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Cost functions are important because there are many ways to design a machine learning algorithm, as well as interpret its performance. This cost functions series will help the analyst discover the underpinnings of the algorithms and usefulness of each algorithm’s functionality. Following this series you will have a deep understanding of a slew of cost functions available for classification and regression models, as well as interpret your predictive models from an expert’s point of view.

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- 1. How To Interpret Model Performance With Cost Functions A publication of
- 2. Introduction Cost functions are directly related to the performance of data mining and predictive models. Cost functions are important because there are many ways to design a machine learning algorithm, as well as interpret its performance. This cost function series will help the analyst discover the underpinnings of the algorithms and usefulness of each algorithm’s functionality. We go deep into the statistical properties and mathematical understanding of each cost function and explore their similarities and differences. We hope you will have a deeper understanding of a variety of cost functions available for classification and regression models, and be able to interpret your models from an expert point of view.
- 3. Topics to Cover Cost Functions for Regression Problems: Least Squares Deviation Least Absolute Deviation Huber- M Cost Cost Functions for Classification Problems: Precision and Recall Measuring Performance with ROC Curve Gains and Lift Logistic Function and Cost Multinomial Classification- Expected Cost Multinomial Classification- Log Likelihood
- 4. Understanding Cost Functions The Supervised Learning problem: 1. Collection of n p-dimensional feature vectors 2. Collection of observed responses {xi}, i = 1, n {yi}, i = 1, n Goal: Construct a response surface (hypothesis): h(x)
- 5. Cost Functions describe how well a response surface h(x) fits the available data, in essence, the goodness of fit (on a given data set): J (yi, h(xi)) Things to keep in mind: • • • Smaller values of the cost function correspond to a better fit Machine learning goal: construct h(x) such that J is minimized In regression, h(x) is usually directly interpretable as the predicted response
- 6. Least Squares Deviation Cost for Regression Problems Defined as: Error between what is observed and what is predicted (difference between yi and h(xi)) is used as the prime element of the LSD cost function. Also known as ‘Mean Squared Error’. Advantage: LSD has nice mathematical and statistical properties. Disadvantage: LSD has a known issue of outliers
- 7. Least Absolute Deviation for Regression Problems Defined as: More robust when it comes to outliers than LSD (minimizes the impact outliers make by using absolute value instead of squared residual). Presence of the absolute term in LAD can cause some computational challenges. There is another type of cost function that combines these two (Least Squares Deviation and Least Absolute Deviation) into an intermediate, known as Huber-M cost.
- 8. Huber-M Costs for Regression Problems Defined as: Combines best qualities of LSD and LAD losses. When residuals are small, use quadratic loss. When the residuals exceed a certain threshold δ, Huber-M switches from quadratic penalty (LSD) to linear penalty (LAD), thus combining quadratic penalty for small residuals and linear penalty for large residuals. Parameter δ is usually set automatically to a specific percentile of absolute residuals.
- 9. The Binary Classification Problem Binary classification: predicting a specific outcome that can only have two distinct values (Yes/No, +/-, 0/1) Observed response y takes only two possible values: + and – (+ and – are used here for convention) Need to define a relationship between h(x) and y Use the decision rule: The decision rule is always governed by a user defined threshold, t. When the response exceeds this threshold, a positive prediction will be made. Otherwise, a negative prediction will be made for the response variable.
- 10. The Binary Classification Problem Summary: When constructing a response surface, a threshold needs to be introduced to make predictions. Classification of positives and negatives will depend on what threshold is designated.
- 11. Evaluating Prediction Success with Precision and Recall Evaluate performance by focusing on how well we capture the “+” group (assumed to be the events of interest) for a given threshold. After running a model, a prediction success table (also known as a ‘confusion matrix’) can be created: The table can contain four outcomes: true-positive (tp), false-positive (fp), false-negative (fn), true-negative (tn).
- 12. Evaluating Prediction Success with Precision and Recall Measure of Success #1: Precision The ratio of true-positives divided by true-positives and false-positives Precision focuses on the group of true-positive predictions- what fraction are actually positive within that group. For instance, if fraudulent transactions are being identified and it is predicted that 1,000 transactions are suspected to be fraudulent, precision will tell you the actual fraction of transactions that are fraudulent within that group of predictions. If precision is .5, it would be expected that about 500 transactions are indeed fraudulent.
- 13. Evaluating Prediction Success with Precision and Recall Measure of Success #2: Recall (Sensitivity) Keeping with our example of fraudulent data, recall tells us that we know we have captured some fraudulent transactions in the group predicted as fraud, but what is the actual fraction of total fraudulent transactions that exist that were captured. There is a relationship between precision and recall. When threshold is varied. In an ideal case, precision = 1.0 and recall = 1.0 (meaning there are no misclassifications). If precision is 1.0, false-positives are equal to zero. If recall is 1.0, false-negatives are equal to zero. Unfortunately, precision and recall cannot be maximized simultaneously, when focusing on one side, the other will suffer.
- 14. Measuring Performance with the ROC Curve Receiver Operating Characteristic (ROC) Curve: A curve that characterizes the performance of the classifier in general as we sweep over the range of different thresholds. ROC measures how well you capture positives, negatives, and the balance between these two response groups. A binary classification performance can always be represented by an ROC Curve.
- 15. Measuring Performance with Gains and Lift If you have a good model, for instance, in a direct marketing campaignpeople who got a higher score are more likely to respond. Therefore when you identify the potential targeted group, you will pick up the people who were scored the highest. This is the underlying framework of gains and lift. The plot of sensitivity versus support is called the Gains curve What are the optimal gains that can be achieved? That question involves the concept of base rate, which is represented by the number of true-positives plus false-negatives divided by the sample size. So in our direct marketing example, if you expect a default rate of 1% of responders, then your base rate becomes 1%.
- 16. Direct Interpretation of Response Using Logistic Function Instead of taking the predicted response of the function as a scored value, we focus on the direct interpretation of the probability of a positive outcome. Define: p= Prob (y= “+”) where p is the probability that y has a positive outcome. The positive outcome is the event in focus (e.g. positive responder to direct marketing campaign) Probability has a characteristic of always being defined between 0 and 1. However, probability can be converted to log odds, which no longer has that constraint. Define: Log-odds Log-odds can be positive or negative, which makes it convenient for modeling.
- 17. Probability can be expressed in terms of log-odds: p=1/1+eh When probability is plotted in terms of log-odds, it maps the entire positive and negative infinity range of values to an interval between 0 and 1 that can be interpreted as a probability space. Probability can be converted to log-odds and vice versa. The following graph establishes the nature of this transformability:
- 18. Multinomial ClassificationExpected Cost Multinomial classification is working with more than two classifications. Instead of trying to predict one of two classes, you are working with k classes. This is the most general form of classification problem. One approach focuses on the performance of a classification model as it classifies classes to one type or another, known as expected cost.
- 19. Multinomial Classification- Log Likelihood In applying the log-likelihood cost function, there is a stricter environment in which we are interested in the model performance, but also the actual predicted probability of the responding classes.
- 20. Conclusion There are many different ways to evaluate the performance of a classification problem model. In the end, it depends on the type of data on hand and the goals that are in mind. There are a host of evaluation techniques available but it is up to you, the data analyst, to decide what is ultimately relevant.

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