 ## data transformations - ml.net | microsoft docs

The transformations in this guide return classes that implement the IEstimator interface. Data transformations can be chained together. Each transformation both expects and produces data of specific types and formats, which are specified in the linked reference documentation.

Some data transformations require training data to calculate their parameters. For example: the NormalizeMeanVariance transformer calculates the mean and variance of the training data during the Fit() operation, and uses those parameters in the Transform() operation.

Other data transformations don't require training data. For example: the ConvertToGrayscale transformation can perform the Transform() operation without having seen any training data during the Fit() operation. ## how to report classifier performance with confidence intervals

After the final model has been prepared on the training data, it can be used to make predictions on the validation dataset. These predictions are used to calculate a classification accuracy or classification error.

Where error is the classification error, const is a constant value that defines the chosen probability, sqrt is the square root function, and n is the number of observations (rows) used to evaluate the model. Technically, this is called the Wilson score interval.

This is based on some statistics of sampling theory that takes calculating the error of a classifier as a binomial distribution, that we have sufficient observations to approximate a normal distribution for the binomial distribution, and that via the central limit theorem that the more observations we classify, the closer we will get to the true, but unknown, model skill.

Often standard deviation of CV score is used to capture model skill variance, perhaps that is generally sufficient and we can leave confidence intervals for presenting the final model or specific predictions?

Ah @Simone, by the way, if n is equal to the number of all observations that is a type of cross-validation that is called LOOCV (Leave-one-out cross-validation) and uses a single observation from the original sample as the validation data, and the remaining observations as the training data.

Mach Learn. 2018; 107(12): 18951922. Published online 2018 May 9. doi: 10.1007/s10994-018-5714-4. PMCID: PMC6191021, PMID: 30393425. Bootstrapping the out-of-sample predictions for efficient and accurate cross-validation. Ioannis Tsamardinos, lissavet Greasidou (corresponding author), and Giorgos Borboudakis.

Well @Simone, from the point of view of a developer if you take a look at the scikit-learn documentation and go over the section 3.1.1. Computing cross-validated metrics (https://scikit-learn.org/stable/modules/cross_validation.html#computing-cross-validated-metrics) you will see that the 95% confidence interval of the score estimate is reported as Jason states in this post.

This leads to the fundamental problem that accuracy or classification error itself is often mediocre to useless metric because data sets usually are imbalanced. And hence the confidence on that error is just as useless.

I found this post for a different reason as I wanted to find if anyone else does what i do, namely provide metrics grouped by class probability. What is the precision if the model has 0.9 class probability vs 0.6 for example. That can be very useful information for end users because the metric will often vary greatly based on class probability.

Thomas, I think Ive done what you described. I wrote a function to calculate a hand full of different performance metrics at different probability cutoffs and had it stored in a data frame. This helped me choose a probability cutoff that balanced the needs of the business. I can share the code of its what your looking for.

The classifier is assigning labels as expected. The problem I am facing is that the classifier is also assigning labels or group customer code to the customers although the customer name does not match closely with the training data. It is doing the best possible match. It is problem for me because I need to manually ungroup these customers. Can you suggest how to overcome this problem? Is it possible to know classifier correct probability for each predicted label? If yes, then I can ignore the once with low probability.

Hi Jason, I am not sure if anyone else brought this up but Ive found one issue here. The confidence interval based measure you suggested is not the Wilson score interval. according to the Wikipedia page(which is cited in that link). Its actually Normal approximation interval which is above Wilson score paragraph. Correct me if I am wrong.

With 150 examples I decide to use a 100 repeated 5-fold Cross Validation to understand the behavior of my classifier. At this point I have 1005 results and I can use the mean and std dev of the error rates to estimate the variance of the model skills:

What are the options one has for reporting on final model skill with a range for uncertainty in each case? Should one have still held out a number of datapoints for validation+binomial confidence interval? Is it too late to use the bootstrap confidence intervals as the final model was trained?

Thanks Jason. I found your other post https://machinelearningmastery.com/difference-test-validation-datasets/ very helpful. Can I confirm that the above procedure of reporting classifier performance with confidence intervals is relevant for the final trained model? If that is so, it seems that the validation dataset mentioned should be called test set to align with the definitions of the linked post?

I am running a classifier with a training set of 41 and a validation set of 14 (55 total observations). I rerun this 50 times with different random slices of the data as training and test. Obviously I cannot make confidence intervals with this small validation set.

Thank you for the quick reply and apologies for my late response. I am dealing with social science data and the validation set is rather limited. I am worried about assertaining confidence intervals for a limited validation sample.

A friend of mine came up with a solution in which I keep all the accuracy outputs in a vector and plot them like a histogram (I cant seem to paste one into this reply window but can send it over if necessary by email).

Thank you for the quick reply. The method I am currently using is subsampling. Randomly selecting different observations for the training set and the validation set 50 times, and collecting the accuracy scores to make a distribution of accuracy scores. I believe this is called subsampling. But I am happy to use bootstrapping instead.

Just to clarify, I am using the bootstrap on the data for partitioning the training and validation set correct? This means that an observation in the training set can also end up in the validation set.

But typically when I check the mean of the resample, mean(model\$resample\$Accuracy), the mean is lower than the k=5 accuracy (typically 0.65). Is there a reason for this? I would have thought that the mean accuracy of the best tune resamples would equal the model accuracy in the results.

Probably using on the standard deviation of the error score, from the mean, e.g. +/- 2 or 3 standard deviations will cover the expected behaviour of the system: https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule

I have a neural network(MLP) for binary classification with a logistic output between 0 and 1. With each run, I have to adjust my threshold on test set for minimizing the misclassifications. My question is to present my results, should I run it multiple times, adjust threshold each time and then take the average of other metrics eg F1 score or I dont optimize for the threshold at all?

I would take the test as an evaluation of the system that includes the model and automatic threshold adjusting procedure. In that case, averaging the results of the whole system is reasonable, as long as you clearly state that is what you are doing.

Lets say I have run a repeated (10) 10-cross validation experiment with predictions implemented via a Markov chain model. As a measure of robustness, I want to compute the SD of the AUC across the runs/folders for the test set.

Wilson score is different, the one youre describing is Normal approximation interval according to Wikipedia https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval#Normal_approximation_interval

I have come across a few posts/slides around CLT which state that in order for the sample proportion (or mean or error rate) of a binomial distribution to approximate to normal distribution (to compute confidence interval), it should follow below 2 conditions: np > 10 n(1-p) > 10 ex ref. http://homepages.math.uic.edu/~bpower6/stat101/Sampling%20Distributions.pdf

How to compute CI with cross-validation? Do we use the CI on mean results? the n value is the test portion or the sum of then? Tne formula used in https://scikit-learn.org/stable/modules/cross_validation.html#computing-cross-validated-metrics , by multiplying std * 2 was unclear.

Perhaps a unreliable estimate, there is a reference or paper/book where this formula came from? cross-val will give me [acc1, acc2,acc3, acc30] list with accuracys and I just comput the mean +/- 2 * std to represent the C.I. (CI = 2* std). What about n value, or 1.96 z value?

Step one Split train/validation 80/20 and use the train (80%) into cross-validation to get perfoemance metrics to show as means and std. Step two Train a final model with bootstrap on 20% left and comput performances with confifence intervals. ## statistics - roc plot and area under the curve (auc)

You may want to move this threshold. False positives (legitimate emails erroneously predicted as spam) are likely to cause more harm than false negatives (spam emails that are not identified as spam), as we might miss an important email, while it is easy to delete a spam message. In this case, we could require a higher threshold (probability) that a message is spam before we move it into a spam folder. ## how to measure variance in a classification dataset? - data science stack exchange

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I have a dataset that contains 20 predictor variables (both categorical and numeric) and one target variable (2 classes - Good and Bad). But, there are only 23 observations in the dataset. While I wait to receive significantly more observations, what tests / models can I perform on the available dataset to understand the variance between the good and bad cases, and to understand the variance within the cases classified as 'good'?

I would start with logistic regression, and get a measure of importance for each of your 20 predictor variables. It's a little tough to understand what you mean by 'variance' in your two classes of good and bad. If you mean variance in your predictor variables that lead to your response, you can calculate the variance of a linear combination of predictor variables (in your logistic regression model) using an appropriate theorem from Wikipedia

By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. ## how to reduce variance in a final machine learning model

This means that each time you fit a model, you get a slightly different set of parameters that in turn will make slightly different predictions. Sometimes more and sometimes less skillful than what you expected.

For example, consider a linear regression model with three coefficients [b0, b1, b2]. We could fit a group of linear regression models and calculate a final b0 as the average of b0 parameters in each model, and repeat this process for b1 and b2.

Elie is right. Early stopping should be done on the validation set, which is separate from the hold out (test) set. Unless you dont care to estimate generalization performance because your goal is to deploy the model, not evaluate it, then you may choose not to have a hold out set.

Would you like to give us an easy example to explain your whole ideas explicitly? Because we need to make several decisions during training the final model and making predictions in the real world. I want to learn how you make decisions when you do a real project.

I read your blog for the first time and I guess I became a fan of you. In the section Ensemble Parameters from Final Models when using neural networks, how can I use this approach. Do I need to train multiple neural networks having same size and average their weight values? Will it improve the performance in terms of generalization?

For lack of a better phrase, this seems a little fishy to me. While I certainly am on board that this averaging method makes a lot of sense for straightforward regression, it seems like this would not work for neural networks.

I guess Im a little worried that different trained models (even with the same architecture) could have learned vastly different representations of the input data in the latent space. Even in a simple fully connected feed forward network, wouldnt it be possible for the nodes in one network to be a permuted version of nodes in the other? Then averaging these weight values would not make sense?

Variance in this blog is about a single model trained on a fixed dataset (final dataset). Training the model with the same dataset somewhat yields non-deterministic parameters (different estimations of the unknown target function), hence when used to create predictions, predictions from different models (trained on same dataset) are quantitatively different, measured by the variance of the model. In other words, this blog post is about the stability of training a final model that is less prone to randomness in data/model architecture.

In your other blog post: gentle intro to bias-variance tradeoff, variance here describes the amount that the target function will change if different training data was used. Different learned target functions will yield different predictions, and this is measured by the variance.

The two variances are somewhat a measurement of differences in predictions (by the different approximations of the target function). However, in this post, models are trained on the same dataset, whereas the bias-variance tradeoff blog post describes training over different datasets.

My question is how does the concept of overfitting fit within these two definitions of variance? Do they both apply? In the case of training on different train datasets yielding different approximations of the target functions, I could describe overfitting here as the model picking up noise and overfitting to the specific data points. In the case of this post, can I describe it the same way?

Not sure that overfitting fits into the discussion, it feels like an orthogonal idea, e.g. fitting statistical noise in training data but still another approximation of the target function just a poor approximation.

Why not choose the trained version that performs best (has the lowest error on the test data set) as the final model for using it to perform the predictions? I think that it should deliver better prediction results than using the average of the predictions of the models in the ensamble.

If we want to reduce the amount of variance in a prediction, we must add bias. I dont understand why is this statement true. Making improvements on the model could reduce both variance and bias, isnt it?

I started by reading your previous post How to Train a Final Machine Learning Model and everything was very clear to me: you use e.g. cross-validation to come up with a specific type of model (e.g. linear regression/k-nn etc.), hyper parameters, features (e.g. specific data preparation or simply which are the best features to be used, etc.) to select the final model that gives you the lowest error/highest accuracy. You also said that we should fit this model with all our dataset and we should not be worried that the performance of the model trained on all of the data is different with respect to our previous evaluation during cross-validation because If well designed, the performance measures you calculate using train-test or k-fold cross validation suitably describe how well the finalized model trained on all available historical data will perform in general.

In this post you are talking about a problem related to such a final model, right? I am not sure about it because as I understood the final model is trained on the entire dataset (the original one) but then in the post you wrote that A problem with most final models is that they suffer variance in their predictions. This means that each time you fit a model, you get a slightly different set of parameters that in turn will make slightly different predictions.

1. At the beginning I though that you would fit the final model only once with the entire dataset but here you are referring to each time meaning that you are fitting it several times, and if it is the case with what? Always with the entire dataset? Or parts of it?

2. The previous answer would be important also for this question I have related to the section Measure Variance in the Final Model. I agree with you about the two possible sources of variance but still I cannot understand why you are evaluating it in the final model: you could have done it in principle while doing the cross-validation to find the final model, because in this case you are dividing your entire dataset into folds where you can compute the variance/standard deviation of the model skills, already. In this way you would have selected, perhaps, the model with a low variance/standard deviation of its skills.

3. Finally in a previous answer you gave, you said that the overfitting concept was not properly related to this post, but when you said that one of the source of variance of the final model is the noise in the training data, dont are you referring exactly at the concept of overfitting, since the model is fitting also the noise and thus the final outputs would be different? Why and how overfitting is not related to all this?

Thanks for this great article. A short question (maybe somewhat off topic with respect to this article): In the beginning you mention that the final model is trained on the whole dataset (i.ex. training and validation set). Now Iam wondering how one would train the final model with keras ReduceOnPlateau callback when there is no validation set left. Normally I would monitor the validation loss and reduce the learning rate depending on that. Should I monitor the training loss instead during final model training?

Thus as you increase the sample size n->n+1 yes the variance should go down but the squared mean error value should increase in the sample space. If the avg of estimate is more accurate then wouldnt that imply the distance between avg of the estimate and the observed value decreasing and thus the L2 mean norm distance also going down implying reduced bias?

And I keep the random seed constant for (a) and measure the variance of (a) due to training data noise (by repeating the evaluation of the algorithm on different samples of training data, but with a constant seed).

And then for (b), I keep the random seed constant (for all models within the bagged ensemble) and measure the variance of (b) due to training data noise (again, by repeating the evaluation of the system on different samples of training data, but with a constant seed)

Also, how would you keep the random seed constant for all the models within the bagged ensemble (for example, for a bagged ensemble of neural networks as shown in your How to Create a Bagging Ensemble of Deep Learning Models in Keras tutorial (linked below)).

I agree with you that navigating the bias-variance tradeoff for a final model is to think in samples, not in terms of single models. And in your another posted blog Embrace Randomness in Machine Learning, you listed 5 Randomness in machine learning, in which only the 3rd one is in the algorithm, others are all from data. But I seldom see you give an example to reduce variance by repeating the evaluation of the algorithm with different data order or to say using varied random seed. Do you think it is more important than other randomness from data to reduce variance?

On 2. Ensemble Parameters from Final Models. I am not sure the linear regression example works here. But maybe I have an error in my logic. A linear regression model usually has a convex (ie unique) analytical solution. So if we estimated a linear regression m times on the exact same dataset, we would get the exact same regression coefficients, every single time. That is, we have no inherent randomness in our linear regression model that we can exploit via this sort of ensembling.

I guess the base assumption is that there is a source of variance to begin with, such as a stochastic learning algorithm. Agreed, linear regression does not suffer this problem and would require variance to be sourced elsewhere, such as random samples of the training data. ## understanding naive bayes classifier from scratch

Naive Bayes classifier belongs to a family of probabilistic classifiers that are built upon the Bayes theorem. In naive Bayes classifiers, the number of model parameters increases linearly with the number of features. Moreover, its trained by evaluating a closed-form expression, i.e., a mathematical expression that can be evaluated using finite steps and has one definite solution. This means that naive Bayes classifiers train in linear time compared to the quadratic or cubic time of other iterative approximation based approaches. These two factors make naive Bayes classifiers highly scalable. In this article, well go through the Bayes theorem, make some assumptions and then implement a naive Bayes classifier from scratch.

Bayes theorem is one of the most important formulas in all probability. Its an essential tool for scientific discovery and for creating AI systems; it has also been used to find century-old treasures. It is formulated as

Steve is very shy and withdrawn, invariably helpful but with very little interest in people or in the world of reality. A meek and tidy soul, he has a need for order and structure and a passion for detail.

Given the above description, do you think Steve is more likely to be a librarian or a farmer? The majority of people immediately conclude that Steve must be a librarian since he fits their idea of a librarian. However, as we see the whole picture,we see that there are twenty times as many farmers as librarians (in the United States). Most people arent aware of this statistic and hence cant make an accurate prediction, and thats okay. Also, thats beside the point of this article. However, if you want to learn why we act irrationally and make assumptions like this, I wholeheartedly recommend reading Kahnemans Thinking Fast and Slow.

Back to Bayes theorem, to model this puzzle more accurately, lets start by creating create a representative sample of 420 people, 20 librarians and 400 farmers. And lets say your intuition is that roughly 50% of librarians would fit that description, and 10% of farmers would. So the probability of a random person fitting this description being a librarian becomes 0.2 (10/50). So even if you think a librarian is five times as likely as a farmer to fit this description, thats not enough to overcome the fact that there are way more farmers.

This new evidence doesnt necessarily overrule your past belief but rather updates it. And this is precisely what the Bayes theorem models. The first relevant number is the probability that your beliefs hold true before considering the new evidence. Using the ratio of farmers to librarians in the general population, this came out to be 1/5 in our example. This is known as the prior P(H). In addition to this, we need to consider the proportion of librarians that fit this description; the probability we would see the evidence given that the hypothesis is true, P(E|H). In the context of the Bayes theorem, this value is called the likelihood. This represents a limited view of your initial hypothesis.

Similarly, we need to consider how much of the farmers side of the sample space make up the evidence; the probability of seeing the evidence given that your beliefs dont hold true P(E|H). Using these notations, the accurate probability of your beliefs being right given the evidence, P(H|E), also called the posterior probability, can be formulated as:

This is the original Bayes theorem that we started with. I hope this illustrated the core point of the Bayes theorem representing a changing belief system, not just a bunch of independent probabilities.

The naive Bayes classifier is called naive because it makes the assumption that all features are independent of each other. Another assumption that it makes is that the values of the features are normally (Gaussian) distributed. Using these assumptions, the original Bayes theorem is modified and transformed into a simpler form that is relevant for solving learning problems. We start with

Create a function that calculates the prior probability, P(H), mean and variance of each class. The mean and variance are later used to calculate the likelihood, P(E|H), using the Gaussian distribution. ## classification of variance: 3 categories | standard costing

It is the difference between the actual quantities of materials used in a mixture at standard price, and the total quantity of materials used at the weighted average price per unit of materials as shown by the Standard Cost Sheet. In short, it is the difference between the standard and the actual composition of a mixture.

Needless to mention that this variance is applicable only when direct materials are physically mixed. So, where a standard mix is specified, a material mix variance will appear. This happened due to the temporary shortage or increase in cost of material used.

This is the difference between the standard yield of the actual materials input and the actual yield, both valued at the standard material cost of the product, i.e., in short, it is the difference between the standard yield specified and the actual yield obtained. This is particularly applicable in process industries where loss is almost a must.

After taking into account the normal loss, it becomes possible to set a standard yield or output. Standard Yield is the output which is expected from the standard input of raw materials. But, in practice, actual output differs from standard output and the said difference is known as yield variance.

Calculate (i) usage variance, (ii) price variance when variance is accumulated at point of purchase, (iii) price variance when variance is accumulated at point of issue on FIFO basis and (iv) price variance when variance is accumulated at point of issue on LIFO basis.

According to ICMA, London, it is that portion of the wage variance which is due to the differences between the standard rate specified and the actual rate paid, i.e., it is nothing but the difference between the standard and the actual direct labour rate per hour for the total hours that are worked. This is actually the responsibility of the personnel department and it is similar to materials prices variances.

It should be remembered that the variance will be a favourable one if actual time is less than the standard time or actual production is more than the standard production, and vice versa in the opposite case.

This arises due to the idleness of the workers for causes like breakdown of machinery, power failure, lock-out etc. which are not controlled. Thus, its effect should be shown separately. Otherwise the workers will be blamed for the same although they are not responsible in efficiency variance.

This variance is similar to the direct material mix variance. This variance appears if, during a particular period, the grades of labour that are used in production are different from those budgeted. This situation arises only when there is a shortage or non-availability of particular grade of labour. This variances does not appear in the new ICMA, London, Terminology.

Labour Cost Variance = Labour Rate Variance + Labour Efficiency Variance + Idle Time Variance Rs. 3,875 (Unfavourable) = Rs. 2,187.50 (Unfavourable) + Rs. 1,462.50 (Unfavourable) + Rs. 225 (Unfavourable)

In a normal working week of 40 hours, the gang is expected to produce 2,000 units of output. During the week ended 31st December 2005, the gang consisted of 40 men, 10 women and 5 boys. The actual wages paid were @ Re. 0.70, 0.65 and 0.30 respectively. 4 hours were lost due to abnormal idle time and 1,600 units were produced.

It must be remembered in this respect that variable overhead is that element of cost which varies directly with output. We should also remember that if it is assumed that variable overheads vary strictly with output, then a change in the production will actually not affect the variable overhead rate per unit. Thus, an expenditure variance will arise if there is a change in the rate per unit.

It is that portion of the fixed production overhead variance which is the difference between the Standard Cost absorbed in the production achieved, whether completed or not, and the budget cost allowance for a specified control period. In short, it is the difference between the budgeted level of output and the actual level of output attained.

It is that portion of the fixed production volume variance which is due to working at higher or lower capacity than standard. Thus, the variance is the difference between the budget cost allowance and the actual direct labour hours worked. It is related to under- or over-utilisation of plant capacity. This variance expresses the information that the factory capacity has been utilised properly or not.

It is that portion of the volume variance which is due to the difference between the number of working days in the budget period and the actual number of working days in the budget period. This variance arises due to the fact that fixed costs remain same for each period, whatever the number of working days and it can be eliminated by apportioning standard allowances and fixed cost on the basis of a working day.

We know the companies operating a yearly budget divide the year into 13 budget periods of 4 weeks on an average. But there are some firms who divide yearly budget by the 12 budget periods according to the calendar months.

Needless to mention that if the latter method is followed, it becomes necessary to operate a calendar variance. This variance is to be computed in order to show the effect on fixed overhead while changing the number of working days. ## understanding the ensemble method bagging and boosting

For the same set of data, different algorithms behave differently. For example, if we want to predict the price of houses given for some dataset, some of the algorithms that can be used are Linear Regression and Decision Tree Regressor. Both of these algorithms will interpret the dataset in different ways and thus make different predictions. One of the key distinctions is how much bias and variance they produce.

There are 3 types of prediction error: bias, variance, and irreducible error. Irreducible error, also known as noise, cant be reduced by the choice of algorithm. The other two types of errors, however, can be reduced because they stem from your algorithm choice.

In the given figure, we use a linear model, such as linear regression, to learn from the model. As we can see, the regression line fails to fit the majority of the data points and thus, this model has high bias and low learning power. Generally, models with low bias are preferred.

variance is an error from sensitivity to small fluctuations in the training set. High variance can cause an algorithm to model the random noise in the training data, rather than the intended outputs (overfitting).

Variance defines the deviation in prediction when switching from one dataset to another. In other words, it defines how much the predictions of a model will change from one dataset to another. It can also be defined as the amount that the estimate of the target function will change if different training data is used.

In the given figure, we can see a non-linear model such as SVR (Support Vector Regressor) tries to generate a polynomial function that passes through all the data points. This may seem like the perfect model, but such models are not able to generalize the data well and perform poorly on data that has not been seen before. Ideally, we want a model with low variance.

The general principle of an ensemble method in Machine Learning to combine the predictions of several models. These are built with a given learning algorithm in order to improve robustness over a single model. Ensemble methods can be divided into two groups:

Most ensemble methods use a single base learning algorithm to produce homogeneous base learners, i.e. learners of the same type, leading to homogeneous ensembles. For example, Random forests (Parallel ensemble method) and Adaboost(Sequential ensemble methods).

Some methods use heterogeneous learners, i.e. learners of different types. This leads to heterogeneous ensembles. For ensemble methods to be more accurate than any of its members, the base learners have to be as accurate and as diverse as possible. In Scikit-learn, there is a model known as a voting classifier. This is an example of heterogeneous learners.

Bagging, a Parallel ensemble method (stands for Bootstrap Aggregating), is a way to decrease the variance of the prediction model by generating additional data in the training stage. This is produced by random sampling with replacement from the original set. By sampling with replacement, some observations may be repeated in each new training data set. In the case of Bagging, every element has the same probability to appear in a new dataset. By increasing the size of the training set, the models predictive force cant be improved. It decreases the variance and narrowly tunes the prediction to an expected outcome.

These multisets of data are used to train multiple models. As a result, we end up with an ensemble of different models. The average of all the predictions from different models is used. This is more robust than a model. Prediction can be the average of all the predictions given by the different models in case of regression. In the case of classification, the majority vote is taken into consideration.

For example, Decision tree models tend to have a high variance. Hence, we apply bagging to them. Usually, the Random Forest model is used for this purpose. It is an extension over-bagging. It takes the random selection of features rather than using all features to grow trees. When you have many random trees. Its called Random Forest.

Boosting is a sequential ensemble method that in general decreases the bias error and builds strong predictive models. The term Boosting refers to a family of algorithms which converts a weak learner to a strong learner.

In each iteration, data points that are mispredicted are identified and their weights are increased so that the next learner pays extra attention to get them right. The following figure illustrates the boosting process.

During training, the algorithm allocates weights to each resulting model. A learner with good prediction results on the training data will be assigned a higher weight than a poor one. So when evaluating a new learner, Boosting also needs to keep track of learners errors.

Some of the Boosting techniques include an extra-condition to keep or discard a single learner. For example, in AdaBoost an error of less than 50% is required to maintain the model; otherwise, the iteration is repeated until achieving a learner better than a random guess.

Theres no outright winner, it depends on the data, the simulation, and the circumstances. Bagging and Boosting decrease the variance of a single estimate as they combine several estimates from different models. As a result, the performance of the model increases, and the predictions are much more robust and stable.

But how do we measure the performance of a model? One of the ways is to compare its training accuracy with its validation accuracy which is done by splitting the data into two sets, viz- training set and validation set.

The model is trained on the training set and evaluated on the validation set. Thus, the training accuracy is evaluated on the training set and gives us a measure of how good the model can fit the training data. On the other hand, validation accuracy is evaluated on the validation set and reveals the generalization ability of the model. A models ability to generalize is crucial to the success of a model. Thus, we can say that the performance of a model is good if it can fit the training data well and also predict the unknown data points accurately.

If a single model gets a low performance, Bagging rarely gets a better bias. However, Boosting can generate a combined model with lower errors. As it optimizes the advantages and reduces the pitfalls of the single model. On the other hand, Bagging can increase the generalization ability of the model and help it better predict the unknown samples. Let us see an example of this in the next section.

From the above plot, we see that the RandomForest algorithm softens the decision boundary, hence decreases the variance of the decision tree model whereas AdaBoost fits the training data in a better way and hence increases the bias of the model.

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Unlike in a standard conformal prediction setting, the p-values of the combined classifier h:m need to be interpreted. It has been assumed that the base classifier h is not based on the conformal prediction framework and is therefore not capable of providing p-values as output for instance classifications. Example classifiers of h are human experts, decision rules, and such [141,231]141231. On the other hand, the metaclassifier is conducive to use with the conformal prediction framework and is capable of providing p-values as outputs for the positive metaclass and the negative metaclass. Example classifiers include nearest-neighbor classifiers , Support Vector Machines , and other classifiers listed in earlier chapters. Given this setting, an instance xX is classified by the combined classifier h:m as follows.

The base classifier h assigns a label yY to the instance x. The meta classifier m acts upon the instance x and estimates the p-value pp for the positive metaclass and the p-value pn for the negative meta class. The positive metaclass indicates that the assigned label y is correct and the negative metaclass indicates that the label y is incorrect. Based upon this understanding, two assumptions are arrived at:

These intuitive assumptions (A1) and (A2) are the basis for how the combined classifier h:m is interpreted. The score pppn is considered to decide if the particular instance should be classified (as in ). A reliable threshold T is determined on the score pppn to decide if a classification made by h on an instance x is reliable. If the score is greater than the threshold, the classification of x is reliable; otherwise x is left unclassified. The threshold T imposes a certain accuracy on the instances that h:m can classify and the rejection of the combined classifier.

Given m base classifiers Lj=1,,m, each classifier Lj is induced on each subset Pi. Hence, m n models are computed. Similar to , given an instance Ik, two error count variables are defined: Skle, local error count and Skge, global error count (step 8). The values of these counters are incremented as follows:

It is assumed that noisy instances receive larger values for Skle and Skge than clean instances . By selecting m classifiers from distinct data mining families, the biases of the learners will cancel out one another and make the filter more efficient in detecting exceptions . Moreover, splitting the training dataset makes the filter more adequate to large and/or distributed datasets, partially because the induction of the base classifiers is less time-consuming .

Given multiview feature representation, one can observe that a representation generated from a specific template can be regarded as a profile for the brain, and can be used to provide supplementary or side information for other representations generated from the other templates (ie, views). Accordingly, Liu et al. (2015) develop a view-centralized multitemplate (VCM) classification method, with flowchart illustrated in Fig. 9.7.

Fig. 9.7. The framework of the view-centralized multitemplate classification method, which includes four main steps: (1) preprocessing and template selection, (2) feature extraction, (3) feature selection, and (4) ensemble classification.

As can be seen from Fig. 9.7, brain images are first nonlinearly registered to multiple templates individually, and then their volumetric features are extracted within each template space. In this way, multiple feature representations can be generated from different templates for each specific subject. Based on such representations, the proposed VCM FS method can be applied to select the most discriminative features, by focusing on the main-view template along with the extra guidance from side-view templates. Finally, multiple SVM classifiers are constructed based on multiple sets of selected features, followed by a classifier ensemble strategy to combine multiple outputs from all SVM classifiers for making the final decision.

Given N training images that have been registered to K templates, we denote X={xi}i=1NRDN (D = M K in this chapter) as the training data, where xiRD is the feature representation generated from K templates for the ith training image. Let Y={yi}i=1NRN be the class labels of N training data, and wRD be the weight vector for the FS task. For clarity, we divide the feature representations from multiple templates into a main-view group and a side-view group, as illustrated in Fig. 9.8. As can be seen from Fig. 9.8, the main-view group (corresponding to the main template) contains features from a certain template, while the side-view group (corresponding to other supplementary templates) contains features from all other (supplementary) templates.

Fig. 9.8. Illustration of group information for feature representations generated from multiple templates. The first group G1 (ie, the main-view group) consists of features from a certain template, while the second group G2 (ie, the side-view group) contains features from all other (supplementary) templates.

Denote a(1) as the weighting value for the main-view (ie, main template) group and a(2) as the weighting value for the side-view (ie, supplementary templates) group. By setting different weighting values for features from the main view and the side views, we can incorporate the prior information into the following learning model:

where w(g) represents the weight vector for the gth group. The first term in Eq. (9.10) is the empirical loss on the training data, and the second one is the l1-norm regularization term that enforces some elements of w to be zero. It is worth noting that the last term in Eq. (9.10) is a view-centralized regularization term, which treats features in the main-view group and the side-view group differently by using different weighting values (ie, a(1) and a(2)). For example, a small a(1) (as well as a large a(2)) implies that the coefficients for features in the main-view group will be penalized lightly, while features in the side-view group will be penalized severely, because the goal of the model defined in Eq. (9.10) is to minimize the objective function. Accordingly, most elements in the weight vector corresponding to the side-view group will be zero, while those corresponding to the main-view group will not. In this way, the prior knowledge that one focuses on the representation from the main template (ie, main view) with extra guidance from other templates can be incorporated into the learning model naturally. In addition, two constraints in Eq. (9.10) are used to ensure that the weighting values for different groups are greater than 0 and not greater than 1. By introducing such constraints, one can efficiently reduce the degrees of freedom of the proposed model, and avoid overfitting with limited training samples.

Based on the VCM FS model defined in Eq. (9.10), one can obtain a feature subset by selecting features with nonzero coefficients in w. Each time, one performs the above-mentioned FS procedure by focusing on one of multiple templates, with other templates used as extra guidance. Accordingly, given K templates, one can get K selected feature subsets, with each of them reflecting the information learned from a certain main template and corresponding supplementary templates.

After obtaining K feature subsets by using the view-centralized FS algorithm, one can then learn K base classifiers individually. In this study, a linear SVM classifier is used to identify AD patients from NCs, and progressive MCI patients from stable MCI patients, since the linear SVM model has good generalization capability across different training data, as shown in extensive studies (Zhang and Shen, 2012; Burges, 1998; Pereira et al., 2009). Finally, a classifier ensemble strategy is used to combine these K base classifiers to construct a more accurate and robust learning model, where the majority voting strategy is employed for the fusion of multiple classifiers. Thus the class label of an unseen test sample can be determined by majority voting for the outputs of base classifiers.

Bagging, boosting and random forests are ensemble learning algorithms. Their common property is that they generate ensembles of base classifiers and ensure their diversity by providing them with different sets of learning examples.

Bagging. The term bagging is short for bootstrap aggregating. Bootstrap (Section 3.3.4) is a method for replication of learning examples when the learning set is small. In bagging, a series of different learning sets is generated. When the total number of learning examples is n, each new learning set is generated by n-times randomly (with replication) selecting an example from the original learning set. Some examples may therefore occur more than once, and some examples may not occur in a new learning set (on average, 36.8% of examples). Each newly generated learning set is used as an input to the learning algorithm. Thus we get a series of potentially different hypotheses that are used for predicting the value of a dependent variable by combining all generated hypotheses.

Bagging excels especially when unstable base algorithms with high variance are used, such as decision or regression trees. Bagging is robust, as increasing the number of generated hypotheses does not lead to overfitting.

Boosting. Boosting is a theoretically well-founded (see Section 14.8.3) ensemble learning method. Its basic idea is to weight learning examples according to how difficult they are. It assumes that the base learning algorithm is able to deal with weighted learning examples. If this is not the case, weighted learning examples are simulated by sampling the learning set similarly as in bagging. The only difference is that the probability of selecting an example is not uniform, but proportional to its weight.

Boosting requires several iterations of the learning process. In the first iteration the learning algorithm builds a hypothesis from equally weighted learning examples (each examples weight is equal to 1). A hypothesis predicts the value of a dependent variable (the class label) for each learning example. Afterwards, all the weights are adjusted. For examples with correct predictions, weights are decreased, and for examples with incorrect predictions, weights are increased.

The most frequently used approach for weighting of examples is as follows. Let e be the (normalized) hypothesis error on a particular learning example. The examples weight is adjusted by multiplication with e/(le). Lesser errors therefore yield lesser weights. After all weights have been adjusted, they are normalized so that they sum up to n (for n learning examples).

The following iterations of the learning process therefore focus on difficult learning examples. The process is reiterated until the overall error f becomes sufficiently small or large, the latter meaning that the remaining difficult examples are not solvable. Such (last) hypotheses are rejected as they do not contribute any useful knowledge. Also rejected are hypotheses with overall error f too close to 0, as they tend to overfit the learning data. The remaining hypotheses form an ordered chain.

All the remaining hypotheses are used for final predictions. Each hypothesis prediction is weighted with its performance (e.g., classification accuracy) on a weighted learning set used for its generation. Accurate hypotheses therefore carry more weight. For voting, a weighting scheme log(f / (l f)) is frequently used, with f being a hypothesis error on the corresponding weighted learning set.

Boosting frequently yields better performance than bagging. Contrary to bagging, it can also be used with stable base learning algorithms which have small variance. Sometimes, however, overfitting can occur. In such (rare) cases, the performance of a combined hypothesis is worse than the performance of a single hypothesis.

Random forests. Random forests are intended to improve the predictive accuracy of tree-based learning algorithms. Originally, they were developed exclusively for improving decision trees but can also be used for improving regression trees. The basic idea is to generate a series of decision trees that limit the selection of the best attribute in each node to a relatively small subset of randomly chosen candidate attributes. If the number of attributes is a, a typical number of candidate attributes is log a+1. This number can also be 1, meaning a completely random attribute selection in each node of a decision tree. The size of the forest (the number of generated trees) is normally at least 100, but can be considerably higher.

Each decision tree is generated from the whole learning set, and is subsequently used for classification of new examples by the uniform voting principle. Each decision tree gives its vote for the class label to the new example. All votes together form a class probability distribution.

The random forests method is robust as it reduces the variance of tree-based algorithms. Decision trees combined in a random forest achieve the classification accuracy of the state-of-the-art algorithms. The downside of the method is the incomprehensibility of its decisions, as interpreting the combined answer of the set of 100 or more decision trees is rather difficult (the same problem also plagues bagging and boosting).

where (x;k){1,1} is the base classifier at iteration k, defined in terms of a set of parameters, k,k=1,2,,K, to be estimated. The base classifier is selected to be a binary one. The set of unknown parameters is obtained in a step-wise approach and in a greedy way; that is, at each iteration step i, we only optimize with respect to a single pair, (ai,i), by keeping the parameters ak,k,k=1,2,,i1, obtained from the previous steps, fixed. Note that ideally, one should optimize with respect to all the unknown parameters, ak,k,k=1,2,,K, simultaneously; however, this would lead to a very computationally demanding optimization task. Greedy algorithms are very popular, due to their computational simplicity, and lead to a very good performance in a wide range of learning tasks. Greedy algorithms will also be discussed in the context of sparsity-aware learning in Chapter 10.

starting from an initial condition. According to the greedy rationale, Fi1() is assumed to be known and the goal is to optimize with respect to the set of parameters ai,i. For optimization, a loss function has to be adopted. No doubt different options are available, giving different names to the derived algorithm. A popular loss function used for classification is the exponential loss, defined as

and it gives rise to the adaptive boosting (AdaBoost) algorithm. The exponential loss function is shown in Fig. 7.14, together with the 0-1 loss function. The former can be considered a (differentiable) upper bound of the (nondifferentiable) 0-1 loss function. Note that the exponential loss weighs misclassified (yF(x)<0) points more heavily compared to the correctly identified ones (yF(x)>0). Employing the exponential loss function, the set ai,i is obtained via the respective empirical cost function, in the following manner:

Figure 7.14. The 0-1, exponential, log-loss, and squared error loss functions. They have all been normalized to cross the point (0,1). The horizontal axis for the squared error corresponds to yF(x).

Observe that wn(i) depends neither on a nor on (xn;); hence it can be considered a weight associated with sample n. Moreover, its value depends entirely on the results obtained from the previous recursions.

We now turn our focus on the cost in (7.89). The optimization depends on the specific form of the base classifier. Note, however, that the loss function is of an exponential form. Furthermore, the base classifier is a binary one, so that (x;){1,1}. If we assume that a>0 (we will come back to it soon) optimization of (7.89) is readily seen to be equivalent to optimizing the following cost:

and [,0]() is the 0-1 loss function.5 In other words, only misclassified points (i.e., those for which yn(xn;)<0) contribute. Note that Pi is the weighted empirical classification error. Obviously, when the misclassification error is minimized, the cost in (7.89) is also minimized, because the exponential loss weighs the misclassified points heavier. To guarantee that Pi remains in the [0,1] interval, the weights are normalized to unity by dividing by the respective sum; note that this does not affect the optimization process. In other words, i can be computed in order to minimize the empirical misclassification error committed by the base classifier. For base classifiers of very simple structure, such a minimization is computationally feasible.

Looking at the way the weights are formed, one can grasp one of the major secrets underlying the AdaBoost algorithm: The weight associated with a training sample xn is increased (decreased) with respect to its value at the previous iteration, depending on whether the pattern has failed (succeeded) in being classified correctly. Moreover, the percentage of the decrease (increase) depends on the value of ai, which controls the relative importance in the buildup of the final classifier. Hard samples, which keep failing over successive iterations, gain importance in their participation in the weighted empirical error value. For the case of the AdaBoost, it can be shown that the training error tends to zero exponentially fast (Problem 7.18). The scheme is summarized in Algorithm 7.1.

Initialize: wn(1)=1N,i=1,2,,NInitialize: i=1Repeat Compute the optimum i in (;i) by minimizing Pi;(7.91)Compute the optimum Pi;(7.92)ai=12ln1PiPiZi=0For n=1 to N Down(i+1)=wn(i)exp(ynai(xn;i))Zi=Zi+wn(i+1)End ForFor n=1 to N Down(i+1)=wn(i+1)/ZiEnd ForK=ii=i+1Until a termination criterion is met.f()=sgn(k=1Kak(,k))

Repeat Compute the optimum i in (;i) by minimizing Pi;(7.91)Compute the optimum Pi;(7.92)ai=12ln1PiPiZi=0For n=1 to N Down(i+1)=wn(i)exp(ynai(xn;i))Zi=Zi+wn(i+1)End ForFor n=1 to N Down(i+1)=wn(i+1)/ZiEnd ForK=ii=i+1

Bagging, boosting, and random forests are examples of ensemble methods (Figure 8.21). An ensemble combines a series of k learned models (or base classifiers), M1,M2,,Mk, with the aim of creating an improved composite classification model, M. A given data set, D, is used to create k training sets, D1,D2,,Dk, where Di(1ik1) is used to generate classifier Mi. Given a new data tuple to classify, the base classifiers each vote by returning a class prediction. The ensemble returns a class prediction based on the votes of the base classifiers.

Figure 8.21. Increasing classifier accuracy: Ensemble methods generate a set of classification models, M1, M2, , Mk. Given a new data tuple to classify, each classifier votes for the class label of that tuple. The ensemble combines the votes to return a class prediction.

An ensemble tends to be more accurate than its base classifiers. For example, consider an ensemble that performs majority voting. That is, given a tuple X to classify, it collects the class label predictions returned from the base classifiers and outputs the class in majority. The base classifiers may make mistakes, but the ensemble will misclassify X only if over half of the base classifiers are in error. Ensembles yield better results when there is significant diversity among the models. That is, ideally, there is little correlation among classifiers. The classifiers should also perform better than random guessing. Each base classifier can be allocated to a different CPU and so ensemble methods are parallelizable.

To help illustrate the power of an ensemble, consider a simple two-class problem described by two attributes, x1 and x2. The problem has a linear decision boundary. Figure 8.22(a) shows the decision boundary of a decision tree classifier on the problem. Figure 8.22(b) shows the decision boundary of an ensemble of decision tree classifiers on the same problem. Although the ensemble's decision boundary is still piecewise constant, it has a finer resolution and is better than that of a single tree.

Figure 8.22. Decision boundary by (a) a single decision tree and (b) an ensemble of decision trees for a linearly separable problem (i.e., where the actual decision boundary is a straight line). The decision tree struggles with approximating a linear boundary. The decision boundary of the ensemble is closer to the true boundary.

There are numerous ensemble approaches besides bagging and boosting. The key difference is that the sense predictions of the base models are combined. The predictions of some base classifiers as features learn to produce a metamodel, which combines their predictions. Learning a linear metamodel is known as stacking. It is also feasible to combine different base models into a heterogeneous ensemble to achieve base model diversity such that base models are trained by diverse learning algorithms by employing the same training set. Hence the model ensembles are composed of a set of base models and a metamodel, which is trained to decide how base model predictions must be combined (Flach,2012).

AdaBoost is one of the most popular implementations of the boosting ensemble approach. It is adaptive because it assigns weights for base models () based on the accuracy of the model, and changes weights of the training records (w) based on the accuracy of the prediction. Here is the framework of the AdaBoost ensemble model with m base classifiers and n training records ((x1,y1), (x2,y2), , (xn,yn)). Following are the steps involved in AdaBoost:

Hence, the AdaBoost model updates the weights based on the prediction and the error rate of the base classifier. If the error rate is more than 50%, the record weight is not updated and reverted back to the next round.

AdaBoost is one of the most popular implementations of the boosting ensemble approach. It is adaptive because it assigns weights for base models () based on the accuracy of the model and changes the weights of the training records (w) based on the accuracy of the prediction. Here is the framework of the AdaBoost ensemble model with m base classifiers and n training records ((x1,y1), (x2,y2), , (xn,yn)). The steps involved in AdaBoost are:

Hence, the AdaBoost model updates the weights the training records based on the prediction and the error rate of the base classifier. If the error rate is more than 50%, the record weight is not updated and reverted back to the next round.

Obviously, Adaboost is not the only boosting algorithm available. For example, one can come up with other algorithms by adopting alternatives to (4.129) cost functions or growing mechanisms to build up the final classifier. In fact, it has been observed that in difficult tasks corresponding to relatively high Bayesian error probabilities (i.e., attained by using the optimal Bayesian classifier), the performance of the AdaBoost can degrade dramatically. An explanation for it is that the exponential cost function over-penalizes bad samples that correspond to large negative margins, and this affects the overall performance. More on these issues can be obtained from [Hast 01, Frie 00] and the references therein.

A variant of the AdaBoost has been proposed in [Viol 01] and later generalized in [Yin 05]. Instead of training a single base classifier, a number of base classifiers are trained simultaneously, each on a different set of features. At each iteration step, the classifier () results by combining these base classifiers. In principle, any of the combination rules can be used. [Scha 05] presents a modification of the AdaBoost that allows for incorporation of prior knowledge into boosting as a means of compensating for insufficient data. The so called AdaBoost*v version was introduced in [Rats 05], where the margin is explicitly brought into the game and the algorithm maximizes the minimum margin of the training setup. The algorithm incorporates a current estimate of the achievable margin which is used for computation of the optimal combining coefficients of the base classifiers.

Multiple additive regression trees (MART) is a possible alternative that overcomes some of the drawbacks related to AdaBoost. In this case, the additive model in (4.128) consists of an expansion in a series of classification trees (CART), and the place of the exponential cost in (4.129) can be taken by any differentiable function. MART classifiers have been reported to perform well in a number of real cases, such as in [Hast 01, Meye 03].

For the multiclass case problem there are several extensions of AdaBoost. A straightforward extension is given in [Freu 97, Eibl 06]. However, this extension fails if the base classifier results in error rates higher than 50%. This means that the base classifier will not be a weak one, since in the multiclass case random guessing means a success rate equal to 1/M, where M is the number of classes. Thus, for large M 50% rate of correct classification can be a strong requirement. To overcome this difficulty, other (more sophisticated) extensions have been proposed. See [Scha 99, Diet 95].

Let us consider a two-class classification task. The data reside in the 20-dimensional space and obey a Gaussian distribution of unit covariance matrix and mean values [-a, a,, a]T, [a,a,, a]T, respectively, for each class, where a=2/20. The training set consists of 200 points (100 from each class) and the test set of 400 points (200 from each class) independently generated from the points of the training set.

To design a classifier using the AdaBoost algorithm, we chose as a seed the weak classifier known as stump. This is a very naive type of tree, consisting of a single node, and classification of a feature vector x is achieved on the basis of the value of only one of its features, say, xi. Thus, if xi, < 0, x is assigned to class A. If xi > 0, it is assigned to class B. The decision about the choice of the specific feature, xi, to be used in the classifier was randomly made. Such a classifier results in a training error rate slightly better than 0.5.

The AdaBoost algorithm was run on the training data for 2000 iteration steps. Figure 4.30 verifies the fact that the training error rate converges to zero very fast. The test error rate keeps decreasing even after the training error rate becomes zero and then levels off at around 0.05.

FIGURE 4.30. Training and test error rate curves as functions of the number of iteration steps for the AdaBoost algorithm, using a stump as the weak base classifier. The test error keeps decreasing even after the training error becomes zero.

Figure 4.31 shows the margin distributions, over the training data points, for four different training iteration steps. It is readily observed that the algorithm is indeed greedy in increasing the margin. Even when only 40 iteration steps are used for the AdaBoost training, the resulting classifier classifies the majority of the training samples with large margins. Using 200 iteration steps, all points are correctly classified (positive margin values), and the majority of them with large margin values. From then on, more iteration steps further improve the margin distribution by pushing it to higher values.

FIGURE 4.31. Margin distribution for the AdaBoost classifier corresponding to different numbers of training iteration steps. Even when only 40 iteration steps are used, the resulting classifier classifies the majority of the training samples with large margins. ## machine learning - why does a decision tree have low bias & high variance? - cross validated

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As the name implies, is an error component that we cannot correct, regardless of algorithm and it's parameter selection. Irreducible error is due to complexities which are simply not captured in the training set. This could be attributes which we don't have in a learning set but they affect the mapping to outcome regardless.

Bias error is due to our assumptions about target function. The more assumptions(restrictions) we make about target functions, the more bias we introduce. Models with high bias are less flexible because we have imposed more rules on the target functions.

Variance error is variability of a target function's form with respect to different training sets. Models with small variance error will not change much if you replace couple of samples in training set. Models with high variance might be affected even with small changes in training set.

On the other hand, due to low variance if you change couple of data samples, it's unlikely that this will cause major changes in the overall mapping the target function performs. On the other hand, algorithm such as k-nearest-neighbors have high variance and low bias. It's easy to imagine how different samples might affect K-N-N decision surface.

Now that we have these definitions in place, it's also straightforward to see that decision trees are example of model with low bias and high variance. The tree makes almost no assumptions about target function but it is highly susceptible to variance in data.

Intuitively, it can be understood in this way. When there are too many decision nodes to go through before arriving at the result i.e number of nodes to traverse before reaching the leaf nodes is high, the conditions that you are checking against becomes multiplicative. That is, the computation becomes (condition 1)&&(condition 2)&&(condition 3)&&(condition 4)&&(condition5).

Only if all the conditions are satisfied, a decision is reached. As you can see, this will work very well for the training set as you are continuously narrowing down on the data. The tree becomes highly tuned to the data present in the training set.

Why does a decision tree have low bias & high variance? Does it depend on whether the tree is shallow or deep? Or can we say this irrespective of the depth/levels of the tree? Why is bias low & variance high? Please explain intuitively and mathematically.

I want to start by saying that everything is relative. Decision Tree in general has low bias and high variance that let's say random forests. Similarly, a shallower tree would have higher bias and lower variance that the same tree with higher depth.

Now with that ironed out, let's think why decision trees would be worse in variance (higher variance and lower bias) than let's say random forests. The way a decision tree algorithm works is that the data is split again and again as we go down in the tree, so the actual predictions would be made by fewer and fewer data points. Compared to that, random forests aggregate the decisions of multiple trees, and that too, less-correlated trees through randomization, hence the model generalizes better (=> performs more reliably across different datasets = lower variance). Similarly, we are making more simplifying assumptions on random forests to consult only a subset of data and features to fit a single tree, hence higher bias. BTW, similary, a tree with lower height = less reliant on fewer data points generalizes better and and has less variance compared to a deep tree.

A. If the tree is shallow then we're not checking a lot of conditions/constrains ie the logic is simple or less complex, hence it automatically reduces over-fitting. This introduces more bias compared to deeper trees where we overfit the data. It can be imagined as we're deliberately not calculating more conditions means we're making some assumption (introduces bias) while creating the tree.

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