ETC3250/5250

Introduction to Machine Learning

Logistic regression

Lecturer: Emi Tanaka

Department of Econometrics and Business Statistics

Classification problems

Classification problems

  • In the previous three lectures, our outcome of interest was numeric.
  • In classification problems, the response y is a categorical variable:
    • Loan approval \in \{\text{successful}, \text{unsuccessful}\}
    • Bankruptcy \in\{\text{paid}, \text{default}\}.
    • Preferred beverage \in\{\text{Coca cola},\text{Pepsi},\text{Fanta}\}.

Breast cancer diagnosis

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  • We use the Wisconsin breast cancer data set to build a model to predict if the breast mass sample is malignant (M) or benign (B).
  • Here the response is categorical with two classes (M and B).
Code
library(tidyverse)
cancer <- read_csv("https://emitanaka.org/iml/data/cancer.csv") %>% 
  mutate(diagnosis_malignant = ifelse(diagnosis=="M", 1, 0),
         diagnosis = factor(diagnosis, levels = c("B", "M"))) %>% 
  janitor::clean_names()

skimr::skim(cancer, diagnosis, radius_mean, concave_points_mean)
Data summary
Name cancer
Number of rows 568
Number of columns 34
_______________________
Column type frequency:
factor 1
numeric 2
________________________
Group variables None

Variable type: factor

skim_variable n_missing complete_rate ordered n_unique top_counts
diagnosis 0 1 FALSE 2 B: 356, M: 212

Variable type: numeric

skim_variable n_missing complete_rate mean sd p0 p25 p50 p75 p100 hist
radius_mean 0 1 14.14 3.52 6.98 11.71 13.38 15.80 28.11 ▂▇▃▁▁
concave_points_mean 0 1 0.05 0.04 0.00 0.02 0.03 0.07 0.20 ▇▃▂▁▁
Code
cancer %>% 
  select(diagnosis, radius_mean, concave_points_mean) %>% 
  GGally::ggpairs(mapping = aes(color = diagnosis))

Why not linear regression?

Code
cancer %>% 
  ggplot(aes(radius_mean, diagnosis_malignant)) +
  geom_point(alpha = 0.25, size = 2, aes(color = diagnosis)) +
  geom_smooth(method = "lm",
              formula = y ~ x,
              se = FALSE,
              color = "#027EB6",
              linewidth = 1.2) +
  scale_y_continuous(breaks = c(0, 1)) +
  labs(y = "diagnosis") +
  scale_color_manual(values = c("forestgreen", "red2")) +
  guides(color = "none")

  • How would we model this then?

Concepts

Propensity score

  • Suppose we consider y_i as a binary category: y_i = \begin{cases} 1 & \text{ if $i$-th observation is in class 1}\\ 0 & \text{ if $i$-th observation is in class 2}\\ \end{cases}
  • Instead of modelling the outcome directly, we consider the conditional probability, say P(y_i = 1|\boldsymbol{x}_i), also known as the propensity score of class 1.
  • The propensity score of class 2 is P(y_i=0|\boldsymbol{x}_i) = 1 - P(y_i=1|\boldsymbol{x}_i).

Odds of an event

  • The odds of an event is defined as \text{odds} = \color{#006DAE}{\frac{p}{1-p}} = \frac{\text{probability that the event will occur}}{\text{probability that the event will not occur}}, where p is the probability of an event occuring.
  • The ratio of the propensity scores of the two classes is the odds of being in class 1:

\text{odds} = \frac{P(y_i=1|\boldsymbol{x}_i)}{1-P(y_i=1|\boldsymbol{x}_i)}.

Logistic function

  • The logistic function:

g(z) = \frac{e^z}{1+e^z} = \frac{1}{1+e^{-z}}

  • Notice that 0 < g(z) < 1 for all finite values of z.

Logit function

  • The logit function:

f(p) = \log_e \left(\frac{p}{1- p}\right)

  • Here -\infty < f(p) < \infty for all p \in (0, 1).
  • Note that logit and logistic functions are inverse functions of one another, i.e. f(g(z)) = z and g(f(p)) = p.

Logistic regression

Logistic regression for binary response

  • Logistic regression is a generalised linear model where it models the log odds as a linear combination of predictors: \text{logit}(p_i) = \log_e \left(\frac{p_i}{1-p_i}\right) = \sum_{j=0}^p\beta_jx_{ij}, \quad p_i = \frac{e^{\sum_{j=0}^p\beta_jx_{ij}}}{1+e^{\sum_{j=0}^p\beta_jx_{ij}}}
  • We assume that y_i \sim B(1, p_i) where p_i = P(y_i=1|\boldsymbol{x}_i).
  • In generalised linear models, the link function links the predictors to the model parameters.
  • In a logistic regression, the link function is the logit function.

Maximum likelihood estimation

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  • We maximise the likelihoood:

L(\boldsymbol{\beta}|\boldsymbol{y}, \mathbf{X}) = \prod_{i=1}^n p_i^{y_i}(1-p_i)^{1-y_i}.

  • It is more convenient to maximize the log-likelihood:
\log L(\boldsymbol{\beta}|\boldsymbol{y}, \mathbf{X}) = \sum_{i = 1}^n \left( y_i\log p_i + (1-y_i)\log(1-p_i)\right).
Mathematical details

\begin{align*} \log L(\boldsymbol{\beta}) &= \sum_{i = 1}^n \left(y_i \log\left( \frac{e^{\sum_{j=0}^p x_{ij}\beta_j}}{1 + e^{\sum_{j=0}^p x_{ij}\beta_j}}\right) + (1 - y_i) \log\left( \frac{1}{1 + e^{\sum_{j=0}^p x_{ij}\beta_j}}\right) \right) \\ &= \sum_{i = 1}^n \left(y_i \log\left(e^{\sum_{j=0}^p x_{ij}\beta_j}\right) - y_i \log\left(1 + e^{\sum_{j=0}^p x_{ij}\beta_j}\right) - (1 - y_i) \log\left(1 + e^{\sum_{j=0}^p x_{ij}\beta_j}\right) \right) \\ &= \sum_{i = 1}^n \left(y_i \sum_{j=0}^p x_{ij}\beta_j - \log\left(1 + e^{\sum_{j=0}^p x_{ij}\beta_j}\right) \right) \end{align*}

We then solve the following to find the MLE of \beta_j

\frac{\partial \log L(\boldsymbol{\beta})}{\partial\beta_j} = \sum_{i = 1}^n \left(y_i x_{ij} - \frac{x_{ij}e^{\sum_{j=0}^p x_{ij}\beta_j}}{1 + e^{\sum_{j=0}^p x_{ij}\beta_j}} \right) = 0 assuming function below is concave.

There is no closed-form solution of the above, so we use analytical approaches (e.g. Newton-Raphson method).

Logistic regression with a binary response

  • When response is a binary value (0 or 1):
str(cancer$diagnosis_malignant)
 num [1:568] 1 1 1 1 1 1 1 1 1 1 ...
table(cancer$diagnosis_malignant)

  0   1 
356 212 
  • Fit the logistic model in R as
cancer_fit <- glm(diagnosis_malignant ~ radius_mean + concave_points_mean, 
                  data = cancer, 
                  family = binomial(link = "logit"))

coef(cancer_fit)
        (Intercept)         radius_mean concave_points_mean 
        -13.6987232           0.6389131          84.2228367 

Logistic regression with a factor response

  • When response is a factor:
str(cancer$diagnosis)
 Factor w/ 2 levels "B","M": 2 2 2 2 2 2 2 2 2 2 ...
  • Watch out for the order of the levels!
cancer_fit2 <- glm(diagnosis ~ radius_mean + concave_points_mean, 
                   data = cancer, 
                   family = binomial(link = "logit"))
coef(cancer_fit2)
        (Intercept)         radius_mean concave_points_mean 
        -13.6987232           0.6389131          84.2228367 
cancer_fit3 <- glm(diagnosis ~ radius_mean + concave_points_mean, 
                   data = cancer %>% mutate(diagnosis = factor(diagnosis, levels = c("M", "B"))),  
                   family = binomial(link = "logit"))
coef(cancer_fit3)
        (Intercept)         radius_mean concave_points_mean 
         13.6987232          -0.6389131         -84.2228367 

Interpretation of logistic models

library(broom)
tidy(cancer_fit) # coefficients
# A tibble: 3 × 5
  term                estimate std.error statistic  p.value
  <chr>                  <dbl>     <dbl>     <dbl>    <dbl>
1 (Intercept)          -13.7       1.57      -8.70 3.20e-18
2 radius_mean            0.639     0.108      5.94 2.91e- 9
3 concave_points_mean   84.2       9.96       8.46 2.75e-17
  • Increasing radius_mean by one unit changes the log odds by \hat{\beta}_1, 0.639, or equivalently it multiplies the odds by e^{\hat\beta_1}, 1.894, provided concave_points_mean is held fixed.

Threshold

  • We choose the threshold q such that P(y_i=1|\boldsymbol{x}_i) \ge q is considered to be in class 1.
  • This gives us the decision boundary: \text{log} \left(\frac{q}{1-q}\right) =\sum_{j = 0}^p\beta_jx_{ij}.
  • E.g., if p = 2, the boundary corresponds to: x_{i2} = \underbrace{\frac{1}{\beta_2}\left[\text{log} \left(\frac{q}{1-q}\right)-\beta_0\right]}_{\text{intercept}}\underbrace{-\frac{\beta_1}{\beta_2}}_{\text{slope}}x_{i1}.

Linear classifier

  • Logistic regression is a linear classifier.
  • The separation in class is a point for one variable, line for two variables and a hyperplane for more than two variables.

Out-of-sample prediction

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data splitting code
library(rsample)
set.seed(2023)
cancer_split <- initial_split(cancer, prop = 3/4)
cancer_train <- training(cancer_split)
cancer_test <- testing(cancer_split)
  • Using threshold q = 0.5:
cancer_logistic <- glm(diagnosis ~ radius_mean + concave_points_mean, 
                       data = cancer_train, 
                       family = binomial(link = "logit"))
cancer_pred <- cancer_test %>% 
  mutate(propensity = predict(cancer_logistic, ., type = 'response'),
         pred50 = factor(as.numeric(propensity > 0.5), labels = c("B", "M"), levels = c(0, 1)),
         index = 1:n())

cancer_pred %>% select(propensity, pred50, diagnosis)
# A tibble: 142 × 3
   propensity pred50 diagnosis
        <dbl> <fct>  <fct>    
 1      0.890 M      M        
 2      0.694 M      M        
 3      0.486 B      M        
 4      0.897 M      M        
 5      0.331 B      M        
 6      0.999 M      M        
 7      0.829 M      M        
 8      0.496 B      M        
 9      0.246 B      B        
10      0.999 M      M        
# … with 132 more rows

Assessing prediction results

  • We see some observations are wrongly classified – how do we summarise how well a model is in classifying?

Metrics for classification problems

Confusion matrix

  • The confusion matrix, also known as classification table, tabulates the number of correct/incorrect predictions by classes of the response variable.
library(yardstick)
cancer_pred %>% 
  # get the confusion matrix
  conf_mat(diagnosis, pred50) %>% 
  # get the table 
  pluck("table") %>% 
  # add the total for each group
  addmargins()
          Truth
Prediction   B   M Sum
       B    76  11  87
       M     5  50  55
       Sum  81  61 142

What is a good classification metric?

cancer_pred %>% 
  conf_mat(diagnosis, pred50) %>% 
  autoplot(type = "heatmap")

  • We want higher numbers along the diagonal entries of the confusion matrix.
  • But what is a single number that can summarise how good the classification is?

Classification metrics

  • Note that these metrics depend on the chosen threshold q.

Sensitivity and specificity

\text{sensitivity} = \frac{\text{TP}}{\text{TP} + \text{FN}}, \quad \text{specificity} = \frac{\text{TN}}{\text{TN} + \text{FP}}

cancer_pred %>% 
  metric_set(sensitivity, specificity)(., truth = diagnosis, estimate = pred50)
# A tibble: 2 × 3
  .metric     .estimator .estimate
  <chr>       <chr>          <dbl>
1 sensitivity binary         0.938
2 specificity binary         0.820
  • Often used in the context of medical diagnostics.
  • Wrongful negative diagnosis could be costly for the patient! E.g. undetected cancer, missing out on early treatment.

Precision and recall

\text{precision} = \frac{\text{TP}}{\text{TP} + \text{FP}}, \quad \text{recall} = \frac{\text{TP}}{\text{TP} + \text{FN}}

cancer_pred %>% 
  metric_set(precision, recall)(., truth = diagnosis, estimate = pred50)
# A tibble: 2 × 3
  .metric   .estimator .estimate
  <chr>     <chr>          <dbl>
1 precision binary         0.874
2 recall    binary         0.938
  • Terminology used more often in the context of information retrieval.
  • Note : recall is the same sensitivity.
  • E.g. search engine retrieves documents – precision measures proportion of retrieved documents that are relevant.

Precision-recall curve

cancer_pred %>% 
  pr_curve(truth = diagnosis, propensity, event_level = "second") %>% 
  ggplot(aes(recall, precision)) +
  geom_path() +
  geom_point(color = "pink", size = 1.3, data = ~filter(., .threshold >= 0.2) %>% arrange(.threshold) %>% slice(1)) +
  geom_point(color = "red", size = 1.3, data = ~filter(., .threshold >= 0.5) %>% arrange(.threshold) %>% slice(1)) +
  geom_point(color = "maroon", size = 1.3, data = ~filter(., .threshold >= 0.8) %>% arrange(.threshold) %>% slice(1)) +
  coord_equal()

Area under the precision-recall curve

cancer_pred %>% 
  pr_auc(truth = diagnosis, propensity, event_level = "second")
# A tibble: 1 × 3
  .metric .estimator .estimate
  <chr>   <chr>          <dbl>
1 pr_auc  binary         0.968

Receiver operating characteristic (ROC) curve

cancer_pred %>% 
  roc_curve(truth = diagnosis, propensity, event_level = "second") %>%
  ggplot(aes(1 - specificity, sensitivity)) +
  geom_path() +
  geom_point(color = "pink", size = 1.3, data = ~filter(., .threshold >= 0.2) %>% arrange(.threshold) %>% slice(1)) +
  geom_point(color = "red", size = 1.3, data = ~filter(., .threshold >= 0.5) %>% arrange(.threshold) %>% slice(1)) +
  geom_point(color = "maroon", size = 1.3, data = ~filter(., .threshold >= 0.8) %>% arrange(.threshold) %>% slice(1)) +
  geom_abline(linetype = "dashed") +
  coord_equal()

Area under the ROC curve

cancer_pred %>% 
  roc_auc(truth = diagnosis, propensity, event_level = "second")
# A tibble: 1 × 3
  .metric .estimator .estimate
  <chr>   <chr>          <dbl>
1 roc_auc binary         0.976

F1 Score

F_{\beta} = (1 + \beta^2) \times \frac{\text{precision} \times \text{recall}}{\beta^2\times\text{precision} + \text{recall}} F_1 = 2 \times \frac{\text{precision} \times \text{recall}}{\text{precision} + \text{recall}}

cancer_pred %>% 
  f_meas(truth = diagnosis, estimate = pred50, beta = 1)
# A tibble: 1 × 3
  .metric .estimator .estimate
  <chr>   <chr>          <dbl>
1 f_meas  binary         0.905

Detection prevalence

\text{detection prevalence} = \frac{\text{TP} + \text{FP}}{\text{TP} + \text{FN} + \text{FP} + \text{TN}}

cancer_pred %>% 
  detection_prevalence(truth = diagnosis, estimate = pred50, event_level = "second")
# A tibble: 1 × 3
  .metric              .estimator .estimate
  <chr>                <chr>          <dbl>
1 detection_prevalence binary         0.387
  • Note: this is not a measure of how good a classification is!

Prevalence

  • Prevalence is the proportion of a particular population with the condition.
  • Assuming we have a representative sample, then we can estimate the prevalence as:

\text{prevalence} = \frac{\text{TP} + \text{FN}}{\text{TP} + \text{FN} + \text{FP} + \text{TN}}

table(cancer_pred$diagnosis)[["M"]]/nrow(cancer_pred)
[1] 0.4295775
  • This is clearly not a representative sample of the population!

Accuracy

\text{accuracy} = \frac{\text{TP} + \text{TN}}{\text{TP} + \text{TN} + \text{FN} + \text{FP}}

cancer_pred %>% 
  accuracy(truth = diagnosis, estimate = pred50)
# A tibble: 1 × 3
  .metric  .estimator .estimate
  <chr>    <chr>          <dbl>
1 accuracy binary         0.887
  • Accuracy is the proportion of the data that are predicted correctly.

Balanced accuracy

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\text{balanced accuracy} = \frac{1}{2}(\text{sensitivity} + \text{specificity})

cancer_pred %>% 
  bal_accuracy(truth = diagnosis, estimate = pred50)
# A tibble: 1 × 3
  .metric      .estimator .estimate
  <chr>        <chr>          <dbl>
1 bal_accuracy binary         0.879
  • This metric works better if there is an imbalance in the response class.
Prediction \ Truth Positive Negative
Positive 20 70
Negative 30 5000
TP <- 20; FP <- 30; TN <- 5000; FN <- 70
(accuracy <- (TP + TN) / (TP + TN + FP + FN))
[1] 0.9804688
(sensitivity <- TP / (TP + FN))
[1] 0.2222222
(specificity <- TN / (TN + FP))
[1] 0.9940358
(balanced_accuracy <- 1/2 * (sensitivity + specificity))
[1] 0.608129

Cohen’s kappa coefficient

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\kappa = \frac{2\times (\text{TP}\times \text{TN} - \text{FN}\times \text{FP})}{(\text{TP} + \text{FP})(\text{TN} + \text{FP}) + (\text{TP} + \text{FN})(\text{TN} + \text{FN})}

cancer_pred %>% 
  kap(truth = diagnosis, estimate = pred50)
# A tibble: 1 × 3
  .metric .estimator .estimate
  <chr>   <chr>          <dbl>
1 kap     binary         0.767
  • Cohen’s kappa coefficient is similar to accuracy but takes into account that some true positives and true negatives are by chance.

Case A

Prediction \ Truth Positive Negative
Positive 45 15
Negative 25 15

Case B

Prediction \ Truth Positive Negative
Positive 25 35
Negative 5 35
# Case A
TP <- 45; FP <- 25; TN <- 15; FN <- 15
(accuracy <- (TP + TN) / (TP + TN + FP + FN))
[1] 0.6
(kappa <- 2 * (TP * TN - FN * FP) / ((TP + FP) * (TN + FP) + (TP + FN) * (TN + FN)))
[1] 0.1304348
# Case B
TP <- 25; FP <- 5; TN <- 35; FN <- 35
(accuracy <- (TP + TN) / (TP + TN + FP + FN))
[1] 0.6
(kappa <- 2 * (TP * TN - FN * FP) / ((TP + FP) * (TN + FP) + (TP + FN) * (TN + FN)))
[1] 0.2592593

Matthew’s correlation coefficient

  • Also called the phi coefficient.

\phi = \frac{\text{TP}\times\text{TN} - \text{FP}\times\text{FN}}{\sqrt{(\text{TP} + \text{FP})(\text{TN} + \text{FN})(\text{TP} + \text{FN})(\text{FP} + \text{TN})}}

cancer_pred %>% 
  mcc(truth = diagnosis, estimate = pred50)
# A tibble: 1 × 3
  .metric .estimator .estimate
  <chr>   <chr>          <dbl>
1 mcc     binary         0.770

Postive and negative predictive values

  • Positive/negative predictive value (PPV/NPV) is the proportion of cases with a positive/negative classification that are actually correct.

\text{PPV} = \frac{\text{TP}}{\text{TP} + \text{FP}}, \quad\text{NPV} = \frac{\text{TN}}{\text{TN} + \text{FN}}

cancer_pred %>% 
  metric_set(ppv, npv)(., truth = diagnosis, estimate = pred50)
# A tibble: 2 × 3
  .metric .estimator .estimate
  <chr>   <chr>          <dbl>
1 ppv     binary         0.874
2 npv     binary         0.909

Youden’s J-index

J = \text{specificity} + \text{sensitivity} - 1

cancer_pred %>% 
  j_index(truth = diagnosis, estimate = pred50)
# A tibble: 1 × 3
  .metric .estimator .estimate
  <chr>   <chr>          <dbl>
1 j_index binary         0.758
  • J index is between 0 and 1 (inclusive).

Classification metrics

cancer_pred %>% 
  metric_set(sensitivity, specificity, precision, recall, pr_auc, roc_auc, 
             f_meas, accuracy, bal_accuracy, kap, mcc, ppv, npv, j_index)(.,
                                                                  truth = diagnosis,
                                                                  propensity,
                                                                  estimate = pred50,
                                                                  event_level = "second")
# A tibble: 14 × 3
   .metric      .estimator .estimate
   <chr>        <chr>          <dbl>
 1 sensitivity  binary         0.820
 2 specificity  binary         0.938
 3 precision    binary         0.909
 4 recall       binary         0.820
 5 f_meas       binary         0.862
 6 accuracy     binary         0.887
 7 bal_accuracy binary         0.879
 8 kap          binary         0.767
 9 mcc          binary         0.770
10 ppv          binary         0.909
11 npv          binary         0.874
12 j_index      binary         0.758
13 pr_auc       binary         0.968
14 roc_auc      binary         0.976

Modelling for count data of binary category

Survival on titanic

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  • Response does not have to be categorical to fit a logistic regression.
  • Observation may be the count for each category as below:
Code
titanic <- datasets::Titanic %>% 
  as.data.frame() %>% 
  pivot_wider(c(Class, Sex, Age), 
              names_from = Survived, 
              values_from = Freq, 
              names_prefix = "Survived_")

titanic
# A tibble: 16 × 5
   Class Sex    Age   Survived_No Survived_Yes
   <fct> <fct>  <fct>       <dbl>        <dbl>
 1 1st   Male   Child           0            5
 2 2nd   Male   Child           0           11
 3 3rd   Male   Child          35           13
 4 Crew  Male   Child           0            0
 5 1st   Female Child           0            1
 6 2nd   Female Child           0           13
 7 3rd   Female Child          17           14
 8 Crew  Female Child           0            0
 9 1st   Male   Adult         118           57
10 2nd   Male   Adult         154           14
11 3rd   Male   Adult         387           75
12 Crew  Male   Adult         670          192
13 1st   Female Adult           4          140
14 2nd   Female Adult          13           80
15 3rd   Female Adult          89           76
16 Crew  Female Adult           3           20
Code
skimr::skim(titanic)
Data summary
Name titanic
Number of rows 16
Number of columns 5
_______________________
Column type frequency:
factor 3
numeric 2
________________________
Group variables None

Variable type: factor

skim_variable n_missing complete_rate ordered n_unique top_counts
Class 0 1 FALSE 4 1st: 4, 2nd: 4, 3rd: 4, Cre: 4
Sex 0 1 FALSE 2 Mal: 8, Fem: 8
Age 0 1 FALSE 2 Chi: 8, Adu: 8

Variable type: numeric

skim_variable n_missing complete_rate mean sd p0 p25 p50 p75 p100 hist
Survived_No 0 1 93.12 183.88 0 0.0 8.5 96.25 670 ▇▁▁▁▁
Survived_Yes 0 1 44.44 56.09 0 9.5 14.0 75.25 192 ▇▂▁▁▁
Code
datasets::Titanic %>% 
  as.data.frame() %>% 
  ggplot(aes(Survived, Freq)) +
  geom_col(aes(fill = Sex, group = Age), position = "fill") +
  facet_grid(Class ~ Age) +
  labs(y = "Proportion")

Logistic regression with count data

titanic_fit <- glm(cbind(Survived_No, Survived_Yes) ~ Class + Age + Sex,
                   data = titanic, 
                   family = binomial(link = "logit"))

tidy(titanic_fit)
# A tibble: 6 × 5
  term        estimate std.error statistic  p.value
  <chr>          <dbl>     <dbl>     <dbl>    <dbl>
1 (Intercept)   -0.685     0.273     -2.51 1.21e- 2
2 Class2nd       1.02      0.196      5.19 2.05e- 7
3 Class3rd       1.78      0.172     10.4  3.69e-25
4 ClassCrew      0.858     0.157      5.45 5.00e- 8
5 AgeAdult       1.06      0.244      4.35 1.36e- 5
6 SexFemale     -2.42      0.140    -17.2  1.43e-66

Multi-class logistic regression

Digit recognition with MNIST data

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  • Images of single digits are rescaled to 28 \times 28 = 784 pixels.
  • Labels are 0, 1, …, 9.
  • This is a multi-class classification problem.
Code
dat_mnist <- dslabs::read_mnist()
mnist <- dat_mnist$train$images %>% 
  as.data.frame() %>% 
  mutate(label = as.factor(dat_mnist$train$label))

glimpse(mnist)
Rows: 60,000
Columns: 785
$ V1    <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V2    <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V3    <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V4    <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V5    <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V6    <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V7    <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V8    <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V9    <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V10   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V11   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V12   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V13   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V14   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V15   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V16   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V17   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V18   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V19   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V20   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V21   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V22   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V23   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V24   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V25   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V26   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V27   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V28   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V29   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V30   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V31   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V32   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V33   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V34   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V35   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V36   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V37   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V38   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V39   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V40   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V41   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V42   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V43   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V44   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V45   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V46   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V47   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V48   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V49   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V50   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V51   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V52   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V53   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V54   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V55   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V56   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V57   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V58   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V59   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V60   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V61   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V62   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V63   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V64   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V65   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V66   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V67   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V68   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V69   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V70   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V71   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V72   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V73   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 38, 0, 0, 0, 0, 0, 0, 0, …
$ V74   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 222, 0, 0, 0, 0, 0, 0, 0,…
$ V75   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 225, 0, 0, 0, 0, 0, 0, 0,…
$ V76   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V77   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V78   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V79   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V80   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V81   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V82   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V83   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V84   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V85   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V86   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V87   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V88   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V89   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V90   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V91   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V92   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V93   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V94   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V95   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V96   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V97   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V98   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V99   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V100  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 147, 0, 0, 0, 0, 0, 0, 0,…
$ V101  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 234, 0, 0, 0, 0, 34, 0, 0…
$ V102  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 252, 0, 0, 0, 0, 169, 0, …
$ V103  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 176, 0, 0, 0, 0, 250, 0, …
$ V104  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 40, 0, 0, …
$ V105  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V106  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V107  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V108  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V109  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V110  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V111  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V112  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V113  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V114  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V115  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V116  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V117  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V118  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V119  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V120  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V121  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V122  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V123  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V124  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ V125  <int> 0, 0, 0, 0, 0, 0, 145, 0, 0, 0, 118, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ V126  <int> 0, 0, 0, 0, 0, 0, 255, 0, 0, 0, 219, 0, 0, 23, 0, 0, 0, 0, 0, 0,…
$ V127  <int> 0, 0, 0, 0, 0, 0, 211, 0, 0, 0, 166, 0, 0, 197, 0, 0, 0, 0, 0, 0…
$ V128  <int> 0, 51, 0, 0, 0, 0, 31, 0, 0, 0, 118, 0, 0, 253, 0, 0, 0, 0, 58, …
$ V129  <int> 0, 159, 0, 0, 0, 0, 0, 0, 0, 0, 118, 0, 0, 252, 0, 0, 0, 0, 242,…
$ V130  <int> 0, 253, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 208, 0, 0, 0, 0, 221, 0…
$ V131  <int> 0, 159, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 19, 0, 0, 0, 0, 143, 0,…
$ V132  <int> 0, 50, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 25…
$ V133  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6…
$ V134  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V135  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 189, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V136  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 190, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V137  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V138  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V139  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V140  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V141  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V142  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V143  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V144  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 13,…
$ V145  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 99, 0, 0, 0, 0, 0, 0, 0, 25,…
$ V146  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 91, 0, 0, 0, 0, 0, 0, 0, 10,…
$ V147  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 142, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V148  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 155, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V149  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 246, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V150  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 182, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V151  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 103, 0, 155, 0, 0, 0, 0, 0, 0, 0, …
$ V152  <int> 0, 0, 0, 0, 0, 0, 32, 38, 0, 0, 242, 0, 155, 0, 1, 0, 93, 0, 0, …
$ V153  <int> 3, 0, 0, 0, 0, 0, 237, 43, 5, 0, 254, 0, 155, 38, 168, 0, 164, 0…
$ V154  <int> 18, 0, 0, 0, 0, 0, 253, 105, 63, 0, 254, 0, 155, 178, 242, 0, 21…
$ V155  <int> 18, 48, 0, 0, 0, 0, 252, 255, 197, 0, 254, 0, 131, 252, 28, 0, 2…
$ V156  <int> 18, 238, 0, 0, 0, 13, 71, 253, 0, 0, 254, 0, 52, 253, 0, 0, 250,…
$ V157  <int> 126, 252, 0, 0, 0, 25, 0, 253, 0, 0, 254, 0, 0, 117, 0, 0, 194, …
$ V158  <int> 136, 252, 0, 0, 0, 100, 0, 253, 0, 0, 66, 0, 0, 65, 0, 0, 15, 0,…
$ V159  <int> 175, 252, 0, 124, 0, 122, 0, 253, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ V160  <int> 26, 237, 0, 253, 0, 7, 0, 253, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0,…
$ V161  <int> 166, 0, 67, 255, 0, 0, 0, 174, 0, 0, 0, 0, 0, 0, 0, 0, 0, 203, 0…
$ V162  <int> 255, 0, 232, 63, 0, 0, 0, 6, 0, 143, 0, 0, 0, 0, 0, 0, 0, 229, 0…
$ V163  <int> 247, 0, 39, 0, 0, 0, 0, 0, 0, 247, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0…
$ V164  <int> 127, 0, 0, 0, 0, 0, 0, 0, 0, 153, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ V165  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V166  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V167  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V168  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V169  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V170  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V171  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V172  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 138, 0, 0, 0, 0, 0, 0, 0, 13…
$ V173  <int> 0, 0, 62, 0, 0, 0, 0, 0, 0, 0, 0, 0, 254, 0, 0, 0, 0, 0, 0, 0, 2…
$ V174  <int> 0, 0, 81, 0, 0, 0, 0, 0, 0, 0, 0, 0, 254, 0, 0, 0, 0, 0, 0, 0, 1…
$ V175  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 254, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V176  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 254, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V177  <int> 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 254, 0, 0, 0, 0, 0, 0, 0, 0…
$ V178  <int> 36, 0, 0, 0, 0, 0, 0, 43, 0, 0, 0, 0, 254, 0, 0, 0, 20, 0, 0, 0,…
$ V179  <int> 94, 0, 0, 0, 0, 0, 0, 139, 0, 0, 18, 0, 254, 0, 0, 0, 176, 0, 0,…
$ V180  <int> 154, 0, 0, 0, 0, 0, 11, 224, 0, 0, 232, 0, 254, 0, 10, 0, 253, 0…
$ V181  <int> 170, 0, 0, 0, 0, 0, 175, 226, 20, 0, 254, 0, 254, 57, 228, 0, 23…
$ V182  <int> 253, 54, 0, 0, 0, 33, 253, 252, 254, 0, 254, 0, 254, 252, 254, 0…
$ V183  <int> 253, 227, 0, 0, 0, 151, 252, 253, 230, 0, 254, 0, 254, 252, 100,…
$ V184  <int> 253, 253, 0, 0, 0, 208, 71, 252, 24, 0, 254, 0, 252, 253, 0, 0, …
$ V185  <int> 253, 252, 0, 0, 0, 252, 0, 252, 0, 0, 254, 0, 210, 89, 0, 0, 254…
$ V186  <int> 253, 239, 0, 96, 0, 252, 0, 252, 0, 0, 238, 0, 122, 0, 0, 0, 214…
$ V187  <int> 225, 233, 0, 244, 0, 252, 0, 252, 0, 0, 70, 0, 33, 0, 0, 0, 0, 3…
$ V188  <int> 172, 252, 0, 251, 0, 146, 0, 252, 0, 0, 0, 0, 0, 0, 0, 0, 0, 95,…
$ V189  <int> 253, 57, 120, 253, 0, 0, 0, 252, 0, 136, 0, 0, 0, 0, 0, 0, 0, 25…
$ V190  <int> 242, 6, 180, 62, 0, 0, 0, 158, 0, 247, 0, 0, 0, 0, 0, 0, 0, 215,…
$ V191  <int> 195, 0, 39, 0, 0, 0, 0, 14, 0, 242, 0, 0, 0, 0, 0, 0, 0, 13, 0, …
$ V192  <int> 64, 0, 0, 0, 0, 0, 0, 0, 0, 86, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V193  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V194  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V195  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V196  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V197  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V198  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V199  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V200  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 220, 0, 0, 0, 0, 0, 0, 0, 13…
$ V201  <int> 0, 0, 126, 0, 0, 0, 0, 0, 0, 0, 0, 0, 254, 0, 0, 0, 0, 0, 0, 0, …
$ V202  <int> 0, 0, 163, 0, 0, 0, 0, 0, 0, 0, 0, 0, 254, 0, 0, 0, 0, 0, 0, 0, …
$ V203  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 254, 0, 0, 0, 0, 0, 0, 0, 79…
$ V204  <int> 49, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 235, 0, 0, 0, 0, 0, 0, 0, 0…
$ V205  <int> 238, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 189, 0, 0, 0, 0, 0, 0, 0, …
$ V206  <int> 253, 0, 0, 0, 0, 0, 0, 178, 0, 0, 0, 0, 189, 0, 0, 0, 204, 0, 0,…
$ V207  <int> 253, 0, 0, 0, 0, 0, 0, 252, 0, 0, 0, 0, 189, 0, 0, 0, 236, 0, 0,…
$ V208  <int> 253, 10, 0, 0, 0, 40, 0, 252, 0, 0, 104, 0, 189, 38, 0, 0, 135, …
$ V209  <int> 253, 60, 0, 0, 55, 152, 144, 252, 20, 0, 244, 0, 150, 222, 190, …
$ V210  <int> 253, 224, 0, 0, 148, 244, 253, 252, 254, 0, 254, 0, 189, 253, 25…
$ V211  <int> 253, 252, 0, 0, 210, 252, 252, 253, 254, 0, 224, 0, 205, 253, 12…
$ V212  <int> 253, 253, 0, 0, 253, 253, 71, 252, 48, 0, 254, 0, 254, 79, 0, 11…
$ V213  <int> 253, 252, 0, 0, 253, 224, 0, 252, 0, 0, 254, 0, 254, 0, 0, 121, …
$ V214  <int> 251, 202, 0, 127, 113, 211, 0, 252, 0, 0, 254, 0, 254, 0, 0, 162…
$ V215  <int> 93, 84, 0, 251, 87, 252, 0, 252, 0, 0, 141, 0, 75, 0, 0, 253, 12…
$ V216  <int> 82, 252, 2, 251, 148, 232, 0, 252, 0, 0, 0, 0, 0, 0, 0, 253, 0, …
$ V217  <int> 82, 253, 153, 253, 55, 40, 0, 252, 0, 192, 0, 31, 0, 0, 0, 213, …
$ V218  <int> 56, 122, 210, 62, 0, 0, 0, 252, 0, 252, 0, 40, 0, 0, 0, 0, 0, 18…
$ V219  <int> 39, 0, 40, 0, 0, 0, 0, 59, 0, 187, 0, 129, 0, 0, 0, 0, 0, 19, 0,…
$ V220  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 234, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V221  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 234, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V222  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 159, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V223  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V224  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V225  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V226  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V227  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V228  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, 0, 0, 132…
$ V229  <int> 0, 0, 220, 0, 0, 0, 0, 0, 0, 0, 0, 0, 74, 0, 0, 0, 0, 0, 0, 0, 2…
$ V230  <int> 0, 0, 163, 0, 0, 0, 0, 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, 0, 0, 2…
$ V231  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, 0, 0, 238…
$ V232  <int> 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 52…
$ V233  <int> 219, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 69, 0, 0, 0, 0…
$ V234  <int> 253, 0, 0, 0, 0, 15, 0, 109, 0, 0, 0, 0, 0, 0, 0, 0, 253, 0, 0, …
$ V235  <int> 253, 0, 0, 0, 0, 152, 0, 252, 0, 0, 0, 0, 0, 0, 0, 0, 167, 0, 0,…
$ V236  <int> 253, 163, 0, 0, 87, 239, 16, 252, 0, 0, 0, 0, 0, 131, 0, 0, 0, 1…
$ V237  <int> 253, 252, 0, 0, 232, 252, 191, 230, 20, 62, 207, 0, 0, 252, 83, …
$ V238  <int> 253, 252, 0, 0, 252, 252, 253, 132, 254, 185, 254, 0, 0, 179, 25…
$ V239  <int> 198, 252, 0, 0, 253, 252, 252, 133, 255, 18, 210, 0, 13, 27, 162…
$ V240  <int> 182, 253, 0, 0, 189, 216, 71, 132, 48, 0, 254, 0, 224, 0, 0, 251…
$ V241  <int> 247, 252, 0, 68, 210, 31, 0, 132, 0, 0, 254, 0, 254, 0, 0, 252, …
$ V242  <int> 241, 252, 0, 236, 252, 37, 0, 189, 0, 0, 254, 0, 254, 0, 0, 252,…
$ V243  <int> 0, 96, 0, 251, 252, 252, 0, 252, 0, 0, 34, 68, 153, 0, 0, 252, 2…
$ V244  <int> 0, 189, 27, 211, 253, 252, 0, 252, 0, 89, 0, 150, 0, 0, 0, 252, …
$ V245  <int> 0, 253, 254, 31, 168, 60, 0, 252, 0, 236, 0, 239, 0, 0, 0, 250, …
$ V246  <int> 0, 167, 162, 8, 0, 0, 0, 252, 0, 217, 0, 254, 0, 0, 0, 214, 0, 9…
$ V247  <int> 0, 0, 0, 0, 0, 0, 0, 59, 0, 47, 0, 253, 0, 0, 0, 0, 0, 0, 0, 0, …
$ V248  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 253, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V249  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 253, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V250  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 215, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V251  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V252  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V253  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V254  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V255  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V256  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 99, …
$ V257  <int> 0, 0, 222, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25…
$ V258  <int> 0, 0, 163, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25…
$ V259  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 252,…
$ V260  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 181,…
$ V261  <int> 80, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 74, 0, 0, 0, 1…
$ V262  <int> 156, 0, 0, 0, 4, 96, 0, 4, 0, 0, 0, 0, 0, 0, 0, 192, 217, 0, 0, …
$ V263  <int> 107, 51, 0, 0, 57, 252, 0, 29, 0, 0, 0, 0, 0, 198, 0, 226, 79, 1…
$ V264  <int> 253, 238, 0, 0, 242, 252, 26, 29, 0, 0, 0, 0, 0, 246, 0, 226, 0,…
$ V265  <int> 253, 253, 0, 0, 252, 252, 221, 24, 20, 216, 84, 0, 0, 220, 29, 2…
$ V266  <int> 205, 253, 0, 0, 190, 252, 253, 0, 254, 253, 206, 0, 0, 37, 254, …
$ V267  <int> 11, 190, 0, 0, 65, 217, 252, 0, 254, 60, 254, 0, 90, 0, 248, 253…
$ V268  <int> 0, 114, 0, 60, 5, 29, 124, 0, 57, 0, 254, 0, 254, 0, 25, 202, 0,…
$ V269  <int> 43, 253, 0, 228, 12, 0, 31, 0, 0, 0, 254, 156, 254, 0, 0, 252, 4…
$ V270  <int> 154, 228, 0, 251, 182, 37, 0, 14, 0, 0, 254, 201, 247, 0, 0, 252…
$ V271  <int> 0, 47, 0, 251, 252, 252, 0, 226, 0, 0, 41, 254, 53, 0, 0, 252, 2…
$ V272  <int> 0, 79, 183, 94, 253, 252, 0, 252, 0, 212, 0, 254, 0, 0, 0, 252, …
$ V273  <int> 0, 255, 254, 0, 116, 60, 0, 252, 0, 255, 0, 254, 0, 0, 0, 252, 0…
$ V274  <int> 0, 168, 125, 0, 0, 0, 0, 172, 0, 81, 0, 241, 0, 0, 0, 225, 0, 8,…
$ V275  <int> 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 150, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V276  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 98, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ V277  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V278  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V279  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V280  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V281  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V282  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V283  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V284  <int> 0, 0, 46, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, …
$ V285  <int> 0, 0, 245, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12…
$ V286  <int> 0, 0, 163, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25…
$ V287  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 252,…
$ V288  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 68, 0, 0, 0, 0, 252…
$ V289  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 223, 8, 0, 0, 0, 10…
$ V290  <int> 14, 48, 0, 0, 96, 181, 0, 0, 0, 0, 0, 0, 0, 79, 0, 252, 10, 0, 0…
$ V291  <int> 1, 238, 0, 0, 252, 252, 0, 0, 0, 0, 0, 0, 0, 253, 0, 252, 0, 139…
$ V292  <int> 154, 252, 0, 0, 252, 252, 0, 0, 0, 0, 0, 0, 0, 252, 0, 252, 0, 2…
$ V293  <int> 253, 252, 0, 0, 183, 220, 125, 0, 20, 206, 0, 0, 6, 135, 29, 252…
$ V294  <int> 90, 179, 0, 0, 14, 167, 253, 0, 254, 252, 24, 0, 152, 28, 255, 2…
$ V295  <int> 0, 12, 0, 0, 0, 30, 252, 0, 254, 68, 209, 19, 246, 0, 254, 39, 0…
$ V296  <int> 0, 75, 0, 155, 0, 0, 252, 0, 108, 0, 254, 154, 254, 0, 103, 19, …
$ V297  <int> 0, 121, 0, 253, 92, 0, 108, 0, 0, 0, 254, 254, 254, 0, 0, 39, 39…
$ V298  <int> 0, 21, 0, 253, 252, 77, 0, 85, 0, 0, 254, 236, 49, 0, 0, 65, 254…
$ V299  <int> 0, 0, 0, 189, 252, 252, 0, 243, 0, 48, 171, 203, 0, 0, 0, 224, 2…
$ V300  <int> 0, 0, 198, 0, 225, 252, 0, 252, 0, 242, 0, 83, 0, 0, 0, 252, 104…
$ V301  <int> 0, 253, 254, 0, 21, 60, 0, 252, 0, 253, 0, 39, 0, 0, 0, 252, 0, …
$ V302  <int> 0, 243, 56, 0, 0, 0, 0, 144, 0, 89, 0, 30, 0, 0, 0, 183, 0, 0, 0…
$ V303  <int> 0, 50, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ V304  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V305  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V306  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V307  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V308  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V309  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V310  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V311  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V312  <int> 0, 0, 120, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V313  <int> 0, 0, 254, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15…
$ V314  <int> 0, 0, 163, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21…
$ V315  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 252,…
$ V316  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 186, 0, 0, 0, 0, 25…
$ V317  <int> 0, 38, 0, 0, 132, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 252, 0, 0, 0, 0…
$ V318  <int> 0, 165, 0, 0, 253, 26, 0, 0, 0, 0, 0, 0, 0, 140, 0, 252, 0, 0, 0…
$ V319  <int> 0, 253, 0, 0, 252, 128, 0, 0, 0, 0, 0, 0, 0, 253, 0, 252, 0, 55,…
$ V320  <int> 139, 233, 0, 0, 146, 58, 0, 0, 0, 131, 0, 0, 66, 252, 0, 245, 0,…
$ V321  <int> 253, 208, 0, 0, 14, 22, 0, 0, 16, 251, 91, 0, 158, 118, 29, 108,…
$ V322  <int> 190, 84, 0, 0, 0, 0, 253, 0, 239, 212, 137, 0, 254, 0, 254, 53, …
$ V323  <int> 2, 0, 0, 20, 0, 0, 252, 0, 254, 21, 253, 144, 254, 0, 254, 0, 0,…
$ V324  <int> 0, 0, 0, 253, 0, 0, 252, 0, 143, 0, 254, 253, 249, 0, 109, 0, 0,…
$ V325  <int> 0, 0, 0, 251, 215, 0, 108, 88, 0, 0, 254, 145, 103, 0, 0, 0, 5, …
$ V326  <int> 0, 0, 0, 235, 252, 100, 0, 189, 0, 11, 254, 12, 8, 111, 0, 150, …
$ V327  <int> 0, 0, 23, 66, 252, 252, 0, 252, 0, 167, 112, 0, 0, 140, 0, 252, …
$ V328  <int> 0, 0, 231, 0, 79, 252, 0, 252, 0, 252, 0, 0, 0, 140, 0, 252, 141…
$ V329  <int> 0, 253, 254, 0, 0, 60, 0, 252, 0, 197, 0, 0, 0, 0, 0, 220, 0, 0,…
$ V330  <int> 0, 252, 29, 0, 0, 0, 0, 14, 0, 5, 0, 0, 0, 0, 0, 20, 0, 0, 0, 0,…
$ V331  <int> 0, 165, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V332  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V333  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V334  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V335  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V336  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V337  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V338  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V339  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V340  <int> 0, 0, 159, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V341  <int> 0, 0, 254, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V342  <int> 0, 0, 120, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15…
$ V343  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 70, 0, 0, 0, 0, 252…
$ V344  <int> 0, 7, 0, 0, 126, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 242, 0, 0, 0, 0, …
$ V345  <int> 0, 178, 0, 0, 253, 0, 0, 0, 0, 0, 0, 0, 0, 13, 0, 252, 0, 0, 0, …
$ V346  <int> 0, 252, 0, 0, 247, 0, 0, 0, 0, 0, 0, 0, 0, 191, 0, 252, 0, 0, 0,…
$ V347  <int> 0, 240, 0, 0, 176, 0, 0, 0, 0, 29, 40, 0, 54, 255, 0, 222, 0, 7,…
$ V348  <int> 11, 71, 0, 0, 9, 0, 0, 0, 0, 232, 214, 10, 251, 253, 0, 59, 0, 1…
$ V349  <int> 190, 19, 0, 0, 0, 0, 0, 0, 0, 247, 250, 129, 254, 56, 29, 0, 0, …
$ V350  <int> 253, 28, 0, 32, 0, 0, 255, 0, 178, 63, 254, 222, 254, 0, 254, 0,…
$ V351  <int> 70, 0, 0, 205, 8, 0, 253, 91, 254, 0, 254, 78, 254, 0, 254, 0, 0…
$ V352  <int> 0, 0, 0, 253, 78, 0, 253, 212, 143, 0, 254, 79, 248, 114, 109, 0…
$ V353  <int> 0, 0, 0, 251, 245, 0, 108, 247, 0, 0, 254, 8, 74, 113, 0, 0, 0, …
$ V354  <int> 0, 0, 0, 126, 253, 157, 0, 252, 0, 153, 254, 0, 5, 222, 0, 178, …
$ V355  <int> 0, 0, 163, 0, 129, 252, 0, 252, 0, 252, 34, 0, 0, 253, 0, 252, 2…
$ V356  <int> 0, 0, 254, 0, 0, 252, 0, 252, 0, 226, 0, 0, 0, 253, 0, 252, 141,…
$ V357  <int> 0, 253, 216, 0, 0, 60, 0, 204, 0, 0, 0, 0, 0, 255, 0, 141, 0, 0,…
$ V358  <int> 0, 252, 16, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 27, 0, 0, 0, 0, 0, 0, …
$ V359  <int> 0, 195, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V360  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V361  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V362  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V363  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V364  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V365  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V366  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V367  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V368  <int> 0, 0, 159, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V369  <int> 0, 0, 254, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V370  <int> 0, 0, 67, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100…
$ V371  <int> 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 185, 0, 0, 0, 0, 2…
$ V372  <int> 0, 57, 0, 0, 232, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 252, 0, 0, 0, 0,…
$ V373  <int> 0, 252, 0, 0, 252, 0, 0, 0, 0, 0, 0, 0, 0, 76, 0, 252, 0, 0, 0, …
$ V374  <int> 0, 252, 0, 0, 176, 0, 0, 32, 0, 45, 0, 0, 0, 252, 0, 194, 0, 0, …
$ V375  <int> 0, 63, 0, 0, 0, 0, 0, 125, 0, 219, 81, 0, 140, 253, 0, 67, 0, 0,…
$ V376  <int> 0, 0, 0, 0, 0, 0, 0, 193, 0, 252, 247, 134, 254, 223, 0, 0, 0, 1…
$ V377  <int> 35, 0, 0, 0, 0, 0, 0, 193, 0, 143, 254, 253, 254, 37, 29, 0, 0, …
$ V378  <int> 241, 0, 0, 104, 36, 110, 253, 193, 178, 0, 254, 167, 254, 0, 254…
$ V379  <int> 225, 0, 0, 251, 201, 121, 252, 253, 254, 0, 254, 8, 254, 48, 254…
$ V380  <int> 160, 0, 14, 253, 252, 122, 252, 252, 143, 0, 254, 0, 254, 174, 1…
$ V381  <int> 108, 0, 86, 184, 252, 121, 108, 252, 0, 116, 254, 0, 254, 252, 0…
$ V382  <int> 1, 0, 178, 15, 169, 202, 0, 252, 0, 249, 254, 0, 202, 252, 0, 24…
$ V383  <int> 0, 0, 248, 0, 11, 252, 0, 238, 0, 252, 146, 0, 125, 242, 0, 252,…
$ V384  <int> 0, 0, 254, 0, 0, 194, 0, 102, 0, 103, 0, 0, 45, 214, 0, 194, 128…
$ V385  <int> 0, 253, 91, 0, 0, 3, 0, 28, 0, 0, 0, 0, 0, 253, 0, 67, 0, 0, 0, …
$ V386  <int> 0, 252, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 199, 0, 0, 0, 0, 0, 0, …
$ V387  <int> 0, 195, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 31, 0, 0, 0, 0, 0, 0, 0…
$ V388  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V389  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V390  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V391  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V392  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V393  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V394  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V395  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V396  <int> 0, 0, 159, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V397  <int> 0, 0, 254, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V398  <int> 0, 0, 85, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ V399  <int> 0, 0, 0, 0, 22, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 83, 0, 0, 0, 0, 16…
$ V400  <int> 0, 198, 0, 0, 252, 0, 0, 0, 0, 4, 0, 0, 0, 13, 0, 205, 0, 0, 0, …
$ V401  <int> 0, 253, 0, 0, 252, 0, 0, 45, 0, 96, 0, 0, 0, 109, 0, 190, 0, 0, …
$ V402  <int> 0, 190, 47, 0, 30, 0, 0, 222, 0, 253, 0, 0, 0, 252, 0, 24, 0, 0,…
$ V403  <int> 0, 0, 49, 0, 22, 10, 0, 252, 0, 255, 0, 0, 58, 228, 0, 0, 0, 0, …
$ V404  <int> 0, 0, 116, 0, 119, 53, 0, 252, 0, 253, 110, 255, 181, 130, 0, 0,…
$ V405  <int> 0, 0, 144, 80, 197, 179, 0, 252, 0, 200, 246, 254, 234, 0, 29, 0…
$ V406  <int> 81, 0, 150, 240, 241, 253, 253, 252, 178, 122, 254, 78, 254, 38,…
$ V407  <int> 240, 0, 241, 251, 253, 253, 252, 253, 254, 7, 254, 0, 254, 165, …
$ V408  <int> 253, 0, 243, 193, 252, 255, 252, 252, 162, 25, 254, 0, 254, 253,…
$ V409  <int> 253, 0, 234, 23, 251, 253, 108, 252, 0, 201, 254, 0, 254, 233, 0…
$ V410  <int> 119, 0, 179, 0, 77, 253, 0, 252, 0, 250, 254, 0, 254, 164, 0, 25…
$ V411  <int> 25, 0, 241, 0, 0, 228, 0, 177, 0, 158, 171, 0, 254, 49, 0, 209, …
$ V412  <int> 0, 0, 252, 0, 0, 35, 0, 0, 0, 0, 0, 0, 252, 63, 0, 24, 56, 0, 72…
$ V413  <int> 0, 255, 40, 0, 0, 0, 0, 0, 0, 0, 0, 0, 140, 253, 0, 0, 0, 0, 0, …
$ V414  <int> 0, 253, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22, 214, 0, 0, 0, 0, 0, 0,…
$ V415  <int> 0, 196, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 31, 0, 0, 20, 0, 0, 0, …
$ V416  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 67, 0, 0, 0, 0, …
$ V417  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 124, 0, 0, 0, 0,…
$ V418  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 39, 0, 0, 0, 0, …
$ V419  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V420  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V421  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V422  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V423  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V424  <int> 0, 0, 150, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V425  <int> 0, 0, 253, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V426  <int> 0, 0, 237, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V427  <int> 0, 76, 207, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ V428  <int> 0, 246, 207, 0, 231, 0, 0, 0, 0, 92, 0, 0, 0, 73, 0, 0, 0, 0, 0,…
$ V429  <int> 0, 252, 207, 0, 252, 5, 0, 45, 0, 252, 0, 0, 0, 252, 0, 0, 0, 0,…
$ V430  <int> 0, 112, 253, 0, 253, 54, 0, 223, 0, 252, 0, 0, 0, 252, 0, 0, 0, …
$ V431  <int> 0, 0, 254, 0, 252, 227, 0, 253, 0, 253, 0, 0, 0, 126, 0, 0, 0, 3…
$ V432  <int> 0, 0, 250, 32, 252, 252, 0, 253, 0, 217, 0, 201, 0, 0, 0, 0, 0, …
$ V433  <int> 0, 0, 240, 253, 252, 243, 0, 253, 0, 252, 73, 253, 30, 23, 29, 0…
$ V434  <int> 0, 0, 198, 253, 226, 228, 253, 253, 178, 252, 89, 226, 50, 178, …
$ V435  <int> 45, 0, 143, 253, 227, 170, 252, 255, 254, 200, 89, 69, 73, 252, …
$ V436  <int> 186, 0, 91, 159, 252, 242, 252, 253, 240, 227, 93, 0, 155, 240, …
$ V437  <int> 253, 0, 28, 0, 231, 252, 108, 253, 0, 252, 240, 0, 253, 148, 0, …
$ V438  <int> 253, 0, 5, 0, 0, 252, 0, 253, 0, 231, 254, 0, 254, 7, 0, 248, 25…
$ V439  <int> 150, 0, 233, 0, 0, 231, 0, 253, 0, 0, 171, 0, 254, 44, 0, 106, 2…
$ V440  <int> 27, 0, 250, 0, 0, 117, 0, 74, 0, 0, 0, 0, 254, 215, 0, 0, 208, 0…
$ V441  <int> 0, 253, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 254, 240, 0, 0, 157, 0, 0,…
$ V442  <int> 0, 252, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 191, 148, 0, 0, 207, 0, 0,…
$ V443  <int> 0, 148, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 225, 0, 0, 0, …
$ V444  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 254, 0, 0, 0, 0,…
$ V445  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 241, 0, 0, 0, 0,…
$ V446  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 160, 0, 0, 0, 0,…
$ V447  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V448  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V449  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V450  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V451  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V452  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V453  <int> 0, 0, 119, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V454  <int> 0, 0, 177, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V455  <int> 0, 85, 177, 0, 0, 0, 0, 0, 0, 87, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ V456  <int> 0, 252, 177, 0, 55, 6, 0, 0, 0, 251, 0, 0, 0, 119, 0, 0, 0, 0, 0…
$ V457  <int> 0, 230, 177, 0, 235, 78, 0, 0, 0, 247, 0, 55, 0, 252, 0, 0, 9, 0…
$ V458  <int> 0, 25, 177, 0, 253, 252, 0, 31, 0, 231, 0, 6, 0, 252, 0, 0, 31, …
$ V459  <int> 0, 0, 98, 0, 217, 252, 0, 123, 0, 65, 0, 0, 0, 0, 0, 0, 82, 171,…
$ V460  <int> 0, 0, 56, 151, 138, 125, 0, 52, 0, 48, 0, 18, 0, 0, 0, 0, 137, 2…
$ V461  <int> 0, 0, 0, 251, 42, 59, 0, 44, 0, 189, 0, 128, 0, 197, 29, 0, 203,…
$ V462  <int> 0, 0, 0, 251, 24, 0, 255, 44, 113, 252, 0, 253, 0, 252, 254, 0, …
$ V463  <int> 0, 0, 0, 251, 192, 18, 253, 44, 254, 252, 0, 241, 0, 252, 254, 2…
$ V464  <int> 16, 0, 0, 39, 252, 208, 253, 44, 240, 253, 1, 41, 0, 63, 63, 252…
$ V465  <int> 93, 0, 0, 0, 143, 252, 170, 143, 0, 252, 128, 0, 91, 0, 0, 252, …
$ V466  <int> 252, 0, 102, 0, 0, 252, 0, 252, 0, 251, 254, 0, 200, 57, 0, 102,…
$ V467  <int> 253, 7, 254, 0, 0, 252, 0, 252, 0, 227, 219, 0, 254, 252, 0, 0, …
$ V468  <int> 187, 135, 220, 0, 0, 252, 0, 74, 0, 35, 31, 0, 254, 252, 0, 0, 2…
$ V469  <int> 0, 253, 0, 0, 0, 87, 0, 0, 0, 0, 0, 0, 254, 140, 0, 0, 223, 0, 0…
$ V470  <int> 0, 186, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 254, 0, 0, 0, 223, 0, 0, 0…
$ V471  <int> 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 118, 0, 0, 0, 127, 0, 0, 0,…
$ V472  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 52, 0, 0, 0, 0, …
$ V473  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 33, 0, 0, 0, 0, …
$ V474  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V475  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V476  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V477  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V478  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V479  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V480  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V481  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V482  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1…
$ V483  <int> 0, 85, 0, 0, 0, 5, 0, 0, 0, 190, 0, 25, 0, 0, 0, 0, 9, 0, 0, 0, …
$ V484  <int> 0, 252, 0, 0, 0, 135, 0, 0, 0, 221, 0, 205, 0, 135, 0, 0, 137, 0…
$ V485  <int> 0, 223, 0, 0, 0, 252, 0, 0, 0, 98, 0, 235, 0, 253, 0, 0, 214, 26…
$ V486  <int> 0, 0, 0, 0, 0, 252, 0, 0, 0, 0, 0, 92, 0, 174, 0, 0, 254, 123, 6…
$ V487  <int> 0, 0, 0, 48, 0, 180, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 254, 254, 254…
$ V488  <int> 0, 0, 0, 221, 0, 16, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 254, 253, 13…
$ V489  <int> 0, 0, 0, 251, 0, 0, 0, 0, 0, 42, 0, 20, 0, 229, 29, 0, 254, 203,…
$ V490  <int> 0, 0, 0, 251, 62, 21, 253, 0, 83, 196, 0, 253, 0, 253, 254, 134,…
$ V491  <int> 0, 0, 0, 172, 255, 203, 252, 0, 254, 252, 0, 253, 0, 112, 254, 2…
$ V492  <int> 0, 0, 0, 0, 253, 253, 252, 0, 245, 253, 7, 58, 0, 0, 28, 253, 25…
$ V493  <int> 0, 0, 0, 0, 109, 247, 252, 15, 31, 252, 254, 0, 0, 38, 0, 253, 2…
$ V494  <int> 249, 7, 169, 0, 0, 129, 42, 252, 0, 252, 254, 0, 4, 222, 0, 39, …
$ V495  <int> 253, 131, 254, 0, 0, 173, 0, 252, 0, 162, 214, 0, 192, 253, 0, 0…
$ V496  <int> 249, 252, 137, 0, 0, 252, 0, 74, 0, 0, 28, 0, 254, 112, 0, 0, 50…
$ V497  <int> 64, 225, 0, 0, 0, 252, 0, 0, 0, 0, 0, 0, 254, 0, 0, 0, 0, 0, 0, …
$ V498  <int> 0, 71, 0, 0, 0, 184, 0, 0, 0, 0, 0, 0, 254, 0, 0, 0, 0, 0, 0, 0,…
$ V499  <int> 0, 0, 0, 0, 0, 66, 0, 0, 0, 0, 0, 0, 154, 0, 0, 0, 0, 0, 0, 0, 0…
$ V500  <int> 0, 0, 0, 0, 0, 49, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ V501  <int> 0, 0, 0, 0, 0, 49, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ V502  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V503  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V504  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V505  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V506  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V507  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V508  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V509  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V510  <int> 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1…
$ V511  <int> 0, 85, 0, 0, 0, 136, 0, 0, 0, 111, 0, 231, 0, 0, 0, 0, 185, 0, 0…
$ V512  <int> 0, 252, 0, 0, 0, 252, 0, 0, 0, 29, 0, 245, 0, 135, 0, 0, 254, 0,…
$ V513  <int> 0, 145, 0, 0, 0, 241, 0, 0, 0, 0, 0, 108, 0, 252, 0, 0, 247, 93,…
$ V514  <int> 0, 0, 0, 0, 0, 106, 0, 0, 0, 0, 0, 0, 0, 173, 0, 0, 179, 253, 68…
$ V515  <int> 0, 0, 0, 234, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 146, 254, 255…
$ V516  <int> 0, 0, 0, 251, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 67, 121, 13, …
$ V517  <int> 0, 0, 0, 251, 0, 53, 0, 0, 0, 62, 0, 132, 0, 227, 29, 6, 60, 13,…
$ V518  <int> 0, 0, 0, 196, 71, 200, 149, 0, 79, 239, 0, 253, 0, 252, 254, 183…
$ V519  <int> 46, 0, 0, 12, 253, 252, 252, 0, 254, 252, 0, 185, 0, 158, 254, 2…
$ V520  <int> 130, 0, 0, 0, 252, 216, 252, 0, 246, 86, 138, 14, 0, 226, 28, 25…
$ V521  <int> 183, 48, 0, 0, 21, 65, 252, 86, 38, 42, 254, 0, 0, 234, 0, 107, …
$ V522  <int> 253, 165, 169, 0, 0, 0, 144, 252, 0, 42, 254, 0, 0, 201, 0, 2, 2…
$ V523  <int> 253, 252, 254, 0, 0, 14, 0, 252, 0, 14, 116, 0, 141, 27, 0, 0, 1…
$ V524  <int> 207, 173, 57, 0, 0, 72, 0, 74, 0, 0, 0, 0, 254, 12, 0, 0, 0, 0, …
$ V525  <int> 2, 0, 0, 0, 0, 163, 0, 0, 0, 0, 0, 0, 254, 0, 0, 0, 0, 0, 0, 0, …
$ V526  <int> 0, 0, 0, 0, 0, 241, 0, 0, 0, 0, 0, 0, 254, 0, 0, 0, 0, 0, 0, 0, …
$ V527  <int> 0, 0, 0, 0, 0, 252, 0, 0, 0, 0, 0, 0, 116, 0, 0, 0, 0, 0, 0, 0, …
$ V528  <int> 0, 0, 0, 0, 0, 252, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V529  <int> 0, 0, 0, 0, 0, 223, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V530  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V531  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V532  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V533  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V534  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V535  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V536  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V537  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V538  <int> 0, 0, 0, 0, 0, 105, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V539  <int> 0, 86, 0, 0, 0, 252, 0, 5, 0, 0, 19, 121, 0, 0, 0, 0, 255, 0, 0,…
$ V540  <int> 0, 253, 0, 0, 0, 242, 0, 75, 0, 0, 177, 245, 0, 57, 0, 0, 222, 6…
$ V541  <int> 0, 225, 0, 0, 0, 88, 0, 9, 0, 0, 90, 254, 25, 252, 0, 0, 49, 239…
$ V542  <int> 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 254, 126, 252, 0, 0, 0, 253, 6…
$ V543  <int> 0, 0, 0, 253, 0, 73, 0, 0, 0, 0, 0, 254, 86, 57, 0, 0, 0, 76, 25…
$ V544  <int> 0, 0, 0, 251, 0, 170, 0, 0, 0, 15, 0, 217, 0, 104, 0, 10, 0, 8, …
$ V545  <int> 39, 0, 0, 251, 0, 244, 0, 0, 0, 148, 0, 254, 0, 240, 29, 102, 0,…
$ V546  <int> 148, 0, 0, 89, 0, 252, 109, 0, 0, 253, 0, 223, 0, 252, 254, 252,…
$ V547  <int> 229, 0, 0, 0, 253, 126, 252, 0, 214, 218, 25, 50, 0, 252, 254, 2…
$ V548  <int> 253, 114, 0, 0, 252, 29, 252, 98, 254, 0, 240, 0, 0, 253, 35, 16…
$ V549  <int> 253, 238, 0, 0, 21, 0, 252, 242, 150, 0, 254, 0, 0, 233, 0, 16, …
$ V550  <int> 253, 253, 169, 0, 0, 0, 144, 252, 0, 0, 254, 0, 3, 74, 0, 0, 50,…
$ V551  <int> 250, 162, 254, 0, 0, 0, 0, 252, 0, 0, 34, 0, 188, 0, 0, 0, 0, 0,…
$ V552  <int> 182, 0, 57, 0, 0, 0, 0, 74, 0, 0, 0, 0, 254, 0, 0, 0, 0, 0, 147,…
$ V553  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 254, 0, 0, 0, 0, 0, 0, 0, 25…
$ V554  <int> 0, 0, 0, 0, 0, 89, 0, 0, 0, 0, 0, 0, 250, 0, 0, 0, 0, 0, 0, 0, 2…
$ V555  <int> 0, 0, 0, 0, 0, 180, 0, 0, 0, 0, 0, 0, 61, 0, 0, 0, 0, 0, 0, 0, 0…
$ V556  <int> 0, 0, 0, 0, 0, 180, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V557  <int> 0, 0, 0, 0, 0, 37, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ V558  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V559  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V560  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V561  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V562  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V563  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V564  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V565  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V566  <int> 0, 0, 0, 0, 0, 231, 0, 61, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V567  <int> 0, 85, 0, 0, 0, 252, 0, 183, 0, 0, 164, 0, 0, 0, 0, 0, 254, 0, 0…
$ V568  <int> 0, 252, 0, 0, 0, 245, 0, 252, 0, 0, 254, 116, 24, 51, 0, 0, 206,…
$ V569  <int> 0, 249, 0, 0, 0, 205, 0, 29, 0, 0, 215, 165, 209, 242, 0, 0, 4, …
$ V570  <int> 0, 146, 0, 159, 0, 216, 0, 0, 0, 0, 63, 233, 254, 252, 0, 0, 0, …
$ V571  <int> 24, 48, 0, 255, 0, 252, 0, 0, 0, 0, 36, 233, 15, 253, 0, 13, 0, …
$ V572  <int> 114, 29, 0, 253, 0, 252, 0, 0, 0, 121, 0, 234, 0, 252, 0, 168, 0…
$ V573  <int> 221, 85, 0, 253, 0, 252, 0, 0, 0, 252, 51, 180, 0, 252, 29, 252,…
$ V574  <int> 253, 178, 0, 31, 71, 124, 0, 18, 0, 231, 89, 39, 0, 252, 254, 25…
$ V575  <int> 253, 225, 0, 0, 253, 3, 218, 92, 144, 28, 206, 3, 0, 252, 254, 1…
$ V576  <int> 253, 253, 0, 0, 252, 0, 253, 239, 241, 0, 254, 0, 0, 240, 109, 2…
$ V577  <int> 253, 223, 0, 0, 21, 0, 253, 252, 8, 0, 254, 0, 23, 148, 0, 0, 64…
$ V578  <int> 201, 167, 169, 0, 0, 0, 255, 252, 0, 0, 139, 0, 137, 0, 0, 0, 0,…
$ V579  <int> 78, 56, 255, 0, 0, 0, 35, 243, 0, 0, 8, 0, 254, 0, 0, 0, 0, 0, 2…
$ V580  <int> 0, 0, 94, 0, 0, 0, 0, 65, 0, 0, 0, 0, 254, 0, 0, 0, 0, 0, 20, 0,…
$ V581  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 254, 0, 0, 0, 0, 0, 0, 0, 25…
$ V582  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 209, 0, 0, 0, 0, 0, 0, 0, 15…
$ V583  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0…
$ V584  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V585  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V586  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V587  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V588  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V589  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V590  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V591  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V592  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V593  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V594  <int> 0, 0, 0, 0, 0, 207, 0, 208, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ V595  <int> 0, 85, 0, 0, 0, 252, 0, 252, 0, 0, 57, 0, 0, 0, 0, 0, 216, 0, 0,…
$ V596  <int> 0, 252, 0, 0, 0, 252, 0, 252, 0, 0, 197, 0, 168, 0, 0, 0, 254, 1…
$ V597  <int> 23, 252, 0, 48, 0, 252, 0, 147, 0, 0, 254, 0, 254, 75, 0, 0, 158…
$ V598  <int> 66, 252, 0, 228, 0, 252, 0, 134, 0, 0, 254, 0, 254, 189, 0, 0, 1…
$ V599  <int> 213, 229, 0, 253, 0, 178, 0, 134, 0, 31, 222, 0, 48, 253, 0, 41,…
$ V600  <int> 253, 215, 0, 247, 0, 116, 0, 134, 0, 221, 180, 0, 9, 252, 0, 252…
$ V601  <int> 253, 252, 0, 140, 0, 36, 0, 134, 0, 251, 241, 0, 0, 252, 6, 252,…
$ V602  <int> 253, 252, 0, 8, 106, 4, 0, 203, 0, 129, 254, 0, 0, 157, 212, 217…
$ V603  <int> 253, 252, 0, 0, 253, 0, 175, 253, 144, 0, 254, 0, 9, 112, 254, 0…
$ V604  <int> 198, 196, 0, 0, 252, 0, 252, 252, 240, 0, 253, 0, 127, 63, 109, …
$ V605  <int> 81, 130, 0, 0, 21, 0, 252, 252, 2, 0, 213, 0, 241, 0, 0, 0, 0, 0…
$ V606  <int> 2, 0, 169, 0, 0, 0, 253, 188, 0, 0, 11, 0, 254, 0, 0, 0, 0, 0, 2…
$ V607  <int> 0, 0, 254, 0, 0, 0, 35, 83, 0, 0, 0, 0, 254, 0, 0, 0, 0, 0, 27, …
$ V608  <int> 0, 0, 96, 0, 0, 0, 0, 0, 0, 0, 0, 0, 255, 0, 0, 0, 0, 0, 0, 0, 2…
$ V609  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 242, 0, 0, 0, 0, 0, 0, 0, 25…
$ V610  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 63, 0, 0, 0, 0, 0, 0, 0, 252…
$ V611  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, …
$ V612  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V613  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V614  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V615  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V616  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V617  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V618  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V619  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V620  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V621  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V622  <int> 0, 0, 0, 0, 0, 13, 0, 208, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V623  <int> 18, 28, 0, 0, 0, 93, 0, 252, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ V624  <int> 171, 199, 0, 0, 0, 143, 0, 252, 0, 0, 140, 0, 101, 0, 0, 0, 131,…
$ V625  <int> 219, 252, 0, 64, 0, 121, 0, 252, 0, 0, 105, 0, 254, 0, 0, 0, 247…
$ V626  <int> 253, 252, 0, 251, 0, 23, 0, 252, 0, 0, 254, 0, 254, 0, 0, 40, 24…
$ V627  <int> 253, 253, 0, 253, 0, 6, 0, 252, 0, 218, 254, 0, 254, 0, 0, 155, …
$ V628  <int> 253, 252, 0, 220, 0, 0, 0, 252, 0, 252, 254, 0, 205, 0, 0, 252, …
$ V629  <int> 253, 252, 0, 0, 0, 0, 0, 252, 0, 160, 254, 0, 190, 0, 0, 214, 17…
$ V630  <int> 195, 233, 0, 0, 45, 0, 0, 252, 0, 0, 254, 0, 190, 0, 203, 31, 72…
$ V631  <int> 80, 145, 0, 0, 255, 0, 73, 253, 144, 0, 254, 0, 205, 0, 254, 0, …
$ V632  <int> 9, 0, 0, 0, 253, 0, 252, 230, 254, 0, 236, 0, 254, 0, 178, 0, 0,…
$ V633  <int> 0, 0, 0, 0, 21, 0, 252, 153, 82, 0, 0, 0, 254, 0, 0, 0, 0, 0, 13…
$ V634  <int> 0, 0, 169, 0, 0, 0, 253, 8, 0, 0, 0, 0, 254, 0, 0, 0, 0, 0, 6, 0…
$ V635  <int> 0, 0, 254, 0, 0, 0, 35, 0, 0, 0, 0, 0, 254, 0, 0, 0, 0, 0, 0, 0,…
$ V636  <int> 0, 0, 153, 0, 0, 0, 0, 0, 0, 0, 0, 0, 242, 0, 0, 0, 0, 0, 0, 0, …
$ V637  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 67, 0, 0, 0, 0, 0, 0, 0, 252…
$ V638  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 252,…
$ V639  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 63, …
$ V640  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V641  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V642  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V643  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V644  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V645  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V646  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V647  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V648  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V649  <int> 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ V650  <int> 172, 0, 0, 0, 0, 0, 0, 49, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V651  <int> 226, 0, 0, 0, 0, 0, 0, 157, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
$ V652  <int> 253, 25, 0, 0, 0, 0, 0, 252, 0, 0, 0, 0, 33, 0, 0, 0, 0, 112, 0,…
$ V653  <int> 253, 128, 0, 64, 0, 0, 0, 252, 0, 0, 7, 0, 166, 0, 0, 0, 0, 253,…
$ V654  <int> 253, 252, 0, 251, 0, 0, 0, 252, 0, 0, 117, 0, 254, 0, 0, 165, 0,…
$ V655  <int> 253, 253, 0, 253, 0, 0, 0, 252, 0, 122, 117, 0, 254, 0, 0, 252, …
$ V656  <int> 244, 252, 0, 220, 0, 0, 0, 252, 0, 252, 165, 0, 254, 0, 0, 252, …
$ V657  <int> 133, 141, 0, 0, 0, 0, 0, 217, 0, 82, 254, 0, 254, 0, 0, 106, 0, …
$ V658  <int> 11, 37, 0, 0, 0, 0, 0, 207, 0, 0, 254, 0, 254, 0, 155, 0, 0, 9, …
$ V659  <int> 0, 0, 0, 0, 218, 0, 31, 146, 230, 0, 239, 0, 254, 0, 254, 0, 0, …
$ V660  <int> 0, 0, 0, 0, 252, 0, 211, 45, 247, 0, 50, 0, 254, 0, 190, 0, 0, 0…
$ V661  <int> 0, 0, 0, 0, 56, 0, 252, 0, 40, 0, 0, 0, 254, 0, 0, 0, 0, 0, 0, 0…
$ V662  <int> 0, 0, 169, 0, 0, 0, 253, 0, 0, 0, 0, 0, 250, 0, 0, 0, 0, 0, 0, 0…
$ V663  <int> 0, 0, 255, 0, 0, 0, 35, 0, 0, 0, 0, 0, 138, 0, 0, 0, 0, 0, 0, 0,…
$ V664  <int> 0, 0, 153, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55, 0, 0, 0, 0, 0, 0, 0, 9…
$ V665  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 206,…
$ V666  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 131,…
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$ V777  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V778  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V779  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V780  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V781  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V782  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V783  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ V784  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ label <fct> 5, 0, 4, 1, 9, 2, 1, 3, 1, 4, 3, 5, 3, 6, 1, 7, 2, 8, 6, 9, 4, 0…
Code
mnist %>% 
  mutate(id = 1:n()) %>% 
  filter(id %in% 1:10) %>% 
  pivot_longer(starts_with("V")) %>% 
  mutate(row = rep(rep(1:28, each = 28), max(id)),
         col = rep(rep(1:28, times = 28), max(id))) %>% 
  ggplot(aes(col, row)) +
  geom_tile(aes(fill = value)) + 
  facet_wrap(~id, nrow = 2) +
  scale_y_reverse() +
  theme_void(base_size = 18) +
  guides(fill = "none")

Multi-class logistic regression

  • For multi-class problems, we can fit a logistic model for every class.
mnist_digit_preds <- map_dfc(0:9, function(digit) {
  mnist_data <- mnist %>% 
    mutate(target_digit = as.numeric(label == digit)) %>% 
    select(-label)
  
  mnist_fit <- glm(target_digit ~ ., data = mnist_data, family = binomial())
  
  predict(mnist_fit, mnist_data, type = "response")
}) %>% 
  mutate(id = 1:n(),
         label = mnist$label) %>% 
  pivot_longer(-c(id, label), names_to = "name", values_to = "pred") 
  • The predicted class then can be determined by the highest probability out of all classes.

Takeaways

  • Logistic regression allows us to predict binary categorical variables.
  • It is estimated via maximum likelihood methods.
  • Logistic regression is a linear classifier and such, it cannot deal with complex classification patterns.
  • In problems with multiple predictors, logistic regression separates the points using a hyper-plane.
  • Variable selection approaches such as lasso and ridge regression can still be considered (covered in tutorial).