ETC3250/5250

Introduction to Machine Learning

Logistic regression

Lecturer: Emi Tanaka

Department of Econometrics and Business Statistics

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	complete_rate	ordered	n_unique	top_counts
Class	1	FALSE	4	1st: 4, 2nd: 4, 3rd: 4, Cre: 4
Sex	1	FALSE	2	Mal: 8, Fem: 8
Age	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).