Predictive Analytics: Homework 4 Texas
Applied Predictive Modeling - Exercises 3.1 & 3.2
Dariusz
2026-10-04
Exercise 3.1
The UC Irvine Machine Learning Repository6 contains a data set related to glass identification. The data consist of 214 glass samples labeled as one of seven class categories. There are nine predictors, including the refractive index and percentages of eight elements: Na, Mg, Al, Si, K, Ca, Ba, and Fe.
The data can be accessed via:
## 'data.frame': 214 obs. of 10 variables:
## $ RI : num 1.52 1.52 1.52 1.52 1.52 ...
## $ Na : num 13.6 13.9 13.5 13.2 13.3 ...
## $ Mg : num 4.49 3.6 3.55 3.69 3.62 3.61 3.6 3.61 3.58 3.6 ...
## $ Al : num 1.1 1.36 1.54 1.29 1.24 1.62 1.14 1.05 1.37 1.36 ...
## $ Si : num 71.8 72.7 73 72.6 73.1 ...
## $ K : num 0.06 0.48 0.39 0.57 0.55 0.64 0.58 0.57 0.56 0.57 ...
## $ Ca : num 8.75 7.83 7.78 8.22 8.07 8.07 8.17 8.24 8.3 8.4 ...
## $ Ba : num 0 0 0 0 0 0 0 0 0 0 ...
## $ Fe : num 0 0 0 0 0 0.26 0 0 0 0.11 ...
## $ Type: Factor w/ 6 levels "1","2","3","5",..: 1 1 1 1 1 1 1 1 1 1 ...Note that the echo = FALSE parameter was added to the
code chunk to prevent printing of the R code that generated the
plot.
- Using visualizations, explore the predictor variables to understand their distributions as well as the relationships between predictors.
- Do there appear to be any outliers in the data? Are any predictors skewed?
- Are there any relevant transformations of one or more predictors that might improve the classification model?
Exercise 3.1: Glass Identification Dataset
The UC Irvine Machine Learning Repository contains a dataset related
to glass identification with 214 glass samples and 9 numeric predictors:
refractive index (RI) and elemental percentages
(Na, Mg, Al, Si,
K, Ca, Ba, Fe).
## 'data.frame': 214 obs. of 10 variables:
## $ RI : num 1.52 1.52 1.52 1.52 1.52 ...
## $ Na : num 13.6 13.9 13.5 13.2 13.3 ...
## $ Mg : num 4.49 3.6 3.55 3.69 3.62 3.61 3.6 3.61 3.58 3.6 ...
## $ Al : num 1.1 1.36 1.54 1.29 1.24 1.62 1.14 1.05 1.37 1.36 ...
## $ Si : num 71.8 72.7 73 72.6 73.1 ...
## $ K : num 0.06 0.48 0.39 0.57 0.55 0.64 0.58 0.57 0.56 0.57 ...
## $ Ca : num 8.75 7.83 7.78 8.22 8.07 8.07 8.17 8.24 8.3 8.4 ...
## $ Ba : num 0 0 0 0 0 0 0 0 0 0 ...
## $ Fe : num 0 0 0 0 0 0.26 0 0 0 0.11 ...
## $ Type: Factor w/ 6 levels "1","2","3","5",..: 1 1 1 1 1 1 1 1 1 1 ...(a) Exploratory Visualizations & Predictor Relationships
We inspect the univariate distributions of the continuous predictors using histograms and examine bivariate correlation patterns using a correlation matrix.
Observations:
- Refractive Index (
RI) & Calcium (Ca): Exhibit a strong positive linear correlation (\(r \approx 0.81\)). - Refractive Index (
RI) & Silicon (Si): Show a notable inverse/negative correlation. - Multimodal / Sparse Elements:
Ba,Fe, andKare heavily concentrated at 0, indicating that many glass types do not contain detectable traces of these trace elements.Mgdisplays a pronounced bimodal distribution.
(b) Outliers and Skewness
We calculate skewness metrics across all 9 numerical features and render boxplots to inspect potential extreme values.
| Predictor | Skewness | Assessment | |
|---|---|---|---|
| K | K | 6.460 | Highly Skewed |
| Ba | Ba | 3.369 | Highly Skewed |
| Ca | Ca | 2.018 | Highly Skewed |
| Fe | Fe | 1.730 | Highly Skewed |
| RI | RI | 1.603 | Highly Skewed |
| Mg | Mg | -1.136 | Highly Skewed |
| Al | Al | 0.895 | Moderately Skewed |
| Si | Si | -0.720 | Moderately Skewed |
| Na | Na | 0.448 | Roughly Symmetric |
Outlier & Skewness Findings:
- Heavy Outliers:
K,Ba,Fe, andCaexhibit severe upper-tail outliers. For example,Kcontains isolated observations above \(6.0\) while the vast majority cluster below \(1.0\). - High Skewness:
K(\(3.56\)),Ba(\(3.37\)),Ca(\(2.02\)),Fe(\(1.73\)), andRI(\(1.61\)) all exhibit substantial right-skewness (\(\vert{}skew\vert{} > 1\)).Mgis negatively skewed (\(-1.14\)) due to its structural bimodal cluster at zero.
(c) Recommended Transformations
Due to strong right-skewness, large scale disparities, and high collinearity:
- Centering & Scaling: Elements like
SiandCaspan much larger ranges than trace minerals (Fe,Ba). Standardizing them to \(\mu = 0, \sigma = 1\) is necessary for distance-based or regularized models. - Box-Cox / Yeo-Johnson Transformations: Continuous
strictly positive features (
RI,Na,Al,Si,Ca) benefit from standard Box-Cox. Predictors with zero values (Ba,Fe,K,Mg) require either the Yeo-Johnson transformation or spatial sign filtering. - PCA / Collinearity Reduction: Because
RIandCaare strongly collinear (\(r > 0.8\)), applying Principal Component Analysis (PCA) or pruning redundant features helps tree-naive algorithms like linear discriminant analysis.
| Predictor | Original_Skew | Transformed_Skew | |
|---|---|---|---|
| RI | RI | 1.603 | 1.566 |
| Na | Na | 0.448 | 0.034 |
| Mg | Mg | -1.136 | -1.136 |
| Al | Al | 0.895 | 0.091 |
| Si | Si | -0.720 | -0.651 |
| K | K | 6.460 | 6.460 |
| Ca | Ca | 2.018 | -0.194 |
| Ba | Ba | 3.369 | 3.369 |
| Fe | Fe | 1.730 | 1.730 |
Exercise 3.2
The soybean data can also be found at the UC Irvine Machine Learning Repository. Data were collected to predict disease in 683 soybeans. The 35 predictors are mostly categorical and include information on the environmental conditions (e.g., temperature, precipitation) and plant conditions (e.g., left spots, mold growth). The outcome labels consist of 19 distinct classes.
Note that the echo = FALSE parameter was added to the
code chunk to prevent printing of the R code that generated the
plot.
- Investigate the frequency distributions for the categorical predictors. Are any of the distributions degenerate in the ways discussed earlier in this chapter?
- Roughly 18% of the data are missing. Are there particular predictors that are more likely to be missing? Is the pattern of missing data related to the classes?
- Develop a strategy for handling missing data, either by eliminating predictors or imputation.
Exercise 3.2: Soybean Disease Dataset
The Soybean dataset contains 683 samples and 35
categorical predictors (soil, weather, leaf, and plant symptoms)
categorized into 19 disease classes.
(a) Frequency Distributions & Degenerate Predictors
A predictor is degenerate (or near-zero variance) if it contains only one unique value, or if the ratio of the most frequent value to the second most frequent value is exceptionally large (\(> 95/5\) rule) and the percentage of unique values is small.
| Predictor | freqRatio | percentUnique | zeroVar | nzv |
|---|---|---|---|---|
| leaf.mild | 26.75 | 0.4392387 | FALSE | TRUE |
| mycelium | 106.50 | 0.2928258 | FALSE | TRUE |
| sclerotia | 31.25 | 0.2928258 | FALSE | TRUE |
Findings:
leaf.mild,mycelium, andsclerotiaexhibit extreme frequency imbalances (frequency ratios \(> 15\) to over \(100\)) and almost zero variance.- When splitting into training/testing folds or building linear/tree models, these features can become invariant constants within specific cross-validation resamples, destabilizing parameter estimation.
(b) Missing Data Patterns and Class Relationships
Approximately 18% of the data values are missing across the observations. We evaluate missingness rates per predictor and assess if missingness correlates with specific disease classes.
| Predictor | Percent_Missing |
|---|---|
| hail | 17.71596 |
| sever | 17.71596 |
| seed.tmt | 17.71596 |
| lodging | 17.71596 |
| germ | 16.39824 |
| leaf.mild | 15.81259 |
| fruiting.bodies | 15.51977 |
| fruit.spots | 15.51977 |
| seed.discolor | 15.51977 |
| shriveling | 15.51977 |
| Class | Total_Cases | Samples_With_Missing | Pct_Cases_Incomplete | Mean_Missing_Fields |
|---|---|---|---|---|
| 2-4-d-injury | 16 | 16 | 100.0 | 28.1 |
| cyst-nematode | 14 | 14 | 100.0 | 24.0 |
| diaporthe-pod-&-stem-blight | 15 | 15 | 100.0 | 11.8 |
| herbicide-injury | 8 | 8 | 100.0 | 20.0 |
| phytophthora-rot | 88 | 68 | 77.3 | 13.8 |
| alternarialeaf-spot | 91 | 0 | 0.0 | 0.0 |
| anthracnose | 44 | 0 | 0.0 | 0.0 |
| bacterial-blight | 20 | 0 | 0.0 | 0.0 |
| bacterial-pustule | 20 | 0 | 0.0 | 0.0 |
| brown-spot | 92 | 0 | 0.0 | 0.0 |
| brown-stem-rot | 44 | 0 | 0.0 | 0.0 |
| charcoal-rot | 20 | 0 | 0.0 | 0.0 |
| diaporthe-stem-canker | 20 | 0 | 0.0 | 0.0 |
| downy-mildew | 20 | 0 | 0.0 | 0.0 |
| frog-eye-leaf-spot | 91 | 0 | 0.0 | 0.0 |
| phyllosticta-leaf-spot | 20 | 0 | 0.0 | 0.0 |
| powdery-mildew | 20 | 0 | 0.0 | 0.0 |
| purple-seed-stain | 20 | 0 | 0.0 | 0.0 |
| rhizoctonia-root-rot | 20 | 0 | 0.0 | 0.0 |
Findings:
- Class-Specific Clustering: Missingness is not
Missing Completely at Random (MCAR). The missing values belong almost
entirely to specific classes:
2-4-d-injury,cyst-nematode,diaporthe-stem-canker, andphytophthora-rotconsist of 100% incomplete observations. - In contrast, diseases such as
bacterial-blight,brown-spot, andanthracnosecontain zero missing values across all measurements.
(c) Strategy for Handling Missing Data
Do Not Indiscriminately Drop Incomplete Rows: Dropping all cases with missing data (
na.omit()) would completely eliminate entire diagnostic classes (e.g., eliminating all occurrences of2-4-d-injury).Filter Degenerate Predictors First: Remove uninformative predictors with near-zero variance (
mycelium,sclerotia,leaf.mild) identified in part (a).Imputation Methods:
- K-Nearest Neighbors (KNN) or Tree-Based Imputation (missForest/MICE): Because the predictors are correlated categorical variables, using a distance-weighted donor (KNN) or tree surrogate model preserves discrete structural relationships without distorting distributions.
- Informative Missingness Dummy: Because the missingness pattern itself is strongly predictive of the disease class, a binary indicator variable (\(I_{missing}\)) can be engineered prior to imputation.
Texas
The official state motto of Texas is “Friendship”.
What it Refers To
Origin of the State’s Name: The motto refers to the origin of the word “Texas” (or the Spanish pronunciation, “Tejas”).
Native American Roots: It comes from a Caddo (specifically Hasinai) Native American word—such as teyshas or texias—which translates to “friends” or “allies”.
Adoption: The 41st Texas Legislature officially adopted “Friendship” as the state motto in February 1930 to honor this historic linguistic connection.