# Beggs-Brill correlation

## How do we find the limits of accuracy in the BB correlation

### Get z at selected Ppr and Tpr

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

z.BeggsBrill(pres.pr = 1.5, temp.pr = 2.0)
# HY = 0.9580002
[1] 0.962902

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.BeggsBrill(pres.pr = 1.5, temp.pr = 1.1)
[1] 0.4631501

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 BB 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)

corr <- z.BeggsBrill(pres.pr = ppr, temp.pr = tpr)
print(corr)

# 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 0.9266436 0.7675523 0.6526911 0.6234648 0.6921991 0.7779095 0.8630653
1.5 0.9555248 0.8618306 0.7945385 0.7691830 0.7828753 0.8248905 0.8837555
1.7 0.9719193 0.9159219 0.8728791 0.8521620 0.8556641 0.8800665 0.9198223
2   0.9853337 0.9629020 0.9471826 0.9404180 0.9443010 0.9593080 0.9848256

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 - corr) / 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 -1.16 -1.53 -2.30  1.51 -1.20 -2.49 -2.26
1.50 -0.79 -0.33 -0.07  0.11  0.90  1.33  0.92
1.70 -0.40 -0.21  0.36  0.56  0.96  1.89  2.35
2.00 -0.34 -0.72 -0.66 -0.36  0.07  1.00  1.81

## Error by Ppr and by PPr

print(colSums(err))
  0.5   1.5   2.5   3.5   4.5   5.5   6.5
-2.69 -2.79 -2.67  1.82  0.73  1.73  2.82 
print(rowSums(err))
 1.30  1.50  1.70  2.00
-9.43  2.07  5.51  0.80 

## 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 BB correlation:

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

corr2 <- z.BeggsBrill(pres.pr = ppr2, temp.pr = tpr2)
print(corr2)
           0.5       1.5       2.5       3.5       4.5       5.5
1.05 0.8325491 0.2851494 0.3333796 0.4569204 0.5802310 0.7033638
1.1  0.8639321 0.4631501 0.3848774 0.4993213 0.6131885 0.7266084

Subtract the matrices and calculate the error in percentage:

err2 <- round((sk2 - corr2) / 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 -0.43 -12.71 2.80 2.99 2.97 3.25
1.10 -1.16  -8.72 2.07 0.14 0.29 0.33

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

t_err2 <- t(err2)
t_err2
      1.05  1.10
0.5  -0.43 -1.16
1.5 -12.71 -8.72
2.5   2.80  2.07
3.5   2.99  0.14
4.5   2.97  0.29
5.5   3.25  0.33

A statistical summary by Tpr curve:

sum_t_err2 <- summary(t_err2)
sum_t_err2
      1.05               1.10
Min.   :-12.7100   Min.   :-8.720
1st Qu.:  0.3775   1st Qu.:-0.835
Median :  2.8850   Median : 0.215
Mean   : -0.1883   Mean   :-1.175
3rd Qu.:  2.9850   3rd Qu.: 0.320
Max.   :  3.2500   Max.   : 2.070  

We can see that the errors in z with DAK are less than HY with a Min. :-12.7100 % and Max. : 3.2500 % for Tpr = 1.05, and a Min. :-8.720 %% and Max. : 2.070 %% for Tpr = 1.10.

## Prepare to plot SK chart vs BB correlation

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_corr_2 <- createTidyFromMatrix(ppr2, tpr2, correlation = "BB")
as_tibble(sk_corr_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  0.833 -0.00355
2 1.1     0.5   0.854  0.864 -0.00993
3 1.2     0.5   0.893  0.903 -0.00965
4 1.3     0.5   0.916  0.927 -0.0106
5 1.05    1     0.589  0.586  0.00330
6 1.1     1     0.669  0.687 -0.0184
7 1.2     1     0.779  0.790 -0.0109
8 1.3     1     0.835  0.845 -0.00951
9 1.05    1.5   0.253  0.285 -0.0321
10 1.1     1.5   0.426  0.463 -0.0372
# … with 42 more rows
library(ggplot2)

p <- ggplot(sk_corr_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_corr_3 <- sk_corr_2[sk_corr_2$Tpr==1.05,] sk_corr_3  Tpr Ppr z.chart z.calc dif 1 1.05 0.5 0.829 0.8325491 -0.003549064 5 1.05 1.0 0.589 0.5857006 0.003299370 9 1.05 1.5 0.253 0.2851494 -0.032149397 13 1.05 2.0 0.280 0.2715884 0.008411630 17 1.05 2.5 0.343 0.3333796 0.009620404 21 1.05 3.0 0.407 0.3951833 0.011816743 25 1.05 3.5 0.471 0.4569204 0.014079649 29 1.05 4.0 0.534 0.5186005 0.015399484 33 1.05 4.5 0.598 0.5802310 0.017769048 37 1.05 5.0 0.663 0.6418172 0.021182758 41 1.05 5.5 0.727 0.7033638 0.023636157 45 1.05 6.0 0.786 0.7648744 0.021125603 49 1.05 6.5 0.846 0.8263519 0.019648067 p <- ggplot(sk_corr_3, aes(x=Ppr, y=z.calc, group=Tpr, color=Tpr)) + geom_line(size = 1) + geom_point(shape = 21, fill = "white", size = 3) + geom_errorbar(aes(ymin=z.calc-dif, ymax=z.calc+dif), width=0.2, size = 0., position=position_dodge(0.05), color = "black") print(p) summary(sk_corr_3) # dif DAK # 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. :0.2716 Class :character 1st Qu.:2.0 1st Qu.:0.4070 1st Qu.:0.3952 Mode :character Median :3.5 Median :0.5890 Median :0.5802 Mean :3.5 Mean :0.5635 Mean :0.5535 3rd Qu.:5.0 3rd Qu.:0.7270 3rd Qu.:0.7034 Max. :6.5 Max. :0.8460 Max. :0.8325 dif Min. :-0.032149 1st Qu.: 0.008412 Median : 0.014080 Mean : 0.010022 3rd Qu.: 0.019648 Max. : 0.023636  With this information there is no much we can say about Beggs-Brill. ## Analyzing performance of the BB correlation for all the Tpr curves 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 = "BB") 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: 112 x 5 Tpr Ppr z.chart z.calc dif <chr> <dbl> <dbl> <dbl> <dbl> 1 1.05 0.5 0.829 0.833 -0.00355 2 1.1 0.5 0.854 0.864 -0.00993 3 1.2 0.5 0.893 0.903 -0.00965 4 1.3 0.5 0.916 0.927 -0.0106 5 1.4 0.5 0.936 0.943 -0.00730 6 1.5 0.5 0.948 0.956 -0.00752 7 1.6 0.5 0.959 0.965 -0.00578 8 1.7 0.5 0.968 0.972 -0.00392 9 1.8 0.5 0.974 0.977 -0.00349 10 1.9 0.5 0.978 0.982 -0.00386 # … with 102 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, RMSE= sqrt(mean((z.chart-z.calc)^2)), MSE = sum((z.calc - z.chart)^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() + ggtitle("Beggs-Brill", subtitle = "BB") The errors with Beggs and Brill are just so big and some z values are even negative. We have to be very careful when using this Beggs and Brill correlation. ## Plotting the Tpr and Ppr values that show more error 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) + geom_errorbar(aes(ymin=z.calc-abs(dif), ymax=z.calc+abs(dif)), position=position_dodge(0.5)) library(dplyr) sk_corr_all %>% filter(Tpr %in% c("2.6", "2.8")) %>% 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) + geom_errorbar(aes(ymin=z.calc-abs(dif), ymax=z.calc+abs(dif)), position=position_dodge(0.5)) Let’s see which observations (rows) have z values that are negative: sk_corr_all[which(sk_corr_all$z.calc < 0), ]
    Tpr Ppr z.chart    z.calc      dif
80    3 4.5   1.041 -0.889701 1.930701
96    3 5.5   1.056 -2.517895 3.573895
112   3 6.5   1.075 -4.882513 5.957513

Or see which rows contain z values that show an error greater than 15%:

sk_corr_all[which(abs(sk_corr_all\$dif) > 0.15), ]
    Tpr Ppr z.chart     z.calc       dif
48    3 2.5   1.018  0.7082371 0.3097629
63  2.8 3.5   1.016  0.8188474 0.1971526
64    3 3.5   1.029  0.1399229 0.8890771
79  2.8 4.5   1.030  0.6645377 0.3654623
80    3 4.5   1.041 -0.8897010 1.9307010
95  2.8 5.5   1.049  0.4548462 0.5941538
96    3 5.5   1.056 -2.5178952 3.5738952
111 2.8 6.5   1.069  0.1860147 0.8829853
112   3 6.5   1.075 -4.8825128 5.9575128

You can also see that there are three rows with error greater than 100% !

## Looking numerically at the errors in BB vs SK chart

# get all lp Tpr curves
tpr <- getStandingKatzTpr(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_corr <- z.BeggsBrill(pres.pr = ppr, temp.pr = tpr)
cat("Calculated from the correlation \n")
print(all_corr)

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_corr) / 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
1.05 0.8325491 0.2851494 0.3333796 0.4569204  0.5802310  0.7033638
1.1  0.8639321 0.4631501 0.3848774 0.4993213  0.6131885  0.7266084
1.2  0.9026461 0.6758659 0.4977865 0.5605500  0.6589953  0.7567099
1.3  0.9266436 0.7675523 0.6526911 0.6234648  0.6921991  0.7779095
1.4  0.9432988 0.8224273 0.7388187 0.7103492  0.7343367  0.7966791
1.5  0.9555248 0.8618306 0.7945385 0.7691830  0.7828753  0.8248905
1.6  0.9647818 0.8920527 0.8376331 0.8139976  0.8214333  0.8532880
1.7  0.9719193 0.9159219 0.8728791 0.8521620  0.8556641  0.8800665
1.8  0.9774854 0.9350585 0.9022222 0.8855750  0.8874829  0.9067399
1.9  0.9818623 0.9504907 0.9267646 0.9149047  0.9171073  0.9333359
2    0.9853337 0.9629020 0.9471826 0.9404180  0.9443010  0.9593080
2.2  0.9904020 0.9803535 0.9768810 0.9795480  0.9887829  1.0049837
2.4  0.9940266 0.9894204 0.9909094 0.9974289  1.0086828  1.0246183
2.6  0.9971532 0.9902007 0.9817596 0.9719460  0.9608415  0.9485126
2.8  1.0004100 0.9802574 0.9222416 0.8188474  0.6645377  0.4548462
3    1.0040392 0.9514557 0.7082371 0.1399229 -0.8897010 -2.5178952
6.5
1.05  0.8263519
1.1   0.8396647
1.2   0.8538299
1.3   0.8630653
1.4   0.8717561
1.5   0.8837555
1.6   0.9011787
1.7   0.9198223
1.8   0.9400764
1.9   0.9619430
2     0.9848256
2.2   1.0282971
2.4   1.0452638
2.6   0.9350177
2.8   0.1860147
3    -4.8825128

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 -0.43 -12.71  2.80  2.99   2.97   3.25   2.32
1.10 -1.16  -8.72  2.07  0.14   0.29   0.33   0.16
1.20 -1.08  -2.87  4.09  0.79  -1.38  -2.12  -1.53
1.30 -1.16  -1.53 -2.30  1.51  -1.20  -2.49  -2.26
1.40 -0.78  -0.79 -1.63 -0.76  -0.05  -0.59  -0.78
1.50 -0.79  -0.33 -0.07  0.11   0.90   1.33   0.92
1.60 -0.60  -0.46  0.16  0.25   0.91   1.69   1.83
1.70 -0.40  -0.21  0.36  0.56   0.96   1.89   2.35
1.80 -0.36  -0.22  0.31  0.61   1.50   2.40   2.78
1.90 -0.39  -0.58 -0.30  0.12   0.75   1.65   2.34
2.00 -0.34  -0.72 -0.66 -0.36   0.07   1.00   1.81
2.20 -0.14  -0.76 -1.44 -1.72  -1.31  -0.50   0.07
2.40 -0.10  -0.55 -1.11 -1.47  -0.97  -0.16   0.36
2.60 -0.02   0.38  1.23  2.81   5.43   8.62  11.96
2.80 -0.14   2.17  8.51 19.40  35.48  56.64  82.60
3.00 -0.20   5.70 30.43 86.40 185.47 338.44 554.19

Errors in Ppr
0.5               1.5                2.5               3.5
Min.   :-1.1600   Min.   :-12.7100   Min.   :-2.3000   Min.   :-1.7200
1st Qu.:-0.7825   1st Qu.: -0.9750   1st Qu.:-0.7725   1st Qu.:-0.0075
Median :-0.3950   Median : -0.5650   Median : 0.2350   Median : 0.4050
Mean   :-0.5056   Mean   : -1.3875   Mean   : 2.6531   Mean   : 6.9612
3rd Qu.:-0.1850   3rd Qu.: -0.2175   3rd Qu.: 2.2525   3rd Qu.: 1.8350
Max.   :-0.0200   Max.   :  5.7000   Max.   :30.4300   Max.   :86.4000
4.5               5.5               6.5
Min.   : -1.380   Min.   : -2.490   Min.   : -2.2600
1st Qu.: -0.280   1st Qu.: -0.245   1st Qu.:  0.1375
Median :  0.825   Median :  1.490   Median :  1.8200
Mean   : 14.364   Mean   : 25.711   Mean   : 41.1950
3rd Qu.:  1.867   3rd Qu.:  2.612   3rd Qu.:  2.4575
Max.   :185.470   Max.   :338.440   Max.   :554.1900

Errors for the transposed matrix: Tpr
1.05              1.10              1.20              1.30
Min.   :-12.710   Min.   :-8.7200   Min.   :-2.8700   Min.   :-2.490
1st Qu.:  0.945   1st Qu.:-0.5100   1st Qu.:-1.8250   1st Qu.:-2.280
Median :  2.800   Median : 0.1600   Median :-1.3800   Median :-1.530
Mean   :  0.170   Mean   :-0.9843   Mean   :-0.5857   Mean   :-1.347
3rd Qu.:  2.980   3rd Qu.: 0.3100   3rd Qu.:-0.1450   3rd Qu.:-1.180
Max.   :  3.250   Max.   : 2.0700   Max.   : 4.0900   Max.   : 1.510
1.40              1.50              1.60            1.70
Min.   :-1.6300   Min.   :-0.7900   Min.   :-0.60   Min.   :-0.4000
1st Qu.:-0.7850   1st Qu.:-0.2000   1st Qu.:-0.15   1st Qu.: 0.0750
Median :-0.7800   Median : 0.1100   Median : 0.25   Median : 0.5600
Mean   :-0.7686   Mean   : 0.2957   Mean   : 0.54   Mean   : 0.7871
3rd Qu.:-0.6750   3rd Qu.: 0.9100   3rd Qu.: 1.30   3rd Qu.: 1.4250
Max.   :-0.0500   Max.   : 1.3300   Max.   : 1.83   Max.   : 2.3500
1.80             1.90              2.00              2.20
Min.   :-0.360   Min.   :-0.5800   Min.   :-0.7200   Min.   :-1.7200
1st Qu.: 0.045   1st Qu.:-0.3450   1st Qu.:-0.5100   1st Qu.:-1.3750
Median : 0.610   Median : 0.1200   Median :-0.3400   Median :-0.7600
Mean   : 1.003   Mean   : 0.5129   Mean   : 0.1143   Mean   :-0.8286
3rd Qu.: 1.950   3rd Qu.: 1.2000   3rd Qu.: 0.5350   3rd Qu.:-0.3200
Max.   : 2.780   Max.   : 2.3400   Max.   : 1.8100   Max.   : 0.0700
2.40              2.60             2.80            3.00
Min.   :-1.4700   Min.   :-0.020   Min.   :-0.14   Min.   : -0.20
1st Qu.:-1.0400   1st Qu.: 0.805   1st Qu.: 5.34   1st Qu.: 18.07
Median :-0.5500   Median : 2.810   Median :19.40   Median : 86.40
Mean   :-0.5714   Mean   : 4.344   Mean   :29.24   Mean   :171.49
3rd Qu.:-0.1300   3rd Qu.: 7.025   3rd Qu.:46.06   3rd Qu.:261.95
Max.   : 0.3600   Max.   :11.960   Max.   :82.60   Max.   :554.19  `