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Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally close to its intrinsic dimension.
Dimensionality Reduction and feature extraction.pptxSivam Chinna
Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally close to its intrinsic dimension.
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Predicting Moscow Real Estate Prices with Azure Machine LearningLeo Salemann
With only three months' instruction, a five-person team uses Azure Machine Learning Studio to predict Moscow real estate prices based on property descriptors, macroeconomic indicators, and geospatial data.
Predicting Moscow Real Estate Prices with Azure Machine LearningKarunakar Kotha
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Have you ever been curious about using Machine Learning and statistical techniques in quantitative finance? This paper gives a brief overview step by step of variable selection, model evaluation, and other insights for when working with financial data
Statistical theory is a branch of mathematics and statistics that provides the foundation for understanding and working with data, making inferences, and drawing conclusions from observed phenomena. It encompasses a wide range of concepts, principles, and techniques for analyzing and interpreting data in a systematic and rigorous manner. Statistical theory is fundamental to various fields, including science, social science, economics, engineering, and more.
Predicting Moscow Real Estate Prices with Azure Machine LearningLeo Salemann
With only three months' instruction, a five-person team uses Azure Machine Learning Studio to predict Moscow real estate prices based on property descriptors, macroeconomic indicators, and geospatial data.
Predicting Moscow Real Estate Prices with Azure Machine LearningKarunakar Kotha
With only three months' instruction, a five-person team uses Azure Machine Learning Studio to predict Moscow real estate prices based on property descriptors, macroeconomic indicators, and geospatial data
Have you ever been curious about using Machine Learning and statistical techniques in quantitative finance? This paper gives a brief overview step by step of variable selection, model evaluation, and other insights for when working with financial data
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Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
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2. Principal Component Analysis (PCA)
The central idea of principal component analysis (PCA) is to reduce the
dimensionality of a data set consisting of a large number of interrelated
features while retaining as much as possible of the variation present in the
data set.
This is achieved by transforming to a new set of features, the principal
components (PCs), which are uncorrelated, and which are ordered so that
the first few retain most of the variation present in all of the original
features.
3. Principal Component Analysis (PCA)
Mathematics Behind PCA
PCA can be thought of as an unsupervised learning problem.
The whole process of PCA can be summarized as follows:
• Standardize the given set of d-dimensional samples using Z-score Normalization.
• Compute the covariance matrix of the standardized dataset.
• Compute the eigenvectors and the corresponding eigenvalues of the
covariance matrix.
• Sort the eigenvectors by decreasing order of eigenvalues and choose the
eigenvectors corresponding to the largest k eigenvalues to form a d × k
dimensional matrix W
• Use this d × k eigenvector matrix W to transform the samples onto the new subspace
4. Consider the following two-dimensional dataset with features x1 and x2:
Our goal is to use PCA to reduce the dimensions of our dataset from
two to one, that is, from to R2 to R
Principal Component Analysis (PCA)
5. Lets visualize the given data set:
Principal Component Analysis (PCA)
6. Principal Component Analysis (PCA)
1. Standardization of the given dataset
Suppose we want to perform PCA on two features - a person's age and weight. If the unit of
weight is in grams, then the magnitude of its spread or variance will be much larger than that
of the age feature.
The variance of the weight would be in the order of magnitude of say 10,000 while that of
age would be say 10.
As PCA uses the variance of each features to reduce the dimensionality, it would focus more
on extracting information from features with higher variances and ignore the other features of
less variance.
The way to overcome this is to initially perform standardization such that all the features are
transformed to the same unitless scale. In PCA, we perform Z-score normalization
7. Principal Component Analysis (PCA)
1. Standardization of the given dataset
Z-score normalization is performed as follows:
After performing this standardization, the transformed features would each have a mean
of 0 and a standard deviation of 1.
8. Principal Component Analysis (PCA)
1. Standardization of the given dataset
As an example, let's manually standardize our first feature x1. We first need to compute
the mean and the variance of this feature. We begin with the mean:
Next, let's compute the standard deviation:
9. Principal Component Analysis (PCA)
1. Standardization of the given dataset
Now, we can compute each scaled value of x1 as :
10. Principal Component Analysis (PCA)
1. Standardization of the given dataset
For example, to compute the first value:
We repeat this process for the rest of the values in the feature to finally obtain the scaled
feature z1.
11. Principal Component Analysis (PCA)
1. Standardization of the given dataset
Remember that we've only standardized the feature x1 - we need to repeat the entire process
(starting with computing the mean and standard deviation) for the feature x2 also. The data set
after Z-score normalization will be as follows:
Z1 Z2
1 1.650 0.990
2 0.889 0.078
3 -0.637 0.990
4 0.126 0.534
5 -1.019 -0.835
6 -1.019 -1.749
12. Principal Component Analysis (PCA)
1. Standardization of the given dataset
Our dataset visually looks like the following after standardization
As we can see, the layout of the points still looks similar even after standardization,
and they are now centered around the origin!
13. Principal Component Analysis (PCA)
The next step of PCA is to find a line (also known as principal components ) on which to
project the given samples that captures the relationship between the two features well.
How well the relationship is captured is based on how much variance is preserved when the
samples are projected onto the line.
Finding the principal components
14. Principal Component Analysis (PCA)
Finding the principal components
Let's intuitively understand what is meant by finding the principal components,
consider the following example. Suppose we have the following samples: