Applied Geostatistics in R: 3. Linear Discriminant Analysis (LDA) for Multinomial Lithofacies Classification in a Sandstone Formation
Watheq J. Al-Mudhafar, Craft and Hawkins Department of Petroleum Engineering, Louisiana State University
August 4, 2015
The Linear Discriminant Analysis (LDA) tries to produce a linear combination of features that separate two or more classes after projection the data to the specified direction. To validating a LDA model and preserve the integrity of the statistical inference, cross-validation is used for estimating generalization error of a given model among various models and choose the optimal one that has the smallest generalization error. In order to fit the the data through LDA, the data should be splitted into approximate equal size k subsets. Then, one of the subsets is trained and the other is leaving out from testing. Leave-one-out cross-validation is used to estimate generalization error; however, the jackknife is used to estimate the bias or the standard error of a statistic by comparing the average of the subset statistics with the corresponding statistic computed from the entire sample in order to estimate the bias of the latter.
In this work, LDA was adopted here to model Lithofacies given core analysis and well logs data in order to predict discrete and posterior probability distributions of the Lithofacies in Karpur Dataset.
Install the packages required to implement LDA algorithm with their functions.
#First, install the required packages.
require(caTools)## Loading required package: caToolsrequire(MASS)## Loading required package: MASSrequire(lattice)## Loading required package: latticelibrary(caTools)
library(MASS)
library(lattice)Call the dataset and show the dataset head: -
## depth caliper ind.deep ind.med gamma phi.N R.deep R.med SP
## 1 5667.0 8.685 618.005 569.781 98.823 0.410 1.618 1.755 -56.587
## 2 5667.5 8.686 497.547 419.494 90.640 0.307 2.010 2.384 -61.916
## 3 5668.0 8.686 384.935 300.155 78.087 0.203 2.598 3.332 -55.861
## 4 5668.5 8.686 278.324 205.224 66.232 0.119 3.593 4.873 -41.860
## 5 5669.0 8.686 183.743 131.155 59.807 0.069 5.442 7.625 -34.934
## 6 5669.5 8.686 109.512 75.633 57.109 0.048 9.131 13.222 -39.769
## density.corr density phi.core k.core Facies
## 1 -0.033 2.205 33.9000 2442.590 F1
## 2 -0.067 2.040 33.4131 3006.989 F1
## 3 -0.064 1.888 33.1000 3370.000 F1
## 4 -0.053 1.794 34.9000 2270.000 F1
## 5 -0.054 1.758 35.0644 2530.758 F1
## 6 -0.058 1.759 35.3152 2928.314 F1Summary of the dataset.
## depth caliper ind.deep ind.med
## Min. :5667 Min. :8.487 Min. : 6.532 Min. : 9.386
## 1st Qu.:5769 1st Qu.:8.556 1st Qu.: 28.799 1st Qu.: 27.892
## Median :5872 Median :8.588 Median :217.849 Median :254.383
## Mean :5873 Mean :8.622 Mean :275.357 Mean :273.357
## 3rd Qu.:5977 3rd Qu.:8.686 3rd Qu.:566.793 3rd Qu.:544.232
## Max. :6083 Max. :8.886 Max. :769.484 Max. :746.028
##
## gamma phi.N R.deep R.med
## Min. : 16.74 Min. :0.0150 Min. : 1.300 Min. : 1.340
## 1st Qu.: 40.89 1st Qu.:0.2030 1st Qu.: 1.764 1st Qu.: 1.837
## Median : 51.37 Median :0.2450 Median : 4.590 Median : 3.931
## Mean : 53.42 Mean :0.2213 Mean : 24.501 Mean : 21.196
## 3rd Qu.: 62.37 3rd Qu.:0.2640 3rd Qu.: 34.724 3rd Qu.: 35.853
## Max. :112.40 Max. :0.4100 Max. :153.085 Max. :106.542
##
## SP density.corr density phi.core
## Min. :-73.95 Min. :-0.067000 Min. :1.758 Min. :15.70
## 1st Qu.:-42.01 1st Qu.:-0.016000 1st Qu.:2.023 1st Qu.:23.90
## Median :-32.25 Median :-0.007000 Median :2.099 Median :27.60
## Mean :-30.98 Mean :-0.008883 Mean :2.102 Mean :26.93
## 3rd Qu.:-19.48 3rd Qu.: 0.002000 3rd Qu.:2.181 3rd Qu.:30.70
## Max. : 25.13 Max. : 0.089000 Max. :2.387 Max. :36.30
##
## k.core Facies
## Min. : 0.42 F8 :184
## 1st Qu.: 657.33 F9 :172
## Median : 1591.22 F10 :171
## Mean : 2251.91 F1 :111
## 3rd Qu.: 3046.82 F5 :109
## Max. :15600.00 F3 : 55
## (Other): 17Visualize the dataset:
In order to use LDA, I need to first split the data into a part used to train the classifier, and another part to test the classifier. For this problem, I considered an 80:20 split, approximately.
The following shows the modeling Lithofacies given well logs and core data by LDA. Also, predict the 1st 10 observations of discrete and posterior distribution in addtion to plot the boxplot of the predicted lthofacies by LDA Algorithm.
Training data:
## [1] 655Test data:
## [1] 164## Length Class Mode
## prior 8 -none- numeric
## counts 8 -none- numeric
## means 104 -none- numeric
## scaling 91 -none- numeric
## lev 8 -none- character
## svd 7 -none- numeric
## N 1 -none- numeric
## call 3 -none- call## [1] "prior" "counts" "means" "scaling" "lev" "svd" "N"
## [8] "call"Prior distribution of Facies:
## F1 F10 F2 F3 F5 F7
## 0.169465649 0.192366412 0.012213740 0.083969466 0.134351145 0.003053435
## F8 F9
## 0.207633588 0.196946565Means of the Well logs and core data given each Facies:
## depth caliper ind.deep ind.med gamma phi.N R.deep
## F1 5694.581 8.685847 47.68522 46.57178 56.48741 0.09327027 37.563919
## F10 5776.806 8.769365 261.51079 287.18687 73.30020 0.24548413 4.518214
## F2 5725.750 8.823875 142.76850 165.96738 60.97350 0.25100000 7.658375
## F3 5772.318 8.771855 260.11147 262.34173 64.82567 0.21241818 4.391618
## F5 5838.341 8.573489 21.44975 25.82666 34.85292 0.16743182 76.979716
## F7 5855.750 8.588000 149.84850 110.46250 36.36050 0.23000000 6.796500
## F8 5939.551 8.524154 227.67058 216.68392 40.70689 0.26244118 38.182978
## F9 6016.128 8.568403 602.58357 564.45968 53.83036 0.25566667 1.661171
## R.med SP density.corr density phi.core k.core
## F1 34.091045 -47.00009 -0.037396396 1.908054 31.60519 2334.6776
## F10 3.907310 -26.03356 0.002880952 2.255119 20.57050 833.2559
## F2 6.576875 -39.66725 0.010125000 2.203375 17.85878 338.5453
## F3 4.204400 -27.99718 -0.005290909 2.182418 23.47783 1098.5071
## F5 62.922545 -40.03389 -0.015545455 2.055750 27.53975 4957.0287
## F7 9.283500 7.69650 -0.002500000 2.025500 27.65975 3908.6753
## F8 30.980191 -27.79474 -0.007897059 2.036934 30.85639 3192.2026
## F9 1.774806 -25.17478 -0.006000000 2.118899 27.17926 774.3533## F1 F10 F2 F3 F5 F7 F8 F9
## 111 126 8 55 88 2 136 129
LDA Modeling Validation by computing the total correct percent.
## F1 F10 F2 F3 F5 F7 F8
## 0.9729730 0.8128655 0.7500000 0.6000000 0.7706422 0.7777778 0.9402174
## F9
## 1.0000000Total percent correct:
## [1] 0.8815629LDA Cross-validation: Rather than splitting the data into a training and testing split, an alternative way is to measure the performance of the model is to ask R to perform cross-validation. The “lda” function is achieved by incorporating the argument CV=TRUE:
Facies Discrimination by LDA:
Visualizing the predicted posterior distribution of the Eight Facies.
Combining the posterior distribution of the eight Lithofacies in one plot.
## Warning in bxp(structure(list(stats = structure(c(5667, 5680.75, 5694.5, :
## some notches went outside hinges ('box'): maybe set notch=FALSE## Warning in bxp(structure(list(stats = structure(c(5667.5, 5681.75,
## 5696.25, : some notches went outside hinges ('box'): maybe set notch=FALSE
References
Pires, A.M. and J.A. Branco, Projection-pursuit approach to robust linear discriminant analysis. Journal of Multivariate Analysis 101 24642485 (2010).
Al-Mudhafar, W. J. (2015). Integrating Component Analysis & Classification Techniques for Comparative Prediction of Continuous & Discrete Lithofacies Distributions. Offshore Technology Conference. doi:10.4043/25806-MS.
Karpur, L., L. Lake, and K. Sepehrnoori. (2000). Probability Logs for Facies Classification. In Situ 24(1): 57.
Al-Mudhafer, W. J. (2014). Multinomial Logistic Regression for Bayesian Estimation of Vertical Facies Modeling in Heterogeneous Sandstone Reservoirs. Offshore Technology Conference. doi:10.4043/24732-MS.
Al-Mudhafar, W. J. (2015). Applied Geostatistics in R: 1. Naive Bayes Classifier for Lithofacies Modeling in a Sandstone Formation. RPubs.
Al-Mudhafar, W. J. (2015). Applied Geostatistics in R: 2. Applied Geostatistics in R: 2. Logistic Boosting Regression (LogitBoost) for Multinomial Lithofacies Classification in a Sandstone Formation. RPubs.