Statistics for Machine Learning

Machine learning is the branch of computer science that utilizes past experience to learn from and use its knowledge to make future decisions. Machine learning is at the intersection of computer science, engineering, and statistics. The goal of machine learning is to generalize a detectable pattern or to create an unknown rule from given examples. An overview of machine learning landscape is as follows:

Machine learning is broadly classified into three categories but nonetheless, based on the situation, these categories can be combined to achieve the desired results for particular applications:

In some cases, we initially perform unsupervised learning to reduce the dimensions followed by supervised learning when the number of variables is very high. Similarly, in some artificial intelligence applications, supervised learning combined with reinforcement learning could be utilized for solving a problem; an example is self-driving cars in which, initially, images are converted to some numeric format using supervised learning and combined with driving actions (left, forward, right, and backward).

Though there are inherent similarities between statistical modeling and machine learning methodologies, sometimes it is not obviously apparent for many practitioners. In the following table, we explain the differences succinctly to show the ways in which both streams are similar and the differences between them:

The development and deployment of machine learning models involves a series of steps that are almost similar to the statistical modeling process, in order to develop, validate, and implement machine learning models. The steps are as follows:

Statistics itself is a vast subject on which a complete book could be written; however, here the attempt is to focus on key concepts that are very much necessary with respect to the machine learning perspective. In this section, a few fundamentals are covered and the remaining concepts will be covered in later chapters wherever it is necessary to understand the statistical equivalents of machine learning.

Predictive analytics depends on one major assumption: that history repeats itself!

By fitting a predictive model on historical data after validating key measures, the same model will be utilized for predicting future events based on the same explanatory variables that were significant on past data.

The first movers of statistical model implementers were the banking and pharmaceutical industries; over a period, analytics expanded to other industries as well.

Statistical models are a class of mathematical models that are usually specified by mathematical equations that relate one or more variables to approximate reality. Assumptions embodied by statistical models describe a set of probability distributions, which distinguishes it from non-statistical, mathematical, or machine learning models

Statistical models always start with some underlying assumptions for which all the variables should hold, then the performance provided by the model is statistically significant. Hence, knowing the various bits and pieces involved in all building blocks provides a strong foundation for being a successful statistician.

In the following section, we have described various fundamentals with relevant codes:

The Python code for the calculation of mean, median, and mode using a numpy array and the stats package is as follows:

>>> import numpy as np 
>>> from scipy import stats 
 
>>> data = np.array([4,5,1,2,7,2,6,9,3]) 
 
# Calculate Mean 
>>> dt_mean = np.mean(data) ; print ("Mean :",round(dt_mean,2)) 
 
# Calculate Median 
>>> dt_median = np.median(data) ; print ("Median :",dt_median)          
 
# Calculate Mode 
>>> dt_mode =  stats.mode(data); print ("Mode :",dt_mode[0][0])   

The output of the preceding code is as follows:

The R code for descriptive statistics (mean, median, and mode) is given as follows:

data <- c(4,5,1,2,7,2,6,9,3) 
dt_mean = mean(data) ; print(round(dt_mean,2)) 
dt_median = median (data); print (dt_median) 
 
func_mode <- function (input_dt) { 
  unq <- unique(input_dt)  unq[which.max(tabulate(match(input_dt,unq)))] 
} 
 
dt_mode = func_mode (data); print (dt_mode)

The Python code is as follows:

>>> from statistics import variance, stdev 
>>> game_points = np.array([35,56,43,59,63,79,35,41,64,43,93,60,77,24,82]) 
 
# Calculate Variance 
>>> dt_var = variance(game_points) ; print ("Sample variance:", round(dt_var,2)) 
 
# Calculate Standard Deviation 
>>> dt_std = stdev(game_points) ; print ("Sample std.dev:", round(dt_std,2)) 
                
# Calculate Range 
>>> dt_rng = np.max(game_points,axis=0) - np.min(game_points,axis=0) ; print ("Range:",dt_rng) 
 
#Calculate percentiles 
>>> print ("Quantiles:") 
>>> for val in [20,80,100]: 
>>>      dt_qntls = np.percentile(game_points,val)  
>>>      print (str(val)+"%" ,dt_qntls) 
                                 
# Calculate IQR                             
>>> q75, q25 = np.percentile(game_points, [75 ,25]); print ("Inter quartile range:",q75-q25)

The output of the preceding code is as follows:

The R code for dispersion (variance, standard deviation, range, quantiles, and IQR) is as follows:

game_points <- c(35,56,43,59,63,79,35,41,64,43,93,60,77,24,82) 
dt_var = var(game_points); print(round(dt_var,2)) 
dt_std = sd(game_points); print(round(dt_std,2)) 
range_val<-function(x) return(diff(range(x)))  
dt_range = range_val(game_points); print(dt_range) 
dt_quantile = quantile(game_points,probs = c(0.2,0.8,1.0)); print(dt_quantile) 
dt_iqr = IQR(game_points); print(dt_iqr)

The steps involved in hypothesis testing are as follows:

The null hypothesis is that μ0 ≥ 1000 (all chocolates weigh more than 1,000 g).

Collected sample:

Calculate test statistic:

t = (990 - 1000) / (12.5/sqrt(30)) = - 4.3818

Critical t value from t tables = t0.05, 30 = 1.699 => - t0.05, 30 = -1.699

P-value = 7.03 e-05

Test statistic is -4.3818, which is less than the critical value of -1.699. Hence, we can reject the null hypothesis (your friend's claim) that the mean weight of a chocolate is above 1,000 g.

Also, another way of deciding the claim is by using the p-value. A p-value less than 0.05 means both claimed values and distribution mean values are significantly different, hence we can reject the null hypothesis:

The Python code is as follows:

>>> from scipy import stats  
>>> xbar = 990; mu0 = 1000; s = 12.5; n = 30 
 
# Test Statistic 
>>> t_smple  = (xbar-mu0)/(s/np.sqrt(float(n))); print ("Test Statistic:",round(t_smple,2)) 
 
# Critical value from t-table 
>>> alpha = 0.05 
>>> t_alpha = stats.t.ppf(alpha,n-1); print ("Critical value from t-table:",round(t_alpha,3))           
 
#Lower tail p-value from t-table                          
>>> p_val = stats.t.sf(np.abs(t_smple), n-1); print ("Lower tail p-value from t-table", p_val) 

The R code for T-distribution is as follows:

xbar = 990; mu0 = 1000; s = 12.5 ; n = 30 
t_smple = (xbar - mu0)/(s/sqrt(n));print (round(t_smple,2)) 
 
alpha = 0.05 
t_alpha = qt(alpha,df= n-1);print (round(t_alpha,3)) 
 
p_val = pt(t_smple,df = n-1);print (p_val)

Example: Assume that the test scores of an entrance exam fit a normal distribution. Furthermore, the mean test score is 52 and the standard deviation is 16.3. What is the percentage of students scoring 67 or more in the exam?

The Python code is as follows:

>>> from scipy import stats 
>>> xbar = 67; mu0 = 52; s = 16.3 
 
# Calculating z-score 
>>> z = (67-52)/16.3  
 
# Calculating probability under the curve     
>>> p_val = 1- stats.norm.cdf(z) 
>>> print ("Prob. to score more than 67 is ",round(p_val*100,2),"%")

The R code for normal distribution is as follows:

xbar = 67; mu0 = 52; s = 16.3 
pr = 1- pnorm(67, mean=52, sd=16.3) 
print(paste("Prob. to score more than 67 is ",round(pr*100,2),"%"))

The test is usually performed by calculating χ2 from the data and χ2 with (m-1, n-1) degrees from the table. A decision is made as to whether both variables are independent based on the actual value and table value, whichever is higher:

Example: In the following table, calculate whether the smoking habit has an impact on exercise behavior:

The Python code is as follows:

>>> import pandas as pd 
>>> from scipy import stats 
 
>>> survey = pd.read_csv("survey.csv")   
 
# Tabulating 2 variables with row & column variables respectively 
>>> survey_tab = pd.crosstab(survey.Smoke, survey.Exer, margins = True)

While creating a table using the crosstab function, we will obtain both row and column totals fields extra. However, in order to create the observed table, we need to extract the variables part and ignore the totals:

# Creating observed table for analysis 
>>> observed = survey_tab.ix[0:4,0:3] 

The chi2_contingency function in the stats package uses the observed table and subsequently calculates its expected table, followed by calculating the p-value in order to check whether two variables are dependent or not. If p-value < 0.05, there is a strong dependency between two variables, whereas if p-value > 0.05, there is no dependency between the variables:

>>> contg = stats.chi2_contingency(observed= observed) 
>>> p_value = round(contg[1],3) 
>>> print ("P-value is: ",p_value)

The p-value is 0.483, which means there is no dependency between the smoking habit and exercise behavior.

The R code for chi-square is as follows:

survey = read.csv("survey.csv",header=TRUE) 
tbl = table(survey$Smoke,survey$Exer) 
p_val = chisq.test(tbl)

Example: A fertilizer company developed three new types of universal fertilizers after research that can be utilized to grow any type of crop. In order to find out whether all three have a similar crop yield, they randomly chose six crop types in the study. In accordance with the randomized block design, each crop type will be tested with all three types of fertilizer separately. The following table represents the yield in g/m². At the 0.05 level of significance, test whether the mean yields for the three new types of fertilizers are all equal:

The Python code is as follows:

>>> import pandas as pd 
>>> from scipy import stats 
>>> fetilizers = pd.read_csv("fetilizers.csv")

Calculating one-way ANOVA using the stats package:

>>> one_way_anova = stats.f_oneway(fetilizers["fertilizer1"], fetilizers["fertilizer2"], fetilizers["fertilizer3"]) 
 
>>> print ("Statistic :", round(one_way_anova[0],2),", p-value :",round(one_way_anova[1],3))

Result: The p-value did come as less than 0.05, hence we can reject the null hypothesis that the mean crop yields of the fertilizers are equal. Fertilizers make a significant difference to crops.

The R code for ANOVA is as follows:

fetilizers = read.csv("fetilizers.csv",header=TRUE) 
r = c(t(as.matrix(fetilizers))) 
f = c("fertilizer1","fertilizer2","fertilizer3") 
k = 3; n = 6 
tm = gl(k,1,n*k,factor(f)) 
blk = gl(n,k,k*n) 
av = aov(r ~ tm + blk) 
smry = summary(av)

Some terms used in a confusion matrix are:

(TP/TP+FP)

(TP/TP+FN)

(TN/TN+FP)

Area under curve is utilized for setting the threshold of cut-off probability to classify the predicted probability into various classes; we will be covering how this method works in upcoming chapters.

In order to answer this question, a probability of default model (or behavioral scorecard in technical terms) needs to be developed by using independent variables from the past 24 months and a dependent variable from the next 12 months. After preparing data with X and Y variables, it will be split into 70 percent - 30 percent as train and test data randomly; this method is called in-time validation as both train and test samples are from the same time period:

Here, R2 = sample R-squared value, n = sample size, k = number of predictors (or) variables.

Adjusted R-squared value is the key metric in evaluating the quality of linear regressions. Any linear regression model having the value of R2 adjusted >= 0.7 is considered as a good enough model to implement.

Example: The R-squared value of a sample is 0.5, with a sample size of 50 and the independent variables are 10 in number. Calculated adjusted R-squared:

Here, k = number of predictors or variables

The idea of AIC is to penalize the objective function if extra variables without strong predictive abilities are included in the model. This is a kind of regularization in logistic regression.

Here, n = number of classes. Entropy is maximal at the middle, with the value of 1 and minimal at the extremes as 0. A low value of entropy is desirable as it will segregate classes better:

Example: Given two types of coin in which the first one is a fair one (1/2 head and 1/2 tail probabilities) and the other is a biased one (1/3 head and 2/3 tail probabilities), calculate the entropy for both and justify which one is better with respect to modeling:

From both values, the decision tree algorithm chooses the biased coin rather than the fair coin as an observation splitter due to the fact the value of entropy is less.

Information gain = Entropy of parent - sum (weighted % * Entropy of child)

Weighted % = Number of observations in particular child / sum (observations in all child nodes)

Here, i = number of classes. The similarity between Gini and entropy is shown as follows:

Every model has both bias and variance error components in addition to white noise. Bias and variance are inversely related to each other; while trying to reduce one component, the other component of the model will increase. The true art lies in creating a good fit by balancing both. The ideal model will have both low bias and low variance.

Errors from the bias component come from erroneous assumptions in the underlying learning algorithm. High bias can cause an algorithm to miss the relevant relations between features and target outputs; this phenomenon causes an underfitting problem.

On the other hand, errors from the variance component come from sensitivity to change in the fit of the model, even a small change in training data; high variance can cause an overfitting problem:

An example of a high bias model is logistic or linear regression, in which the fit of the model is merely a straight line and may have a high error component due to the fact that a linear model could not approximate underlying data well.

An example of a high variance model is a decision tree, in which the model may create too much wiggly curve as a fit, in which even a small change in training data will cause a drastic change in the fit of the curve.

At the moment, state-of-the-art models are utilizing high variance models such as decision trees and performing ensemble on top of them to reduce the errors caused by high variance and at the same time not compromising on increases in errors due to the bias component. The best example of this category is random forest, in which many decision trees will be grown independently and ensemble in order to come up with the best fit; we will cover this in upcoming chapters:

In practice, data usually will be split randomly 70-30 or 80-20 into train and test datasets respectively in statistical modeling, in which training data utilized for building the model and its effectiveness will be checked on test data:

In the following code, we split the original data into train and test data by 70 percent - 30 percent. An important point to consider here is that we set the seed values for random numbers in order to repeat the random sampling every time we create the same observations in training and testing data. Repeatability is very much needed in order to reproduce the results:

# Train & Test split 
>>> import pandas as pd       
>>> from sklearn.model_selection import train_test_split 
 
>>> original_data = pd.read_csv("mtcars.csv")     

In the following code, train size is 0.7, which means 70 percent of the data should be split into the training dataset and the remaining 30% should be in the testing dataset. Random state is seed in this process of generating pseudo-random numbers, which makes the results reproducible by splitting the exact same observations while running every time:

>>> train_data,test_data = train_test_split(original_data,train_size = 0.7,random_state=42)

The R code for the train and test split for statistical modeling is as follows:

full_data = read.csv("mtcars.csv",header=TRUE) 
set.seed(123) 
numrow = nrow(full_data) 
trnind = sample(1:numrow,size = as.integer(0.7*numrow)) 
train_data = full_data[trnind,] 
test_data = full_data[-trnind,]

In the following code, a comparison has been made between applying linear regression in a statistical way and gradient descent in a machine learning way on the same dataset:

>>> import numpy as np 
>>> import pandas as pd

The following code describes reading data using a pandas DataFrame:

>>> train_data = pd.read_csv("mtcars.csv") 

Converting DataFrame variables into NumPy arrays in order to process them in scikit learn packages, as scikit-learn is built on NumPy arrays itself, is shown next:

>>> X = np.array(train_data["hp"])  ; y = np.array(train_data["mpg"])  
>>> X = X.reshape(32,1); y = y.reshape(32,1)

Importing linear regression from the scikit-learn package; this works on the least squares method:

>>> from sklearn.linear_model import LinearRegression 
>>> model = LinearRegression(fit_intercept = True)

Fitting a linear regression model on the data and display intercept and coefficient of single variable (hp variable):

>>> model.fit(X,y) 
>>> print ("Linear Regression Results" ) 
>>> print ("Intercept",model.intercept_[0] ,"Coefficient", model.coef_[0])

Now we will apply gradient descent from scratch; in future chapters, we can use the scikit-learn built-in modules rather than doing it from first principles. However, here, an illustration has been provided on the internal workings of the optimization method on which the whole machine learning has been built.

Defining the gradient descent function gradient_descent with the following:

>>> def gradient_descent(x, y,learn_rate, conv_threshold,batch_size, max_iter): 
...    converged = False 
...    iter = 0 
...    m = batch_size   
...    t0 = np.random.random(x.shape[1]) 
...    t1 = np.random.random(x.shape[1])

The following code states, run the algorithm until it does not meet the convergence criteria:

...      while not converged:         
...          grad0 = 1.0/m * sum([(t0 + t1*x[i] - y[i]) for i in range(m)])   
...          grad1 = 1.0/m * sum([(t0 + t1*x[i] - y[i])*x[i] for i in range(m)])  
...          temp0 = t0 - learn_rate * grad0 
...          temp1 = t1 - learn_rate * grad1     
...          t0 = temp0 
...          t1 = temp1

Calculate a new error with updated parameters, in order to check whether the new error changed more than the predefined convergence threshold value; otherwise, stop the iterations and return parameters:

...          MSE_New = (sum( [ (t0 + t1*x[i] - y[i])**2 for i in range(m)] ) / m) 
...          if abs(MSE - MSE_New ) <= conv_threshold: 
...              print 'Converged, iterations: ', iter 
...              converged = True     
...          MSE = MSE_New    
...          iter += 1      
...          if iter == max_iter: 
...              print 'Max interactions reached' 
...              converged = True 
...          return t0,t1

The following code describes running the gradient descent function with defined values. Learn rate = 0.0003, convergence threshold = 1e-8, batch size = 32, maximum number of iteration = 1500000:

>>> if __name__ == '__main__': 
...      Inter, Coeff = gradient_descent(x = X,y = y,learn_rate=0.00003 , conv_threshold = 1e-8, batch_size=32,max_iter=1500000) 
...      print ('Gradient Descent Results')  
...      print (('Intercept = %s Coefficient = %s') %(Inter, Coeff))

The R code for linear regression versus gradient descent is as follows:

# Linear Regression 
train_data = read.csv("mtcars.csv",header=TRUE) 
model <- lm(mpg ~ hp, data = train_data) 
print (coef(model)) 
 
# Gradient descent 
gradDesc <- function(x, y, learn_rate, conv_threshold, batch_size, max_iter) { 
  m <- runif(1, 0, 1) 
  c <- runif(1, 0, 1) 
  ypred <- m * x + c 
  MSE <- sum((y - ypred) ^ 2) / batch_size 
   
  converged = F 
  iterations = 0 
   
  while(converged == F) { 
    m_new <- m - learn_rate * ((1 / batch_size) * (sum((ypred - y) * x))) 
    c_new <- c - learn_rate * ((1 / batch_size) * (sum(ypred - y))) 
     
    m <- m_new 
    c <- c_new
  ypred <- m * x + c
    MSE_new <- sum((y - ypred) ^ 2) / batch_size

    if(MSE - MSE_new <= conv_threshold) {
      converged = T
      return(paste("Iterations:",iterations,"Optimal intercept:", c, "Optimal slope:", m))
    }
    iterations = iterations + 1

    if(iterations > max_iter) {
      converged = T
      return(paste("Iterations:",iterations,"Optimal intercept:", c, "Optimal slope:", m))
    }
    MSE = MSE_new
  }
}
gradDesc(x = train_data$hp,y =  train_data$mpg, learn_rate = 0.00003, conv_threshold = 1e-8, batch_size = 32, max_iter = 1500000)

The loss function or cost function in machine learning is a function that maps the values of variables onto a real number intuitively representing some cost associated with the variable values. Optimization methods are applied to minimize the loss function by changing the parameter values, which is the central theme of machine learning.

Zero-one loss is L0-1 = 1 (m <= 0); in zero-one loss, value of loss is 0 for m >= 0 whereas 1 for m < 0. The difficult part with this loss is it is not differentiable, non-convex, and also NP-hard. Hence, in order to make optimization feasible and solvable, these losses are replaced by different surrogate losses for different problems.

Surrogate losses used for machine learning in place of zero-one loss are given as follows. The zero-one loss is not differentiable, hence approximated losses are being used instead:

Some loss functions are as follows:

When to stop tuning the hyperparameters in a machine learning model is a million-dollar question. This problem can be mostly solved by keeping tabs on training and testing errors. While increasing the complexity of a model, the following stages occur:

Cross-validation is not popular in the statistical modeling world for many reasons; statistical models are linear in nature and robust, and do not have a high variance/overfitting problem. Hence, the model fit will remain the same either on train or test data, which does not hold true in the machine learning world. Also, in statistical modeling, lots of tests are performed at the individual parameter level apart from aggregated metrics, whereas in machine learning we do not have visibility at the individual parameter level:

In the following code, both the R and Python implementation has been provided. If none of the percentages are provided, the default parameters are 50 percent for train data, 25 percent for validation data, and 25 percent for the remaining test data.

Python implementation has only one train and test split functionality, hence we have used it twice and also used the number of observations to split rather than the percentage (as shown in the previous train and test split example). Hence, a customized function is needed to split into three datasets:

>>> import pandas as pd       
>>> from sklearn.model_selection import train_test_split               
                         
>>> original_data = pd.read_csv("mtcars.csv")                    
  
>>> def data_split(dat,trf = 0.5,vlf=0.25,tsf = 0.25): 
...      nrows = dat.shape[0]     
...      trnr = int(nrows*trf) 
...      vlnr = int(nrows*vlf)    

The following Python code splits the data into training and the remaining data. The remaining data will be further split into validation and test datasets:

...      tr_data,rmng = train_test_split(dat,train_size = trnr,random_state=42) 
...      vl_data, ts_data = train_test_split(rmng,train_size = vlnr,random_state=45)     
...      return (tr_data,vl_data,ts_data)

Implementation of the split function on the original data to create three datasets (by 50 percent, 25 percent, and 25 percent splits) is as follows:

>>> train_data, validation_data, test_data = data_split (original_data ,trf=0.5, vlf=0.25,tsf=0.25)

The R code for the train, validation, and test split is as follows:

# Train Validation & Test samples 
trvaltest <- function(dat,prop = c(0.5,0.25,0.25)){ 
  nrw = nrow(dat) 
  trnr = as.integer(nrw *prop[1]) 
  vlnr = as.integer(nrw*prop[2]) 
  set.seed(123) 
  trni = sample(1:nrow(dat),trnr) 
  trndata = dat[trni,] 
  rmng = dat[-trni,] 
  vlni = sample(1:nrow(rmng),vlnr) 
  valdata = rmng[vlni,] 
  tstdata = rmng[-vlni,] 
  mylist = list("trn" = trndata,"val"= valdata,"tst" = tstdata) 
  return(mylist) 
} 
outdata = trvaltest(mtcars,prop = c(0.5,0.25,0.25)) 
train_data = outdata$trn; valid_data = outdata$val; test_data = outdata$tst

Cross-validation is another way of ensuring robustness in the model at the expense of computation. In the ordinary modeling methodology, a model is developed on train data and evaluated on test data. In some extreme cases, train and test might not have been homogeneously selected and some unseen extreme cases might appear in the test data, which will drag down the performance of the model.

On the other hand, in cross-validation methodology, data was divided into equal parts and training performed on all the other parts of the data except one part, on which performance will be evaluated. This process repeated as many parts user has chosen.

Example: In five-fold cross-validation, data will be divided into five parts, subsequently trained on four parts of the data, and tested on the one part of the data. This process will run five times, in order to cover all points in the data. Finally, the error calculated will be the average of all the errors:

Grid search in machine learning is a popular way to tune the hyperparameters of the model in order to find the best combination for determining the best fit:

In the following code, implementation has been performed to determine whether a particular user will click an ad or not. Grid search has been implemented using a decision tree classifier for classification purposes. Tuning parameters are the depth of the tree, the minimum number of observations in terminal node, and the minimum number of observations required to perform the node split:

# Grid search 
>>> import pandas as pd 
>>> from sklearn.tree import DecisionTreeClassifier 
>>> from sklearn.model_selection import train_test_split 
>>> from sklearn.metrics import classification_report,confusion_matrix,accuracy_score 
>>> from sklearn.pipeline import Pipeline 
>>> from sklearn.grid_search import GridSearchCV 
 
>>> input_data = pd.read_csv("ad.csv",header=None)                        
 
>>> X_columns = set(input_data.columns.values) 
>>> y = input_data[len(input_data.columns.values)-1] 
>>> X_columns.remove(len(input_data.columns.values)-1) 
>>> X = input_data[list(X_columns)]

Split the data into train and testing:

>>> X_train, X_test,y_train,y_test = train_test_split(X,y,train_size = 0.7,random_state=33)

Create a pipeline to create combinations of variables for the grid search:

>>> pipeline = Pipeline([ 
...      ('clf', DecisionTreeClassifier(criterion='entropy')) ])

Combinations to explore are given as parameters in Python dictionary format:

>>> parameters = { 
...      'clf__max_depth': (50,100,150), 
...      'clf__min_samples_split': (2, 3), 
...      'clf__min_samples_leaf': (1, 2, 3)}

The n_jobs field is for selecting the number of cores in a computer; -1 means it uses all the cores in the computer. The scoring methodology is accuracy, in which many other options can be chosen, such as precision, recall, and f1:

>>> grid_search = GridSearchCV(pipeline, parameters, n_jobs=-1, verbose=1, scoring='accuracy') 
>>> grid_search.fit(X_train, y_train) 

Predict using the best parameters of grid search:

>>> y_pred = grid_search.predict(X_test)

The output is as follows:

>>> print ('\n Best score: \n', grid_search.best_score_) 
>>> print ('\n Best parameters set: \n')   
>>> best_parameters = grid_search.best_estimator_.get_params() 
>>> for param_name in sorted(parameters.keys()): 
>>>     print ('\t%s: %r' % (param_name, best_parameters[param_name])) 
>>> print ("\n Confusion Matrix on Test data \n",confusion_matrix(y_test,y_pred)) 
>>> print ("\n Test Accuracy \n",accuracy_score(y_test,y_pred)) 
>>> print ("\nPrecision Recall f1 table \n",classification_report(y_test, y_pred))

The R code for grid searches on decision trees is as follows:

# Grid Search on Decision Trees 
library(rpart) 
input_data = read.csv("ad.csv",header=FALSE) 
input_data$V1559 = as.factor(input_data$V1559) 
set.seed(123) 
numrow = nrow(input_data) 
trnind = sample(1:numrow,size = as.integer(0.7*numrow)) 
 
train_data = input_data[trnind,];test_data = input_data[-trnind,] 
minspset = c(2,3);minobset = c(1,2,3) 
initacc = 0 
 
for (minsp in minspset){ 
  for (minob in minobset){ 
    tr_fit = rpart(V1559 ~.,data = train_data,method = "class",minsplit = minsp, minbucket = minob) 
    tr_predt = predict(tr_fit,newdata = train_data,type = "class") 
    tble = table(tr_predt,train_data$V1559) 
    acc = (tble[1,1]+tble[2,2])/sum(tble) 
    acc 
    if (acc > initacc){ 
      tr_predtst = predict(tr_fit,newdata = test_data,type = "class") 
      tblet = table(test_data$V1559,tr_predtst) 
      acct = (tblet[1,1]+tblet[2,2])/sum(tblet) 
      acct 
      print(paste("Best Score")) 
      print( paste("Train Accuracy ",round(acc,3),"Test Accuracy",round(acct,3))) 
      print( paste(" Min split ",minsp," Min obs per node ",minob)) 
      print(paste("Confusion matrix on test data")) 
      print(tblet) 
      precsn_0 = (tblet[1,1])/(tblet[1,1]+tblet[2,1]) 
      precsn_1 = (tblet[2,2])/(tblet[1,2]+tblet[2,2]) 
      print(paste("Precision_0: ",round(precsn_0,3),"Precision_1: ",round(precsn_1,3))) 
      rcall_0 = (tblet[1,1])/(tblet[1,1]+tblet[1,2]) 
      rcall_1 = (tblet[2,2])/(tblet[2,1]+tblet[2,2]) 
      print(paste("Recall_0: ",round(rcall_0,3),"Recall_1: ",round(rcall_1,3))) 
      initacc = acc 
    } 
  } 
}