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The concept of a hypothesis is fundamental in Machine Learning and data science endeavours. In the realm of machine learning, a hypothesis serves as an initial assumption made by data scientists and ML professionals when attempting to address a problem. Machine learning involves conducting experiments based on past experiences, and these hypotheses are crucial in formulating potential solutions.

It’s important to note that in machine learning discussions, the terms “hypothesis” and “model” are sometimes used interchangeably. However, a hypothesis represents an assumption, while a model is a mathematical representation employed to test that hypothesis. This section on “Hypothesis in Machine Learning” explores key aspects related to hypotheses in machine learning and their significance.

Table of Content

How does a Hypothesis work?

Hypothesis space and representation in machine learning, hypothesis in statistics, faqs on hypothesis in machine learning.

A hypothesis in machine learning is the model’s presumption regarding the connection between the input features and the result. It is an illustration of the mapping function that the algorithm is attempting to discover using the training set. To minimize the discrepancy between the expected and actual outputs, the learning process involves modifying the weights that parameterize the hypothesis. The objective is to optimize the model’s parameters to achieve the best predictive performance on new, unseen data, and a cost function is used to assess the hypothesis’ accuracy.

In most supervised machine learning algorithms, our main goal is to find a possible hypothesis from the hypothesis space that could map out the inputs to the proper outputs. The following figure shows the common method to find out the possible hypothesis from the Hypothesis space:

Hypothesis Space (H)

Hypothesis space is the set of all the possible legal hypothesis. This is the set from which the machine learning algorithm would determine the best possible (only one) which would best describe the target function or the outputs.

Hypothesis (h)

A hypothesis is a function that best describes the target in supervised machine learning. The hypothesis that an algorithm would come up depends upon the data and also depends upon the restrictions and bias that we have imposed on the data.

The Hypothesis can be calculated as:

[Tex]y = mx + b [/Tex]

• m = slope of the lines
• b = intercept

To better understand the Hypothesis Space and Hypothesis consider the following coordinate that shows the distribution of some data:

Say suppose we have test data for which we have to determine the outputs or results. The test data is as shown below:

We can predict the outcomes by dividing the coordinate as shown below:

So the test data would yield the following result:

But note here that we could have divided the coordinate plane as:

The way in which the coordinate would be divided depends on the data, algorithm and constraints.

• All these legal possible ways in which we can divide the coordinate plane to predict the outcome of the test data composes of the Hypothesis Space.
• Each individual possible way is known as the hypothesis.

Hence, in this example the hypothesis space would be like:

The hypothesis space comprises all possible legal hypotheses that a machine learning algorithm can consider. Hypotheses are formulated based on various algorithms and techniques, including linear regression, decision trees, and neural networks. These hypotheses capture the mapping function transforming input data into predictions.

Hypothesis Formulation and Representation in Machine Learning

Hypotheses in machine learning are formulated based on various algorithms and techniques, each with its representation. For example:

• Linear Regression : [Tex] h(X) = \theta_0 + \theta_1 X_1 + \theta_2 X_2 + … + \theta_n X_n[/Tex]
• Decision Trees : [Tex]h(X) = \text{Tree}(X)[/Tex]
• Neural Networks : [Tex]h(X) = \text{NN}(X)[/Tex]

In the case of complex models like neural networks, the hypothesis may involve multiple layers of interconnected nodes, each performing a specific computation.

Hypothesis Evaluation:

The process of machine learning involves not only formulating hypotheses but also evaluating their performance. This evaluation is typically done using a loss function or an evaluation metric that quantifies the disparity between predicted outputs and ground truth labels. Common evaluation metrics include mean squared error (MSE), accuracy, precision, recall, F1-score, and others. By comparing the predictions of the hypothesis with the actual outcomes on a validation or test dataset, one can assess the effectiveness of the model.

Hypothesis Testing and Generalization:

Once a hypothesis is formulated and evaluated, the next step is to test its generalization capabilities. Generalization refers to the ability of a model to make accurate predictions on unseen data. A hypothesis that performs well on the training dataset but fails to generalize to new instances is said to suffer from overfitting. Conversely, a hypothesis that generalizes well to unseen data is deemed robust and reliable.

The process of hypothesis formulation, evaluation, testing, and generalization is often iterative in nature. It involves refining the hypothesis based on insights gained from model performance, feature importance, and domain knowledge. Techniques such as hyperparameter tuning, feature engineering, and model selection play a crucial role in this iterative refinement process.

In statistics , a hypothesis refers to a statement or assumption about a population parameter. It is a proposition or educated guess that helps guide statistical analyses. There are two types of hypotheses: the null hypothesis (H0) and the alternative hypothesis (H1 or Ha).

• Null Hypothesis(H 0 ): This hypothesis suggests that there is no significant difference or effect, and any observed results are due to chance. It often represents the status quo or a baseline assumption.
• Aternative Hypothesis(H 1 or H a ): This hypothesis contradicts the null hypothesis, proposing that there is a significant difference or effect in the population. It is what researchers aim to support with evidence.

Q. How does the training process use the hypothesis?

The learning algorithm uses the hypothesis as a guide to minimise the discrepancy between expected and actual outputs by adjusting its parameters during training.

Q. How is the hypothesis’s accuracy assessed?

Usually, a cost function that calculates the difference between expected and actual values is used to assess accuracy. Optimising the model to reduce this expense is the aim.

Q. What is Hypothesis testing?

Hypothesis testing is a statistical method for determining whether or not a hypothesis is correct. The hypothesis can be about two variables in a dataset, about an association between two groups, or about a situation.

Q. What distinguishes the null hypothesis from the alternative hypothesis in machine learning experiments?

The null hypothesis (H0) assumes no significant effect, while the alternative hypothesis (H1 or Ha) contradicts H0, suggesting a meaningful impact. Statistical testing is employed to decide between these hypotheses.

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1. Introduction

Machine-learning algorithms come with implicit or explicit assumptions about the actual patterns in the data. Mathematically, this means that each algorithm can learn a specific family of models, and that family goes by the name of the hypothesis space.

In this tutorial, we’ll talk about hypothesis spaces and how to choose the right one for the data at hand.

2. Hypothesis Spaces

Let’s say that we have a binary classification task and that the data are two-dimensional. Our goal is to find a model that classifies objects as positive or negative. Applying Logistic Regression , we can get the models of the form:

which estimate the probability that the object at hand is positive.

2.1. Hypotheses and Assumptions

The underlying assumption of hypotheses ( 1 ) is that the boundary separating the positive from negative objects is a straight line. So, every hypothesis from this space corresponds to a straight line in a 2D plane. For instance:

2.2. Regression

3. expressivity of a hypothesis space.

We could informally say that one hypothesis space is more expressive than another if its hypotheses are more diverse and complex.

We may underfit the data if our algorithm’s hypothesis space isn’t expressive enough. For instance, linear hypotheses aren’t particularly good options if the actual data are extremely non-linear:

So, training an algorithm that has a very expressive space increases the chance of completely capturing the patterns in the data. However, it also increases the risk of overfitting. For instance, a space containing the hypotheses of the form:

would start modelling the noise, which we see from its decision boundary:

Such models would generalize poorly to unseen data.

3.1. Expressivity vs. Interpretability

Additionally, even if a complex hypothesis has a good generalization capability, it may be unusable in practice because it’s too complicated to understand or compute. What’s more, intricated hypotheses offer limited insight into the real-world process that generated the data. For example, a quadratic model:

4. How to Choose the Hypothesis Space?

We need to find the right balance between expressivity and simplicity. Unfortunately, that’s easier said than done. Most of the time, we need to rely on our intuition about the data.

So, we should start by exploring the dataset, using visualizations as much as possible. For instance, we can conclude that a straight line isn’t likely to be an adequate boundary for the above classification data. However, a high-order curve would probably be too complex even though it might split the dataset into two classes without an error.

A second-degree curve might be the compromise we seek, but we aren’t sure. So, we start with the space of quadratic hypotheses:

We get a model whose decision boundary appears to be a good fit even though it misclassifies some objects:

Since we’re satisfied with the model, we can stop here. If that hadn’t been the case, we could have tried a space of cubic models. The idea would be to iteratively try incrementally complex families until finding a model that both performs well and is easy to understand.

4. Conclusion

In this article, we talked about hypotheses spaces in machine learning. An algorithm’s hypothesis space contains all the models it can learn from any dataset.

The algorithms with too expressive spaces can generalize poorly to unseen data and be too complex to understand, whereas those with overly simple hypotheses may underfit the data. So, when applying machine-learning algorithms in practice, we need to find the right balance between expressivity and simplicity.

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Best Guesses: Understanding The Hypothesis in Machine Learning

• February 22, 2024
• General , Supervised Learning , Unsupervised Learning

Machine learning is a vast and complex field that has inherited many terms from other places all over the mathematical domain.

It can sometimes be challenging to get your head around all the different terminologies, never mind trying to understand how everything comes together.

In this blog post, we will focus on one particular concept: the hypothesis.

While you may think this is simple, there is a little caveat regarding machine learning.

The statistics side and the learning side.

Don’t worry; we’ll do a full breakdown below.

You’ll learn the following:

What Is a Hypothesis in Machine Learning?

• Is This any different than the hypothesis in statistics?
• What is the difference between the alternative hypothesis and the null?
• Why do we restrict hypothesis space in artificial intelligence?
• Example code performing hypothesis testing in machine learning

In machine learning, the term ‘hypothesis’ can refer to two things.

First, it can refer to the hypothesis space, the set of all possible training examples that could be used to predict or answer a new instance.

Second, it can refer to the traditional null and alternative hypotheses from statistics.

Since machine learning works so closely with statistics, 90% of the time, when someone is referencing the hypothesis, they’re referencing hypothesis tests from statistics.

Is This Any Different Than The Hypothesis In Statistics?

In statistics, the hypothesis is an assumption made about a population parameter.

The statistician’s goal is to prove it true or disprove it.

This will take the form of two different hypotheses, one called the null, and one called the alternative.

Usually, you’ll establish your null hypothesis as an assumption that it equals some value.

For example, in Welch’s T-Test Of Unequal Variance, our null hypothesis is that the two means we are testing (population parameter) are equal.

This means our null hypothesis is that the two population means are the same.

We run our statistical tests, and if our p-value is significant (very low), we reject the null hypothesis.

This would mean that their population means are unequal for the two samples you are testing.

Usually, statisticians will use the significance level of .05 (a 5% risk of being wrong) when deciding what to use as the p-value cut-off.

What Is The Difference Between The Alternative Hypothesis And The Null?

The null hypothesis is our default assumption, which we are trying to prove correct.

The alternate hypothesis is usually the opposite of our null and is much broader in scope.

For most statistical tests, the null and alternative hypotheses are already defined.

You are then just trying to find “significant” evidence we can use to reject our null hypothesis.

These two hypotheses are easy to spot by their specific notation. The null hypothesis is usually denoted by H₀, while H₁ denotes the alternative hypothesis.

Example Code Performing Hypothesis Testing In Machine Learning

Since there are many different hypothesis tests in machine learning and data science, we will focus on one of my favorites.

This test is Welch’s T-Test Of Unequal Variance, where we are trying to determine if the population means of these two samples are different.

There are a couple of assumptions for this test, but we will ignore those for now and show the code.

You can read more about this here in our other post, Welch’s T-Test of Unequal Variance .

We see that our p-value is very low, and we reject the null hypothesis.

What Is The Difference Between The Biased And Unbiased Hypothesis Spaces?

The difference between the Biased and Unbiased hypothesis space is the number of possible training examples your algorithm has to predict.

The unbiased space has all of them, and the biased space only has the training examples you’ve supplied.

Since neither of these is optimal (one is too small, one is much too big), your algorithm creates generalized rules (inductive learning) to be able to handle examples it hasn’t seen before.

Here’s an example of each:

Example of The Biased Hypothesis Space In Machine Learning

The Biased Hypothesis space in machine learning is a biased subspace where your algorithm does not consider all training examples to make predictions.

This is easiest to see with an example.

Let’s say you have the following data:

Happy  and  Sunny  and  Stomach Full  = True

Whenever your algorithm sees those three together in the biased hypothesis space, it’ll automatically default to true.

This means when your algorithm sees:

Sad  and  Sunny  And  Stomach Full  = False

It’ll automatically default to False since it didn’t appear in our subspace.

This is a greedy approach, but it has some practical applications.

Example of the Unbiased Hypothesis Space In Machine Learning

The unbiased hypothesis space is a space where all combinations are stored.

We can use re-use our example above:

This would start to breakdown as

Happy  = True

Happy  and  Sunny  = True

Happy  and  Stomach Full  = True

Let’s say you have four options for each of the three choices.

This would mean our subspace would need 2^12 instances (4096) just for our little three-word problem.

This is practically impossible; the space would become huge.

So while it would be highly accurate, this has no scalability.

More reading on this idea can be found in our post, Inductive Bias In Machine Learning .

Why Do We Restrict Hypothesis Space In Artificial Intelligence?

We have to restrict the hypothesis space in machine learning. Without any restrictions, our domain becomes much too large, and we lose any form of scalability.

This is why our algorithm creates rules to handle examples that are seen in production. 

This gives our algorithms a generalized approach that will be able to handle all new examples that are in the same format.

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Hypothesis Space

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• Hendrik Blockeel  

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Model space

The hypothesis space used by a machine learning system is the set of all hypotheses that might possibly be returned by it. It is typically defined by a Hypothesis Language , possibly in conjunction with a Language Bias .

Motivation and Background

Many machine learning algorithms rely on some kind of search procedure: given a set of observations and a space of all possible hypotheses that might be considered (the “hypothesis space”), they look in this space for those hypotheses that best fit the data (or are optimal with respect to some other quality criterion).

To describe the context of a learning system in more detail, we introduce the following terminology. The key terms have separate entries in this encyclopedia, and we refer to those entries for more detailed definitions.

A learner takes observations as inputs. The Observation Language is the language used to describe these observations.

The hypotheses that a learner may produce, will be formulated in...

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De Raedt, L. (1992). Interactive theory revision: An inductive logic programming approach . London: Academic Press.

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Nédellec, C., Adé, H., Bergadano, F., & Tausend, B. (1996). Declarative bias in ILP. In L. De Raedt (Ed.), Advances in inductive logic programming . Frontiers in artificial intelligence and applications (Vol. 32, pp. 82–103). Amsterdam: IOS Press.

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Blockeel, H. (2011). Hypothesis Space. In: Sammut, C., Webb, G.I. (eds) Encyclopedia of Machine Learning. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30164-8_373

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Introduction to the hypothesis space and the bias-variance tradeoff in machine learning.

In this post, we introduce the hypothesis space and discuss how machine learning models function as hypotheses. Furthermore, we discuss the challenges encountered when choosing an appropriate machine learning hypothesis and building a model, such as overfitting, underfitting, and the bias-variance tradeoff.

The hypothesis space in machine learning is a set of all possible models that can be used to explain a data distribution given the limitations of that space. A linear hypothesis space is limited to the set of all linear models. If the data distribution follows a non-linear distribution, the linear hypothesis space might not contain a model that is appropriate for our needs.

To understand the concept of a hypothesis space, we need to learn to think of machine learning models as hypotheses.

The Machine Learning Model as Hypothesis

Generally speaking, a hypothesis is a potential explanation for an outcome or a phenomenon. In scientific inquiry, we test hypotheses to figure out how well and if at all they explain an outcome. In supervised machine learning, we are concerned with finding a function that maps from inputs to outputs.

But machine learning is inherently probabilistic. It is the art and science of deriving useful hypotheses from limited or incomplete data. Our functions are not axioms that explain the data perfectly, and for most real-life problems, we will never have all the data that exists. Accordingly, we will not find the one true function that perfectly describes the data. Instead, we find a function through training a model to map from known training input to known training output. This way, the model gradually approximates the assumed true function that describes the distribution of the data. So we treat our model as a hypothesis that needs to be tested as to how well it explains the output from a given input. We do this using a test or validation data set.

The Hypothesis Space

During the training process, we select a model from a hypothesis space that is subject to our constraints. For example, a linear hypothesis space only provides linear models. We can approximate data that follows a quadratic distribution using a model from the linear hypothesis space.

Of course, a linear model will never have the same predictive performance as a quadratic model, so we can adjust our hypothesis space to also include non-linear models or at least quadratic models.

The Data Generating Process

The data generating process describes a hypothetical process subject to some assumptions that make training a machine learning model possible. We need to assume that the data points are from the same distribution but are independent of each other. When these requirements are met, we say that the data is independent and identically distributed (i.i.d.).

Independent and Identically Distributed Data

How can we assume that a model trained on a training set will perform better than random guessing on new and previously unseen data? First of all, the training data needs to come from the same or at least a similar problem domain. If you want your model to predict stock prices, you need to train the model on stock price data or data that is similarly distributed. It wouldn’t make much sense to train it on whether data. Statistically, this means the data is identically distributed . But if data comes from the same problem, training data and test data might not be completely independent. To account for this, we need to make sure that the test data is not in any way influenced by the training data or vice versa. If you use a subset of the training data as your test set, the test data evidently is not independent of the training data. Statistically, we say the data must be independently distributed .

Overfitting and Underfitting

We want to select a model from the hypothesis space that explains the data sufficiently well. During training, we can make a model so complex that it perfectly fits every data point in the training dataset. But ultimately, the model should be able to predict outputs on previously unseen input data. The ability to do well when predicting outputs on previously unseen data is also known as generalization. There is an inherent conflict between those two requirements.

If we make the model so complex that it fits every point in the training data, it will pick up lots of noise and random variation specific to the training set, which might obscure the larger underlying patterns. As a result, it will be more sensitive to random fluctuations in new data and predict values that are far off. A model with this problem is said to overfit the training data and, as a result, to suffer from high variance .

To avoid the problem of overfitting, we can choose a simpler model or use regularization techniques to prevent the model from fitting the training data too closely. The model should then be less influenced by random fluctuations and instead, focus on the larger underlying patterns in the data. The patterns are expected to be found in any dataset that comes from the same distribution. As a consequence, the model should generalize better on previously unseen data.

But if we go too far, the model might become too simple or too constrained by regularization to accurately capture the patterns in the data. Then the model will neither generalize well nor fit the training data well. A model that exhibits this problem is said to underfit the data and to suffer from high bias . If the model is too simple to accurately capture the patterns in the data (for example, when using a linear model to fit non-linear data), its capacity is insufficient for the task at hand.

When training neural networks, for example, we go through multiple iterations of training in which the model learns to fit an increasingly complex function to the data. Typically, your training error will decrease during learning the more complex your model becomes and the better it learns to fit the data. In the beginning, the training error decreases rapidly. In later training iterations, it typically flattens out as it approaches the minimum possible error. Your test or generalization error should initially decrease as well, albeit likely at a slower pace than the training error. As long as the generalization error is decreasing, your model is underfitting because it doesn’t live up to its full capacity. After a number of training iterations, the generalization error will likely reach a trough and start to increase again. Once it starts to increase, your model is overfitting, and it is time to stop training.

Ideally, you should stop training once your model reaches the lowest point of the generalization error. The gap between the minimum generalization error and no error at all is an irreducible error term known as the Bayes error that we won’t be able to completely get rid of in a probabilistic setting. But if the error term seems too large, you might be able to reduce it further by collecting more data, manipulating your model’s hyperparameters, or altogether picking a different model.

Bias Variance Tradeoff

We’ve talked about bias and variance in the previous section. Now it is time to clarify what we actually mean by these terms.

Understanding Bias and Variance

In a nutshell, bias measures if there is any systematic deviation from the correct value in a specific direction. If we could repeat the same process of constructing a model several times over, and the results predicted by our model always deviate in a certain direction, we would call the result biased.

Variance measures how much the results vary between model predictions. If you repeat the modeling process several times over and the results are scattered all across the board, the model exhibits high variance.

In their book “Noise” Daniel Kahnemann and his co-authors provide an intuitive example that helps understand the concept of bias and variance. Imagine you have four teams at the shooting range.

Team B is biased because the shots of its team members all deviate in a certain direction from the center. Team B also exhibits low variance because the shots of all the team members are relatively concentrated in one location. Team C has the opposite problem. The shots are scattered across the target with no discernible bias in a certain direction. Team D is both biased and has high variance. Team A would be the equivalent of a good model. The shots are in the center with little bias in one direction and little variance between the team members.

Generally speaking, linear models such as linear regression exhibit high bias and low variance. Nonlinear algorithms such as decision trees are more prone to overfitting the training data and thus exhibit high variance and low bias.

A linear model used with non-linear data would exhibit a bias to predict data points along a straight line instead of accomodating the curves. But they are not as susceptible to random fluctuations in the data. A nonlinear algorithm that is trained on noisy data with lots of deviations would be more capable of avoiding bias but more prone to incorporate the noise into its predictions. As a result, a small deviation in the test data might lead to very different predictions.

To get our model to learn the patterns in data, we need to reduce the training error while at the same time reducing the gap between the training and the testing error. In other words, we want to reduce both bias and variance. To a certain extent, we can reduce both by picking an appropriate model, collecting enough training data, selecting appropriate training features and hyperparameter values. At some point, we have to trade-off between minimizing bias and minimizing variance. How you balance this trade-off is up to you.

The Bias Variance Decomposition

Mathematically, the total error can be decomposed into the bias and the variance according to the following formula.

Remember that Bayes’ error is an error that cannot be eliminated.

Our machine learning model represents an estimating function \hat f(X) for the true data generating function f(X) where X represents the predictors and y the output values.

Now the mean squared error of our model is the expected value of the squared difference of the output produced by the estimating function \hat f(X) and the true output Y.

The bias is a systematic deviation from the true value. We can measure it as the squared difference between the expected value produced by the estimating function (the model) and the values produced by the true data-generating function.

Of course, we don’t know the true data generating function, but we do know the observed outputs Y, which correspond to the values generated by f(x) plus an error term.

The variance of the model is the squared difference between the expected value and the actual values of the model.

Now that we have the bias and the variance, we can add them up along with the irreducible error to get the total error.

A machine learning model represents an approximation to the hypothesized function that generated the data. The chosen model is a hypothesis since we hypothesize that this model represents the true data generating function.

We choose the hypothesis from a hypothesis space that may be subject to certain constraints. For example, we can constrain the hypothesis space to the set of linear models.

When choosing a model, we aim to reduce the bias and the variance to prevent our model from either overfitting or underfitting the data. In the real world, we cannot completely eliminate bias and variance, and we have to trade-off between them. The total error produced by a model can be decomposed into the bias, the variance, and irreducible (Bayes) error.

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What is the difference between hypothesis space and representational capacity?

I am reading Goodfellow et al Deeplearning Book . I found it difficult to understand the difference between the definition of the hypothesis space and representation capacity of a model.

In Chapter 5 , it is written about hypothesis space:

One way to control the capacity of a learning algorithm is by choosing its hypothesis space, the set of functions that the learning algorithm is allowed to select as being the solution.

And about representational capacity:

The model specifies which family of functions the learning algorithm can choose from when varying the parameters in order to reduce a training objective. This is called the representational capacity of the model.

If we take the linear regression model as an example and allow our output $y$ to takes polynomial inputs, I understand the hypothesis space as the ensemble of quadratic functions taking input $x$ , i.e $y = a_0 + a_1x + a_2x^2$ .

How is it different from the definition of the representational capacity, where parameters are $a_0$ , $a_1$ and $a_2$ ?

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3 Answers 3

Consider a target function $f: x \mapsto f(x)$ .

A hypothesis refers to an approximation of $f$ . A hypothesis space refers to the set of possible approximations that an algorithm can create for $f$ . The hypothesis space consists of the set of functions the model is limited to learn. For instance, linear regression can be limited to linear functions as its hypothesis space, or it can be expanded to learn polynomials.

The representational capacity of a model determines the flexibility of it, its ability to fit a variety of functions (i.e. which functions the model is able to learn), at the same. It specifies the family of functions the learning algorithm can choose from.

• 1 $\begingroup$ Does it mean that the set of functions described by the representational capacity is strictly included in the hypothesis space ? By definition, is it possible to have functions in the hypothesis space NOT described in the representational capacity ? $\endgroup$ –  Qwarzix Commented Aug 23, 2018 at 8:43
• $\begingroup$ It's still pretty confusing to me. Most sources say that a "model" is an instance (after execution/training on data) of a "learning algorithm". How, then, can a model specify the family of functions the learning algorithm can choose from? It doesn't make sense to me. The authors of the book should've explained these concepts in more depth. $\endgroup$ –  Talendar Commented Oct 9, 2020 at 13:09

A hypothesis space is defined as the set of functions $\mathcal H$ that can be chosen by a learning algorithm to minimize loss (in general).

$$\mathcal H = \{h_1, h_2,....h_n\}$$

The hypothesis class can be finite or infinite, for example a discrete set of shapes to encircle certain portion of the input space is a finite hypothesis space, whereas hpyothesis space of parametrized functions like neural nets and linear regressors are infinite.

Although the term representational capacity is not in the vogue a rough definition woukd be: The representational capacity of a model, is the ability of its hypothesis space to approximate a complex function, with 0 error, which can only be approximated by infinitely many hypothesis spaces whose representational capacity is equal to or exceed the representational capacity required to approximate the complex function.

The most popular measure of representational capacity is the $\mathcal V$ $\mathcal C$ Dimension of a model. The upper bound for VC dimension ( $d$ ) of a model is: $$d \leq \log_2| \mathcal H|$$ where $|H|$ is the cardinality of the set of hypothesis space.

A hypothesis space/class is the set of functions that the learning algorithm considers when picking one function to minimize some risk/loss functional.

The capacity of a hypothesis space is a number or bound that quantifies the size (or richness) of the hypothesis space, i.e. the number (and type) of functions that can be represented by the hypothesis space. So a hypothesis space has a capacity. The two most famous measures of capacity are VC dimension and Rademacher complexity.

In other words, the hypothesis class is the object and the capacity is a property (that can be measured or quantified) of this object, but there is not a big difference between hypothesis class and its capacity, in the sense that a hypothesis class naturally defines a capacity, but two (different) hypothesis classes could have the same capacity.

Note that representational capacity (not capacity , which is common!) is not a standard term in computational learning theory, while hypothesis space/class is commonly used. For example, this famous book on machine learning and learning theory uses the term hypothesis class in many places, but it never uses the term representational capacity .

Your book's definition of representational capacity is bad , in my opinion, if representational capacity is supposed to be a synonym for capacity , given that that definition also coincides with the definition of hypothesis class, so your confusion is understandable.

• 1 $\begingroup$ I agree with you. The authors of the book should've explained these concepts in more depth. Most sources say that a "model" is an instance (after execution/training on data) of a "learning algorithm". How, then, can a model specify the family of functions the learning algorithm can choose from? Also, as you pointed out, the definition of the terms "hypothesis space" and "representational capacity" given by the authors are practically the same, although they use the terms as if they represent different concepts. $\endgroup$ –  Talendar Commented Oct 9, 2020 at 13:18

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Could anyone explain these terms, "input space", "feature space", "sample space", "hypothesis space", "parameter space" with a concrete example?

People use these terms "input space", "feature space", "sample space", "hypothesis space", "parameter space" in machine learning.

Could anyone explain these terms with a concrete example, such as sklearn MNIST dataset? , which has 1797 Samples, 10 Classes, 8*8 Dimensionality and 17 Features.

Please do NOT talk about in general.

For example, in this particular case, is the feature space a set of 17 elements {0, 1, ..., 16}?

• machine-learning
• terminology

• $\begingroup$ A sample is the pair of known input and output. Thus sample space simply feature space plus "label" space. $\endgroup$ –  否开河 Commented Jan 9, 2022 at 16:08

We'll discuss each of the terms.

Input Space

It contains all the possible inputs for a model. Suppose the model takes in a vector, $input = [ x_1 , x_2]$ , where $x_1 , x_2 \in [ 1 , 10 ]$ , then we can have $10^{2}$ inputs. This constitutes the "input space". See here .

For the MNIST dataset, the dimensions of the image are 8 * 8 meaning 64 points. Now each point can have a value lying in the interval $[ 0 , 16 ]$ , so it can have 16 values. So the input space has a size of $16^{64}$ .

Feature Space

The multidimensional space in which are features is defined. Considering the above examples, we can have three samples,

$a_1 = [ 2 , 3 ] \\ a_2 = [ 7 , 4.5 ] \\ a_3 = [ 3.67 , 2 ]$

These vectors could be included in an n-dimensional space ( here n=2 for our case ). Hence, in our case, the 2D space where we can plot our features constitutes our "feature space".

For the MNIST dataset, the input vector has 64 elements which correspond to a 64-dimensional space ( feature space ).

See this answer.

Difference between input space and feature space. Input spaces include all possible inputs for our model. Feature spaces, on the other hand, include the feature vectors from a given set of data. They may not contain all the possible inputs for a model.

Hypothesis Space

Space which contains all the functions produced by a model. The functions map the inputs to their respective outputs. A model can output various functions ( or rather relationships between the inputs and outputs ) based on its learning. If you have a larger hypothesis space, the model cannot find the "best" one. See this answer .

For the MNIST dataset, as we calculated earlier, the size of the input space is $16^{64}$ . Each one of them can have any one of the 10 labels ( classes ). Hence, the size of the hypothesis space is $10^{16^{64}}.$

Parameter Space

For each model in ML, we have some parameters for the model. The space in which we can define these parameters ( or hyperparameters ) is our "parameter space". From Wikipedia's example , we can understand it,

The parameter space would differ for every model.

In a sine wave model ${\displaystyle y(t)=A\cdot \sin(\omega t+\phi > ),}y(t)=A\cdot \sin(\omega t+\phi )$ , the parameters are amplitude A > 0, angular frequency ω > 0, and phase φ ∈ S1. Thus the parameter space is ${\displaystyle R^{+}\times R^{+}\times S^{1}}$ .

• $\begingroup$ where x1,x2∈[0,10], then we can have 2pow10 inputs. Was it supposed to be 10pow2 ? $\endgroup$ –  Naveen Meka Commented Sep 25, 2019 at 4:42

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Version Space and List-Then-Eliminate Algorithm

Computer graphics opengl mini projects, download final year projects, consistent hypothesis, version space and list-then-eliminate algorithm.

An hypothesis h is said to be consistent hypothesis with a set of training examples D iff  h ( x ) = c ( x ) for each example in D ,

Video Tutorial on Consistent Hypothesis, Version Space and List-Then-Eliminate Algorithm

For Example:

h1 = (?, ?, No, ?, Many) – Consistent Hypothesis as it is consistent with all the training examples

h2 = (?, ?, No, ?, ?) – Inconsistent Hypothesis as it is inconsistent with first training example

Version Space

The version space VS H,D is the subset of the hypothesis from H consistent with the training example in D ,

List-Then-Eliminate algorithm

Steps in list-then-eliminate algorithm.

1. V ersionSpace = a list containing every hypothesis in H

2. For each training example, <a(x), c(x)> Remove from VersionSpace any hypothesis h for which h ( x ) != c ( x )

3. Output the list of hypotheses in VersionSpace .

F1  – > A, B

F2  – > X, Y

Here F1 and F2 are two features (attributes) with two possible values for each feature or attribute.

Instance Space: (A, X), (A, Y), (B, X), (B, Y) – 4 Examples

Hypothesis Space: (A, X), (A, Y), (A, ø ), (A, ?), (B, X), (B, Y), (B, ø ), (B, ?), ( ø , X), ( ø , Y), ( ø , ø ), ( ø , ?), ( ? , X), ( ? , Y), ( ? , ø ), ( ? , ?)  – 16 Hypothesis

Semantically Distinct Hypothesis : (A, X), (A, Y), (A, ?), (B, X), (B, Y), (B, ?), ( ? , X), ( ? , Y ( ? , ?), ( ø , ø ) – 10

List-Then-Eliminate Algorithm Steps

Version Space: (A, X), (A, Y), (A, ?), (B, X), (B, Y), (B, ?), (?, X), (?, Y) (?, ?), ( ø , ø ), •Training Instances

F1  F2  Target

A  X      Yes

A  Y      Yes

Consistent Hypothesis are (Version Space):   (A, ?), (?, ?)

Problems with List-Then-Eliminate Algorithm

The hypothesis space must be finite

Enumeration of all the hypothesis, rather inefficient

This tutorial discusses the Consistent Hypothesis, Version Space, and List-Then-Eliminate Algorithm in Machine Learning. If you like the tutorial share with your friends. Like the Facebook page for regular updates and YouTube channel for video tutorials.

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2 thoughts on “version space and list-then-eliminate algorithm”.

h1 = (?, ?, No, ?, Many) – Consistent Hypothesis as it is consistent with all the training examples can you please explain how was this hypothesis written why it is “many” why not ‘?’ please explain

See this video you’ll understand https://youtu.be/_FMDyEoIX3A

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Genetic algorithm: Hypothesis space search

As already understood from our illustrative example, it is clear that genetic algorithms employ a randomized beam search method to seek maximally fit hypotheses. In the hypothesis space search method, we can see that the gradient descent search in backpropagation moves smoothly from one hypothesis to another. On the other hand, the genetic algorithm search can move much more abruptly. It replaces the parent hypotheses with an offspring that can be very different from the parent. Due to this reason, genetic algorithm search has lower chances of it falling into the same kind of local minima that plaques the gradient descent methods.

There is one practical difficulty that is often encountered in genetic algorithms, it is crowding. Crowding can be defined as the phenomenon in which some individuals that are more fit in comparison to others, reproduce quickly, therefore the copies of this individual take over a larger fraction of the population. Most of the strategies used in the genetic algorithms are inspired by biological evolution. One such other strategy used is fitness sharing, in which the measured fitness of an individual is decreased by the presence of another individual of a similar kind. The third method is to restrict all the individuals to combine to form offspring. To better understand we can say that by allowing individuals of the same kind to recombine, clusters of similar individuals are formed, forming multiple subspecies in the population.

Another method would be to spatially distribute individuals and allow only nearby individuals to combine.

Population evolution and schema theorem.

The schema theorem of Holland is used to mathematically characterize the evolution over time of the population with respect to time. It is based on the concept of schema. So, what is schema? Schema is any string composed of 0s, and 1s, and *s, where * represents null, so a schema 0*10, is the same as 0010 and 0110. The schema theorem characterizes the evolution within a genetic algorithm on the basis of the number of instances representing each schema. Let us assume the m(s, t) to denote the number of instances of schema denoted by ‘s’, in the population at the time ‘t’, the expected value in the schema theorem is described as m(s, t+1), in terms of m(s, t), and the other parameters of the population, schema, and GA.

In a genetic algorithm, the evolution of the population depends on the selection step, the recombination step, and the mutation step. The schema theorem is one of the most widely used theorems in the characterization of population evolution within a genetic algorithm. If it fails to consider the positive effects of crossover and mutation, it is in a way incomplete. There are many other recent theoretical analyses that have been proposed, many of these analogies are based on models such as Markov chain models and the statistical mechanical model.

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Sample space for hypothseis, training data of bayes theorem

I am learning about Bayes theorem in machine learning .

$p(h/D) = \frac{p(D/h)p(h)}{p(D)}$

$p(h) = $prior probability of hypothesis h

$p(D)$ = prior probability of training data D

$p(h/D)$ = probability of h given D

$p(D/h)$ = probability of D given h

I am from mathematical background , so generally I calculates probability by using sets or area . I mean

$p(h)$ = cardinality of h / cardinality of sample space

$p(h)$ = area covered by h / total area

But when comes to machine learning $h$ is hypothesis and $D$ is training data , how it has to be imagined as a set or area and what is sample space ?

$D$ = Training Data = input to machine

$h$ = hypothesis = output given by machine

That is all i know .

Another doubt is its stated prior probability of h , whats make difference between the "probability of hypothesis h " and "probability of getting hypothesis h" (Since h is hypothesis output given by machine)

• machine-learning

The best way to think of it may be as follows:

$\Pr(D)$: This represents the probability of having observed the training data. Consider the sample space to be the set of possible sets of observed data. Each will be observed with some probability and that probability for the training set is represented by $\Pr(D)$.

$\Pr(h)$: I am not entirely sure I understand your last doubt, but the prior probability of the hypothesis is the probability ascribed to the hypothesis $h$ being true prior to drawing the sample. Perhaps you can consider the sample space as the set of possible hypotheses.

• $\begingroup$ so $Pr(h)$ = version space / hypothesis space ? $\endgroup$ –  hanugm Commented Apr 7, 2014 at 6:29

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Machine learning with monotonic constraint for geotechnical engineering applications: an example of slope stability prediction

Machine learning (ML) algorithms have been widely applied to analyze geotechnical engineering problems due to recent advances in data science. However, flexible ML models trained with limited data can exhibit unexpected behaviors, leading to low interpretability and physical inconsistency, thus, reducing the reliability and robustness of ML models for risk forecasting and engineering applications. As input features for geotechnical engineering applications often represent physical parameters following intrinsic and often monotonic relationships, incorporating monotonicity into ML models can help ensure the physical realism of model outputs. In this study, monotonicity was introduced as a soft constraint into artificial neural network (ANN) models, and their results were compared with several benchmark ML models. During the training process, data augmentation and point-wise gradient were used to evaluate the monotonicity of model predictions, and monotonicity violations were minimized through a modified loss function. A compilation of slope stability case histories from the literature was used for model development, benchmarking their performance, and evaluating the effects of monotonicity constraints. Cross-validation procedures were used for all model performance evaluations to reduce bias in sample selections. Results showed that unconstrained ML models produced predictions that violate monotonicity in many parts of the input space. However, by adding monotonicity constraints into ANN models, monotonicity violations were effectively reduced while maintaining relatively high performance, thus providing a more robust and interpretable prediction. Using slope stability prediction as a proxy, the methods developed in this study to incorporate monotonicity constraints into ML models can be applied to many geotechnical engineering applications. The proposed approach enhances the reliability and interpretability of ML models, resulting in more accurate and consistent outcomes for real-world applications.

• Interpretability;
• Machine learning;
• Monotonicity;
• Slope stability

News and views for the UB community

• Stories >

Eight UB researchers awarded over $4.7 million in NSF CAREER awards

research news

By ELIZABETH EGAN, PETER MURPHY and LAURIE KAISER

Published June 27, 2024

Eight UB researchers — seven from the School of Engineering and Applied Sciences (SEAS) and one from the School of Pharmacy and Pharmaceutical Sciences (SPPS) — have received National Science Foundation CAREER awards, one of the nation’s most prestigious honors for early-career engineers and scientists.

CAREER grants provide scholars with funding to conduct research and develop educational programming for K-12 students, university students and members of the public.

The SEAS recipients are Courtney Faber, Luis Herrera, Craig Snoeyink, Kang Sun, Yinyin Ye, Zhuoyue Zhao and Shaofeng Zou. The SPPS recipient is Jason Sprowl.

Together, the eight grantees will receive more than $4.7 million for projects that address pressing societal problems such as the need for more reliable artificial intelligence algorithms, preventing deaths from bacterial infections, mapping air pollution and better understanding how glucose moves throughout the human body.

“We take great pride in our eight faculty members who have been honored with this prestigious NSF award,” says Venu Govindaraju, vice president for research and economic development. “Their exceptional research is integral to UB’s mission of fostering a better world for all.”

Among the support that awardees receive is guidance from UB’s Office of Research Advancement, which is overseen by Chitra Rajan, associate vice president for research advancement. The office is managed by three co-directors — Joanna Tate, Maggie Shea and Menna Mbah — and provides a comprehensive suite of services, including proposal management, scientific editing, graphics and help with non-technical parts of the proposal. 

These services, Rajan says, play a critical role in assisting faculty members in submitting high-quality proposals.

UB’s awardees are:

Courtney Faber , assistant professor of engineering education; award amount: $590,963.

When a research team is made up of people with various engineering and education backgrounds, different ideas of what knowledge is and how it is acquired can hinder team members’ ability to work cohesively.

Having firsthand experience with this issue, Faber’s goal is to support engineering education researchers who find themselves in a similar situation. 

She will facilitate interdisciplinary work by identifying barriers that research teams face related to differences in thinking and creating ways to bring them to the surface for discussion before they become a problem.

“It’s important for the field of engineering education to be able to do this type of interdisciplinary work,” Faber says. “The problems we are trying to solve are very complex and require an interdisciplinary approach to make space for diversity of thinking.”

The project will involve observing research teams and conducting interviews to see how they function together, as well as how individual members think independently of the group.

Faber plans to develop trainings that new and established engineering education researchers can freely access.

She also hopes to create a tool that assists research groups in integrating approaches and goals that might otherwise be problematic for a group. The tool could be as simple as a one-page guide that provides questions to be considered throughout the research process to help identify where a team’s ideas might differ across various aspects of their research. 

Luis Herrera , assistant professor of electrical engineering; award amount: $500,000.

Herrera’s research lies at the intersection of power electronics, power systems and control theory.

With this grant, he is developing different control methods to promote the wider adoption of direct current (DC) microgrids, which can run more efficiently than the more commonly used AC (alternating current) microgrids.

“Currently, DC electrical systems are primarily used in applications such as electric aircrafts, including the Boeing 787 Dreamliner; navy ships; and data centers,” Herrera says. “However, most renewable energy sources are interfaced to the AC power grid through an intermediate DC stage.”

More networks operated through DC grids could significantly increase energy efficiency, reduce losses and improve the overall operation of electrical systems, he explains.

This potential creates motivation for DC systems to be implemented in commonly used structures, such as residential and office buildings.

Graduate students will participate in a summer internship at the Air Force Research Laboratory through a partnership with the University of Dayton Research Institute.

Herrera also plans to create demonstrations of the research and present them to elementary, middle school and high school students, aiming to get students excited about STEM early in their academic careers. 

Craig Snoeyink , assistant professor of mechanical and aerospace engineering; award amount: $581,088 .

Water filtration, whiskey distillation and blood-based diagnostics are just a few of the potential applications of dielectrophoretic molecular transport (DMT), a process that uses strong electric fields to push solutes out of water. This even includes those such as sugar and alcohol that do not have an electrical charge.

DMT is not used, however, due to the inaccuracy of current mathematical models.

With his grant, Snoeyink will develop and validate models for DMT for use in these applications. With one of the first accurate models of DMT, the process could be used, for example, to clean water as effectively as a water filter that never needs to be changed.

Snoeyink notes that point-of-care diagnostics are another significant application. 

“Down the line, we could use this technology to separate blood into components we want to test and stuff we don’t, making medical diagnostics cheaper and more sensitive,” he says.

To help with testing and to offer students research opportunities that could propel them into graduate school, Snoeyink will teach a course for students to do research for the project as part of their curriculum. With his guidance, students will run tests and create their own hypothesis. He hopes students will have papers based on their research that will bolster their graduate school applications.

Jason A. Sprowl , assistant professor of pharmaceutical sciences; award amount: $746,886.

Sodium-glucose-linked transporters (SGLT) work like little doors in human cells that help bring in glucose, an important type of sugar that fuels the human body. Without the right amount of glucose, an individual can experience nutrient deficiencies and other health issues.

Unfortunately, cellular events that regulate SGLT activity are poorly understood. This is particularly true for tyrosine phosphorylation, a form of modification that can change protein structure and function.

For his research project, Sprowl will study how tyrosine phosphorylation regulates changes in glucose movement into cells. He’ll use techniques like genetic manipulation and mass spectrometry to see how changing the tyrosine phosphorylation state of SLGTs affects its ability to let glucose into a cell. Finally, he will try to figure out which tyrosine kinases are responsible for phosphorylating SGLTs.

The project also includes several strategies for educational improvements at the middle school, high school and university levels. They include highlighting the biological importance of SGLTs, as well as the training and recruitment of junior scientists who will lead future research efforts. Collectively, the project is expected to impact many scientific disciplines, including molecular, cellular and systems biology.

To improve basic scientific knowledge, generate a passion for research and improve leadership capabilities in the field of biological sciences, Sprowl plans to establish an annual summer research position for underprivileged high school students. He also will work with middle school educators to increase recognition of reproducible and high-quality science, and develop online content that will increase familiarity with transporter proteins. 

Kang Sun , assistant professor of civil, structural and environmental engineering; award amount: $643,562.

Sun has been interested in astronomy since he was a young child. He’s currently fascinated by the idea of pointing a space telescope toward Earth and imaging emission sources like celestial objects.

With the research grant, Sun will map global emission sources of gaseous air pollutants and greenhouse gases. Such gases are invisible to the human eye. While they can be detected by satellites, their images are naturally smeared due to wind dispersion.

“This research removes the smearing effect using a simple and elegant equation that originates from mass balance,” Sun explains. “The results are timely and precise estimates of emissions that can inform policy and scientific studies.”

Currently, the two mainstream, emission-estimating methods are bottom-up, accounting for activities on the ground and how they emit, and top-down, inferring emissions with observations, numerical models and complicated frameworks that are usually region-specific.

Sun’s method will fall within the scope of the latter but will work faster, be globally applicable and provide the high spatial resolutions that are more commonly achieved by the bottom-up method.

The results will resemble a space-telescope image, with significant emission sources standing out like galaxies and smaller sources, such as towns and power plants, sprinkled about like star clusters.

By the end of the five-year study, Sun hopes students and educators may use his open-source algorithms to generate satellite-based concentration and emission maps on their personal computers. 

Yinyin Ye , assistant professor of civil, structural and environmental engineering; award amount: $580,393.

Bacterial infections cause more than 300,000 deaths annually in the United States. Many of these infections are triggered by proteins secreted from bacteria in lipid-containing particles called extracellular vesicles (EV). These harmful materials move from the human body through feces into the sewer systems, where their fate is not fully understood.

With the research grant, Ye will monitor EV persistence and stability in wastewater and throughout the wastewater-treatment process. She will analyze functions of environmental EV and what contents are packed in them, and develop an analysis method that integrates genome sequencing and proteomic analysis.

“If the vesicles preserve the function of virulence proteins in wastewater, we need to better understand the fate of the vesicles when they go through the treatment chain,” Ye says. “How are we able to minimize the health risks of vesicles after the treatment at the wastewater treatment plants? If they escape the treatment process and are still active, that can have certain health impacts.”

Ye’s project will focus on wastewater samples. However, these approaches can be applied to analyzing vesicles and their potential health risks in air dust, drinking water and rainwater, she notes. Ultimately, this work will help determine what harmful materials — if any — are still present after the wastewater-treatment process and how to remove them most effectively through disinfection.

She will also create hands-on activities to engage K-12 and undergraduate students in learning about wastewater microbiome analysis and microbial risk mitigation for public health and potentially build their interest in environmental engineering.

Zhuoyue Zhao , assistant professor of computer science and engineering; award amount: $599,977.

Today’s internet databases hold large volumes of data that are processed at higher speeds than ever before.

A new type of database system, hybrid transactional/analytical processing (HTAP), allows for real-time data analytics on databases that undergo constant updates.

“While real-time data analytics can provide valuable insights for applications such as marketing, fraud detection and supply chain analytics, it is increasingly hard to ensure a sufficiently low response time of query answering in existing HTAP systems,” Zhao says.

Approximate query processing (AQP) is a faster alternative that uses random sampling. However, many AQP prototypes and adopted systems sacrifice query efficiency or the ability to handle rapid updates correctly.

With the research grant, Zhao aims to support real-time data analytics on large and rapidly growing databases by enabling reliable AQP capabilities in HTAP systems, leading to increasingly demanding, real-time analytics applications.

“If this problem is solved, it will potentially make it possible to finally adopt AQP in many existing database systems and create sizable impacts on real-world data analytics applications,” Zhao explains.

Zhao will incorporate new material into existing UB undergraduate- and graduate-level courses, as well as offer tutorials and projects in various K-12 outreach and undergraduate experiential learning programs. 

Shaofeng Zou , assistant professor of electrical engineering; award amount: $520,000.

Reinforcement learning (RL) is a type of machine learning that trains autonomous robots, self-driving cars and other intelligent agents to make sequential decisions while interacting with an environment.

Many RL approaches assume the learned policy will be deployed in the same — or similar — environment as the one it was trained in. In most cases, however, the simulated environment is vastly different from the real world — such as when a real-world environment is mobile while a simulated environment is stationary. These differences often lead to major disruptions in industries using RL, including health care, critical infrastructure, transportation systems, education and more.

Zou’s award will fund his work developing RL algorithms that do not require excessive resources, and that will perform effectively under the most challenging conditions, including those outside of the training environment. According to Zou, the project could have a significant impact on both the theory and practice of sequential decision-making associated with RL in special education, intelligent transportation systems, wireless communication networks, power systems and drone networks.

“The activities in this project will provide concrete principles and design guidelines to achieve robustness in the face of model uncertainty,” Zou says. “Advances in machine learning and data science will transform modern humanity across nearly every industry. They are already the main driver of emerging technologies.

“The overarching goal of my research is to make machine learning and data science provably competent.”

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Exploring the nexus between green space availability, connection with nature, and pro-environmental behavior in the urban landscape.

1. Introduction

• To explore the impact of UGS availability on different levels of perceived CN.
• To understand the association between perceived CN and engagement in environmental activities, reflecting PEB.
• To investigate the relationships between demographic factors such as gender, age, education, and work status on EB.

2.1. Study Area

2.2. data collection, 2.3. measures and data variables, 2.4. statistical analysis, 3.1. respondent demographic characteristics, 3.2. connection with nature against ugs availability, 3.3. connection with nature and engagement in environmental activities, 3.4. factors influencing engagement in environmental activities (ea), 4. discussion, 4.1. ugss’ role in enhancing perceived cn, 4.2. connection to nature and its association with pro-environmental behavior, 4.3. influencing demographic factors of pro-environmental behavior, 4.4. implications, limitations, and future research, author contributions, informed consent statement, data availability statement, acknowledgments, conflicts of interest.

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Lahoti, S.A.; Dhyani, S.; Sahle, M.; Kumar, P.; Saito, O. Exploring the Nexus between Green Space Availability, Connection with Nature, and Pro-Environmental Behavior in the Urban Landscape. Sustainability 2024 , 16 , 5435. https://doi.org/10.3390/su16135435

Lahoti SA, Dhyani S, Sahle M, Kumar P, Saito O. Exploring the Nexus between Green Space Availability, Connection with Nature, and Pro-Environmental Behavior in the Urban Landscape. Sustainability . 2024; 16(13):5435. https://doi.org/10.3390/su16135435

Lahoti, Shruti Ashish, Shalini Dhyani, Mesfin Sahle, Pankaj Kumar, and Osamu Saito. 2024. "Exploring the Nexus between Green Space Availability, Connection with Nature, and Pro-Environmental Behavior in the Urban Landscape" Sustainability 16, no. 13: 5435. https://doi.org/10.3390/su16135435

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