Data Mining Process flow – Easy Understanding

February 4, 2021/in Artificial Intelligence, Data Mining, Data Science, Deep Learning, Machine Learning, Main Category, Predictive Analytics, Statistics/by Ahamed Al Farabi

1 Overview

Development of computer processing power, network and automated software completely change and give new concept of each business. And data mining play the vital part to solve, finding the hidden patterns and relationship from large dataset with business by using sophisticated data analysis tools like methodology, method, process flow etc.

On this paper, proposed a process flow followed CRISP-DM methodology and has six steps where data understanding does not considered.

Phase of new process flow given below:-

Phase 1: Involved with collection, outliner treatment, imputation, transformation, scaling, and partition dataset in to two sub-frames (Training and Testing). Here as an example for outliner treatment, imputation, transformation, scaling consider accordingly Z score, mean, One hot encoding and Min Max Scaler.

Phase 2: On this Phase training and testing data balance with same balancing algorithm but separately. As an example here SMOTE (synthetic minority oversampling technique) is considered.

Phase 3: This phase involved with reduction, selection, aggregation, extraction. But here for an example considering same feature reduction algorithm (LDA -Linear Discriminant analysis) on training and testing data set separately.

Phase 4: On this Phase Training data set again partition into two more set (Training and Validation).

Phase 5: This Phase considering several base algorithms as a base model like CNN, RNN, Random forest, MLP, Regression, Ensemble method. This phase also involve to find out best hyper parameter and sub-algorithm for each base algorithm. As an example on this paper consider two class classification problems and also consider Random forest (Included CART – Classification and Regression Tree and GINI index impurity) and MLP classifier (Included (Relu, Sigmoid, binary cross entropy, Adam – Adaptive Moment Estimation) as base algorithms.

Phase 6: First, Prediction with validation data then evaluates with Test dataset which is fully unknown for these (Random forest, MLP classifier) two base algorithms. Then calculate the confusion matrix, ROC, AUC to find the best base algorithm.

New method from phase 1 to phase 4 followed CRISP-DM methodology steps such as data collection, data preparation then phase 5 followed modelling and phase 6 followed evaluation and implementation steps.

Structure of proposed process flow for two class problem combined with algorithm and sub-algorithm display on figure – 1.

These articles mainly focus to describe all algorithms which are going to implementation for better understanding.

Figure 1 – Data Mining Process Flow

2 Phase 1: Outlier treatment, Transform, Scaling, Imputation

This phase involved with outlier treatment, imputation, scaling, and transform data.

2.1 Outliner treatment: – Z score

Outlier is a data point which lies far from all other data point in a data set. Outlier need to treat because it may bias the entire result. Outlier treatment with Z score is a common technique. Z score is a standard score in statistics. Z score provides information about data value is smaller or grater then mean that means how many standard deviations away from the mean value. Z score equation display below:

$Z = \frac{(x - \mu)}{\sigma}$

Here x = data point
σ = Standard deviation
μ = mean value

Equation- 1 Z-Score

In a normal distribution Z score represent 68% data lies on +/- 1, 95% data point lies on +/- 2, 99.7% data point lies on +/- 3 standard deviation.

2.2 Imputation data: – mean

Imputation is a way to handle missing data by replacing substituted value. There are many imputation technique represent like mean, median, mode, k-nearest neighbours. Mean imputation is the technique to replacing missing information with mean value. On the mean imputation first calculate the particular features mean value and then replace the missing value with mean value. The next equation displays the mean calculation:

$\mu = \frac{(\sum x)}{n}$

Here x = value of each point
n = number of values
μ = mean value

Equation- 2 Mean

2.3 Transform: – One hot encoding

Encoding is a pre-processing technique which represents data in such a way that computer can understand. For understanding of machine learning algorithm categorical columns convert to numerical columns, this process called categorical encoding. There are multiple way to handle categorical variable but most widely used techniques are label encoding and one host encoding. On label encoding give a numeric (integer number) for each category. Suppose there are 3 categories of foods like apples, orange, banana. When label encoding is used then 3 categories will get a numerical value like apples = 1, banana = 2 and orange = 3. But there is very high probability that machine learning model can capture the relationship in between categories such as apple < banana < orange or calculate average across categories like 1 +3 = 4 / 2 = 2 that means model can understand average of apple and orange together is banana which is not acceptable because model correlation calculation is wrong. For solving this problem one hot encoding appear. The following table displays the label encoding is transformed into one hot encoding.

Table- 1 Encoding example

On hot encoding categorical value split into columns and each column contains 0 or 1 according to columns placement.

2.4 Scaling data: – Min Max Scaler

Feature scaling method is standardized or normalization the independent variable that means it is used to scale the data in a particular range like -1 to +1 or depending on algorithm. Generally normalization used where data distribution does not follow Gaussian distribution and standardization used where data distribution follow Gaussian distribution. On standardization techniques transform data values are cantered around the mean and unit is standard deviation. Formula for standardization given below:

$Standardization X = \frac{(X - \mu)}{\sigma}$

Equation-3 Equations for Standardization

X represent the feature value, µ represent mean of the feature value and σ represent standard deviation of the feature value. Standardized data value does not restrict to a particular range.

Normalization techniques shifted and rescaled data value range between 0 and 1. Normalization techniques also called Min-Max scaling. Formula for normalization given below:

$Normalization X = \frac{(X - X_{min})}{X_{max} - X_{min}}$

Equation – 4 Equations for Normalization

Above X, X_min, X_max are accordingly feature values, feature minimum value and feature maximum value. On above formula when X is minimums then numerator will be 0 ( is 0) or if X is maximums then the numerator is equal to the denominator ( is 1). But when X data value between minimum and maximum then is between 0 and 1. If ranges value of data does not normalized then bigger range can influence the result.

3 Phase 2: – Balance Data

3.1 SMOTE

SMOTE (synthetic minority oversampling technique) is an oversampling technique where synthetic observations are created based on existing minority observations. This technique operates in feature space instead of data space. Under SMOTE each minority class observation calculates k nearest neighbours and randomly chose the neighbours depending on over-sampling requirements. Suppose there are 4 data point on minority class and 10 data point on majority class. For this imbalance data set, balance by increasing minority class with synthetic data point. SMOTE creating synthetic data point but it is necessary to consider k nearest neighbours first. If k = 3 then SMOTE consider 3 nearest neighbours. Figure-2 display SMOTE with k = 3 and x = x1, x2, x3, x4 data point denote minority class. And all circles represent majority class.

Figure- 2 SMOTE example

4 Phase 3: – Feature Reduction

4.1 LDA

LDA stands for Linear Discriminant analysis supervised technique are commonly used for classification problem. On this feature reduction account continuous independent variable and output categorical variable. It is multivariate analysis technique. LDA analyse by comparing mean of the variables. Main goal of LDA is differentiate classes in low dimension space. LDA is similar to PCA (Principal component analysis) but in addition LDA maximize the separation between multiple classes. LDA is a dimensionality reduction technique where creating synthetic feature from linear combination of original data set then discard less important feature. LDA calculate class variance, it maximize between class variance and minimize within class variance. Table-2 display the process steps of LDA.

Table- 2 LDA process

5 Phase 5: – Base Model

Here we consider two base model ensemble random forest and MLP classifier.

5.1 Random Forest

Random forest is an ensemble (Bagging) method where group of weak learner (decision tree) come together to form a strong leaner. Random forest is a supervised algorithm which is used for regression and classification problem. Random forests create several decisions tree for predictions and provide solution by voting (classification) or mean (regression) value. Working process of Random forest given below (Table -3).

Table-3 Random Forest process

When training a Random forest root node contains a sample of bootstrap dataset and the feature is as same as original dataset. Suppose the dataset is D and contain d record and m number of columns. From the dataset D random forest first randomly select sample of rows (d^’) with replacement and sample of features (n^’) and give it to the decision tree. Suppose Random forest created several decision trees like T₁, T₂, T₃, T₄ . . . T_n. Then randomly selected dataset D^’ = d^’ + n^’ is given to the decision tree T₁, T₂, T₃, T₄ . . . T_n where D^’ < D, m > n^’ and d > d^’. After taking the dataset decision tree give the prediction for binary classification 1 or 0 then aggregating the decision and select the majority voted result. Figure-3 describes the structure of random forest process.

Figure- 3 Random Forest process

On Random forest base learner Decision Tree grows complete depth where bias (properly train on training dataset) is low and variance is high (when implementing test data give big error) called overfitting. On Random forest using multiple decision trees where each Decision tree is high variance but when combining all decision trees with the respect of majority vote then high variance converted into low variance because using row and feature sampling with replacement and taking the majority vote where decision is not depend on one decision tree.

CART (Classification and Regression Tree) is binary segmentation technique. CART is a Gini’s impurity index based classical algorithm to split a dataset and build a decision tree. By splitting a selected dataset CART created two child nodes repeatedly and builds a tree until the data no longer be split. There are three steps CART algorithm follow:

Find best split for each features. For each feature in binary split make two groups of the ordered classes. That means possibility of split for k classes is k-1. Find which split is maximized and contain best splits (one for each feature) result.
Find the best split for nodes. From step 1 find the best one split (from all features) which maximized the splitting criterion.
Split the best node from step 2 and repeat from step 1 until fulfil the stopping criterion.

For splitting criteria CART use GINI index impurity algorithm to calculate the purity of split in a decision tree. Gini impurity randomly classified the labels with the same distribution in the dataset. A Gini impurity of 0 (lowest) is the best possible impurity and it is achieve when everything is in a same class. Gini index varies from 0 to 1. 0 indicate the purity of class where only one class exits or all element under a specific class. 1 indicates that elements are randomly distributed across various classes. And 0.5 indicate equal elements distributed over classes. Gini index (GI) described by mathematically that sum of squared of probabilities of each class (p_i) deducted from one (Equation-5).

Equation – 5 Gini impurities

Here (Equation-5) p_i represent the probability (probability of p+ or yes and probability of p- or no) of distinct class with classified element. Suppose randomly selected feature (a1) which has 8 yes and 4 no. After the split right had side (b1 on equation-6) has 4 yes and 4 no and left had side (b2 on equation – 7) has 4 yes and 0 no. here b2 is a pure split (leaf node) because only one class yes is present. By using the GI (Gini index) formula for b1 and b2:-

Equation- 6 & 7 – Gini Impurity b1 & Gini Impurity b2

Here for b1 value 0.5 indicates that equal element (yes and no) distribute over classes which is not pure split. And b2 value 0 indicates pure split. On GINI impurity indicates that when probability (yes or no) increases GINI value also increases. Here 0 indicate pure split and .5 indicate equal split that means worst situation. After calculating the GINI index for b1 and b2 now calculate the reduction of impurity for data point a1. Here total yes 8 (b1 and b2 on Equation – 8) and total no 4 (b1) so total data is 12 on a1. Below display the weighted GINI index for feature a1:

Total data point on b1 with Gini index (m) = 8/12 * 0.5 = 0.3333

Total data point on b2 with Gini index (n) = 4/12 * 0 = 0

Weighted Gini index for feature a1 = m + n = 0.3333

Equation- 8 Gini Impurity b1 & b2

After computing the weighted Gini value for every feature on a dataset taking the highest value feature as first node and split accordingly in a decision tree. Gini is less costly to compute.

5.2 Multilayer Perceptron Classifier (MLP Classifier)

Multilayer perceptron classifier is a feedforward neural network utilizes supervised learning technique (backpropagation) for training. MLP Classifier combines with multiple perceptron (hidden) layers. For feedforward taking input send combining with weight bias and then activation function from one hidden layer output goes to other hidden and this process continuing until reached the output. Then output calculates the error with error algorithm. These errors send back with backpropagation for weight adjustment by decreasing the total error and process is repeated, this process is call epoch. Number of epoch is determined with the hyper-parameter and reduction rate of total error.

5.2.1 Back-Propagation

Backpropagation is supervised learning algorithm that is used to train neural network. A neural network consists of input layer, hidden layer and output layer and each layer consists of neuron. So a neural network is a circuit of neurons. Backpropagation is a method to train multilayer neural network the updating of the weights of neural network and is done in such a way so that the error observed can be reduced here, error is only observed in the output layer and that error is back propagated to the previous layers and previous layer is proportionally updated weight. Backpropagation maintain chain rule to update weight. Mainly three steps on backpropagation are (Table-4):

Step	Process
Step 1	Forward Pass
Step 2	Backward Pass
Step 3	Sum of all values and calculate updated weight value with Chain – rules.

Table-4 Back-Propagation process

5.2.2 Forward pass/ Forward propagation

Forward propagation is the process where input layer send the input value with randomly selected weight and bias to connected neuron and inside neuron selected activation function combine them and forward to other connected neuron layer after layer then give an output with the help of output layer. Below (Figure-4) display the forward propagation.

Figure-4 Forward passes

Input layer take the input of X (X₁, X₂) combine with randomly selected weight for each connection and with fixed bias (different hidden layer has different bias) send it to first hidden layer where first multiply the input with corresponding weight and added all input with single bias then selected activation function (may different form other layer) combine all input and give output according to function and this process is going on until reach in output layer. Output layer give the output like Y (Y₁, Y₂) (here output is binary classification as an example) according to selected activation function.

5.2.3 Backward Pass

After calculating error (difference between Forward pass output and actual output) backward pass try to minimize the error with optimisation function by sending backward with proportionally distribution and maintain a chain rule. Backward pass distribution the error in such a way where weighted value is taking under consideration. Below (Figure-5) diagram display the Backward pass process.

Figure-5 Backward passes

Backpropagation push back the error which is calculated with error function or loss function for update proportional distribution with the help of optimisation algorithm. Division of Optimisation algorithm given below on Figure – 6

Figure -6 Division of Optimisation algorithms

Gradient decent calculate gradient and update value by increases or decreases opposite direction of gradients unit and try to find the minimal value. Gradient decent update just one time for whole dataset but stochastic gradient decent update on each training sample and it is faster than normal gradient decent. Gradient decent can be improve by tuning parameter like learning rate (0 to 1 mostly use 0.5). Adagrad use time step based parameter to compute learning rate for every parameter. Adam is Adaptive Moment Estimation. It calculates different parameter with different learning rate. It is faster and performance rate is higher than other optimization algorithm. On the other way Adam algorithm is squares the calculated exponential weighted moving average of gradient.

5.2.4 Chain – rules

Backpropagation maintain chain-rules to update weighted value. On chain-rules backpropagation find the derivative of error respect to any weight. Suppose E is output error. w is weight for input a and bias b and ac neuron output respect of activation function and summation of bias with weighted input (w*a) input to neuron is net. So partial derivative for error respect to weight is ∂E / ∂w display the process on figure-7.

Figure- 7 Partial derivative for error respect to weight

On the chain rules for backward pass to find (error respect to weight) ∂E / ∂w = ∂E / ∂ac * ∂ac / ∂net * ∂net / ∂w. here find to error respect to weight are error respect to output of activation function multiply by activation function output respect to input in a neuron multiply by input in a neuron respect to weight.

5.2.5 Activation function

Activation function is a function which takes the decision about neuron to activate or deactivate. If the activate function activate the neuron then it will give an output on the basis of input. Input in a activation function is sum of input multiply with corresponding weight and adding the layered bias. The main function of a activate function is non-linearity output of a neuron.

Figure-8 Activation function

Figure – 8 display a neuron in a hidden layer. Here several input (1, 2, 3) with corresponding weight (w1, w2, w3) putting in a neuron input layer where layer bias add with summation of multiplication with input and weight. Equation-9 display the output of an activate function.

Output from activate function y = Activate function (Ʃ (weight * input) + bias)

y = f (Ʃ (w*x) +b)

Equation- 9 Activate function

There are many activation functions like linear function for regression problem, sigmoid function for binary classification problem where result either 0 or 1, Tanh function which is based on sigmoid function but mathematically shifted version and values line -1 to 1. RELU function is Rectified linear unit. RELU is less expensive to compute.

5.2.6 Sigmoid

Sigmoid is a squashing activate function where output range between 0 and 1. Sigmoidal name comes from Greek letter sigma which looks like letter S when graphed. Sigmoid function is a logistic type function, it mainly use in output layer in neural network. Sigmoid is non-linear, fixed output range (between 0 and 1), monotonic (never decrees or never increases) and continuously differentiated function. Sigmoid function is good at classification and output from sigmoid is nonlinear. But Sigmoid has a vanishing gradient problem because output variable is very less to change in input variable. Figure- 9 displays the output of a Sigmoid and derivative of Sigmoid. Here x is any number (positive or negative). On sigmoid function 1 is divided by exponential negative input with adding 1.

Figure – 9 Sigmoid Functions

4.5.2.7 RELU

RELU stands for Rectified Linear Units it is simple, less expensive in computation and rectifies the gradient vanishing problem. RELU is nonlinear activation function. It gives output either positive (infinity) or 0. RELU has a dying problem because if neurons stop for responding to variation because of gradient is 0 or nothing has to change. Figure- 10 displays the output of an RELU and derivative of RELU. Here x is any positive input and if x is grater then 0 give the output as x or give output 0. RELU function gives the output maximum value of input, here max (0, x).

Figure – 10 RELU Function

4.5.2.8 Cost / loss function (Binary Cross-Entropy)

Cost or loss function compare the predictive value (model outcome) with actual value and give a quantitative value which give the indication about how much good or bad the prediction is.

Figure- 11 Cost function work process

Figure-11 x1 and x2 are input in a activate function f(x) and output y1_out which is sum of weighted input added with bias going through activate function. After model output activate function compare the output with actual output and give a quantitative value which indicate how good or bad the prediction is.

There are many type of loss function but choosing of optimal loss function depends on the problem going to be solved such as regression or classification. For binary classification problem binary cross entropy is used to calculate cost. Equation-10 displays the binary cross entropy where y is actual binary value and y^p predictive outcome range 0 and 1. And i is scalar vale range between 1 to model output size (N).

Equation-10 displays the binary cross entropy

6 Phase 6: – Evaluation

6.1 Confusion matrix

In a classification confusion matrix describe the performance of actual value against predictive value. Confusion Matrix does the performance measurement. So confusion matrix classifies and display predicted and actual value (Visa, S., Ramsay 2011).

Table- 5 Confusion Matrix

Confusion Matrix (Table-5) combines with True Positive (TP), True Negative (TN), False Positive (FP), and False Negative (FN). True Positive is prediction positive and true. True Negative is prediction negative and that is true. False positive is prediction positive and it’s false. False negative is prediction negative and that is false. False positive is known as Type1 error and false negative is known as Type 2 error. Confusion matrix can able to calculate several list of rates which are given below on Table- 6.

Here N = Total number of observation, TP = True Positive, TN = True Negative

FP = False Positive, FN = False Negative, Total Actual No (AN) = TN + FP,

Total Predictive Yes (PY) = FP + TP. Total Actual Yes (AY) = FN + TP

Rate	Description	Mathematical Description
Accuracy	Classifier, overall how often correctly identified	(TP+TN) / N
Misclassification Rate	Classifier, overall how often wrongly identified	(FP + FN) / N
True Positive Rate (Sensitivity / Recall)	Classifier, how often predict correctly yes when it is actually yes.	TP / AY
False Positive Rate	Classifier, how often predict wrongly yes when it is actually no.	FP / AN
True Negative Rate (Specificity)	Classifier, how often predict correctly no when it is actually no.	TN / AN
Precision	Classifier how often predict yes when it is correct.	TP / PY
Prevalence	Yes conditions how often occur in a sample.	AY / N

Table – 6 Confusion matrixes Calculation

From confusion matrix F1 score can be calculated because F1 score related to precision and recall. Higher F1 score is better. If precision or recall any one goes down F1 score also go down.

$F1 = \frac{2 * Precision * Recall}{Precision + Recall}$

4.6.2 ROC (Receiver Operating Characteristic) curve

In statistics ROC is represent in a graph with plotting a curve which describe a binary classifiers performance as its differentiation threshold is varied. ROC (Equation-11) curve created true positive rate (TPR) against false positive rate (FPR). True positive rate also called as Sensitivity and False positive rate also known as Probability of false alarm. False positive rate also called as a probability of false alarm and it is calculated as 1 – Specificity.

$True Positive Rate = \frac{True Positive}{True Positive + False Negative} = Recall or Sensivity$

$False Positive Rate = \frac{True Negative}{True Negative + False Positive} = 1 - Specificity$

Equation- 11 ROC

So ROC (Receiver Operation Characteristic) curve allows visual representation between sensitivity and specificity associated with different values of the test result (Grzybowski, M. and Younger, J.G., 1997)

On ROC curve each point has different Threshold level. Below (Figure – 12) display the ROC curve. Higher the area curve covers is better that means high sensitivity and high specificity represent more accuracy. ROC curve also represent that if classifier predict more often true than it has more true positive and also more false positive. If classifier predict true less often then fewer false positive and also fewer true positive.

ROC Curve

Figure – 12 ROC curve description

4.6.3 AUC (Area under Curve)

Area under curve (AUC) is the area surrounded by the ROC curve and AUC also represent the degree of separability that means how good the model to distinguished between classes. Higher the AUC value represents better the model performance to separate classes. AUC = 1 for perfect classifier, AUC = 0 represent worst classifier, and AUC = 0.5 means has no class separation capacity. Suppose AUC value is 0.6 that means 60% chance that model can classify positive and negative class.

Figure- 13 to Figure – 16 displays an example of AUC where green distribution curve for positive class and blue distribution curve for negative class. Here threshold or cut-off value is 0.5 and range between ‘0’ to ‘1’. True negative = TN, True Positive = TP, False Negative = FN, False Positive = FP, True positive rate = TPR (range 0 to 1), False positive rate = FPR (range 0 to 1).

On Figure – 13 left distribution curve where two class curves does not overlap that means both class are perfectly distinguished. So this is ideal position and AUC value is 1. On the left side ROC also display that TPR for positive class is 100% occupied.

ROC distributions (perfectly distinguished

Figure – 14 two class overlap each other and raise false positive (Type 1), false negative (Type 2) errors. Here error could be minimize or maximize according to threshold. Suppose here AUC = 0.6, that means chance of a model to distinguish two classes is 60%. On ROC curve also display the curve occupied for positive class is 60%.

ROC distributions (class partly overlap distinguished)

Figure- 15 displayed that positive and negative overlap each other. Here AUC value is 0.5 or near to 0.5. On this position classifier model does not able distinguish positive and negative classes. On left side ROC curve become straight that means TPR and FPR are equal.

ROC distributions (class fully overlap distinguished)

Figure- 16 positive and negative class swap position and on this position AUC = 0. That means classified model predict positive as a negative and negative as a positive. On the left ROC curve display that curve on FPR side fully fitted.

ROC distributions (class swap position distinguished)

7 Summaries

This paper describes a data mining process flow and related model and its algorithm with textual representation. One hot encoding create dummy variable for class features and min-max scaling scale the data in a single format. Balancing by SMOTE data where Euclidian distance calculates the distance in-between nearest neighbour to produce synthetic data under minority class. LDA reduce the distance inside class and maximise distance in-between class and for two class problem give a single dimension features which is less costly to calculate accuracy by base algorithm (random forest and MLP classifier). Confusion matrix gives the accuracy, precision, sensitivity, specificity which is help to take a decision about base algorithm. AUC and ROC curve also represent true positive rate against false positive rate which indicate base algorithm performance.

Base algorithm Random forest using CART with GINI impurity for feature selection to spread the tree. Here CART is selected because of less costly to run. Random forest algorithm is using bootstrap dataset to grow trees, and aggregation using majority vote to select accuracy.

MLP classifier is a neural network algorithm using backpropagation chain-rule to reducing error. Here inside layers using RLU activation function. Output layers using Sigmoid activation function and binary cross entropy loss function calculate the loss which is back propagate with Adam optimizer to optimize weight and reduce loss.

References:

Visa, S., Ramsay, B., Ralescu, A.L. and Van Der Knaap, E., 2011. Confusion Matrix-based Feature Selection. MAICS, 710, pp.120-127.
Grzybowski, M. and Younger, J.G., 1997. Statistical methodology: III. Receiver operating characteristic (ROC) curves. Academic Emergency Medicine, 4(8), pp.818-826.

Hypothesis Test for real problems

August 26, 2020/in Data Science, Main Category, Statistics/by Saurav Singla

Hypothesis tests are significant for evaluating answers to questions concerning samples of data.

A statistical hypothesis is a belief made about a population parameter. This belief may or might not be right. In other words, hypothesis testing is a proper technique utilized by scientist to support or reject statistical hypotheses. The foremost ideal approach to decide if a statistical hypothesis is correct is examine the whole population.

Since that’s frequently impractical, we normally take a random sample from the population and inspect the equivalent. Within the event sample data set isn’t steady with the statistical hypothesis, the hypothesis is refused.

Types of hypothesis:

There are two sort of hypothesis and both the Null Hypothesis (Ho) and Alternative Hypothesis (Ha) must be totally mutually exclusive events.

• Null hypothesis is usually the hypothesis that the event wont’t happen.

• Alternative hypothesis is a hypothesis that the event will happen.

Why we need Hypothesis Testing?

Suppose a specific cosmetic producing company needs to launch a new Shampoo in the market. For this situation they will follow Hypothesis Testing all together decide the success of new product in the market.

Where likelihood of product being ineffective in market is undertaken as Null Hypothesis and likelihood of product being profitable is undertaken as Alternative Hypothesis. By following the process of Hypothesis testing they will foresee the accomplishment.

How to Calculate Hypothesis Testing?

State the two theories with the goal that just one can be correct, to such an extent that the two occasions are totally unrelated.
Now figure a study plan, that will lay out how the data will be assessed.
Now complete the plan and genuinely investigate the sample dataset.
Finally examine the outcome and either accept or reject the null hypothesis.

Another example

Assume, Person have gone after a typing job and he has expressed in the resume that his composing speed is 70 words per minute. The recruiter might need to test his case. On the off chance that he sees his case as adequate, he will enlist him in any case reject him. Thus, he types an example letter and found that his speed is 63 words a minute. Presently, he can settle on whether to employ him or not. In the event that he meets all other qualification measures. This procedure delineates Hypothesis Testing in layman’s terms.

In statistical terms Hypothesis his typing speed is 70 words per minute is a hypothesis to be tested so-called null hypothesis. Clearly, the alternating hypothesis his composing speed isn’t 70 words per minute.

So, normal composing speed is population parameter and sample composing speed is sample statistics.

The conditions of accepting or rejecting his case is to be chosen by the selection representative. For instance, he may conclude that an error of 6 words is alright to him so he would acknowledge his claim between 64 to 76 words per minute. All things considered, sample speed 63 words per minute will close to reject his case. Furthermore, the choice will be he was producing a fake claim.

In any case, if the selection representative stretches out his acceptance region to positive/negative 7 words that is 63 to 77 words, he would be tolerating his case.

In this way, to finish up, Hypothesis Testing is a procedure to test claims about the population dependent on sample. It is a fascinating reasonable subject with a quite statistical jargon. You have to dive more to get familiar with the details.

Significance Level and Rejection Region for Hypothesis

Type I error probability is normally indicated by α and generally set to 0.05. The value of α is recognized as the significance level.

The rejection region is the set of sample data that prompts the rejection of the null hypothesis. The significance level, α, decides the size of the rejection region. Sample results in the rejection region are labelled statistically significant at level of α .

The impact of differing α is that If α is small, for example, 0.01, the likelihood of a type I error is little, and a ton of sample evidence for the alternative hypothesis is needed before the null hypothesis can be dismissed. Though, when α is bigger, for example, 0.10, the rejection region is bigger, and it is simpler to dismiss the null hypothesis.

Significance from p-values

A subsequent methodology is to evade the utilization of a significance level and rather just report how significant the sample evidence is. This methodology is as of now more widespread. It is accomplished by method of a p value. P value is gauge of power of the evidence against null hypothesis. It is the likelihood of getting the observed value of test statistic, or value with significantly more prominent proof against null hypothesis (Ho), if the null hypothesis of an investigation question is true. The less significant the p value, the more proof there is supportive of the alternative hypothesis. Sample evidence is measurably noteworthy at the α level just if the p value is less than α. They have an association for two tail tests. When utilizing a confidence interval to playout a two-tailed hypothesis test, reject the null hypothesis if and just if the hypothesized value doesn’t lie inside a confidence interval for the parameter.

Hypothesis Tests and Confidence Intervals

Hypothesis tests and confidence intervals are cut out of the same cloth. An event whose 95% confidence interval reject the hypothesis is an event for which p<0.05 under the relating hypothesis test, and the other way around. A p value is letting you know the greatest confidence interval that despite everything prohibits the hypothesis. As such, if p<0.03 against the null hypothesis, that implies that a 97% confidence interval does exclude the null hypothesis.

Hypothesis Tests for a Population Mean

We do a t test on the ground that the population mean is unknown. The general purpose is to contrast sample mean with some hypothetical population mean, to assess whether the watched the truth is such a great amount of unique in relation to the hypothesis that we can say with assurance that the hypothetical population mean isn’t, indeed, the real population mean.

Hypothesis Tests for a Population Proportion

At the point when you have two unique populations Z test facilitates you to choose if the proportion of certain features is the equivalent or not in the two populations. For instance, if the male proportion is equivalent between two nations.

Hypothesis Test for Equal Population Variances

F Test depends on F distribution and is utilized to think about the variance of the two impartial samples. This is additionally utilized with regards to investigation of variance for making a decision about the significance of more than two sample.

T test and F test are totally two unique things. T test is utilized to evaluate the population parameter, for example, population mean, and is likewise utilized for hypothesis testing for population mean. However, it must be utilized when we don’t know about population standard deviation. On the off chance that we know the population standard deviation, we will utilize Z test. We can likewise utilize T statistic to approximate population mean. T statistic is likewise utilised for discovering the distinction in two population mean with the assistance of sample means.

Z statistic or T statistic is utilized to assess population parameters such as population mean and population proportion. It is likewise used for testing hypothesis for population mean and population proportion. In contrast to Z statistic or T statistic, where we manage mean and proportion, Chi Square or F test is utilized for seeing if there is any variance inside the samples. F test is the proportion of fluctuation of two samples.

Conclusion

Hypothesis encourages us to make coherent determinations, the connection among variables, and gives the course to additionally investigate. Hypothesis for the most part results from speculation concerning studied behaviour, natural phenomenon, or proven theory. An honest hypothesis ought to be clear, detailed, and reliable with the data. In the wake of building up the hypothesis, the following stage is validating or testing the hypothesis. Testing of hypothesis includes the process that empowers to concur or differ with the expressed hypothesis.

Attribution Models in Marketing

April 17, 2019/in Big Data, Business Analytics, Business Intelligence, Data Mining, Data Science, Insights, Machine Learning, Main Category, Predictive Analytics, Statistics, Use Case, Use Cases/by Barbara Staron

Attribution Models

A Business and Statistical Case

INTRODUCTION

A desire to understand the causal effect of campaigns on KPIs

Advertising and marketing costs represent a huge and ever more growing part of the budget of companies. Studies have found out this share is as high as 10% and increases with the size of companies (CMO study by American Marketing Association and Duke University, 2017). Measuring precisely the impact of a specific marketing campaign on the sales of a company is a critical step towards an efficient allocation of this budget. Would the return be higher for an euro spent on a Facebook ad, or should we better spend it on a TV spot? How much should I spend on Twitter ads given the volume of sales this channel is responsible for?

Attribution Models have lately received great attention in Marketing departments to answer these issues. The transition from offline to online marketing methods has indeed permitted the collection of multiple individual data throughout the whole customer journey, and allowed for the development of user-centric attribution models. In short, Attribution Models use the information provided by Tracking technologies such as Google Analytics or Webtrekk to understand customer journeys from the first click on a Facebook ad to the final purchase and adequately ponderate the different marketing campaigns encountered depending on their responsibility in the final conversion.

Issues on Causal Effects

A key question then becomes: how to declare a channel is responsible for a purchase? In other words, how can we isolate the causal effect or incremental value of a campaign ?

1. A/B-Tests

One method to estimate the pure impact of a campaign is the design of randomized experiments, wherein a control and treated groups are compared. A/B tests belong to this broad category of randomized methods. Provided the groups are a priori similar in every aspect except for the treatment received, all subsequent differences may be attributed solely to the treatment. This method is typically used in medical studies to assess the effect of a drug to cure a disease.

Main practical issues regarding Randomized Methods are:

Assuring that control and treated groups are really similar before treatment. Uually a random assignment (i.e assuring that on a relevant set of observable variables groups are similar) is realized;
Potential spillover-effects, i.e the possibility that the treatment has an impact on the non-treated group as well (Stable unit treatment Value Assumption, or SUTVA in Rubin’s framework);
The costs of conducting such an experiment, and especially the costs linked to the deliberate assignment of individuals to a group with potentially lower results;
The number of such experiments to design if multiple treatments have to be measured;
Difficulties taking into account the interaction effects between campaigns or the effect of spending levels. Indeed, usually A/B tests are led by cutting off temporarily one campaign entirely and measuring the subsequent impact on KPI’s compared to the situation where this campaign is maintained;
The dynamical reproduction of experiments if we assume that treatment effects may change over time.

In the marketing context, multiple campaigns must be tested in a dynamical way, and treatment effect is likely to be heterogeneous among customers, leading to practical issues in the lauching of A/B tests to approximate the incremental value of all campaigns. However, sites with a lot of traffic and conversions can highly benefit from A/B testing as it provides a scientific and straightforward way to approximate a causal impact. Leading companies such as Uber, Netflix or Airbnb rely on internal tools for A/B testing automation, which allow them to basically test any decision they are about to make.

References:

Books:

A/B testing: the most powerful way to turn clicks into customers. Dan Siroker, Pete Koomen; Wiley, 2013.

Blogs:

https://eng.uber.com/xp

https://medium.com/airbnb-engineering/growing-our-host-community-with-online-marketing-9b2302299324

Study:

https://cmosurvey.org/wp-content/uploads/sites/15/2018/08/The_CMO_Survey-Results_by_Firm_and_Industry_Characteristics-Aug-2018.pdf

2. Attribution models

Attribution Models do not demand to create an experimental setting. They take into account existing data and derive insights from the variability of customer journeys. One key difficulty is then to differentiate correlation and causality in the links observed between the exposition to campaigns and purchases. Indeed, selection effects may bias results as exposure to campaigns is usually dependant on user-characteristics and thus may not be necessarily independant from the customer’s baseline conversion probabilities. For example, customers purchasing from a discount price comparison website may be intrinsically different from customers buying from FB ad and this a priori difference may alone explain post-exposure differences in purchasing bahaviours. This intrinsic weakness must be remembered when interpreting Attribution Models results.

2.1 General Issues

The main issues regarding the implementation of Attribution Models are linked to

Causality and fallacious reasonning, as most models do not take into account the aforementionned selection biases.
Their difficult evaluation. Indeed, in almost all attribution models (except for those based on classification, where the accuracy of the model can be computed), the additionnal value brought by the use of a given attribution models cannot be evaluated using existing historical data. This additionnal value can only be approximated by analysing how the implementation of the conclusions of the attribution model have impacted a given KPI.
Tracking issues, leading to an uncorrect reconstruction of customer journeys
- Cross-device journeys: cross-device issue arises from the use of different devices throughout the customer journeys, making it difficult to link datapoints. For example, if a customer searches for a product on his computer but later orders it on his mobile, the AM would then mistakenly consider it an order without prior campaign exposure. Though difficult to measure perfectly, the proportion of cross-device orders can approximate 20-30%.
- Cookies destruction makes it difficult to track the customer his the whole journey. Both regulations and consumers’ rising concerns about data privacy issues mitigate the reliability and use of cookies.1 – From 2002 on, the EU has enacted directives concerning privacy regulation and the extended use of cookies for commercial targeting purposes, which have highly impacted marketing strategies, such as the ‘Privacy and Electronic Communications Directive’ (2002/58/EC). A research was conducted and found out that the adoption of this ‘Privacy Directive’ had led to 64% decrease in advertising methods compared to the rest of the world (Goldfarb et Tucker (2011)). The effect was stronger for generalized sites (Yahoo) than for specialized sites.2 – Users have grown more and more conscious of data privacy issues and have adopted protective measures concerning data privacy, such as automatic destruction of cookies after a session is ended, or simply giving away less personnal information (Goldfarb et Tucker (2012) ) .Valuable user information may be lost, though tracking technologies evolution have permitted to maintain tracking by other means. This issue may be particularly important in countries highly concerned with data privacy issues such as Germany.
- Offline/Online bridge: an Attribution Model should take into account all campaigns to draw valuable insights. However, the exposure to offline campaigns (TV, newspapers) are difficult to track at the user level. One idea to tackle this issue would be to estimate the proportion of conversions led by offline campaigns through AB testing and deduce this proportion from the credit assigned to the online campaigns accounted for in the Attribution Model.
- Touch point information available: clicks are easy to follow but irrelevant to take into account the influence of purely visual campaigns such as display ads or video.

2.2 Today’s main practices

Two main families of Attribution Models exist:

Rule-Based Attribution Models, which have been used for in the last decade but from which companies are gradualy switching.

Attribution depends on the individual journeys that have led to a purchase and is solely based on the rank of the campaign in the journey. Some models focus on a single touch points (First Click, Last Click) while others account for multi-touch journeys (Bathtube, Linear). It can be calculated at the customer level and thus doesn’t require large amounts of data points. We can distinguish two sub-groups of rule-based Attribution Models:

One Touch Attribution Models attribute all credit to a single touch point. The First-Click model attributes all credit for a converion to the first touch point of the customer journey; last touch attributes all credit to the last campaign.
Multi-touch Rule-Based Attribution Models incorporate information on the whole customer journey are thus an improvement compared to one touch models. To this family belong Linear model where credit is split equally between all channels, Bathtube model where 40% of credit is given to first and last clicks and the remaining 20% is distributed equally between the middle channels, or time-decay models where credit assigned to a click diminishes as the time between the click and the order increases..

The main advantages of rule-based models is their simplicity and cost effectiveness. The main problems are:

– They are a priori known and can thus lead to optimization strategies from competitors
– They do not take into account aggregate intelligence on customer journeys and actual incremental values.
– They tend to bias (depending on the model chosen) channels that are over-represented at the beggining or end of the funnel, according to theoretical assumptions that have no observationnal back-ups.

Data-Driven Attribution Models

These models take into account the weaknesses of rule-based models and make a relevant use of available data. Being data-driven, following attribution models cannot be computed using single user level data. On the contrary values are calculated through data aggregation and thus require a certain volume of customer journey information.

References:

https://dspace.mit.edu/handle/1721.1/64920

3. Data-Driven Attribution Models in practice

3.1 Issues

Several issues arise in the computation of campaigns individual impact on a given KPI within a data-driven model.

Selection biases: Exposure to certain types of advertisement is usually highly correlated to non-observable variables which are in turn correlated to consumption practices. Differences in the behaviour of users exposed to different campaigns may thus only be driven by core differences in conversion probabilities between groups whether than by the campaign effect.
Complementarity: it may be that campaigns A and B only have an effect when combined, so that measuring their individual impact would lead to misleading conclusions. The model could then try to assess the effect of combinations of campaigns on top of the effect of individual campaigns. As the number of possible non-ordered combinations of k campaigns is 2k, it becomes clear that inclusing all possible combinations would however be time-consuming.
Order-sensitivity: The effect of a campaign A may depend on the place where it appears in the customer journey, meaning the rank of a campaign and not merely its presence could be accounted for in the model.
Relative Order-sensitivity: it may be that campaigns A and B only have an effect when one is exposed to campaign A before campaign B. If so, it could be useful to assess the effect of given combinations of campaigns as well. And this for all campaigns, leading to tremendous numbers of possible combinations.
All previous phenomenon may be present, increasing even more the potential complexity of a comprehensive Attribution Model. The number of all possible ordered combination of k campaigns is indeed :

3.2 Main models

A) Logistic Regression and Classification models

If non converting journeys are available, Attribition Model can be shaped as a simple classification issue. Campaign types or campaigns combination and volume of campaign types can be included in the model along with customer or time variables. As we are interested in inference (on campaigns effect) whether than prediction, a parametric model should be used, such as Logistic Regression. Non paramatric models such as Random Forests or Neural Networks can also be used though the interpretation of campaigns value would be more difficult to derive from the model results.

A common pitfall is the usual issue of spurious correlations on one hand and the correct interpretation of coefficients in business terms.

An advantage if the possibility to evaluate the relevance of the model using common model validation methods to evaluate its predictive power (validation set \ AUC \pseudo R squared).

B) Shapley Value

Theory

The Shapley Value is based on a Game Theory framework and is named after its creator, the Nobel Price Laureate Lloyd Shapley. Initially meant to calculate the marginal contribution of players in cooperative games, the model has received much attention in research and industry and has lately been applied to marketing issues. This model is typically used by Google Adords and other ad bidding vendors. Campaigns or marketing channels are in this model seen as compementary players looking forward to increasing a given KPI.
Contrarily to Logistic Regressions, it is a non-parametric model. Contrarily to Markov Chains, all results are built using existing journeys, and not simulated ones.

Channels are considered to enter the game sequentially under a certain joining order. Shapley value try to The Shapley value of channel i is the weighted sum of the marginal values that channel i adds to all possible coalitions that don’t contain channel i.
In other words, the main logic is to analyse the difference of gains when a channel i is added after a coalition Ck of k channels, k<=n. We then sum all the marginal contributions over all possible ordered combination Ck of all campaigns excluding i, with k<=n-1.

Subsets framework

A first an most usual way to compute the Shapley Vaue is to consider that when a channel enters coalition, its additionnal value is the same irrelevant of the order in which previous channels have appeared. In other words, journeys (A>B>C) and (B>A>C) trigger the same gains.
Shapley value is computed as the gains associated to adding a channel i to a subset of channels, weighted by the number of (ordered) sequences that the (unordered) subset represents, summed up on all possible subsets of the total set of campaigns where the channel i is not present.
The Shapley value of the channel ???????? is then:

where |S| is the number of campaigns of a coalition S and the sum extends over all subsets S that do not not contain channel j. ????(????) is the value of the coalition S and ????(???? ∪ {????????}) the value of the coalition formed by adding ???????? to coalition S. ????(???? ∪ {????????}) − ????(????) is thus the marginal contribution of channel ???????? to the coalition S.

The formula can be rewritten and understood as:

This method is convenient when data on the gains of on all possible permutations of all unordered k subsets of the n campaigns are available. It is also more convenient if the order of campaigns prior to the introduction of a campaign is thought to have no impact.

Ordered sequences

Let us define ????((A>B)) as the value of the sequence A then B. What is we let ????((A>B)) be different from ????((B>A)) ?
This time we would need to sum over all possible permutation of the S campaigns present before ???????? and the N-(S+1) campaigns after ????????. Doing so we will sum over all possible orderings (i.e all permutations of the n campaigns of the grand coalition containing all campaigns) and we can remove the permutation coefficient s!(p-s+1)!.

This method is convenient when the order of channels prior to and after the introduction of another channel is assumed to have an impact. It is also necessary to possess data for all possible permutations of all k subsets of the n campaigns, and not only on all (unordered) k-subsets of the n campaigns, k<=n. In other words, one must know the gains of A, B, C, A>B, B>A, etc. to compute the Shapley Value.

Differences between the two approaches

We simulate an ordered case where the value for each ordered sequence k for k<=3 is known. We compare it to the usual Shapley value calculated based on known gains of unordered subsets of campaigns. So as to compare relevant values, we have built the gains matrix so that the gains of a subset A, B i.e ????({B,A}) is the average of the gains of ordered sequences made up with A and B (assuming the number of journeys where A>B equals the number of journeys where B>A, we have ????({B,A})=0.5( ????((A>B)) + ????((B>A)) ). We let the value of the grand coalition be different depending on the order of campaigns-keeping the constraints that it averages to the value used for the unordered case.

Note: mvA refers to the marginal value of A in a given sequence.
With traditionnal unordered coalitions:

With ordered sequences used to compute the marginal values:

We can see that the two approaches yield very different results. In the unordered case, the Shapley Value campaign C is the highest, culminating at 20, while A and B have the same Shapley Value mvA=mvB=15. In the ordered case, campaign A has the highest Shapley Value and all campaigns have different Shapley Values.

This example illustrates the inherent differences between the set and sequences approach to Shapley values. Real life data is more likely to resemble the ordered case as conversion probabilities may for any given set of campaigns be influenced by the order through which the campaigns appear.

Advantages

Shapley value has become popular in allocation problems in cooperative games because it is the unique allocation which satisfies different axioms:

Efficiency: Shaple Values of all channels add up to the total gains (here, orders) observed.
Symmetry: if channels A and B bring the same contribution to any coalition of campaigns, then their Shapley Value i sthe same
Null player: if a channel brings no additionnal gains to all coalitions, then its Shapley Value is zero
Strong monotony: the Shapley Value of a player increases weakly if all its marginal contributions increase weakly

These properties make the Shapley Value close to what we intuitively define as a fair attribution.

Issues

The Shapley Value is based on combinatory mathematics, and the number of possible coalitions and ordered sequences becomes huge when the number of campaigns increases.
If unordered, the Shapley Value assumes the contribution of campaign A is the same if followed by campaign B or by C.
If ordered, the number of combinations for which data must be available and sufficient is huge.
Channels rarely present or present in long journeys will be played down.
Generally, gains are supposed to grow with the number of players in the game. However, it is plausible that in the marketing context a journey with a high number of channels will not necessarily bring more orders than a journey with less channels involved.

References:

R package: GameTheoryAllocation

Article:
Zhao & al, 2018 “Shapley Value Methods for Attribution Modeling in Online Advertising “
https://link.springer.com/content/pdf/10.1007/s13278-017-0480-z.pdf
Courses: https://www.lamsade.dauphine.fr/~airiau/Teaching/CoopGames/2011/coopgames-7%5b8up%5d.pdf
Blogs: https://towardsdatascience.com/one-feature-attribution-method-to-supposedly-rule-them-all-shapley-values-f3e04534983d

B) Markov Chains

Markov Chains are used to model random processes, i.e events that occur in a sequential manner and in such a way that the probability to move to a certain state only depends on the past steps. The number of previous steps that are taken into account to model the transition probability is called the memory parameter of the sequence, and for the model to have a solution must be comprised between 0 and 4. A Markov Chain process is thus defined entirely by its Transition Matrix and its initial vector (i.e the starting point of the process).

Markov Chains are applied in many scientific fields. Typically, they are used in weather forecasting, with the sequence of Sunny and Rainy days following a Markov Process of memory parameter 0, so that for each given day the probability that the next day will be rainy or sunny only depends on the weather of the current day. Other applications can be found in sociology to understand the dynamics of social classes intergenerational reproduction. To get more both mathematical and applied illustration, I recommend the reading of this course.

In the marketing context, Markov Chains are an interesting way to model the conversion funnel. To go from the from the Markov Model to the Attribution logic, we calculate the Removal Effect of each channel, i.e the difference in conversions that happen if the channel is removed. Please read below for an introduction to the methodology.

The first step in a Markov Chains Attribution Model is to build the transition matrix that captures the transition probabilities between the campaigns accross existing customer journeys. This Matrix is to be read as a “From state A to state B” table, from the left to the right. A first difficulty is finding the right memory parameter to use. A large memory parameter would allow to take more into account interraction effects within the conversion funnel but would lead to increased computationnal time, a non-readable transition matrix, and be more sensitive to noisy data. Please note that this transition matrix provides useful information on the conversion funnel and on the relationships between campaigns and can be used as such as an analytical tool. I suggest the clear and easily R code which can be found here or here.

Here is an illustration of a Markov Chain with memory Parameter of 0: the probability to go to a certain campaign B in the next step only depend on the campaign we are currently at:

The associated Transition Matrix is then (with null probabilities left as Blank):

The second step is to compute the actual responsibility of a channel in total conversions. As mentionned above, the main philosophy to do so is to calculate the Removal Effect of each channel, i.e the changes in the number of conversions when a channel is entirely removed. All customer journeys which went through this channel are settled out to be unsuccessful. This calculation is done by applying the transition matrix with and without the removed channels to an initial vector that contains the number of desired simulations.

Building on our current example, we can then settle an initial vector with the desired number of simulations, e.g 10 000:

It is possible at this stage to add a constraint on the maximum number of times the matrix is applied to the data, i.e on the maximal number of campaigns a simulated journey is allowed to have.

Advantages

The dynamic journey is taken into account, as well as the transition between two states. The funnel is not assumed to be linear.
It is possile to build a conversion graph that maps the customer journey provides valuable insights.
It is possible to evaluate partly the accuracy of the Attribution Model based on Markov Chains. It is for example possible to see how well the transition matrix help predict the future by analysing the number of correct predictions at any given step over all sequences.

Disadvantages

It can be somewhat difficult to set the memory parameter. Complementarity effects between channels are not well taken into account if the memory is low, but a parameter too high will lead to over-sensitivity to noise in the data and be difficult to implement if customer journeys tend to have a number of campaigns below this memory parameter.
Long journeys with different channels involved will be overweighted, as they will count many times in the Removal Effect. For example, if there are n-1 channels in the customer journey, this journey will be considered as failure for the n-1 channel-RE. If the volume effects (i.e the impact of the overall number of channels in a journey, irrelevant from their type° are important then results may be biased.

References:

R package: ChannelAttribution

Git:

https://github.com/MatCyt/Markov-Chain/blob/master/README.md

Course:

https://www.ssc.wisc.edu/~jmontgom/markovchains.pdf

Article:

“Mapping the Customer Journey: A Graph-Based Framework for Online Attribution Modeling”; Anderl, Eva and Becker, Ingo and Wangenheim, Florian V. and Schumann, Jan Hendrik, 2014. Available at SSRN: https://ssrn.com/abstract=2343077 or http://dx.doi.org/10.2139/ssrn.2343077

“Media Exposure through the Funnel: A Model of Multi-Stage Attribution”, Abhishek & al, 2012

“Multichannel Marketing Attribution Using Markov Chains”, Kakalejčík, L., Bucko, J., Resende, P.A.A. and Ferencova, M. Journal of Applied Management and Investments, Vol. 7 No. 1, pp. 49-60. 2018

Blogs:

https://analyzecore.com/2016/08/03/attribution-model-r-part-1

https://analyzecore.com/2016/08/03/attribution-model-r-part-2

3.3 To go further: Tackling selection biases with Quasi-Experiments

Exposure to certain types of advertisement is usually highly correlated to non-observable variables. Differences in the behaviour of users exposed to different campaigns may thus only be driven by core differences in converison probabilities between groups whether than by the campaign effect. These potential selection effects may bias the results obtained using historical data.

Quasi-Experiments can help correct this selection effect while still using available observationnal data. These methods recreate the settings on a randomized setting. The goal is to come as close as possible to the ideal of comparing two populations that are identical in all respects except for the advertising exposure. However, populations might still diﬀer with respect to some unobserved characteristics.

Common quasi-experimental methods used for instance in Public Policy Evaluation are:

Discontinuity Regressions
Matching Methods, such as Exact Matching, Propensity-score matching or k-nearest neighbourghs.

References:

Article:

“Towards a digital Attribution Model: Measuring the impact of display advertising on online consumer behaviour”, Anindya Ghose & al, MIS Quarterly Vol. 40 No. 4, pp. 1-XX, 2016

https://pdfs.semanticscholar.org/4fa6/1c53f281fa63a9f0617fbd794d54911a2f84.pdf

4. First Steps towards a Practical Implementation

Identify key points of interests

Identify the nature of touchpoints available: is the data based on clicks? If so, is there a way to complement the data with A/B tests to measure the influence of ads without clicks (display, video) ? For example, what happens to sales when display campaign is removed? Analysing this multiplier effect would give the overall responsibility of display on sales, to be deduced from current attribution values given to click-based channels. More interestingly, what is the impact of the removal of display campaign on the occurences of click-based campaigns ? This would give us an idea of the impact of display ads on the exposure to each other campaigns, which would help correct the attribution values more precisely at the campaign level.
Define the KPI to track. From a pure Marketing perspective, looking at purchases may be sufficient, but from a financial perspective looking at profits, though a bit more difficult to compute, may drive more interesting results.
Define a customer journey. It may seem obvious, but the notion needs to be clarified at first. Would it be defined by a time limit? If so, which one? Does it end when a conversion is observed? For example, if a customer makes 2 purchases, would the campaigns he’s been exposed to before the first order still be accounted for in the second order? If so, with a time decay?
Define the research framework: are we interested only in customer journeys which have led to conversions or in all journeys? Keep in mind that successful customer journeys are a non-representative sample of customer journeys. Models built on the analysis of biased samples may be conservative. Take an extreme example: 80% of customers who see campaign A buy the product, VS 1% for campaign B. However, campaign B exposure is great and 100 Million people see it VS only 1M for campaign A. An Attribution Model based on successful journeys will give higher credit to campaign B which is an auguable conclusion. Taking into account costs per campaign (in the case where costs are calculated by clicks) may of course tackle this issue partly, as campaign A could then exhibit higher returns, but a serious fallacious reasonning is at stake here.

Analyse the typical customer journey

Performing a duration analysis on the data may help you improve the definition of the customer journey to be used by your organization. After which days are converison probabilities null? Should we consider the effect of campaigns disappears after x days without orders? For example, if 99% of orders are placed in the 30 days following a first click, it might be interesting to define the customer journey as a 30 days time frame following the first oder.
Look at the distribution of the number of campaigns in a typical journey. If you choose to calculate the effect of campaigns interraction in your Attribution Model, it may indeed help you determine the maximum number of campaigns to be included in a combination. Indeed, you may not need to assess the impact of channel combinations with above than 4 different channels if 95% of orders are placed after less then 4 campaigns.
Transition matrixes: what if a campaign A systematically leads to a campaign B? What happens if we remove A or B? These insights would give clues to ask precise questions for a latter AB test, for example to find out if there is complementarity between channels A and B – (implying none should be removed) or mere substitution (implying one can be given up).
If conversion rates are available: it can be interesting to perform a survival analysis i.e to analyse the likelihood of conversion based on duration since first click. This could help us excluse potential outliers or individuals who have very low conversion probabilities.

Summary

Attribution is a complex topic which will probably never be definitively solved. Indeed, a main issue is the difficulty, or even impossibility, to evaluate precisely the accuracy of the attribution model that we’ve built. Attribution Models should be seen as a good yet always improvable approximation of the incremental values of campaigns, and be presented with their intrinsinc limits and biases.

A common trap when it comes to sampling from a population that intrinsically includes outliers

February 21, 2019/in Business Analytics, Business Intelligence, Data Mining, Data Science, Data Science Hack, Main Category, R Statistics, Statistics/by Moussa El-Zaarah

I will discuss a common fallacy concerning the conclusions drawn from calculating a sample mean and a sample standard deviation and more importantly how to avoid it.

Suppose you draw a random sample $x_1$ , $x_2$ , … $x_N$ of size $N$ and compute the ordinary (arithmetic) sample mean $x_m$ and a sample standard deviation $sd$ from it. Now if (and only if) the (true) population mean µ (first moment) and population variance (second moment) obtained from the actual underlying PDF are finite, the numbers $x_m$ and $sd$ make the usual sense otherwise they are misleading as will be shown by an example.

By the way: The common correlation coefficient will also be undefined (or in practice always point to zero) in the presence of infinite population variances. Hopefully I will create an article discussing this related fallacy in the near future where a suitable generalization to Lévy-stable variables will be proposed.

Drawing a random sample from a heavy tailed distribution and discussing certain measures

As an example suppose you have a one dimensional random walker whose step length is distributed by a symmetric standard Cauchy distribution (Lorentz-profile) with heavy tails, i.e. an alpha-stable distribution with alpha being equal to one. The PDF of an individual independent step is given by $p(x) = \frac{\pi^{-1}}{(1 + x^2)}$ , thus neither the first nor the second moment exist whereby the first exists and vanishes at least in the sense of a principal value due to symmetry.

Still let us generate $N = 3000$ (pseudo) standard Cauchy random numbers in R* to analyze the behavior of their sample mean and standard deviation $sd$ as a function of the reduced sample size $n \leq N$ .

*The R-code is shown at the end of the article.

Here are the piecewise sample mean (in blue) and standard deviation (in red) for the mentioned Cauchy sampling. We see that both the sample mean and $sd$ include jumps and do not converge.

Especially the mean deviates relatively largely from zero even after 3000 observations. The sample $sd$ has no target due to the population variance being infinite.

If the data is new and no prior distribution is known, computing the sample mean and $sd$ will be misleading. Astonishingly enough the sample mean itself will have the (formally exact) same distribution as the single step length $p(x)$ . This means that the sample mean is also standard Cauchy distributed implying that with a different Cauchy sample one could have easily observed different sample means far of the presented values in blue.

What sense does it make to present the usual interval $x_m \pm sd / \sqrt{N}$ in such a case? What to do?

The sample median, median absolute difference (mad) and Inter-Quantile-Range (IQR) are more appropriate to describe such a data set including outliers intrinsically. To make this plausible I present the following plot, whereby the median is shown in black, the mad in green and the IQR in orange.

This example shows that the median, mad and IQR converge quickly against their assumed values and contain no major jumps. These quantities do an obviously better job in describing the sample. Even in the presence of outliers they remain robust, whereby the mad converges more quickly than the IQR. Note that a standard Cauchy sample will contain half of its sample in the interval median $\pm$ mad meaning that the IQR is twice the mad.

Drawing a random sample from a PDF that has finite moments

Just for comparison I also show the above quantities for a standard normal (pseudo) sample labeled with the same color as before as a counter example. In this case not only do both the sample mean and median but also the $sd$ and mad converge towards their expected values (see plot below). Here all the quantities describe the data set properly and there is no trap since there are no intrinsic outliers. The sample mean itself follows a standard normal, so that the $sd$ in deed makes sense and one could calculate a standard error $\frac{sd}{\sqrt{N}}$ from it to present the usual stochastic confidence intervals for the sample mean.

A careful observation shows that in contrast to the Cauchy case here the sampled mean and $sd$ converge more quickly than the sample median and the IQR. However still the sampled mad performs about as well as the $sd$ . Again the mad is twice the IQR.

And here are the graphs of the prementioned quantities for a pseudo normal sample:

The take-home-message:

Just be careful when you observe outliers and calculate sample quantities right away, you might miss something. At best one carefully observes how the relevant quantities change with sample size as demonstrated in this article.

Such curves should become of broader interest in order to improve transparency in the Data Science process and reduce fallacies as well.

Thank you for reading.

P.S.: Feel free to play with the set random seed in the R-code below and observe how other quantities behave with rising sample size. Of course you can also try different PDFs at the beginning of the code. You can employ a Cauchy, Gaussian, uniform, exponential or Holtsmark (pseudo) random sample.

QUIZ: Which one of the recently mentioned random samples contains a trap** and why?

**in the context of this article

R-code used to generate the data and for producing plots:

#R-script for emphasizing convergence and divergence of sample means

####install and load relevant packages ####

#uncomment these lines if necessary
#install.packages(c('ggplot2',’stabledist’))
#library(ggplot2)
#library(stabledist)

#####drawing random samples #####

#Setting a random seed for being able to reproduce results  
set.seed(1234567)   
N= 2000     #sample size

#Choose a PDF from which a sample shall be drawn
#To do so (un)comment the respective lines of following code

data <- rcauchy(N)    # option1(default): standard Cauchy sampling

#data <- rnorm(N)     #option2: standard Gaussian sampling
                               
#data <- rexp(N)    # option3: standard exponential sampling

#data <- rstable(N,alpha=1.5,beta=0)  # option4: standard symmetric Holtsmark sampling

#data <- runif(N)              #option5: standard uniform sample

#####descriptive statistics####
#preparations/declarations

SUM = vector()
sd =vector()
mean = vector()
SQ =vector()
SQUARES = vector()
median = vector()
mad =vector()
quantiles = data.frame()
sem =vector()

#piecewise calculaion of descrptive quantities

for (k in 1:length(data)){              #mainloop
SUM[k] <- sum(data[1:k])            # sum of sample
mean[k] <- mean(data[1:k])          # arithmetic mean
sd[k] <- sd(data[1:k])              # standard deviation
sem[k] <- sd[k]/(sqrt(k))          #standard error of the sample mean (for finite variances)
mad[k] <- mad(data[1:k],const=1)   # median absolute deviation    

for (j in 1:5){
qq <- quantile(data[1:k],na.rm = T)
quantiles[k,j] <- qq[j]         #quantiles of sample
}
colnames(quantiles) <- c('min','Q1','median','Q3','max')

for (i in 1:length(data[1:k])){
SQUARES[i] <- data[i]*data[i]    
}
SQ[k] <- sum(SQUARES[1:k])    #sum of squares of random sample
}  #end of mainloop

#create table containing all relevant data
TABLE <-  as.data.frame(cbind(quantiles,mean,sd,SQ,SUM,sem))




#####plotting results###
x11()
print(ggplot(TABLE,aes(1:N,median))+
geom_point(size=.5)+xlab('sample size n')+ylab('sample median'))
x11()
print(ggplot(TABLE,aes(1:N,mad))+geom_point(size=.5,color ='green')+
xlab('sample size n')+ylab('sample median absolute difference'))
x11()
print(ggplot(TABLE,aes(1:N,sd))+geom_point(size=.5,color ='red')+
xlab('sample size n')+ylab('sample standard deviation'))
x11()
print(ggplot(TABLE,aes(1:N,mean))+geom_point(size=.5, color ='blue')+
xlab('sample size n')+ylab('sample mean'))
x11()
print(ggplot(TABLE,aes(1:N,Q3-Q1))+geom_point(size=.5, color ='blue')+
xlab('sample size n')+ylab('IQR'))

#uncomment the following lines of code to see further plots

#x11()
#print(ggplot(TABLE,aes(1:N,sem))+geom_point(size=.5)+
#xlab('sample size n')+ylab('sample sum of r.v.'))
#x11()
#print(ggplot(TABLE,aes(1:N,SUM))+geom_point(size=.5)+
#xlab('sample size n')+ylab('sample sum of r.v.'))
#x11()
#print(ggplot(TABLE,aes(1:N,SQ))+geom_point(size=.5)+
#xlab('sample size n')+ylab('sample sum of squares'))

Statistical Relational Learning – Part 2

January 18, 2017/in Artificial Intelligence, Data Mining, Data Science, Data Science at the Command Line, Machine Learning, Statistics, Tools, Tutorial/by Vishal Bhalla

In the first part of this series on “An Introduction to Statistical Relational Learning”, I touched upon the basic Machine Learning paradigms, some background and intuition of the concepts and concluded with how the MLN template looks like. In this blog, we will dive in to get an in depth knowledge on the MLN template; again with the help of sample examples. I would then conclude by highlighting the various toolkit available and some of its differentiating features.

MLN Template – explained

A Markov logic network can be thought of as a group of formulas incorporating first-order logic and also tied with a weight. But what exactly does this weight signify?

Weight Learning

According to the definition, it is the log odds between a world where F is true and a world where F is false,

and captures the marginal distribution of the corresponding predicate.

Each formula can be associated with some weight value, that is a positive or negative real number. The higher the value of weight, the stronger the constraint represented by the formula. In contrast to classical logic, all worlds (i.e., Herbrand Interpretations) are possible with a certain probability [1]. The main idea behind this is that the probability of a world increases as the number of formulas it violates decreases.

Markov logic networks with its probabilistic approach combined to logic posit that a world is less likely if it violates formulas unlike in pure logic where a world is false if it violates even a single formula. Consider the case when a formula with high weight i.e. more significance is violated implying that it is less likely in occurrence.

Another important concept during the first phase of Weight Learning while applying an MLN template is “Grounding”. Grounding means to replace each variable/function in predicate with constants from the domain.

Weight Learning – An Example

Note: All examples are highlighted in the Alchemy MLN format

Let us consider an example where we want to identify the relationship between 2 different types of verb-noun pairs i.e noun subject and direct object.

The input predicateFormula.mln file contains

The predicates nsubj(verb, subject) and dobj(verb, object) and
Formula of nsubj(+ver, +s) and dobj(+ver, +o)

These predicates or rules are to learn all possible SVO combinations i.e. what is the probability of a Subject-Verb-Object combination. The + sign ensures a cross product between the domains and learns all combinations. The training database consists of the nsubj and dobj tuples i.e. relations is the evidence used to learn the weights.

When we run the above command for this set of rules against the training evidence, we learn the weights as here:

Note that the formula is now grounded by all occurrences of nsubj and dobj tuples from the training database or evidence and the weights are attached to it at the start of each such combination.

But it should be noted that there is no network yet and this is just a set of weighted first-order logic formulas. The MLN template we created so far will generate Markov networks from all of our ground formulas. Internally, it is represented as a factor graph.where each ground formula is a factor and all the ground predicates found in the ground formula are linked to the factor.

Inference

The definition goes as follows:

Estimate probability distribution encoded by a graphical model, for a given data (or observation).

Out of the many Inference algorithms, the two major ones are MAP & Marginal Inference. For example, in a MAP Inference we find the most likely state of world given evidence, where y is the query and x is the evidence.

which is in turn equivalent to this formula.

Another is the Marginal Inference which computes the conditional probability of query predicates, given some evidence. Some advanced inference algorithms are Loopy Belief Propagation, Walk-SAT, MC-SAT, etc.

The probability of a world is given by the weighted sum of all true groundings of a formula i under an exponential function, divided by the partition function Z i.e. equivalent to the sum of the values of all possible assignments. The partition function acts a normalization constant to get the probability values between 0 and 1.

Inference – An Example

Let us draw inference on the the same example as earlier.

After learning the weights we run inference (with or without partial evidence) and query the relations of interest (nsubj here), to get inferred values.

Tool-kits

Let’s look at some of the MLN tool-kits at disposal to do learning and large scale inference. I have tried to make an assorted list of all tools here and tried to highlight some of its main features & problems.

For example, BUGS i.e. Bayesian Logic uses a Swift Compiler but is Not relational! ProbLog has a Python wrapper and is based on Horn clauses but has No Learning feature. These tools were invented in the initial days, much before the present day MLN looks like.

ProbCog developed at Technical University of Munich (TUM) & the AI Lab at Bremen covers not just MLN but also Bayesian Logic Networks (BLNs), Bayesian Networks & ProLog. In fact, it is now GUI based. Thebeast gives a shell to analyze & inspect model feature weights & missing features.

Alchemy from University of Washington (UoW) was the 1st First Order (FO) probabilistic logic toolkit. RockIt from University of Mannheim has an online & rest based interface and uses only Conjunctive Normal Forms (CNF) i.e. And-Or format in its formulas.

Tuffy scales this up by using a Relational Database Management System (RDBMS) whereas Felix allows Large Scale inference! Elementary makes use of secondary storage and Deep Dive is the current state of the art. All of these tools are part of the HAZY project group at Stanford University.

Lastly, LoMRF i.e. Logical Markov Random Field (MRF) is Scala based and has a feature to analyse different hypothesis by comparing the difference in .mln files!

Hope you enjoyed the read. The content starts from basic concepts and ends up highlighting key tools. In the final part of this 3 part blog series I would explain an application scenario and highlight the active research and industry players. Any feedback as a comment below or through a message is more than welcome!

Back to Part I – Statistical Relational Learning

Additional Links:

[1] Knowledge base files in Logical Markov Random Fields (LoMRF)

[2] (still) nothing clever Posts categorized “Machine Learning” – Markov Logic Networks

[3] A gentle introduction to statistical relational learning: maths, code, and examples

Statistical Relational Learning

August 17, 2016/in Artificial Intelligence, Data Science, Machine Learning, Statistics/by Vishal Bhalla

An Introduction to Statistical Relational Learning – Part 1

Statistical Relational Learning (SRL) is an emerging field and one that is taking centre stage in the Data Science age. Big Data has been one of the primary reasons for the continued prominence of this relational learning approach given, the voluminous amount of data available now to learn interesting and unknown patterns from data. Moreover, the tools have also improved their processing prowess especially, in terms of scalability.

This introductory blog is a prelude on SRL and later on I would also touch base on more advanced topics, specifically Markov Logic Networks (MLN). To start off, let’s look at how SRL fits into one of the 5 different Machine Learning paradigms.

Five Machine Learning Paradigms

Lets look at the 5 Machine Learning Paradigms: Each of which is inspired by ideas from a different field!

Connectionists as they are called and led by Geoffrey Hinton (University of Toronto & Google and one of the major names in the Deep Learning community) think that a learning algorithm should mimic the brain! After all it is the brain that does all the complex actions for us and, this idea stems from Neuroscience.
Another group of Evolutionists whose leader is the late John Holland (from the University of Michigan) believed it is not the brain but evolution that was precedent and hence the master algorithm to build anything. And using this approach of having the fittest ones program the future they are currently building 3D prints of future robots.
Another thought stems from Philosophy where Analogists like Douglas R. Hofstadter an American writer and author of popular and award winning book – Gödel, Escher, Bach: an Eternal Golden Braid believe that Analogy is the core of Cognition.
Symbolists like Stephen Muggleton (Imperial College London) think Psychology is the base and by developing Rules in deductive reasoning they built Adam – a robot scientist at the University of Manchester!
Lastly we have a school of thought which has its foundations rested on Statistics & Logic, which is the focal point of interest in this blog. This emerging field has started to gain prominence with the invention of Bayesian networks 2011 by Judea Pearl (University of California Los Angeles – UCLA) who was awarded with the Turing award (the highest award in Computer Science). Bayesians as they are called, are the most fanatical of the lot as they think everything can be represented by the Bayes theorem using hypothesis which can be updated based on new evidence.

SRL fits into the last paradigm of Statistics and Logic. As such it offers another alternative to the now booming Deep Learning approach inspired from Neuroscience.

Background

In many real world scenario and use cases, often the underlying data is assumed to be independent and identically distributed (i.i.d.). However, real world data is not and instead consists of many relationships. SRL as such attempts to represent, model, and learn in the relational domain!

There are 4 main Models in SRL

Probabilistic Relational Models (PRM)
Markov Logic Networks (MLN)
Relational Dependency Networks (RDN)
Bayesian Logic Programs (BLP)

It is difficult to cover all major models and hence the focus of this blog is only on the emerging field of Markov Logic Networks.
MLN is a powerful framework that combines statistics (i.e. it uses Markov Random Fields) and logical reasoning (first order logic).

Academia

Some of the prominent names in academic and the research community in MLN include:

Professor Pedro Domingos from the University of Washington is credited with introducing MLN in his paper from 2006. His group created the tool called Alchemy which was one of the first, First Order Logic tools.
Another famous name – Professor Luc De Raedt from the AI group at University of Leuven in Belgium, and their team created the tool ProbLog which also has a Python Wrapper.
HAZY Project (Stanford University) led by Prof. Christopher Ré from the InfoLab is doing active research in this field and Tuffy, Felix, Elementary, Deep Dive are some of the tools developed by them. More on it later!
Talking about academia close by i.e. in Germany, Prof. Michael Beetz and his entire team moved from TUM to TU Bremen. Their group invented the tool – ProbCog
At present, Prof. Volker Tresp from Ludwig Maximilians University (LMU), Munich & Dr. Matthias Nickles at Technical University of Munich (TUM) have research interests in SRL.

Theory & Formulation

A look at some background and theoretical concepts to understand MLN better.

A. Basics – Probabilistic Graphical Models (PGM)

The definition of a PGM goes as such:

A PGM encodes a joint p(x,y) or conditional p(y|x) probability distribution such that given some observations we are provided with a full probability distribution over all feasible solutions.

A PGM helps to encode relationships between a set of random variables. And it achieves this by making use of a graph! These graphs can be either be Directed or Undirected Graphs.

B. Markov Blanket

A Markov Blanket is a Directed Acyclic graph. It is a Bayesian network and as you can see the central node A highlighted in red is dependent on its parents and parents of descendents (moralization) by the circle drawn around it. Thus these nodes are the only knowledge needed to predict node A.

Image courtesy – Wikipedia

C. Markov Random Fields (MRF)

A MRF is an Undirected graphical model. Every node in an MRF satisfies the Local Markov property of Conditional Independence, i.e. a node is conditionally independent of another node, given its neighbours. And now relating it to Markov Blanket as explained previously, a Markov blanket for a node is simply its adjacent nodes!

Image courtesy – Wikipedia

Intuition

We now that Probability handles uncertainty whereas Logic handles complexity. So why not make use of both of them to model relationships in data that is both uncertain and complex. Markov Logic Networks (MLN) precisely does that for us!

MLN is composed of a set of pairs of <w, F> where F is the formula (written in FO logic) and weights (real numbers identifying the strength of the constraint).

MLN basically provides a template to ground a Markov network. Grounding would be explained in detail in the next but one section on “Weight Learning”.

It can be defined as a Log linear model where probability of a world is given by the weighted sum of all true groundings of a formula i under an exponential function. It is then divided by Z which is termed as the partition function and used to normalize and get probability values between 0 and 1.

The MLN Template

Rules or Predicates

The relation to be learned is expressed in FO logic. Some of the different possible FO logical connectives and quantifiers are And (^), Or (V), Implication (→), and many more. Plus, Formulas may contain one or more predicates, connected to each other with logical connectives and quantified symbols.

Evidence

Evidence represent known facts i.e. the ground predicates. Each fact is expressed with predicates that contain only constants from their corresponding domains.

Weight Learning

Discover the importance of relations based on grounded evidence.

Inference

Query relations, given partial evidence to infer a probabilistic estimate of the world.

More on Weight Learning and Inference in the next part of this series!

Hope you enjoyed the read. I have deliberately kept the content basic and a mix of non technical and technical so as to highlight first the key players and some background concepts and generate the reader’s interest in this topic, the technicalities of which can easily be read in the paper. Any feedback as a comment below or through a message are more than welcome!

Continue reading with Statistical Relational Learning – Part II.

References

Richardson, Matthew, and Pedro Domingos – Markov logic networks. In Machine learning, vol. 62, no. 1-2, pp. 107-136, 2006.
Pedro’s TEDx video at University of Washington: The Quest for the Master Algorithm
Hazy project webpage

Data Mining Process flow – Easy Understanding

Hypothesis Test for real problems

Attribution Models in Marketing

Attribution Models

A Business and Statistical Case

INTRODUCTION

1. A/B-Tests

2. Attribution models

2.1 General Issues

2.2 Today’s main practices

3. Data-Driven Attribution Models in practice

3.1 Issues

3.2 Main models

A) Logistic Regression and Classification models

B) Shapley Value

Theory

Advantages

B) Markov Chains

3.3 To go further: Tackling selection biases with Quasi-Experiments

4. First Steps towards a Practical Implementation

Identify key points of interests

Analyse the typical customer journey

Summary

A common trap when it comes to sampling from a population that intrinsically includes outliers

Statistical Relational Learning – Part 2

MLN Template – explained

Weight Learning

Weight Learning – An Example

Inference

Inference – An Example

Tool-kits

Statistical Relational Learning

An Introduction to Statistical Relational Learning – Part 1

Interesting links

Pages

Categories

Archive