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I wanted to briefly mention that PCA can also take the compressed representation of the data lower dimensional data back to an approximation of the original high dimensional data.

If you are interested in the code that produces the image below, check out my github. This is a post that I could have written on for a lot longer as PCA has many different uses.

I hope this post helps you with whatever you are working on. If you any questions or thoughts on the tutorial, feel free to reach out in the comments below or through Twitter.

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Understand O. Julian Herrera in Towards Data Science. Thus the weight vectors are eigenvectors of X T X. The transpose of W is sometimes called the whitening or sphering transformation.

Columns of W multiplied by the square root of corresponding eigenvalues, that is, eigenvectors scaled up by the variances, are called loadings in PCA or in Factor analysis.

X T X itself can be recognised as proportional to the empirical sample covariance matrix of the dataset X T [9] : 30— The sample covariance Q between two of the different principal components over the dataset is given by:.

However eigenvectors w j and w k corresponding to eigenvalues of a symmetric matrix are orthogonal if the eigenvalues are different , or can be orthogonalised if the vectors happen to share an equal repeated value.

The product in the final line is therefore zero; there is no sample covariance between different principal components over the dataset. Another way to characterise the principal components transformation is therefore as the transformation to coordinates which diagonalise the empirical sample covariance matrix.

However, not all the principal components need to be kept. Keeping only the first L principal components, produced by using only the first L eigenvectors, gives the truncated transformation.

Such dimensionality reduction can be a very useful step for visualising and processing high-dimensional datasets, while still retaining as much of the variance in the dataset as possible.

Similarly, in regression analysis , the larger the number of explanatory variables allowed, the greater is the chance of overfitting the model, producing conclusions that fail to generalise to other datasets.

One approach, especially when there are strong correlations between different possible explanatory variables, is to reduce them to a few principal components and then run the regression against them, a method called principal component regression.

Dimensionality reduction may also be appropriate when the variables in a dataset are noisy. If each column of the dataset contains independent identically distributed Gaussian noise, then the columns of T will also contain similarly identically distributed Gaussian noise such a distribution is invariant under the effects of the matrix W , which can be thought of as a high-dimensional rotation of the co-ordinate axes.

However, with more of the total variance concentrated in the first few principal components compared to the same noise variance, the proportionate effect of the noise is less—the first few components achieve a higher signal-to-noise ratio.

PCA thus can have the effect of concentrating much of the signal into the first few principal components, which can usefully be captured by dimensionality reduction; while the later principal components may be dominated by noise, and so disposed of without great loss.

If the dataset is not too large, the significance of the principal components can be tested using parametric bootstrap , as an aid in determining how many principal components to retain [11].

The principal components transformation can also be associated with another matrix factorization, the singular value decomposition SVD of X ,. This form is also the polar decomposition of T.

Efficient algorithms exist to calculate the SVD of X without having to form the matrix X T X , so computing the SVD is now the standard way to calculate a principal components analysis from a data matrix [ citation needed ] , unless only a handful of components are required.

The truncation of a matrix M or T using a truncated singular value decomposition in this way produces a truncated matrix that is the nearest possible matrix of rank L to the original matrix, in the sense of the difference between the two having the smallest possible Frobenius norm , a result known as the Eckart—Young theorem [].

Given a set of points in Euclidean space , the first principal component corresponds to a line that passes through the multidimensional mean and minimizes the sum of squares of the distances of the points from the line.

The second principal component corresponds to the same concept after all correlation with the first principal component has been subtracted from the points.

Each eigenvalue is proportional to the portion of the "variance" more correctly of the sum of the squared distances of the points from their multidimensional mean that is associated with each eigenvector.

The sum of all the eigenvalues is equal to the sum of the squared distances of the points from their multidimensional mean.

PCA essentially rotates the set of points around their mean in order to align with the principal components. This moves as much of the variance as possible using an orthogonal transformation into the first few dimensions.

The values in the remaining dimensions, therefore, tend to be small and may be dropped with minimal loss of information see below. PCA is often used in this manner for dimensionality reduction.

PCA has the distinction of being the optimal orthogonal transformation for keeping the subspace that has largest "variance" as defined above.

This advantage, however, comes at the price of greater computational requirements if compared, for example, and when applicable, to the discrete cosine transform , and in particular to the DCT-II which is simply known as the "DCT".

Nonlinear dimensionality reduction techniques tend to be more computationally demanding than PCA. PCA is sensitive to the scaling of the variables.

But if we multiply all values of the first variable by , then the first principal component will be almost the same as that variable, with a small contribution from the other variable, whereas the second component will be almost aligned with the second original variable.

This means that whenever the different variables have different units like temperature and mass , PCA is a somewhat arbitrary method of analysis.

Different results would be obtained if one used Fahrenheit rather than Celsius for example. Pearson's original paper was entitled "On Lines and Planes of Closest Fit to Systems of Points in Space" — "in space" implies physical Euclidean space where such concerns do not arise.

One way of making the PCA less arbitrary is to use variables scaled so as to have unit variance, by standardizing the data and hence use the autocorrelation matrix instead of the autocovariance matrix as a basis for PCA.

However, this compresses or expands the fluctuations in all dimensions of the signal space to unit variance.

Mean subtraction a. If mean subtraction is not performed, the first principal component might instead correspond more or less to the mean of the data.

A mean of zero is needed for finding a basis that minimizes the mean square error of the approximation of the data. Mean-centering is unnecessary if performing a principal components analysis on a correlation matrix, as the data are already centered after calculating correlations.

Correlations are derived from the cross-product of two standard scores Z-scores or statistical moments hence the name: Pearson Product-Moment Correlation.

PCA is a popular primary technique in pattern recognition. It is not, however, optimized for class separability. The statistical implication of this property is that the last few PCs are not simply unstructured left-overs after removing the important PCs.

Because these last PCs have variances as small as possible they are useful in their own right. They can help to detect unsuspected near-constant linear relationships between the elements of x , and they may also be useful in regression , in selecting a subset of variables from x , and in outlier detection.

Before we look at its usage, we first look at diagonal elements,. As noted above, the results of PCA depend on the scaling of the variables.

This can be cured by scaling each feature by its standard deviation, so that one ends up with dimensionless features with unital variance.

The applicability of PCA as described above is limited by certain tacit assumptions [16] made in its derivation.

In particular, PCA can capture linear correlations between the features but fails when this assumption is violated see Figure 6a in the reference. In some cases, coordinate transformations can restore the linearity assumption and PCA can then be applied see kernel PCA.

Another limitation is the mean-removal process before constructing the covariance matrix for PCA. In fields such as astronomy, all the signals are non-negative, and the mean-removal process will force the mean of some astrophysical exposures to be zero, which consequently creates unphysical negative fluxes, [17] and forward modeling has to be performed to recover the true magnitude of the signals.

Dimensionality reduction loses information, in general. PCA-based dimensionality reduction tends to minimize that information loss, under certain signal and noise models.

The following is a detailed description of PCA using the covariance method see also here as opposed to the correlation method. The goal is to transform a given data set X of dimension p to an alternative data set Y of smaller dimension L.

Mean subtraction is an integral part of the solution towards finding a principal component basis that minimizes the mean square error of approximating the data.

In some applications, each variable column of B may also be scaled to have a variance equal to 1 see Z-score.

Let X be a d -dimensional random vector expressed as column vector. Without loss of generality, assume X has zero mean.

This is very constructive, as cov X is guaranteed to be a non-negative definite matrix and thus is guaranteed to be diagonalisable by some unitary matrix.

In practical implementations, especially with high dimensional data large p , the naive covariance method is rarely used because it is not efficient due to high computational and memory costs of explicitly determining the covariance matrix.

The covariance-free approach avoids the np 2 operations of explicitly calculating and storing the covariance matrix X T X , instead utilizing one of matrix-free methods , for example, based on the function evaluating the product X T X r at the cost of 2np operations.

One way to compute the first principal component efficiently [33] is shown in the following pseudo-code, for a data matrix X with zero mean, without ever computing its covariance matrix.

This power iteration algorithm simply calculates the vector X T X r , normalizes, and places the result back in r. If the largest singular value is well separated from the next largest one, the vector r gets close to the first principal component of X within the number of iterations c , which is small relative to p , at the total cost 2cnp.

The power iteration convergence can be accelerated without noticeably sacrificing the small cost per iteration using more advanced matrix-free methods , such as the Lanczos algorithm or the Locally Optimal Block Preconditioned Conjugate Gradient LOBPCG method.

Subsequent principal components can be computed one-by-one via deflation or simultaneously as a block. In the former approach, imprecisions in already computed approximate principal components additively affect the accuracy of the subsequently computed principal components, thus increasing the error with every new computation.

The latter approach in the block power method replaces single-vectors r and s with block-vectors, matrices R and S. Every column of R approximates one of the leading principal components, while all columns are iterated simultaneously.

The main calculation is evaluation of the product X T X R. Implemented, for example, in LOBPCG , efficient blocking eliminates the accumulation of the errors, allows using high-level BLAS matrix-matrix product functions, and typically leads to faster convergence, compared to the single-vector one-by-one technique.

Non-linear iterative partial least squares NIPALS is a variant the classical power iteration with matrix deflation by subtraction implemented for computing the first few components in a principal component or partial least squares analysis.

The matrix deflation by subtraction is performed by subtracting the outer product, t 1 r 1 T from X leaving the deflated residual matrix used to calculate the subsequent leading PCs.

In an "online" or "streaming" situation with data arriving piece by piece rather than being stored in a single batch, it is useful to make an estimate of the PCA projection that can be updated sequentially.

This can be done efficiently, but requires different algorithms. In PCA, it is common that we want to introduce qualitative variables as supplementary elements.

For example, many quantitative variables have been measured on plants. For these plants, some qualitative variables are available as, for example, the species to which the plant belongs.

These data were subjected to PCA for quantitative variables. When analyzing the results, it is natural to connect the principal components to the qualitative variable species.

For this, the following results are produced. These results are what is called introducing a qualitative variable as supplementary element.

Few software offer this option in an "automatic" way. In quantitative finance , principal component analysis can be directly applied to the risk management of interest rate derivative portfolios.

Converting risks to be represented as those to factor loadings or multipliers provides assessments and understanding beyond that available to simply collectively viewing risks to individual 30— buckets.

PCA has also been applied to equity portfolios in a similar fashion, [39] both to portfolio risk and to risk return.

One application is to reduce portfolio risk, where allocation strategies are applied to the "principal portfolios" instead of the underlying stocks.

A variant of principal components analysis is used in neuroscience to identify the specific properties of a stimulus that increase a neuron 's probability of generating an action potential.

In a typical application an experimenter presents a white noise process as a stimulus usually either as a sensory input to a test subject, or as a current injected directly into the neuron and records a train of action potentials, or spikes, produced by the neuron as a result.

Presumably, certain features of the stimulus make the neuron more likely to spike. In order to extract these features, the experimenter calculates the covariance matrix of the spike-triggered ensemble , the set of all stimuli defined and discretized over a finite time window, typically on the order of ms that immediately preceded a spike.

The eigenvectors of the difference between the spike-triggered covariance matrix and the covariance matrix of the prior stimulus ensemble the set of all stimuli, defined over the same length time window then indicate the directions in the space of stimuli along which the variance of the spike-triggered ensemble differed the most from that of the prior stimulus ensemble.

Specifically, the eigenvectors with the largest positive eigenvalues correspond to the directions along which the variance of the spike-triggered ensemble showed the largest positive change compared to the variance of the prior.

Since these were the directions in which varying the stimulus led to a spike, they are often good approximations of the sought after relevant stimulus features.

In neuroscience, PCA is also used to discern the identity of a neuron from the shape of its action potential. Spike sorting is an important procedure because extracellular recording techniques often pick up signals from more than one neuron.

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You can use PCA to reduce that 4 dimensional data into 2 or 3 dimensions so that you can plot and hopefully understand the data better.

The Iris dataset is one of datasets scikit-learn comes with that do not require the downloading of any file from some external website. The code below will load the iris dataset.

If you want to see the negative effect not scaling your data can have, scikit-learn has a section on the effects of not standardizing your data. The original data has 4 columns sepal length, sepal width, petal length, and petal width.

In this section, the code projects the original data which is 4 dimensional into 2 dimensions. The new components are just the two main dimensions of variation.

This section is just plotting 2 dimensional data. Notice on the graph below that the classes seem well separated from each other. The explained variance tells you how much information variance can be attributed to each of the principal components.

This is important as while you can convert 4 dimensional space to 2 dimensional space, you lose some of the variance information when you do this.

Together, the two components contain One of the most important applications of PCA is for speeding up machine learning algorithms.

Using the IRIS dataset would be impractical here as the dataset only has rows and only 4 feature columns. The MNIST database of handwritten digits is more suitable as it has feature columns dimensions , a training set of 60, examples, and a test set of 10, examples.

The images that you downloaded are contained in mnist. The labels the integers 0—9 are contained in mnist. The features are dimensional 28 x 28 images and the labels are simply numbers from 0—9.

The text in this paragraph is almost an exact copy of what was written earlier. Note you fit on the training set and transform on the training and test set.

Notice the code below has. Fit PCA on training set. Note: you are fitting PCA on the training set only.

Note: You can find out how many components PCA choose after fitting the model using pca. Step 1: Import the model you want to use. In sklearn, all machine learning models are implemented as Python classes.

Step 2: Make an instance of the Model. In the section ref pca-variable-contributions , we described how to highlight variables according to their contributions to the principal components.

Note also that, the function dimdesc [in FactoMineR], for dimension description, can be used to identify the most significantly associated variables with a given principal component.

It can be used as follow:. Note that, variables are sorted by the p-value of the correlation. To create a simple plot, type this:. As for variables, individuals can be colored by any custom continuous variable by specifying the argument col.

Here, we describe how to color individuals by group. Additionally, we show how to add concentration ellipses and confidence ellipses by groups.

We start by computing principal component analysis as follow:. In the R code below: the argument habillage or col. The argument palette can be used to change group colors.

If you want confidence ellipses instead of concentration ellipses, use ellipse. Here, we present some of these additional arguments to customize the PCA graph of variables and individuals.

The argument geom for geometry and derivatives are used to specify the geometry elements or graphical elements to be used for plotting.

Note that, the argument ellipse. Possible values are:. The argument ellipse. For example, specify ellipse. When coloring individuals by groups section ref color-ind-by-groups , the mean points of groups barycenters are also displayed by default.

The argument axes. To change easily the graphical of any ggplots, you can use the function ggpar [ggpubr package]. Note that, the biplot might be only useful when there is a low number of variables and individuals in the data set; otherwise the final plot would be unreadable.

Note also that, the coordinate of individuals and variables are not constructed on the same space.

Therefore, in the biplot, you should mainly focus on the direction of variables but not on their absolute positions on the plot.

In the following example, we want to color both individuals and variables by groups. This particular point shape can be filled by a color using the argument fill.

To color variable by groups, the argument col. Another complex example is to color individuals by groups discrete color and variables by their contributions to the principal components gradient colors.

As described above section ref pca-data-format , the decathlon2 data sets contain supplementary continuous variables quanti.

Supplementary variables and individuals are not used for the determination of the principal components. To specify supplementary individuals and variables, the function PCA can be used as follow:.

Note that, by default, supplementary quantitative variables are shown in blue color and dashed lines.

Note that, you can add the quanti. An example is shown below. Supplementary individuals are shown in blue. The levels of the supplementary qualitative variable are shown in red color.

Note that, the supplementary qualitative variables can be also used for coloring individuals by groups. This can help to interpret the data.

The data sets decathlon2 contain a supplementary qualitative variable at columns 13 corresponding to the type of competitions. To color individuals by a supplementary qualitative variable, the argument habillage is used to specify the index of the supplementary qualitative variable.

Historically, this argument name comes from the FactoMineR package. To keep consistency between FactoMineR and factoextra, we decided to keep the same argument name.

Recall that, to remove the mean points of groups, specify the argument mean. Allowed values are NULL or a list containing the arguments name, cos2 or contrib:.

The factoextra package produces a ggplot2-based graphs. To save any ggplots, the standard R code is as follow:.

Note that, using the above R code will create the PDF file into your current working directory. To see the path of your current working directory, type getwd in the R console.

Another alternative, to export ggplots, is to use the function ggexport [in ggpubr package]. With one line R code, it allows us to export individual plots to a file pdf, eps or png one plot per page.

It can also arrange the plots 2 plot per page, for example before exporting them. The examples below demonstrates how to export ggplots using ggexport.

Arrange and export. Specify nrow and ncol to display multiple plots on the same page:. Export plots to png files.

If you specify a list of plots, then multiple png files will be automatically created to hold each plot. In conclusion, we described how to perform and interpret principal component analysis PCA.

Next, we used the factoextra R package to produce ggplot2-based visualization of the PCA results. No matter what functions you decide to use, in the list above, the factoextra package can handle the output for creating beautiful plots similar to what we described in the previous sections for FactoMineR:.

For the mathematical background behind CA, refer to the following video courses, articles and books:. Jollife, I. Principal Component Analysis.

New York: Springer-Verlag. Kaiser, Henry F. Peres-Neto, Pedro R. Jackson, and Keith M. Basics Understanding the details of PCA requires knowledge of linear algebra.

The dimensionality of our two-dimensional data can be reduced to a single dimension by projecting each sample onto the first principal component Plot 1B Technically speaking, the amount of variance retained by each principal component is measured by the so-called eigenvalue.

Taken together, the main purpose of principal component analysis is to: identify hidden pattern in a data set, reduce the dimensionnality of the data by removing the noise and redundancy in the data, identify correlated variables.

Active individuals in light blue, rows : Individuals that are used during the principal component analysis.

Supplementary variables : As supplementary individuals, the coordinates of these variables will be predicted also. These can be: Supplementary continuous variables red : Columns 11 and 12 corresponding respectively to the rank and the points of athletes.

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