   :-0.42000   Min.   :-1.15000  
 1st Qu.:-0.11000   1st Qu.:-0.07000  
 Median : 0.18500   Median : 0.07500  
 Mean   : 0.06333   Mean   :-0.08833  
 3rd Qu.: 0.26250   3rd Qu.: 0.20500  
 Max.   : 0.35000   Max.   : 0.34000

# We can see that the errors in `z` with `ANN10` are less than `HY` with a `r sum_t_err2[1,1]`% and `r sum_t_err2[6,1]`% for `Tpr = 1.05`, and a `r sum_t_err2[1,2]`% and `r sum_t_err2[6,2]`% for `Tpr = 1.10`.

Prepare to plot `SK` chart vs `ANN10` model.

library(zFactor)
library(tibble)

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_dpr_2 <- createTidyFromMatrix(ppr2, tpr2, correlation = "N10")
as.tibble(sk_dpr_2)

# A tibble: 52 x 5
     Tpr   Ppr z.chart    z.calc           dif
   <chr> <dbl>   <dbl>     <dbl>         <dbl>
 1  1.05   0.5   0.829 0.8324799 -0.0034799212
 2   1.1   0.5   0.854 0.8547310 -0.0007310197
 3   1.2   0.5   0.893 0.8953923 -0.0023923156
 4   1.3   0.5   0.916 0.9196115 -0.0036115485
 5  1.05   1.0   0.589 0.5896265 -0.0006265401
 6   1.1   1.0   0.669 0.6708697 -0.0018696755
 7   1.2   1.0   0.779 0.7807145 -0.0017144762
 8   1.3   1.0   0.835 0.8363278 -0.0013277938
 9  1.05   1.5   0.253 0.2526076  0.0003924371
10   1.1   1.5   0.426 0.4309125 -0.0049124610
# ... with 42 more rows

library(ggplot2)

p <- ggplot(sk_dpr_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`

Extract only values at Tpr = 1.05.

sk_dpr_3 <- sk_dpr_2[sk_dpr_2$Tpr==1.05,]
sk_dpr_3

    Tpr Ppr z.chart    z.calc           dif
1  1.05 0.5   0.829 0.8324799 -0.0034799212
5  1.05 1.0   0.589 0.5896265 -0.0006265401
9  1.05 1.5   0.253 0.2526076  0.0003924371
13 1.05 2.0   0.280 0.2813986 -0.0013986023
17 1.05 2.5   0.343 0.3420322  0.0009678343
21 1.05 3.0   0.407 0.4046718  0.0023282390
25 1.05 3.5   0.471 0.4693520  0.0016480466
29 1.05 4.0   0.534 0.5347267 -0.0007266737
33 1.05 4.5   0.598 0.5991874 -0.0011874468
37 1.05 5.0   0.663 0.6627276  0.0002723595
41 1.05 5.5   0.727 0.7254470  0.0015529844
45 1.05 6.0   0.786 0.7868394 -0.0008393750
49 1.05 6.5   0.846 0.8464481 -0.0004481081

p <- ggplot(sk_dpr_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=.4,
                  position=position_dodge(0.05))
print(p)

summary(sk_dpr_3)

 #         dif ANN10      
 # Min.   :-0.048404  
 # 1st Qu.:-0.035300  
 # Median :-0.025978  
 # Mean   :-0.023178  
 # 3rd Qu.:-0.009960  
 # Max.   : 0.002325

     Tpr                 Ppr         z.chart           z.calc      
 Length:13          Min.   :0.5   Min.   :0.2530   Min.   :0.2526  
 Class :character   1st Qu.:2.0   1st Qu.:0.4070   1st Qu.:0.4047  
 Mode  :character   Median :3.5   Median :0.5890   Median :0.5896  
                    Mean   :3.5   Mean   :0.5635   Mean   :0.5637  
                    3rd Qu.:5.0   3rd Qu.:0.7270   3rd Qu.:0.7254  
                    Max.   :6.5   Max.   :0.8460   Max.   :0.8464  
      dif            
 Min.   :-0.0034799  
 1st Qu.:-0.0008394  
 Median :-0.0004481  
 Mean   :-0.0001188  
 3rd Qu.: 0.0009678  
 Max.   : 0.0023282

Analyzing performance of the `ANN10` correlation for all the `Tpr` curves

library(ggplot2)
library(tibble)

# get all `lp` Tpr curves
tpr_all <- getCurvesDigitized(pprRange = "lp")
ppr <- c(0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0, 6.5) 
sk_corr_all <- createTidyFromMatrix(ppr, tpr_all, correlation = "N10")
as.tibble(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)

# A tibble: 208 x 5
     Tpr   Ppr z.chart    z.calc           dif
   <chr> <dbl>   <dbl>     <dbl>         <dbl>
 1  1.05   0.5   0.829 0.8324799 -0.0034799212
 2   1.1   0.5   0.854 0.8547310 -0.0007310197
 3   1.2   0.5   0.893 0.8953923 -0.0023923156
 4   1.3   0.5   0.916 0.9196115 -0.0036115485
 5   1.4   0.5   0.936 0.9367118 -0.0007118312
 6   1.5   0.5   0.948 0.9508509 -0.0028509478
 7   1.6   0.5   0.959 0.9607316 -0.0017315940
 8   1.7   0.5   0.968 0.9682749 -0.0002748589
 9   1.8   0.5   0.974 0.9758251 -0.0018251256
10   1.9   0.5   0.978 0.9814269 -0.0034269385
# ... with 198 more rows

# MSE: Mean Squared Error
# RMSE: Root Mean Sqyared 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, 
          #mean=mean(z.calc), 
          #sd=sd(z.calc), 
          RMSE= sqrt(mean((z.chart-z.calc)^2)), 
          MSE = sum((z.calc - z.chart)^2) / n(), 
          # rmse2 = sqrt(sum((z.chart-z.calc)^2)/n()),
          RSS = sum((z.calc - z.chart)^2),
          ARE = sum((z.calc - z.chart) / z.chart) * 100 / n(),
          AARE = sum( abs((z.calc - z.chart) / z.chart)) * 100 / n()
          )

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

Looking numerically at the errors in `DKP` vs `SK` chart

# get all `lp` Tpr curves
tpr <- getCurvesDigitized(pprRange = "lp")
ppr <- c(0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5) 

# calculate HY for the given Tpr
all_dak <- z.Ann10(pres.pr = ppr, temp.pr = tpr)
cat("Calculated from the correlation \n")
print(all_dak) 

cat("\nStanding-Katz chart\n")
all_sk <- getStandingKatzMatrix(ppr_vector = ppr, tpr_vector = tpr)
all_sk

# find the error
cat("\n Errors in percentage \n")
all_err <- round((all_sk - all_dak) / all_sk * 100, 2)  # in percentage
all_err

cat("\n Errors in Ppr\n")
summary(all_err)

# for the transposed matrix
cat("\n Errors for the transposed matrix: Tpr \n")
summary(t(all_err))

Calculated from the correlation 
           0.5       1.5       2.5       3.5       4.5       5.5       6.5
1.05 0.8324799 0.2526076 0.3420322 0.4693520 0.5991874 0.7254470 0.8464481
1.1  0.8547310 0.4309125 0.3930420 0.4983162 0.6136523 0.7278621 0.8417240
1.2  0.8953923 0.6607512 0.5179963 0.5676801 0.6492856 0.7424365 0.8393892
1.3  0.9196115 0.7567070 0.6394479 0.6341957 0.6857549 0.7611212 0.8471371
1.4  0.9367118 0.8179531 0.7301083 0.7058966 0.7360320 0.7938440 0.8650626
1.5  0.9508509 0.8607096 0.7940885 0.7685691 0.7867923 0.8323518 0.8918874
1.6  0.9607316 0.8909372 0.8413772 0.8186001 0.8303206 0.8669610 0.9174184
1.7  0.9682749 0.9146453 0.8767457 0.8581919 0.8672123 0.8978116 0.9413442
1.8  0.9758251 0.9330673 0.9033038 0.8900081 0.8983954 0.9253309 0.9638663
1.9  0.9814269 0.9463194 0.9244863 0.9154178 0.9242331 0.9491918 0.9841183
2    0.9839990 0.9572277 0.9414698 0.9352303 0.9453140 0.9693022 1.0014522
2.2  0.9900357 0.9744007 0.9641093 0.9632802 0.9770073 1.0002489 1.0281919
2.4  0.9948006 0.9851441 0.9796739 0.9835802 0.9995876 1.0223775 1.0476552
2.6  0.9973689 0.9948164 0.9945873 1.0008562 1.0160982 1.0374500 1.0612594
2.8  1.0003231 1.0036769 1.0082887 1.0163706 1.0293895 1.0474565 1.0693476
3    1.0028553 1.0095269 1.0179196 1.0286167 1.0412701 1.0563968 1.0751669

Standing-Katz chart
       0.5   1.5   2.5   3.5   4.5   5.5   6.5
1.05 0.829 0.253 0.343 0.471 0.598 0.727 0.846
1.10 0.854 0.426 0.393 0.500 0.615 0.729 0.841
1.20 0.893 0.657 0.519 0.565 0.650 0.741 0.841
1.30 0.916 0.756 0.638 0.633 0.684 0.759 0.844
1.40 0.936 0.816 0.727 0.705 0.734 0.792 0.865
1.50 0.948 0.859 0.794 0.770 0.790 0.836 0.892
1.60 0.959 0.888 0.839 0.816 0.829 0.868 0.918
1.70 0.968 0.914 0.876 0.857 0.864 0.897 0.942
1.80 0.974 0.933 0.905 0.891 0.901 0.929 0.967
1.90 0.978 0.945 0.924 0.916 0.924 0.949 0.985
2.00 0.982 0.956 0.941 0.937 0.945 0.969 1.003
2.20 0.989 0.973 0.963 0.963 0.976 1.000 1.029
2.40 0.993 0.984 0.980 0.983 0.999 1.023 1.049
2.60 0.997 0.994 0.994 1.000 1.016 1.038 1.062
2.80 0.999 1.002 1.008 1.016 1.030 1.049 1.069
3.00 1.002 1.009 1.018 1.029 1.041 1.056 1.075

 Errors in percentage 
       0.5   1.5   2.5   3.5   4.5   5.5   6.5
1.05 -0.42  0.16  0.28  0.35 -0.20  0.21 -0.05
1.10 -0.09 -1.15 -0.01  0.34  0.22  0.16 -0.09
1.20 -0.27 -0.57  0.19 -0.47  0.11 -0.19  0.19
1.30 -0.39 -0.09 -0.23 -0.19 -0.26 -0.28 -0.37
1.40 -0.08 -0.24 -0.43 -0.13 -0.28 -0.23 -0.01
1.50 -0.30 -0.20 -0.01  0.19  0.41  0.44  0.01
1.60 -0.18 -0.33 -0.28 -0.32 -0.16  0.12  0.06
1.70 -0.03 -0.07 -0.09 -0.14 -0.37 -0.09  0.07
1.80 -0.19 -0.01  0.19  0.11  0.29  0.39  0.32
1.90 -0.35 -0.14 -0.05  0.06 -0.03 -0.02  0.09
2.00 -0.20 -0.13 -0.05  0.19 -0.03 -0.03  0.15
2.20 -0.10 -0.14 -0.12 -0.03 -0.10 -0.02  0.08
2.40 -0.18 -0.12  0.03 -0.06 -0.06  0.06  0.13
2.60 -0.04 -0.08 -0.06 -0.09 -0.01  0.05  0.07
2.80 -0.13 -0.17 -0.03 -0.04  0.06  0.15 -0.03
3.00 -0.09 -0.05  0.01  0.04 -0.03 -0.04 -0.02

 Errors in Ppr
      0.5               1.5               2.5                3.5          
 Min.   :-0.4200   Min.   :-1.1500   Min.   :-0.43000   Min.   :-0.47000  
 1st Qu.:-0.2775   1st Qu.:-0.2100   1st Qu.:-0.09750   1st Qu.:-0.13250  
 Median :-0.1800   Median :-0.1350   Median :-0.04000   Median :-0.03500  
 Mean   :-0.1900   Mean   :-0.2081   Mean   :-0.04125   Mean   :-0.01188  
 3rd Qu.:-0.0900   3rd Qu.:-0.0775   3rd Qu.: 0.01500   3rd Qu.: 0.13000  
 Max.   :-0.0300   Max.   : 0.1600   Max.   : 0.28000   Max.   : 0.35000  
      4.5               5.5               6.5         
 Min.   :-0.3700   Min.   :-0.2800   Min.   :-0.3700  
 1st Qu.:-0.1700   1st Qu.:-0.0525   1st Qu.:-0.0225  
 Median :-0.0300   Median : 0.0150   Median : 0.0650  
 Mean   :-0.0275   Mean   : 0.0425   Mean   : 0.0375  
 3rd Qu.: 0.0725   3rd Qu.: 0.1525   3rd Qu.: 0.1000  
 Max.   : 0.4100   Max.   : 0.4400   Max.   : 0.3200  

 Errors for the transposed matrix: Tpr 
      1.05               1.10               1.20              1.30        
 Min.   :-0.42000   Min.   :-1.15000   Min.   :-0.5700   Min.   :-0.3900  
 1st Qu.:-0.12500   1st Qu.:-0.09000   1st Qu.:-0.3700   1st Qu.:-0.3250  
 Median : 0.16000   Median :-0.01000   Median :-0.1900   Median :-0.2600  
 Mean   : 0.04714   Mean   :-0.08857   Mean   :-0.1443   Mean   :-0.2586  
 3rd Qu.: 0.24500   3rd Qu.: 0.19000   3rd Qu.: 0.1500   3rd Qu.:-0.2100  
 Max.   : 0.35000   Max.   : 0.34000   Max.   : 0.1900   Max.   :-0.0900  
      1.40             1.50               1.60              1.70        
 Min.   :-0.430   Min.   :-0.30000   Min.   :-0.3300   Min.   :-0.3700  
 1st Qu.:-0.260   1st Qu.:-0.10500   1st Qu.:-0.3000   1st Qu.:-0.1150  
 Median :-0.230   Median : 0.01000   Median :-0.1800   Median :-0.0900  
 Mean   :-0.200   Mean   : 0.07714   Mean   :-0.1557   Mean   :-0.1029  
 3rd Qu.:-0.105   3rd Qu.: 0.30000   3rd Qu.:-0.0500   3rd Qu.:-0.0500  
 Max.   :-0.010   Max.   : 0.44000   Max.   : 0.1200   Max.   : 0.0700  
      1.80              1.90               2.00         
 Min.   :-0.1900   Min.   :-0.35000   Min.   :-0.20000  
 1st Qu.: 0.0500   1st Qu.:-0.09500   1st Qu.:-0.09000  
 Median : 0.1900   Median :-0.03000   Median :-0.03000  
 Mean   : 0.1571   Mean   :-0.06286   Mean   :-0.01429  
 3rd Qu.: 0.3050   3rd Qu.: 0.02000   3rd Qu.: 0.06000  
 Max.   : 0.3900   Max.   : 0.09000   Max.   : 0.19000  
      2.20               2.40               2.60         
 Min.   :-0.14000   Min.   :-0.18000   Min.   :-0.09000  
 1st Qu.:-0.11000   1st Qu.:-0.09000   1st Qu.:-0.07000  
 Median :-0.10000   Median :-0.06000   Median :-0.04000  
 Mean   :-0.06143   Mean   :-0.02857   Mean   :-0.02286  
 3rd Qu.:-0.02500   3rd Qu.: 0.04500   3rd Qu.: 0.02000  
 Max.   : 0.08000   Max.   : 0.13000   Max.   : 0.07000  
      2.80               3.00         
 Min.   :-0.17000   Min.   :-0.09000  
 1st Qu.:-0.08500   1st Qu.:-0.04500  
 Median :-0.03000   Median :-0.03000  
 Mean   :-0.02714   Mean   :-0.02571  
 3rd Qu.: 0.01500   3rd Qu.:-0.00500  
 Max.   : 0.15000   Max.   : 0.04000

Artificial Neural Network correlation

Vignette Author

2017-06-27

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

Get `z` at selected `Ppr` and `Tpr`

Get `z` at selected `Ppr` and `Tpr`

Get values of `z` for several `Ppr` and `Tpr`

Error by `Ppr` and by `PPr`

Analyze the error for smaller values of `Tpr`

Prepare to plot `SK` chart vs `ANN10` model.

Analysis at the lowest `Tpr`

Analyzing performance of the `ANN10` correlation for all the `Tpr` curves

Looking numerically at the errors in `DKP` vs `SK` chart

Artificial Neural Network correlation

Vignette Author

2017-06-27

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

Get z at selected Ppr and Tpr

Get z at selected Ppr and Tpr

Get values of z for several Ppr and Tpr

Error by Ppr and by PPr

Analyze the error for smaller values of Tpr

Prepare to plot SK chart vs ANN10 model.

Analysis at the lowest Tpr

Analyzing performance of the ANN10 correlation for all the Tpr curves

Looking numerically at the errors in DKP vs SK chart

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

Get `z` at selected `Ppr` and `Tpr`

Get `z` at selected `Ppr` and `Tpr`

Get values of `z` for several `Ppr` and `Tpr`

Error by `Ppr` and by `PPr`

Analyze the error for smaller values of `Tpr`

Prepare to plot `SK` chart vs `ANN10` model.

Analysis at the lowest `Tpr`

Analyzing performance of the `ANN10` correlation for all the `Tpr` curves

Looking numerically at the errors in `DKP` vs `SK` chart