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My homework solutions for CMU Machine Learning Course (10-601 2018Fall)

puttak/10601-18Fall-Homework

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This repository contains the homework solutions for CMU course Introduction to Machine Learning (10601 2018 Fall). All coding parts are completed in Python3.

  • Homework 1: Background Material
  • Homework 2: Decision Trees
  • Homework 3: KNN, Perceptron, Linear Regression
  • Homework 4: Logistic Regression
  • Homework 5: Neural Networks
  • Homework 6: Generative Models
  • Homework 7: Hidden Markov Models
  • Homework 8: Reinforcement Learning
  • Homework 9: SVMs, K-Means, PCA, Boosting

CS4341 Introduction to Artificial Intelligence  SOLUTIONS Homework - D 2001

By songting chen and carolina ruiz , instructions, problem 1. decision trees (20 points), problem 2. genetic algorithms (20 points), problem 3. neural networks (20 points), problem 4. logic-based systems (20 points), problem 5. planning (20 points), problem 6. machine vision (20 points).

  • Machine Learning Tutorial
  • Data Analysis Tutorial
  • Python - Data visualization tutorial
  • Machine Learning Projects
  • Machine Learning Interview Questions
  • Machine Learning Mathematics
  • Deep Learning Tutorial
  • Deep Learning Project
  • Deep Learning Interview Questions
  • Computer Vision Tutorial
  • Computer Vision Projects
  • NLP Project
  • NLP Interview Questions
  • Statistics with Python
  • 100 Days of Machine Learning
  • ML | Active Learning
  • Difference between Supervised and Unsupervised Learning
  • Overview of ROBERTa model
  • What is Saliency Map?
  • Intuition behind Adagrad Optimizer
  • Differentiate between Support Vector Machine and Logistic Regression
  • Logistic Regression using Python
  • Cost function in Logistic Regression in Machine Learning
  • CNN | Introduction to Padding
  • GrowNet: Gradient Boosting Neural Networks
  • Orthogonal Projections
  • Multiclass classification using scikit-learn
  • Getting started with Dialogflow
  • Rule-Based Classifier - Machine Learning
  • Introduction to Speech Separation Based On Fast ICA
  • Python | Create Test DataSets using Sklearn
  • Early Stopping for Regularisation in Deep Learning
  • ALBERT - A Light BERT for Supervised Learning

Decision Tree

A decision tree is one of the most powerful tools of supervised learning algorithms used for both classification and regression tasks. It builds a flowchart-like tree structure where each internal node denotes a test on an attribute, each branch represents an outcome of the test, and each leaf node (terminal node) holds a class label. It is constructed by recursively splitting the training data into subsets based on the values of the attributes until a stopping criterion is met, such as the maximum depth of the tree or the minimum number of samples required to split a node.

During training, the Decision Tree algorithm selects the best attribute to split the data based on a metric such as entropy or Gini impurity, which measures the level of impurity or randomness in the subsets. The goal is to find the attribute that maximizes the information gain or the reduction in impurity after the split.

What is a Decision Tree?

A decision tree is a flowchart-like tree structure where each internal node denotes the feature, branches denote the rules and the leaf nodes denote the result of the algorithm. It is a versatile supervised machine-learning algorithm, which is used for both classification and regression problems. It is one of the very powerful algorithms. And it is also used in Random Forest to train on different subsets of training data, which makes random forest one of the most powerful algorithms in machine learning .

Decision Tree Terminologies

Some of the common Terminologies used in Decision Trees are as follows:

  • Root Node: It is the topmost node in the tree,  which represents the complete dataset. It is the starting point of the decision-making process.
  • Decision/Internal Node: A node that symbolizes a choice regarding an input feature. Branching off of internal nodes connects them to leaf nodes or other internal nodes.
  • Leaf/Terminal Node: A node without any child nodes that indicates a class label or a numerical value.
  • Splitting: The process of splitting a node into two or more sub-nodes using a split criterion and a selected feature.
  • Branch/Sub-Tree: A subsection of the decision tree starts at an internal node and ends at the leaf nodes.
  • Parent Node: The node that divides into one or more child nodes.
  • Child Node: The nodes that emerge when a parent node is split.
  • Impurity : A measurement of the target variable’s homogeneity in a subset of data. It refers to the degree of randomness or uncertainty in a set of examples. The Gini index and entropy are two commonly used impurity measurements in decision trees for classifications task 
  • Variance : Variance measures how much the predicted and the target variables vary in different samples of a dataset. It is used for regression problems in decision trees. Mean squared error, Mean Absolute Error, friedman_mse, or Half Poisson deviance are used to measure the variance for the regression tasks in the decision tree.
  • Information Gain: Information gain is a measure of the reduction in impurity achieved by splitting a dataset on a particular feature in a decision tree. The splitting criterion is determined by the feature that offers the greatest information gain, It is used to determine the most informative feature to split on at each node of the tree, with the goal of creating pure subsets
  • Pruning : The process of removing branches from the tree that do not provide any additional information or lead to overfitting.

Decision Tree -Geeksforgeeks

Attribute Selection Measures:

Construction of Decision Tree:  A tree can be “learned” by splitting the source set into subsets based on Attribute Selection Measures. Attribute selection measure (ASM) is a criterion used in decision tree algorithms to evaluate the usefulness of different attributes for splitting a dataset. The goal of ASM is to identify the attribute that will create the most homogeneous subsets of data after the split, thereby maximizing the information gain. This process is repeated on each derived subset in a recursive manner called recursive partitioning . The recursion is completed when the subset at a node all has the same value of the target variable, or when splitting no longer adds value to the predictions. The construction of a decision tree classifier does not require any domain knowledge or parameter setting and therefore is appropriate for exploratory knowledge discovery. Decision trees can handle high-dimensional data.

Entropy is the measure of the degree of randomness or uncertainty in the dataset. In the case of classifications, It measures the randomness based on the distribution of class labels in the dataset.

The entropy for a subset of the original dataset having K number of classes for the i th node can be defined as:

H_i = -\sum_{k \epsilon K}^{n} p(i,k)\log_2p(i,k)

  • S is the dataset sample.
  • k is the particular class from K classes

p(k) = \frac{1}{n}\sum{I(y=k)}

  • Here p(i,k) should not be equal to zero.

Important points related to Entropy:

  • The entropy is 0 when the dataset is completely homogeneous, meaning that each instance belongs to the same class. It is the lowest entropy indicating no uncertainty in the dataset sample.
  • when the dataset is equally divided between multiple classes, the entropy is at its maximum value. Therefore, entropy is highest when the distribution of class labels is even, indicating maximum uncertainty in the dataset sample.
  • Entropy is used to evaluate the quality of a split. The goal of entropy is to select the attribute that minimizes the entropy of the resulting subsets, by splitting the dataset into more homogeneous subsets with respect to the class labels.
  • The highest information gain attribute is chosen as the splitting criterion (i.e., the reduction in entropy after splitting on that attribute), and the process is repeated recursively to build the decision tree.

Gini Impurity or index:

Gini Impurity is a score that evaluates how accurate a split is among the classified groups. The Gini Impurity evaluates a score in the range between 0 and 1, where 0 is when all observations belong to one class, and 1 is a random distribution of the elements within classes. In this case, we want to have a Gini index score as low as possible. Gini Index is the evaluation metric we shall use to evaluate our Decision Tree Model.

\text{Gini Impurity} = 1- \sum{p_i^2}

  • p i is the proportion of elements in the set that belongs to the i th category.

Information Gain:

Information gain measures the reduction in entropy or variance that results from splitting a dataset based on a specific property. It is used in decision tree algorithms to determine the usefulness of a feature by partitioning the dataset into more homogeneous subsets with respect to the class labels or target variable. The higher the information gain, the more valuable the feature is in predicting the target variable. 

The information gain of an attribute A, with respect to a dataset S, is calculated as follows:

\text{Information Gain(H, A)}= H - \sum{\frac{|H_V|}{|H|}H_{v}}

  • A is the specific attribute or class label
  • |H| is the entropy of dataset sample S
  • |H V | is the number of instances in the subset S that have the value v for attribute A

Information gain measures the reduction in entropy or variance achieved by partitioning the dataset on attribute A. The attribute that maximizes information gain is chosen as the splitting criterion for building the decision tree.

Information gain is used in both classification and regression decision trees. In classification, entropy is used as a measure of impurity, while in regression, variance is used as a measure of impurity. The information gain calculation remains the same in both cases, except that entropy or variance is used instead of entropy in the formula.

How does the Decision Tree algorithm Work? The decision tree operates by analyzing the data set to predict its classification. It commences from the tree’s root node, where the algorithm views the value of the root attribute compared to the attribute of the record in the actual data set. Based on the comparison, it proceeds to follow the branch and move to the next node. 

The algorithm repeats this action for every subsequent node by comparing its attribute values with those of the sub-nodes and continuing the process further. It repeats until it reaches the leaf node of the tree. The complete mechanism can be better explained through the algorithm given below.

  • Step-1: Begin the tree with the root node, says S, which contains the complete dataset.
  • Step-2: Find the best attribute in the dataset using Attribute Selection Measure (ASM).
  • Step-3: Divide the S into subsets that contains possible values for the best attributes.
  • Step-4: Generate the decision tree node, which contains the best attribute.
  • Step-5: Recursively make new decision trees using the subsets of the dataset created in step -3. Continue this process until a stage is reached where you cannot further classify the nodes and called the final node as a leaf nodeClassification and Regression Tree algorithm.

Advantages of the Decision Tree:

  •  It is simple to understand as it follows the same process which a human follow while making any decision in real-life.
  •  It can be very useful for solving decision-related problems.
  •  It helps to think about all the possible outcomes for a problem.
  •  There is less requirement of data cleaning compared to other algorithms.

Disadvantages of the Decision Tree:

  •  The decision tree contains lots of layers, which makes it complex.
  •  It may have an overfitting issue, which can be resolved using the Random Forest algorithm.
  •  For more class labels, the computational complexity of the decision tree may increase.

What are appropriate problems for Decision tree learning?

Although a variety of decision tree learning methods have been developed with somewhat differing capabilities and requirements, decision tree learning is generally best suited to problems with the following characteristics:

1. Instances are represented by attribute-value pairs:

In the world of decision tree learning, we commonly use attribute-value pairs to represent instances. An instance is defined by a predetermined group of attributes, such as temperature, and its corresponding value, such as hot. Ideally, we want each attribute to have a finite set of distinct values, like hot, mild, or cold. This makes it easy to construct decision trees. However, more advanced versions of the algorithm can accommodate attributes with continuous numerical values, such as representing temperature with a numerical scale.

2. The target function has discrete output values:

The marked objective has distinct outcomes. The decision tree method is ordinarily employed for categorizing Boolean examples, such as yes or no. Decision tree approaches can be readily expanded for acquiring functions with beyond dual conceivable outcome values. A more substantial expansion lets us gain knowledge about aimed objectives with numeric outputs, although the practice of decision trees in this framework is comparatively rare.

3. Disjunctive descriptions may be required:

Decision trees naturally represent disjunctive expressions.

4.The training data may contain errors:

“Techniques of decision tree learning demonstrate high resilience towards discrepancies, including inconsistencies in categorization of sample cases and discrepancies in the feature details that characterize these cases.”

5. The training data may contain missing attribute values:

In certain cases, the input information designed for training might have absent characteristics. Employing decision tree approaches can still be possible despite experiencing unknown features in some training samples. For instance, when considering the level of humidity throughout the day, this information may only be accessible for a specific set of training specimens.

Practical issues in learning decision trees include:

  •  Determining how deeply to grow the decision tree,
  •  Handling continuous attributes,
  •  Choosing an appropriate attribute selection measure,
  •  Handling training data with missing attribute values,
  •  Handling attributes with differing costs, and
  •  Improving computational efficiency.

To build the Decision Tree, CART (Classification and Regression Tree) algorithm is used. It works by selecting the best split at each node based on metrics like Gini impurity or information Gain. In order to create a decision tree. Here are the basic steps of the CART algorithm:

  • The root node of the tree is supposed to be the complete training dataset.
  • Determine the impurity of the data based on each feature present in the dataset. Impurity can be measured using metrics like the Gini index or entropy for classification and Mean squared error, Mean Absolute Error, friedman_mse, or Half Poisson deviance for regression.
  • Then selects the feature that results in the highest information gain or impurity reduction when splitting the data.
  • For each possible value of the selected feature, split the dataset into two subsets (left and right), one where the feature takes on that value, and another where it does not. The split should be designed to create subsets that are as pure as possible with respect to the target variable.
  • Based on the target variable, determine the impurity of each resulting subset.
  • For each subset, repeat steps 2–5 iteratively until a stopping condition is met. For example, the stopping condition could be a maximum tree depth, a minimum number of samples required to make a split or a minimum impurity threshold.
  • Assign the majority class label for classification tasks or the mean value for regression tasks for each terminal node (leaf node) in the tree.

Classification and Regression Tree algorithm for Classification

Let the data available at node m be Q m and it has n m samples. and t m as the threshold for node m. then, The classification and regression tree algorithm for classification can be written as :

G(Q_m, t_m) = \frac{n_m^{Left}}{n_m}H(Q_m^{Left}(t_m)) +  \frac{n_m^{Right}}{n_m}H(Q_m^{Right}(t_m))

  • H is the measure of impurities of the left and right subsets at node m. it can be entropy or Gini impurity. 
  • n m is the number of instances in the left and right subsets at node m.

To select the parameter, we can write as:

t_m = \argmin_{t_m} H(Q_m, t_m)

Decision Tree Classifier

Classification and Regression Tree algorithm for Regression

Let the data available at node m be Q m and it has n m samples. and t m as the threshold for node m. then, The classification and regression tree algorithm for regression can be written as :

G(Q_m, t_m) = \frac{n_m^{Left}}{n_m}MSE(Q_m^{Left}(t_m)) +  \frac{n_m^{Right}}{n_m}MSE(Q_m^{Right}(t_m))

Decision Tree Regression

Strengths and Weaknesses of the Decision Tree Approach  

The strengths of decision tree methods are: 

  • Decision trees are able to generate understandable rules.
  • Decision trees perform classification without requiring much computation.
  • Decision trees are able to handle both continuous and categorical variables.
  • Decision trees provide a clear indication of which fields are most important for prediction or classification.
  • Ease of use: Decision trees are simple to use and don’t require a lot of technical expertise, making them accessible to a wide range of users.
  • Scalability: Decision trees can handle large datasets and can be easily parallelized to improve processing time.
  • Missing value tolerance: Decision trees are able to handle missing values in the data, making them a suitable choice for datasets with missing or incomplete data.
  • Handling non-linear relationships: Decision trees can handle non-linear relationships between variables, making them a suitable choice for complex datasets.
  • Ability to handle imbalanced data: Decision trees can handle imbalanced datasets, where one class is heavily represented compared to the others, by weighting the importance of individual nodes based on the class distribution.

The weaknesses of decision tree methods : 

  • Decision trees are less appropriate for estimation tasks where the goal is to predict the value of a continuous attribute.
  • Decision trees are prone to errors in classification problems with many classes and a relatively small number of training examples.
  • Decision trees can be computationally expensive to train. The process of growing a decision tree is computationally expensive. At each node, each candidate splitting field must be sorted before its best split can be found. In some algorithms, combinations of fields are used and a search must be made for optimal combining weights. Pruning algorithms can also be expensive since many candidate sub-trees must be formed and compared.
  • Decision trees are prone to overfitting the training data, particularly when the tree is very deep or complex. This can result in poor performance on new, unseen data.
  • Small variations in the training data can result in different decision trees being generated, which can be a problem when trying to compare or reproduce results.
  • Many decision tree algorithms do not handle missing data well, and require imputation or deletion of records with missing values.
  • The initial splitting criteria used in decision tree algorithms can lead to biased trees, particularly when dealing with unbalanced datasets or rare classes.
  • Decision trees are limited in their ability to represent complex relationships between variables, particularly when dealing with nonlinear or interactive effects.
  • Decision trees can be sensitive to the scaling of input features, particularly when using distance-based metrics or decision rules that rely on comparisons between values.

Implementation:

In the next post, we will be discussing the ID3 algorithm for the construction of the Decision tree given by J. R. Quinlan. 

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scikit-learn 1.4.1 Other versions

Please cite us if you use the software.

  • 1.10.1. Classification
  • 1.10.2. Regression
  • 1.10.3. Multi-output problems
  • 1.10.4. Complexity
  • 1.10.5. Tips on practical use
  • 1.10.6. Tree algorithms: ID3, C4.5, C5.0 and CART
  • 1.10.7.1. Classification criteria
  • 1.10.7.2. Regression criteria
  • 1.10.8. Missing Values Support
  • 1.10.9. Minimal Cost-Complexity Pruning

1.10. Decision Trees ¶

Decision Trees (DTs) are a non-parametric supervised learning method used for classification and regression . The goal is to create a model that predicts the value of a target variable by learning simple decision rules inferred from the data features. A tree can be seen as a piecewise constant approximation.

For instance, in the example below, decision trees learn from data to approximate a sine curve with a set of if-then-else decision rules. The deeper the tree, the more complex the decision rules and the fitter the model.

../_images/sphx_glr_plot_tree_regression_001.png

Some advantages of decision trees are:

Simple to understand and to interpret. Trees can be visualized.

Requires little data preparation. Other techniques often require data normalization, dummy variables need to be created and blank values to be removed. Some tree and algorithm combinations support missing values .

The cost of using the tree (i.e., predicting data) is logarithmic in the number of data points used to train the tree.

Able to handle both numerical and categorical data. However, the scikit-learn implementation does not support categorical variables for now. Other techniques are usually specialized in analyzing datasets that have only one type of variable. See algorithms for more information.

Able to handle multi-output problems.

Uses a white box model. If a given situation is observable in a model, the explanation for the condition is easily explained by boolean logic. By contrast, in a black box model (e.g., in an artificial neural network), results may be more difficult to interpret.

Possible to validate a model using statistical tests. That makes it possible to account for the reliability of the model.

Performs well even if its assumptions are somewhat violated by the true model from which the data were generated.

The disadvantages of decision trees include:

Decision-tree learners can create over-complex trees that do not generalize the data well. This is called overfitting. Mechanisms such as pruning, setting the minimum number of samples required at a leaf node or setting the maximum depth of the tree are necessary to avoid this problem.

Decision trees can be unstable because small variations in the data might result in a completely different tree being generated. This problem is mitigated by using decision trees within an ensemble.

Predictions of decision trees are neither smooth nor continuous, but piecewise constant approximations as seen in the above figure. Therefore, they are not good at extrapolation.

The problem of learning an optimal decision tree is known to be NP-complete under several aspects of optimality and even for simple concepts. Consequently, practical decision-tree learning algorithms are based on heuristic algorithms such as the greedy algorithm where locally optimal decisions are made at each node. Such algorithms cannot guarantee to return the globally optimal decision tree. This can be mitigated by training multiple trees in an ensemble learner, where the features and samples are randomly sampled with replacement.

There are concepts that are hard to learn because decision trees do not express them easily, such as XOR, parity or multiplexer problems.

Decision tree learners create biased trees if some classes dominate. It is therefore recommended to balance the dataset prior to fitting with the decision tree.

1.10.1. Classification ¶

DecisionTreeClassifier is a class capable of performing multi-class classification on a dataset.

As with other classifiers, DecisionTreeClassifier takes as input two arrays: an array X, sparse or dense, of shape (n_samples, n_features) holding the training samples, and an array Y of integer values, shape (n_samples,) , holding the class labels for the training samples:

After being fitted, the model can then be used to predict the class of samples:

In case that there are multiple classes with the same and highest probability, the classifier will predict the class with the lowest index amongst those classes.

As an alternative to outputting a specific class, the probability of each class can be predicted, which is the fraction of training samples of the class in a leaf:

DecisionTreeClassifier is capable of both binary (where the labels are [-1, 1]) classification and multiclass (where the labels are [0, …, K-1]) classification.

Using the Iris dataset, we can construct a tree as follows:

Once trained, you can plot the tree with the plot_tree function:

../_images/sphx_glr_plot_iris_dtc_002.png

Alternative ways to export trees Click for more details ¶

We can also export the tree in Graphviz format using the export_graphviz exporter. If you use the conda package manager, the graphviz binaries and the python package can be installed with conda install python-graphviz .

Alternatively binaries for graphviz can be downloaded from the graphviz project homepage, and the Python wrapper installed from pypi with pip install graphviz .

Below is an example graphviz export of the above tree trained on the entire iris dataset; the results are saved in an output file iris.pdf :

The export_graphviz exporter also supports a variety of aesthetic options, including coloring nodes by their class (or value for regression) and using explicit variable and class names if desired. Jupyter notebooks also render these plots inline automatically:

../_images/sphx_glr_plot_iris_dtc_001.png

Alternatively, the tree can also be exported in textual format with the function export_text . This method doesn’t require the installation of external libraries and is more compact:

1.10.2. Regression ¶

Decision trees can also be applied to regression problems, using the DecisionTreeRegressor class.

As in the classification setting, the fit method will take as argument arrays X and y, only that in this case y is expected to have floating point values instead of integer values:

1.10.3. Multi-output problems ¶

A multi-output problem is a supervised learning problem with several outputs to predict, that is when Y is a 2d array of shape (n_samples, n_outputs) .

When there is no correlation between the outputs, a very simple way to solve this kind of problem is to build n independent models, i.e. one for each output, and then to use those models to independently predict each one of the n outputs. However, because it is likely that the output values related to the same input are themselves correlated, an often better way is to build a single model capable of predicting simultaneously all n outputs. First, it requires lower training time since only a single estimator is built. Second, the generalization accuracy of the resulting estimator may often be increased.

With regard to decision trees, this strategy can readily be used to support multi-output problems. This requires the following changes:

Store n output values in leaves, instead of 1;

Use splitting criteria that compute the average reduction across all n outputs.

This module offers support for multi-output problems by implementing this strategy in both DecisionTreeClassifier and DecisionTreeRegressor . If a decision tree is fit on an output array Y of shape (n_samples, n_outputs) then the resulting estimator will:

Output n_output values upon predict ;

Output a list of n_output arrays of class probabilities upon predict_proba .

The use of multi-output trees for regression is demonstrated in Multi-output Decision Tree Regression . In this example, the input X is a single real value and the outputs Y are the sine and cosine of X.

../_images/sphx_glr_plot_tree_regression_multioutput_001.png

The use of multi-output trees for classification is demonstrated in Face completion with a multi-output estimators . In this example, the inputs X are the pixels of the upper half of faces and the outputs Y are the pixels of the lower half of those faces.

../_images/sphx_glr_plot_multioutput_face_completion_001.png

References Click for more details ¶

M. Dumont et al, Fast multi-class image annotation with random subwindows and multiple output randomized trees , International Conference on Computer Vision Theory and Applications 2009

1.10.4. Complexity ¶

In general, the run time cost to construct a balanced binary tree is \(O(n_{samples}n_{features}\log(n_{samples}))\) and query time \(O(\log(n_{samples}))\) . Although the tree construction algorithm attempts to generate balanced trees, they will not always be balanced. Assuming that the subtrees remain approximately balanced, the cost at each node consists of searching through \(O(n_{features})\) to find the feature that offers the largest reduction in the impurity criterion, e.g. log loss (which is equivalent to an information gain). This has a cost of \(O(n_{features}n_{samples}\log(n_{samples}))\) at each node, leading to a total cost over the entire trees (by summing the cost at each node) of \(O(n_{features}n_{samples}^{2}\log(n_{samples}))\) .

1.10.5. Tips on practical use ¶

Decision trees tend to overfit on data with a large number of features. Getting the right ratio of samples to number of features is important, since a tree with few samples in high dimensional space is very likely to overfit.

Consider performing dimensionality reduction ( PCA , ICA , or Feature selection ) beforehand to give your tree a better chance of finding features that are discriminative.

Understanding the decision tree structure will help in gaining more insights about how the decision tree makes predictions, which is important for understanding the important features in the data.

Visualize your tree as you are training by using the export function. Use max_depth=3 as an initial tree depth to get a feel for how the tree is fitting to your data, and then increase the depth.

Remember that the number of samples required to populate the tree doubles for each additional level the tree grows to. Use max_depth to control the size of the tree to prevent overfitting.

Use min_samples_split or min_samples_leaf to ensure that multiple samples inform every decision in the tree, by controlling which splits will be considered. A very small number will usually mean the tree will overfit, whereas a large number will prevent the tree from learning the data. Try min_samples_leaf=5 as an initial value. If the sample size varies greatly, a float number can be used as percentage in these two parameters. While min_samples_split can create arbitrarily small leaves, min_samples_leaf guarantees that each leaf has a minimum size, avoiding low-variance, over-fit leaf nodes in regression problems. For classification with few classes, min_samples_leaf=1 is often the best choice.

Note that min_samples_split considers samples directly and independent of sample_weight , if provided (e.g. a node with m weighted samples is still treated as having exactly m samples). Consider min_weight_fraction_leaf or min_impurity_decrease if accounting for sample weights is required at splits.

Balance your dataset before training to prevent the tree from being biased toward the classes that are dominant. Class balancing can be done by sampling an equal number of samples from each class, or preferably by normalizing the sum of the sample weights ( sample_weight ) for each class to the same value. Also note that weight-based pre-pruning criteria, such as min_weight_fraction_leaf , will then be less biased toward dominant classes than criteria that are not aware of the sample weights, like min_samples_leaf .

If the samples are weighted, it will be easier to optimize the tree structure using weight-based pre-pruning criterion such as min_weight_fraction_leaf , which ensure that leaf nodes contain at least a fraction of the overall sum of the sample weights.

All decision trees use np.float32 arrays internally. If training data is not in this format, a copy of the dataset will be made.

If the input matrix X is very sparse, it is recommended to convert to sparse csc_matrix before calling fit and sparse csr_matrix before calling predict. Training time can be orders of magnitude faster for a sparse matrix input compared to a dense matrix when features have zero values in most of the samples.

1.10.6. Tree algorithms: ID3, C4.5, C5.0 and CART ¶

What are all the various decision tree algorithms and how do they differ from each other? Which one is implemented in scikit-learn?

Various decision tree algorithms Click for more details ¶

ID3 (Iterative Dichotomiser 3) was developed in 1986 by Ross Quinlan. The algorithm creates a multiway tree, finding for each node (i.e. in a greedy manner) the categorical feature that will yield the largest information gain for categorical targets. Trees are grown to their maximum size and then a pruning step is usually applied to improve the ability of the tree to generalize to unseen data.

C4.5 is the successor to ID3 and removed the restriction that features must be categorical by dynamically defining a discrete attribute (based on numerical variables) that partitions the continuous attribute value into a discrete set of intervals. C4.5 converts the trained trees (i.e. the output of the ID3 algorithm) into sets of if-then rules. The accuracy of each rule is then evaluated to determine the order in which they should be applied. Pruning is done by removing a rule’s precondition if the accuracy of the rule improves without it.

C5.0 is Quinlan’s latest version release under a proprietary license. It uses less memory and builds smaller rulesets than C4.5 while being more accurate.

CART (Classification and Regression Trees) is very similar to C4.5, but it differs in that it supports numerical target variables (regression) and does not compute rule sets. CART constructs binary trees using the feature and threshold that yield the largest information gain at each node.

scikit-learn uses an optimized version of the CART algorithm; however, the scikit-learn implementation does not support categorical variables for now.

1.10.7. Mathematical formulation ¶

Given training vectors \(x_i \in R^n\) , i=1,…, l and a label vector \(y \in R^l\) , a decision tree recursively partitions the feature space such that the samples with the same labels or similar target values are grouped together.

Let the data at node \(m\) be represented by \(Q_m\) with \(n_m\) samples. For each candidate split \(\theta = (j, t_m)\) consisting of a feature \(j\) and threshold \(t_m\) , partition the data into \(Q_m^{left}(\theta)\) and \(Q_m^{right}(\theta)\) subsets

The quality of a candidate split of node \(m\) is then computed using an impurity function or loss function \(H()\) , the choice of which depends on the task being solved (classification or regression)

Select the parameters that minimises the impurity

Recurse for subsets \(Q_m^{left}(\theta^*)\) and \(Q_m^{right}(\theta^*)\) until the maximum allowable depth is reached, \(n_m < \min_{samples}\) or \(n_m = 1\) .

1.10.7.1. Classification criteria ¶

If a target is a classification outcome taking on values 0,1,…,K-1, for node \(m\) , let

be the proportion of class k observations in node \(m\) . If \(m\) is a terminal node, predict_proba for this region is set to \(p_{mk}\) . Common measures of impurity are the following.

Log Loss or Entropy:

Shannon entropy Click for more details ¶

The entropy criterion computes the Shannon entropy of the possible classes. It takes the class frequencies of the training data points that reached a given leaf \(m\) as their probability. Using the Shannon entropy as tree node splitting criterion is equivalent to minimizing the log loss (also known as cross-entropy and multinomial deviance) between the true labels \(y_i\) and the probabilistic predictions \(T_k(x_i)\) of the tree model \(T\) for class \(k\) .

To see this, first recall that the log loss of a tree model \(T\) computed on a dataset \(D\) is defined as follows:

where \(D\) is a training dataset of \(n\) pairs \((x_i, y_i)\) .

In a classification tree, the predicted class probabilities within leaf nodes are constant, that is: for all \((x_i, y_i) \in Q_m\) , one has: \(T_k(x_i) = p_{mk}\) for each class \(k\) .

This property makes it possible to rewrite \(\mathrm{LL}(D, T)\) as the sum of the Shannon entropies computed for each leaf of \(T\) weighted by the number of training data points that reached each leaf:

1.10.7.2. Regression criteria ¶

If the target is a continuous value, then for node \(m\) , common criteria to minimize as for determining locations for future splits are Mean Squared Error (MSE or L2 error), Poisson deviance as well as Mean Absolute Error (MAE or L1 error). MSE and Poisson deviance both set the predicted value of terminal nodes to the learned mean value \(\bar{y}_m\) of the node whereas the MAE sets the predicted value of terminal nodes to the median \(median(y)_m\) .

Mean Squared Error:

Half Poisson deviance:

Setting criterion="poisson" might be a good choice if your target is a count or a frequency (count per some unit). In any case, \(y >= 0\) is a necessary condition to use this criterion. Note that it fits much slower than the MSE criterion.

Mean Absolute Error:

Note that it fits much slower than the MSE criterion.

1.10.8. Missing Values Support ¶

DecisionTreeClassifier and DecisionTreeRegressor have built-in support for missing values when splitter='best' and criterion is 'gini' , 'entropy ’, or 'log_loss' , for classification or 'squared_error' , 'friedman_mse' , or 'poisson' for regression.

For each potential threshold on the non-missing data, the splitter will evaluate the split with all the missing values going to the left node or the right node.

Decisions are made as follows:

By default when predicting, the samples with missing values are classified with the class used in the split found during training:

If the criterion evaluation is the same for both nodes, then the tie for missing value at predict time is broken by going to the right node. The splitter also checks the split where all the missing values go to one child and non-missing values go to the other:

If no missing values are seen during training for a given feature, then during prediction missing values are mapped to the child with the most samples:

1.10.9. Minimal Cost-Complexity Pruning ¶

Minimal cost-complexity pruning is an algorithm used to prune a tree to avoid over-fitting, described in Chapter 3 of [BRE] . This algorithm is parameterized by \(\alpha\ge0\) known as the complexity parameter. The complexity parameter is used to define the cost-complexity measure, \(R_\alpha(T)\) of a given tree \(T\) :

where \(|\widetilde{T}|\) is the number of terminal nodes in \(T\) and \(R(T)\) is traditionally defined as the total misclassification rate of the terminal nodes. Alternatively, scikit-learn uses the total sample weighted impurity of the terminal nodes for \(R(T)\) . As shown above, the impurity of a node depends on the criterion. Minimal cost-complexity pruning finds the subtree of \(T\) that minimizes \(R_\alpha(T)\) .

The cost complexity measure of a single node is \(R_\alpha(t)=R(t)+\alpha\) . The branch, \(T_t\) , is defined to be a tree where node \(t\) is its root. In general, the impurity of a node is greater than the sum of impurities of its terminal nodes, \(R(T_t)<R(t)\) . However, the cost complexity measure of a node, \(t\) , and its branch, \(T_t\) , can be equal depending on \(\alpha\) . We define the effective \(\alpha\) of a node to be the value where they are equal, \(R_\alpha(T_t)=R_\alpha(t)\) or \(\alpha_{eff}(t)=\frac{R(t)-R(T_t)}{|T|-1}\) . A non-terminal node with the smallest value of \(\alpha_{eff}\) is the weakest link and will be pruned. This process stops when the pruned tree’s minimal \(\alpha_{eff}\) is greater than the ccp_alpha parameter.

L. Breiman, J. Friedman, R. Olshen, and C. Stone. Classification and Regression Trees. Wadsworth, Belmont, CA, 1984.

https://en.wikipedia.org/wiki/Decision_tree_learning

https://en.wikipedia.org/wiki/Predictive_analytics

J.R. Quinlan. C4. 5: programs for machine learning. Morgan Kaufmann, 1993.

T. Hastie, R. Tibshirani and J. Friedman. Elements of Statistical Learning, Springer, 2009.

Decision Tree Examples: Problems With Solutions

On this page:

  • What is decision tree? Definition.
  • 5 solved simple examples of decision tree diagram (for business, financial, personal, and project management needs).
  • Steps to creating a decision tree.

Let’s define it.

A decision tree is a diagram representation of possible solutions to a decision. It shows different outcomes from a set of decisions. The diagram is a widely used decision-making tool for analysis and planning.

The diagram starts with a box (or root), which branches off into several solutions. That’s way, it is called decision tree.

Decision trees are helpful for a variety of reasons. Not only they are easy-to-understand diagrams that support you ‘see’ your thoughts, but also because they provide a framework for estimating all possible alternatives.

In addition, decision trees help you manage the brainstorming process so you are able to consider the potential outcomes of a given choice.

Example 1: The Structure of Decision Tree

Let’s explain the decision tree structure with a simple example.

Each decision tree has 3 key parts:

  • a root node
  • leaf nodes, and

No matter what type is the decision tree, it starts with a specific decision. This decision is depicted with a box – the root node.

Root and leaf nodes hold questions or some criteria you have to answer. Commonly, nodes appear as a squares or circles. Squares depict decisions, while circles represent uncertain outcomes.

As you see in the example above, branches are lines that connect nodes, indicating the flow from question to answer.

Each node normally carries two or more nodes extending from it. If the leaf node results in the solution to the decision, the line is left empty.

How long should the decision trees be?

Now we are going to give more simple decision tree examples.

Example 2: Simple Personal Decision Tree Example

Let’s say you are wondering whether to quit your job or not. You have to consider some important points and questions. Here is an example of a decision tree in this case.

Download  the following decision tree in PDF

Now, let’s deep further and see decision tree examples in business and finance.

Example 3: Project Management Decision Tree Example

Imagine you are an IT project manager and you need to decide whether to start a particular project or not. You need to take into account important possible outcomes and consequences.

The decision tree examples, in this case, might look like the diagram below.

Download  the following decision tree diagram in PDF.

Don’t forget that in each decision tree, there is always a choice to do nothing!

Example 4: Financial Decision Tree Example

When it comes to the finance area, decision trees are a great tool to help you organize your thoughts and to consider different scenarios.

Let’s say you are wondering whether it’s worth to invest in new or old expensive machines. This is a classical financial situation. See the decision tree diagram example below.

Download it.

The above decision tree example representing the financial consequences of investing in old or new machines. It is quite obvious that buying new machines will bring us much more profit than buying old ones.

Need more decision tree diagram examples?

Example 5: Very Simple Desicion Tree Example

As we have the basis, let’ sum the steps for creating decision tree diagrams.

Steps for Creating Decision Trees:

1. Write the main decision.

Begin the decision tree by drawing a box (the root node) on 1 edge of your paper. Write the main decision on the box.

2. Draw the lines 

Draw line leading out from the box for each possible solution or action. Make at least 2, but better no more than 4 lines. Keep the lines as far apart as you can to enlarge the tree later.

3. Illustrate the outcomes of the solution at the end of each line.

A tip: It is a good practice here to draw a circle if the outcome is uncertain and to draw a square if the outcome leads to another problem.

4. Continue adding boxes and lines.

Continue until there are no more problems, and all lines have either uncertain outcome or blank ending.

5. Finish the tree.

The boxes that represent uncertain outcomes remain as they are.

A tip: A very good practice is to assign a score or a percentage chance of an outcome happening. For example, if you know for a certain situation there is 50% chance to happen, place that 50 % on the appropriate branch.

When you finish your decision tree, you’re ready to start analyzing the decisions and problems you face.

How to Create a Decision Tree?

In our IT world, it is a piece of cake to create decision trees. You have a plenty of different options. For example, you can use paid or free graphing software or free mind mapping software solutions such as:

  • Silverdecisions

The above tools are popular online chart creators that allow you to build almost all types of graphs and diagrams from scratch.

Of course, you also might want to use Microsoft products such as:

And finally, you can use a piece of paper and a pen or a writing board.

Advantages and Disadvantages of Decision Trees:

Decision trees are powerful tools that can support decision making in different areas such as business, finance, risk management, project management, healthcare and etc. The trees are also widely used as root cause analysis tools and solutions.

As any other thing in this world, the decision tree has some pros and cons you should know.

Advantages:

  • It is very easy to understand and interpret.
  • The data for decision trees require minimal preparation.
  • They force you to find many possible outcomes of a decision.
  • Can be easily used with many other decision tools.
  • Helps you to make the best decisions and best guesses on the basis of the information you have.
  • Helps you to see the difference between controlled and uncontrolled events.
  • Helps you estimate the likely results of one decision against another.

Disadvantages:

  • Sometimes decision trees can become too complex.
  • The outcomes of decisions may be based mainly on your expectations. This can lead to unrealistic decision trees.
  • The diagrams can narrow your focus to critical decisions and objectives.

Conclusion:

The above decision tree examples aim to make you understand better the whole idea behind. As you see, the decision tree is a kind of probability tree that helps you to make a personal or business decision.

In addition, they show you a balanced picture of the risks and opportunities related to each possible decision.

If you need more examples, our posts fishbone diagram examples and Venn diagram examples might be of help.

About The Author

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Decision Trees Tutoring & Homework Help

Decisions making becomes complex with uncertainties. Complex decisions with significant uncertainty can be better analyzed using decision trees. In addition to many decision options and significant uncertainty, decisions also include dependent uncertainties, and sequential decisions that add to the complexity of the real world. Drawing a decision tree helps capture all aspects and aids in decision making.

Graduate Tutor’s CPA, MBA, or CFA tutors provide live 1-o-1 online tutoring to help you understand, interpret and build decision trees. Decision trees provide a logical approach to decision-making.  Decision trees lay out the order in which decisions need to be made and possible outcomes (events) with their probability of occurrence at every point.

Top Tips to Draw a Decision Tree

The following decision tree facts help students draw decision trees:

  • Decision trees are linked in a sequence from left to right. 
  • Decisions, events, and end outcomes are represented by decision tree nodes and connected by branches.
  • Decision tree nodes can be of two types: Decision notes and chance nodes. 
  • The probability of an event occurring is estimated for every chance/event/outcome.
  • The complete specification of the alternatives that should be selected at all decision nodes in a decision tree is called a decision strategy
  • The expected value helps you make decisions by arriving at a number after evaluating all the possible outcomes for each decision alternative and the probability of each of those outcomes occurring.

When you have completed a decision tree well, you will have a diagram starting with a single “root” on the left side connected with various branches.  These branches represent various paths (decisions notes and chance nodes).  Probabilities of the various events occurring and payoffs for each branch/events are also represented in the decisions trees.

You can learn how to build decision trees with Graduate Tutor’s CPA/MBA/CFA tutors online. Decision trees homework help & tutoring online is convenient as it saves you valuable time traveling.  

Decision Tree Software & Tools

Decision trees can be drawn by hand or using Microsoft Excel or using various decision tree software/tools. Tools include Treeplan Edraw , Smartdraw , Zingtree , IBM’s SPSS ,  Precisiontree from Palisade ,  Lucidchart , etc. Graduate Tutor’s CPA/MBA/CFA tutors provide tutoring for decision tree software such as Treeplan or Palisade’s PrecisionTree online.  You can learn how to build Precisiontree with Graduate Tutor’s tutors online.  Precisiontree homework help & tutoring online is convenient and saves you valuable time.

Examples of Courses with Decision Trees

All MBAs encounter decision trees in their decision-making courses. Note the  difference between decision-making vs. decision science . You will see decision trees taught in operations management and decision-making courses. Some examples include DECISION 611G which is the Decision Models course at the Fuqua School of Business, STA 287 which is the Business Analytics course at Houston-MBA, DMD or Data models and Decisions at MIT’s EMBA program , 355 A/B: Decision Analysis at the University of Virginia, BS1628 which is Decision Analytics at the Imperial College Business School, Decision Analysis at Darden , etc. The Quantitative Methods module in Level 2 of the CFA program also expects you to know how to draw a decision tree. In fact, decision trees are also taught in other non-business and non-statistical/decision-making courses including finance, medical school, engineering, etc. We know because we have tutored decision trees in NYU’s Real Options, Acquisition Valuation, and Value Enhancement which is Professor Aswath Damodaran’s valuation course . We have also seen decision trees taught in NCCW 5030 Weill a Medical College at Cornell University as well as MGMT524 Management Science at the Embry-Riddle Aeronautical University. These courses generally do not go into decision sciences and artificial intelligence. We can tutor you on these decision tree topics such as Classification Trees, Entropy and Information Gain .

Decision Tree: Course Expectations

Graduate-level students are expected to know the following

  • How to draw a decision tree for a case study or question.
  • How to analyze the decision tree.
  • How to calculate expected values with or without decision tree tools/software.
  • How to calculate the expected values of perfect information.
  • How to use utilities in decision trees.
  • How to perform sensitivity analysis on your decision tree.

A decision criterion is a rule for making a decision. Expected value is one criterion for making a decision and the one used in a decision tree.  However, there are other decision-making criteria that can be used in decision making including:

  • Optimistic decision-making framework.
  • Conservative (pessimistic) decision-making framework.
  • Regret minimizing framework.
  • Equally likely frameworks.

While these may not be directly related to solving a decision tree problem, it helps to understand that there are different decision criteria that can be made for decision making.

Textbooks that Cover Decision Trees

Most operations and decision-making textbooks have at least one chapter on decision trees. A variety of textbooks can be used to teach decision trees. For example BS1628: Decision Analytics at Imperial College Business School recommends Powell and Baker’s Business Analytics: The Art of Modeling with Spreadsheets, published by Wiley. Other texts include Introduction to management science written by Taylor, B. W. (ISBN: 0131888099), Quantitative analysis for management written by Render, B., Stair, R. M., & Hanna, M. E. (ISBN: 0136036252), “Making Hard Decisions” by Clemen and Reilly, etc.

Decision Tree Case Studies

There are many wonderful case studies including:

  • CHANCE ENCOUNTERS II UVA-QA-0783 Rev. Jun. 15, 2012, Darden Business Publishing where David Fitzhugh, a respected movie-industry analyst, was hired to evaluate an unusual business idea—the purchasing of the sequel rights associated with a soon-to-be produced movie.
  • Evaluating Pharma Licensing Projects A very interesting article regarding licensing in the pharmaceutical industry can be found at Understanding why licensing works in biotech, and why deals are structured as they are, will help the entrepreneur negotiate. by Richard Mason, Nicos Savva & Stefan Scholtes.
  • Freemark Abbey Winery by William S. Krasker where Freemark Abbey must decide whether to harvest in view of the possibility of rain. Rain could damage the crop but delaying the harvest would be risky. On the other hand, rain could be beneficial and greatly increase the value of the resulting wine. This decision is further complicated by the fact that ripe Riesling grapes can be vinified in two ways, resulting in two different styles of wine. Their relative prices would depend on the uncertain preference of consumers two years later when the wine is bottled and sold. (Harvard Business School, Product #: 181027-PDF-ENG)

A Simple Decision Tree Example

Mrs. Barn is thinking of producing a new cooling widget for hunters during this hunting season. If the summer is extreme he would earn profits of $50,000 but if the summer is mild she would lose $30,000. Mrs. Barn estimates the probability of a mild summer is 30% without a weather forecast. Mrs. Barn has a choice to launch the business right away or hire a weather forecaster. The forecaster’s accuracy when he predicts a strong summer is 80%. However, the forecaster’s accuracy when he predicts a mild summer is only 60%. a) Should Mrs. Barn produce the new cooling widget? b) How much would Mrs. Barn pay for a weather forecast? EVPI?

Most decision tree questions require you to: 

  • Draw the decision tree.
  • Indicating the various decisions and events. 
  • Indicate the payoffs at the end of the final branches. 
  • Indicate the relevant probabilities of the events on the decision tree. 
  • Compute the optimal strategy, using expected monetary value as the decision criteria. 
  • Outline the optimal strategy (in words). 

Graduate Tutor’s CPA, MBA, or CFA tutors provide live 1-o-1 online tutoring to help you understand, interpret and build decision trees. In addition to decision tree homework help and tutoring, graduate tutor’s CPA/MBA/CFA tutors also provide online tutoring and homework help in a variety of other subjects. Other topics that our operations research and decision analysis tutors and  statistics tutors  can assist you with include:

  • Simple Linear Regression
  • Multiple Linear Regression
  • Time Series Forecasting
  • Simulation Using Crystal Ball
  • Simulation Using @Risk from Palisade
  • Financial Modeling using Microsoft Excel

Operations Research Tutoring

Our  operations research tutors  can assist you with tutoring to understand and draw decision trees.  Other operations topics we can assist you include  queuing theory and waiting lines ,  decision trees ,  linear programing using Microsoft Excel’s Solver ,  newsvendor models ,  batch processing ,  Littlefield simulation games . etc. Feel free to call or email if we can be of assistance with live one on one tutoring.

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Making Homework Central to Learning

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Three Reasons Teachers Continue to Grade Homework

"if i don't grade it, they won't do it.", "hard work should be rewarded.", "homework grades help students who test poorly.", what to do instead, practice 1. evaluate each assignment to determine whether to grade it., practice 2. tie homework to assessments., practice 3. focus on demonstration of learning, not task completion..

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U.S. teachers lead the world in their predilection for grading homework. In a study of educational practices in 50 countries, almost 70 percent of U.S. teachers said that they used homework assignments to calculate student grades, compared with 20 percent of teachers in Canada, 14 percent in Japan, and 9 percent in Singapore (Baker &amp; LeTendre, 2005). It's worthwhile to ask whether the hours spent scoring student homework and calculating it into grades pay off. This study said no; in fact, it found a negative correlation between grading homework and increased achievement:Not only did we fail to find any positive relationships, the overall correlations between national average student achievement and national averages in the frequency, total amount, and percentage of teachers who used homework in grading are all negative ! (pp. 127–128)
Even though teachers at Glenn Westlake Middle School in Lombard, Illinois, no longer count homework in students' grades, students still understand that homework must be done, teachers still document which work has been completed and when, and teachers still give learners feedback about their homework. Explaining this fact to parents was a big part of the transition. Glenn Westlake's principal, Phil Wieczorek, met with parent groups several times:The parents had a lot of misconceptions. We had to explain to them this did not mean there was not going to be homework. Homework would still be looked at, kept track of, and given feedback. It just wasn't going to be averaged into the student's grade.
Schools that still wish to grade some homework should separate homework into formative and summative assessments. Formative assessments, such as practice with math problems, spelling, or vocabulary, should not be factored into the overall course grade. Summative assessments, such as research papers or portfolios of student work, may be. Many district policies outline these differences in their grading and homework policies, such as this guideline from the Rockwood School District in Eureka, Missouri:Homework is an important part of teaching, learning, and parent involvement in the Rockwood School District. Student work should always receive feedback to further student learning. Teachers will exclude homework from the course grade if it was assigned for pre-assessment or early learning guided practice. Homework assigned as a summative assessment may be included in the course grade based on curriculum guidelines.
The easiest way to tie homework to assessments in students' minds is to allow them to use homework assignments and notes when taking a test. Another method is to correlate the amount of homework completed with test scores. One teacher does this by writing two numbers at the top of each test or quiz—the student's test score and the student's number of missing homework assignments. This not only helps the students see the connection, but also shows the teacher which students are not benefiting from a specific homework task and which students may know the content so well that they don't need to do homework. Patricia Scriffiny (2008), a teacher at Montrose High School in Colorado, makes the connection explicit:When I assign homework, I discuss with my students where and how it applies to their assessments… Some students don't do all of the homework that I assign, but they know that they are accountable for mastering the standard connected to it. (p. 72)

Figure 1. Sample Monthly Feedback Form for English 11

Making Homework Central to Learning- table

Baker, D. P., &amp; LeTendre, G. K. (2005). National differences, global similarities: World culture and the future of schooling . Stanford, CA: Stanford University Press.

Cushman, K. (2010). Fires in the mind: What kids can tell us about motivation and mastery . San Francisco: Jossey-Bass.

O'Connor, K. (2009). How to grade for learning K–12 . (3rd ed.). Thousand Oaks, CA: Corwin.

O'Donnell, H. (2010, October 8). Grading for learning: Dealing with the student who "won't work" (Revisited) [blog post]. Retrieved from The Thoughtful Teacher at http://repairman.wordpress.com/2010/10/08/grading-for-learning-dealing-with-the-student-who-wont-work-revisited .

Pink, D. (2009). Drive: The surprising truth about what motivates us . New York: Riverhead Books.

Scriffiny, P. L. (2008). Seven reasons for standards-based grading. Educational Leadership, 66 (2), 70–74.

Vatterott, C. (2009). Rethinking homework: Best practices that support diverse needs . Alexandria, VA: ASCD.

decision tree homework

Cathy Vatterott is professor emeritus of education at the University of Missouri–St. Louis. Referred to as the "homework lady," Vatterott has been researching, writing, and speaking about K–12 homework for more than 20 years.

She frequently presents at a variety of state and national educational conferences and also serves as a consultant and workshop presenter for K–12 schools on a variety of topics.

She serves on the Parents magazine advisory board and is author of two ASCD books: Rethinking Homework: Best Practices That Support Diverse Needs, 2nd edition (2018) and Rethinking Grading: Meaningful Assessment for Standards Based Learning   (2015).

ASCD is a community dedicated to educators' professional growth and well-being.

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COMMENTS

  1. PDF Machine Learning Homework 1 : Decision Trees (due Noon Jan 15)

    a dataset with N samples and M binary features. Vary M and N to plot the time taken for: 1) learning the tree, 2) predicting for test data. How do these results compare with theoretical time complexity for decision tree creation and prediction. [2 marks] Some useful references for the homework: 1.Scikit-learn page on decision trees

  2. PDF CSE446 Machine Learning, Winter 2016: Homework 1

    the homework document as a template, putting your solutions inline. We will only accept answers in .pdf format. 1 Probability Review [30 points] ... Build a decision tree for classifying whether a person has a college degree by greedily choosing threshold splits that maximize information gain. What is the depth of your tree and the information ...

  3. puttak/10601-18Fall-Homework

    This repository contains the homework solutions for CMU course Introduction to Machine Learning (10601 2018 Fall). All coding parts are completed in Python3. Homework 1: Background Material; Homework 2: Decision Trees; Homework 3: KNN, Perceptron, Linear Regression; Homework 4: Logistic Regression; Homework 5: Neural Networks; Homework 6 ...

  4. CS 4341 D01

    The deadline for this OPTIONAL homework is Friday, April 27, 2001 at 5 pm. Please turn in a HARD COPY of your homework assignment by this deadline. Homework Problems: This homework consists of six problems: Problem 1: Decision Trees (20 points) Problem 2: Genetic Algorithms (20 points)

  5. PDF CS 446 Machine Learning Fall 2016 SEP 8, 2016 Decision Trees

    Decision tree is a hierarchical data structure that represents data through a di-vide and conquer strategy. In this class we discuss decision trees with categorical labels, but non-parametric classi cation and regression can be performed with decision trees as well. In classi cation, the goal is to learn a decision tree that represents the training

  6. PDF Decision Trees

    Learn how to build and evaluate decision trees for classification and regression problems in this lecture from the University of Pennsylvania. The lecture covers the concepts of entropy, information gain, pruning, and ensemble methods. The lecture is in PDF format and includes examples and exercises.

  7. PDF 10-715 Advanced Introduction to Machine Learning: Homework 1 Decision

    10-715 Advanced Introduction to Machine Learning: Homework 1 Decision Trees and Perceptron Algorithm Released: Wednesday, August 29, 2018 Due: 11:59 p.m. Wednesday, September 5, 2018 Instructions Late homework policy: Homework is worth full credit if submitted before the due date, half credit during the next 48 hours, and zero credit after that.

  8. PDF 10-701 Machine Learning, Spring 2011: Homework 1

    When we construct a decision tree, the next attribute to split is the one with maximum mutual information (a.k.a. information gain), which is deflned in terms of entropy. In this problem, we will ... Report in your homework 1) for the fully grown tree (without post-pruning): the tree size (i.e., the

  9. PDF Machine Learning: Decision Trees

    A Decision Tree • A decision tree has 2 kinds of nodes 1. Each leaf node has a class label, determined by majority vote of training examples reaching that leaf. 2. Each internal node is a question on features. It branches out according to the answers.

  10. PDF Decision Tree Learning

    Decision Tree Learning CS4780/5780 - Machine Learning Fall 2011 Thorsten Joachims Cornell University Reading: Mitchell Sections 2.1, 2.2, 2.5-2.5.2, 2.7, Chapter 3 . Outline • Hypothesis space ... A+Homework complete yes yes clear no yes / +1 complete no yes clear no yes / +1 ...

  11. PDF Homework 1

    which feature to split on at a node in a decision tree. Gain ME(S;a) = MinError(S) X v2values jS vj jSj MinError(S v) Using the version of information gain modi ed to use MinError, compute the full decision tree using the Balloons dataset.2 You can type your decision tree as a series of if statements using Latex's verbatim environment, like ...

  12. PDF CSC 411: Lecture 06

    UofT CSC 411: 06-Decision Trees 28 / 40. Decision Tree Construction Algorithm. Simple, greedy, recursive approach, builds up tree node-by-node 1.pick an attribute to split at a non-terminal node 2.split examples into groups based on attribute value 3.for each group: Iif no examples { return majority from parent.

  13. How To Implement The Decision Tree Algorithm From Scratch In Python

    root = get_split (train) split (root, max_depth, min_size, 1) return root. In this section the "split" function returns "none",Then how the changes made in "split" function are reflecting in the variable "root". To know what values are stored in "root" variable, I run the code as below. # Build a decision tree.

  14. PDF Decision Trees

    Foundations of Machine Learning Homework 5 Decision Trees Exercise 1 : Decision Trees Construct by hand decision trees corresponding to each of the following Boolean formulas. The examples (x,c) ∈ Dconsist of a feature vector x where each component corresponds to one of the Boolean variables

  15. PDF 1 A Simplified Decision Tree

    Canvas. Please check Piazza for updates about the homework. 1 A Simplified Decision Tree You are to implement a decision-tree learner for classification. To simplify your work, this will not be a general purpose decision tree. Instead, your program can assume that •each item has two continuous features x 2R2

  16. Decision Tree

    A decision tree is a flowchart-like tree structure where each internal node denotes the feature, branches denote the rules and the leaf nodes denote the result of the algorithm. It is a versatile supervised machine-learning algorithm, which is used for both classification and regression problems. It is one of the very powerful algorithms.

  17. 1.10. Decision Trees

    Examples: Decision Tree Regression. 1.10.3. Multi-output problems¶. A multi-output problem is a supervised learning problem with several outputs to predict, that is when Y is a 2d array of shape (n_samples, n_outputs).. When there is no correlation between the outputs, a very simple way to solve this kind of problem is to build n independent models, i.e. one for each output, and then to use ...

  18. PDF Exercise 12: Decision Trees, Nearest Neighbor Classifier ...

    Decision Choose arbitrary, here gender. There remains only a single non-pure branch, female, which can be split using time. The final tree is given by area low urban gender high male time high 1-2 low >7 female rural (b)Apply the decision tree to the following drivers: ID time gender area A 1-2 f rural B 2-7 m urban C 1-2 f urban

  19. CIS 520 Homework 1 KNN, Decision Trees

    Recognize tradeoffs in performance between K-NNs and decision trees; Recognize and address overfitting in nonparametric models; Deliverables. This homework can be completed individually or in groups of 2. You need to make one submission per group. Make sure to add your team member's name on Gradescope when submitting the homework's written and ...

  20. Decision Tree Examples: Problems With Solutions

    Example 1: The Structure of Decision Tree. Let's explain the decision tree structure with a simple example. Each decision tree has 3 key parts: a root node. leaf nodes, and. branches. No matter what type is the decision tree, it starts with a specific decision. This decision is depicted with a box - the root node.

  21. Solved In a random forest, attributes/features for every

    Question: In a random forest, attributes/features for every individual decision tree are picked In a random forest, attributes / features for every individual decision tree are picked Here's the best way to solve it.

  22. Decision Trees Homework help & Tutoring from MBA Tutors

    Drawing a decision tree helps capture all aspects and aids in decision making. Graduate Tutor's CPA, MBA, or CFA tutors provide live 1-o-1 online tutoring to help you understand, interpret and build decision trees. Decision trees provide a logical approach to decision-making. Decision trees lay out the order in which decisions need to be made ...

  23. Making Homework Central to Learning

    U.S. teachers lead the world in their predilection for grading homework. In a study of educational practices in 50 countries, almost 70 percent of U.S. teachers said that they used homework assignments to calculate student grades, compared with 20 percent of teachers in Canada, 14 percent in Japan, and 9 percent in Singapore (Baker & LeTendre, 2005).