Partial dependence is a statistical tool used to understand the relationship between input variables and outcomes in machine learning models. Partial dependence uses a combination of feature importance, local interpretability methods, and model-agnostic explanations to uncover the underlying relationships between model inputs and predictions.

**In short, partial dependence works by recalculating the average prediction of a machine learning model after manipulating one or more variables while keeping all other inputs constant. The result gives insights into how much impact each variable has on the model's overall prediction, as well as how sensitive the model is to any given variable.**

Partial dependence plots are often used to evaluate models both before and after training, as well as for feature selection, parameter tuning, and hyperparameter optimization. By examining partial dependence plots, data scientists can gain valuable insight into their models' predictive power without necessarily needing to understand complex mathematical formulas or code.

For instance, partial dependence plots allow for easy interpretation of interactions between two or more features and how they influence the outcome of a model.

## What is the purpose of partial dependence?

The overall goal of partial dependence is to explain and visualize the effect of an input variable on a model’s predictions. By doing this, it becomes easier to identify areas within a dataset that could be improved or help guide data exploration efforts.

By understanding what factors are most influential in predicting an outcome with a machine learning model, data scientists can better focus their modeling efforts on those areas. Partial dependence can also help identify potential problems with a dataset, such as unbalanced classes or noisy features that do not contribute much useful information for making accurate predictions.

## In marketing mix modeling (MMM)

The main thing we want to learn from a marketing mix model is what to spend our budget on. We can't access this directly: we don't know what all the different layers of neurons do in our model.

However, we can feed our neurons inputs and get outputs: if we give it the different spend levels we want to predict, it will return the predictions.

By controlling for all other variables and only changing the spend for one channel, we can show the partial dependence of a channel. This means we can choose how much to spend at different spend levels and use the model to make predictions for us.

## Benefits and uses of partial dependence

Partial dependence (PD) is a powerful tool for understanding the relationship between a set of input variables and an output of interest.

By analyzing the partial dependence, we can gain insight into how certain input variables influence the output without considering all factors simultaneously. This makes PD an invaluable tool in exploratory data analysis and gives us a better understanding of our model works.

### Detailed analysis

One of the greatest benefits of partial dependence is that it gives us a more granular, detailed view of our data than standard regression methods. In addition, partial dependence allows us to isolate individual inputs and measure their importance separately from other inputs, which can help us decide which ones are more influential in determining the model outcome.

This can make the process of feature selection much easier since it highlights which features have more influence on our model output. Furthermore, PD can give us insight into non-linear relationships between our inputs and outputs, which might otherwise be difficult to detect using conventional linear regression techniques.

### Visualization

Another benefit of partial dependence is that it allows for better interpretation and visualization of our models’ performance due to its use of heatmaps and 3D plots. These visualizations are far superior at providing us with information about how important certain inputs are relative to others rather than just providing raw numbers or values.

Furthermore, these visualizations make it easier for us to identify difficult-to-spot areas within our models that may need additional tuning or changes in order for them to perform optimally.

Lastly, using partial dependence can be beneficial when training machine learning algorithms as it helps remove any confounding effects caused by other input variables that might have an impact on the model’s performance.

By removing these effects prior to training we can ensure that our algorithms are relying solely on relevant variables while ignoring correlation noise that may have been created by other unrelated features.

Further, this helps reduce computational costs since we don’t need to re-train our models every time we want to see what effect changing one variable has as compared to all other variables included in the dataset.

## Limitations of partial dependence

When it comes to using partial dependence, data scientists must consider some important limitations.

Firstly, partial dependence does not consider how the various features interact with each other. This means that a small change to one feature could have an unexpected effect on the other features and their impact on the target variable.

Again, this is due to a lack of information about the interactions between features in the data set.

### Partial dependence assumptions

Partial dependence assumes that each feature only has a linear effect on the prediction outcome, which may not always be true. For example, a non-linear relationship between two variables can lead to an inaccurate judgment when using partial dependence. This limitation can also cause issues when dealing with interactions among multiple variables, as these are often non-linear or even chaotic.

Further, partial dependence assumes additive effects for all of the features meaning that all of the variables are assumed to have an equal impact on the outcome regardless of whether they actually do or not. This can lead to incorrect assumptions about the most important drivers of an outcome and lead to poor decisions being made based on those assumptions.

Finally, partial dependence is limited by its generalizability and applicability to new datasets, which may have different structures or contain more complex relationships between features than previously seen before. Additionally, due to its reliance on statistical techniques such as regression algorithms, it is limited by its ability to accurately model highly complex non-linear relationships that may be present in real-world data sets.

Overall, while partial dependence can provide useful insights into how a certain input affects an output, it is still subject to numerous limitations which must be considered when looking at potential applications of the technique. It should therefore be used with caution and understanding in order to avoid any potential pitfalls associated with its application.

## Interpreting partial dependence plots

Interpreting partial dependence plots (PDP) is an invaluable tool used to better understand and interpret the behavior of machine learning models. At their core, PDPs are a graphical representation of how different variables in a dataset contribute to the final prediction made by a model.

PDPs allow for the identification of patterns between features and the output of a model, which is extremely valuable when it comes to understanding how different features interact with one another.

You can create PDPs by calculating the average outcome from a model across all observations in a dataset, while varying one feature at a time. The output from this calculation is then plotted against the varying feature, allowing us to observe how changes in that feature result in different predictions from the model. The shape of the curve can provide insights into how sensitive the model is to changes in each variable, as well as providing indications about potential interactions between features.

The interpretation of PDPs requires an understanding of both statistics and machine learning principles. When reading a PDP plot, it’s important to be aware of what type of modeling technique was used, as certain modeling techniques may require slightly different interpretations or interpretations with extra caution.

For example, when working with tree-based models such as Random Forests, it’s important that one not just reads partial dependence plots literally as they can be affected by correlations present within the data.

### What to look for

In many cases, Partial Dependence Plots can also be useful for detecting whether or not certain signals or relationships exist within data that should not be unnecessarily trusted.

For example, the presence of ‘spikes’ or ‘edges’ on PDPs could indicate that there may be outliers or unbalanced classes present in the data contributing disproportionately more heavily to results than other points within those classes. If this is the case, these points should be removed or addressed prior to further analysis if possible; otherwise, results could be distorted, leading to inaccurate conclusions being drawn.

PDP plots are a powerful tool for interpreting machine learning models, but they need to be used carefully and knowledgeably if accurate results and conclusions are to be obtained from them. As such, if you don’t have a strong background in statistics and machine learning principles then it would be wise to consult someone who does before attempting to use them or make any assumptions based upon them.

## Using partial dependence in machine learning

Partial dependence is a powerful tool for understanding the relationship between machine learning models and their input features, allowing data scientists to assess both the global and local effects of input variables on the prediction task at hand.

By using partial dependence plots to visualize the relationships between one or more input features and the model’s predictions, data scientists can gain crucial insight into which variables are more influential than others, helping them to make informed decisions about feature engineering and feature selection.

### Model training

The most common way of leveraging partial dependence when training models is to apply it during model inference. By carefully studying a wide range of partial dependence plots, data scientists can identify which patterns are being learned by the model and gain a much better understanding of what type of behavior each feature contributes to the overall predictive ability of their model.

This type of analysis can also help them determine how sensitive the model is to certain inputs or interactions between different features, allowing them to make informed decisions about whether certain features should be included or excluded from the dataset.

### Feature engineering

In addition to its use in model inference, partial dependence can also be used for feature engineering. By creating new features that combine two or more existing features, data scientists can create more powerful models with improved predictive accuracy.

For example, by combining two independent variables with linear combinations such as addition or multiplication, data scientists can create derived variables that capture complex relationships between those variables in a compact form. By studying these derived variables through partial dependence plots, they can quickly identify which combinations lead to improved performance on the specific target task they are trying to solve.

Moreover, partial dependence can also be used as part of an iterative modeling process to understand how changes made during training impact overall performance. By running experiments with different training configurations and measuring both global performance scores as well as individual feature importance scores via partial dependence plots, data scientists can more effectively identify which training parameters are most beneficial in achieving their desired goals.

Using partial dependence in machine learning is an incredibly powerful tool that helps data scientists better understand the intricate relationships between their models and their inputs so that they can quickly adjust their training configurations in order to achieve higher accuracy levels. With its combination of graphical visualizations and local explainability metrics, partial dependence has become an essential tool for any modern machine learning workflow.

## Measuring partial dependence

Partial dependence is a method of measuring the relationship between an individual predictor and the outcome in a predictive model.

This measure allows the user to assess how much of the variation in a model is due to a specific input variable.

PD can be measured through several means, including partial Cramer's V coefficient, partial F-test, and permutation importance scores. The Partial Cramer's V coefficient measures the linear correlation between a single predictor and the response variable; this allows us to assess the strength of the relationship between any single predictor and the outcome.

The Partial F-test measures how much of the variance in a model can be explained by a single predictor or combination of predictors. This test can help determine which predictors are most important when it comes to determining an outcome.

Permutation importance scores provide another measure for assessing how a single predictor affects an outcome by calculating changes in accuracy after removing that particular feature from a dataset. By comparing performance before and after permuting (or randomly shuffling) each input feature, we can see which variables are having the most impact on our prediction accuracy.

Overall, measuring PD allows us to determine just how influential each input variable is in predicting an output value with some degree of certainty and accuracy. This information is invaluable when it comes to understanding complex models such as random forests or neural networks, where it’s difficult to interpret what’s driving predictions without directly examining individual feature relationships.

## Types of partial dependence

Partial dependence is a powerful tool that allows us to explore the relationship between an individual variable and the outcome of interest. It provides insight into the way that a single variable impacts our model results and can be used for feature importance or even for interpreting complex models. However, it is important to understand that there are two main types of partial dependence: global and local.

Global partial dependence goes beyond just looking at one variable’s contribution, instead giving us insight into how multiple variables interact to produce a single output.

A global partial dependence plot shows how all model variables interact together to produce an outcome. This type of plot can help identify relationships between variables that might not be obvious when analyzing the data individually.

Local partial dependence focuses on only one variable at a time, allowing us to analyze its contribution to the model independently from other features. This type of plot is useful when you want to get more detail about how an individual variable affects your model outcomes. While global partial dependence gives us a much more comprehensive analysis, local partial dependence can be useful for exploring specific relationships between inputs and outputs in more detail.

So, partial dependence is an effective tool that can help researchers better understand how individual variables contribute to their models' performance. Additionally, global partial dependence provides insight into how multiple variables interact with each other, while local studies focus on one variable at a time.

With both types of studies, we can learn which features are most important in our models and how they contribute to overall accuracy or performance metrics.

## Summary: Partial dependence

Partial dependence is a useful tool in data science and machine learning that can help us explore important relationships between input variables and the predicted output of a model. It provides insight into what factors matter most when making predictions, and it can be used to explore other aspects beyond the interpretation of a model's predictions.

However, there are some limitations to partial dependence, such as its reliance on a linear form for the relationship between input variables and output, as well as its reliance on being able to calculate joint distributions of multiple input variables. Also, caution should be taken when using partial dependence as it is not an exact representation of how inputs affect outputs.

Despite these limitations, partial dependence is still a very powerful tool for gaining insight into prediction models and exploring relationships between input variables and response variables.

Partial dependence can be used in various applications, from the traditional analysis of large datasets to more complex machine learning projects involving neural networks or deep learning. In addition, partial dependence can also be measured with various metrics in order to assess the strength or importance of certain input variables in producing results.

In summary, partial dependence is an effective tool for understanding how inputs influence predictions made by models. Despite its limitations, partial dependence plots offer invaluable insight into predictive models that would not otherwise be available, making them an invaluable asset for data scientists and engineers alike.