Lets start with single input linear regression. Otherwise, the model will start to exhibit an overfitting nature. That way, if we are given a new house and its floor area, we can see whether we are paying a reasonable amount of not. As we learn more details about least squares, and then move onto using these methods in logistic regression and then move onto using all these methods in neural networks, you will be very glad you worked hard to understand these derivations. Why don't math grad schools in the U.S. use entrance exams? 503), Fighting to balance identity and anonymity on the web(3) (Ep. Will it have a bad influence on getting a student visa? Lets import LinearRegression from the Sklearn module. 953.3728792569658, However, if you can push the I BELIEVE button on some important linear algebra properties, itll be possible and less painful. -70.33486068111438, As we already have the code for polynomial regression, we can go ahead with the same code by changing the poly_features declaration as follows. The bias and variance are two of the most important parameters we should be familiar with. Thats right. However, there is an even greater advantage here. Well apply these calculus steps to the matrix form and to the individual equations for extreme clarity. Step 1: Import libraries and dataset Import the important libraries and the dataset we are using to perform Polynomial Regression. 807.1289164086686, But it fails to fit and catch the pattern in non-linear data. Install it using pip as follows. Our matrix and vector format is conveniently clean looking. If the line would not be a nice curve, polynomial regression can learn some more complex trends as well. I am not going to the differential calculus here. Going from engineer to entrepreneur takes more than just good code (Ep. Install it using pip as follows. We can see that the price has been increased proportionally with the available floor area of the house. 75.90910216718271, It is laid out the same as the previous file. Regression is considered to be the Hello World in the machine learning world. We are using each processor to execute the same task and we are recording the time for each CPU to process the task. However, the math, depending on how deep you want to go, is substantial. -143.45684210526315]. ], Then we will create a LinearRegression object and train the model using the polynomial-transformed array. Both however are using the least squares method in determining the best fitting functions. Lets use equation 3.7 on the right side of equation 3.6. However, we are still solving for only one \footnotesize{b} (we still have a single continuous output variable, so we only have one \footnotesize{y} intercept), but weve rolled it conveniently into our equations to simplify the matrix representation of our equations and the one \footnotesize{b}. Now we have to reshape our list since they depict a single feature of our dataset. Because if you multiply 1 with a number it does not change. Now, lets produce some fake data that necessitates using a least squares approach. Our realistic data set was obtained from HERE. In an attempt to best predict that system, we take more data, than is needed to simplymathematically find a model for the system, in the hope that the extra data will help us find the best fit through a lot of noisy error filled data. In this post, we create a clustering algorithm class that uses the same principles as scipy, or sklearn, but without using sklearn or numpy or scipy. It should be kept in mind that when we are training models, we should make sure that we train our model amply but never more than what is required. Then dividing that value by 2 times the number of training examples. If youve never been through the linear algebra proofs for whats coming below, think of this at a very high level. Ill try to get those posts out ASAP. So I divided the process into two steps. 441.51900928792566, How to do gradient descent in python without numpy or scipy. Instead of a b in each equation, we will replace those with x_{10} ~ w_0, x_{20} ~ w_0, and x_{30} ~ w_0. Section 1 prepares the fake data for usage. It states clearly polynomial regression leverages least squares for computation and it can model the expected dependent variables y as an n th degree polynomial, yielding the general polynomial regression model. For those otherwise positioned at the moment, I will still show all the code below. Why do we focus on the derivation for least squares like this? Usually, when we are training machine learning models, it is always good to have them as floating point values. Theta values are initialized randomly. The next step is to apply calculus to find where the error E is minimized. Scipy Odrpack works noramally but it needs a good initial guess for correct results. That is we want find a model that passes through the data with the least of the squares of the errors. Did the words "come" and "home" historically rhyme? The subtraction above results in a vector sticking out perpendicularly from the \footnotesize{\bold{X_2}} column space. We can then calculate the w (slope) and b (intercept) terms using the above formula: w = (n*sum(xy) - sum(x)*sum(y)) / (n*sum(x_sqrt) - sum(x)**2) b = (sum(y) - w*sum(x))/n w 0.4950512786062967 b 31.82863092838909 Least Squares Linear Regression With Python Sklearn If we stretch the spring to integral values of our distance unit, we would have the following data points: Hookes law is essentially the equation of a line and is the application of linear regression to the data associated with force, spring displacement, and spring stiffness (spring stiffness is the inverse of spring compliance). In a good machine learning algorithm, cost should keep going down until the convergence. Note, the way that the least_squares function calls the fitting function is slightly different here. Below is the output from the above code including the output graph. Here we will use the above example and introduce you more ways to do it. In this post, we create a clustering algorithm class that uses the same principles as scipy, or sklearn, but without using sklearn or numpy or scipy. As before, the two tool sets, pure python and scikit learn, have extremely small prediction deltas and the two graph lines, that run through the initial fake data points, follow the same path. The method returns the Polynomial coefficients ordered from low to high. So we finally got our equation that describes the fitted line. We can acquire the root mean squared error as follows to get a better idea. Lets say we are looking to buy houses from the same city. Once we encode each text element to have its own column, where a 1 only occurs when the text element occurs for a record, and it has 0s everywhere else. WITH NO RANDOM NOISE injected into the outputs, the coefficients would have exactly matched the ones that we started with. Block 1 does imports. Not the answer you're looking for? y1 = theta*X We will look at matrix form along with the equations written out as we go through this to keep all the steps perfectly clear for those that arent as versed in linear algebra (or those who know it, but have cold memories on it dont we all sometimes). This next file well go over is named LeastSquaresPolyPractice_2b.py in the repository. Feel free to pick any name for the regressor object. Well cover pandas in detail in future posts. The method of least squares aims to minimise the variance between the values estimated from the polynomial and the expected values from the dataset. Linear regression can perform well only if there is a linear correlation between the input variables and the output variable. The output from the above code is shown below along with its 3D output graph. In this tutorial, we only used the Sklearn library. [759000. Assume that we have a dataset for CPUs with different hypothetical clock rates. The difference in this section is that we are solving for multiple \footnotesize{m}s (i.e. Note that we are flattening the NumPy arrays by creating a list of the predicted values. 218.33, 178.73, 120.21, 80.20, 60.1]. Add the bias column for theta 0. Divide each column by the maximum value of that column. We also havent talked about pandas yet. The model we develop based on this form of the equation is polynomial in nature. Lets walk through this code and then look at the output. Now for a bit more of a challenge. Now, lets subtract \footnotesize{\bold{Y_2}} from both sides of equation 3.4. The least-squares regression method works by minimizing the sum of the square of the errors as small as possible, hence the name least squares. This is why we should be aware of the phenomenon called bias-variance tradeoff. for c in range(0, len(X.columns)): Setting equation 1.10 to 0 gives. Lets revert T, U, V and W back to the terms that they replaced. Well then learn how to use this to fit curved surfaces, which has some great applications on the boundary between machine learning and system modeling and other cool/weird stuff. \footnotesize{\bold{X}} is \footnotesize{4x3} and its transpose is \footnotesize{3x4}. Constraining the least squares fitting in python. Feel free to choose one you like. Browse other questions tagged, Where developers & technologists share private knowledge with coworkers, Reach developers & technologists worldwide. The first one is polynomial transformation and then it is followed by linear regression (Yes, it is linear regression). lin_reg2 = LinearRegression () lin_reg2.fit (X_poly,y) The above code produces the following output: Output 6. This phenomenon is also known as underfitting. Both of these files are in the repo. Ive broken the test files down into sections noted by comments. Instead, we are importing the LinearRegression class from the sklearn.linear_model module. Section 7 compares the outputs and Section 8 shows the final graph. Could we derive a least squares solution using the principles of linear algebra alone? For example, we can use packages as numpy, scipy, statsmodels, sklearn and so on to get a least square solution. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Also, the coefficients were sorted and are identical (once rounded off). array ( [ 0, 0.25, 0.5, 0.75, 1.0 ], float ) # x-values y = np. Import the dataset: import pandas as pd import numpy as np df = pd.read_csv ('position_salaries.csv') df.head () 2. Understanding this will be very important to discussions in upcoming posts when all the dimensions are not necessarily independent, and then we need to find ways to constructively eliminate input columns that are not independent from one of more of the other columns. plt.scatter(x=X['Level'],y= y) In case you werent aware, when we multiply one matrix on another, this transforms the right matrix into the space of the left matrix. We do this by minimizing . Data Scientist, PhD multi-physics engineer, and python loving geek living in the United States. However, as in this scenario, we only have one independent variable. Curve_fit however doesn't seem to work for me as the graph should be heading down in theory but in your pic, it is going up. You dont even need least squares to do this one. 16.6 Summary and Problems. y=180.648x+408735.903 You can use numpy.polyfit to do the fitting and numpy.polyval to get the data to plot. Lets plot the cost we calculated in each epoch in our gradient descent function. The only variables that we must keep visible after these substitutions are m and b. This further confirms that the accuracy of the newer model is much higher. We can isolate b by multiplying equation 1.15 by U and 1.16 by T and then subtracting the later from the former as shown next. Block 3 does the actual fit of the data and prints the resulting coefficients for the model. Our "objective" is to minimize the square errors. This is great! ], All that is left is to algebraically isolate b. More than a video, you. [1998. Theres a lot of good work and careful planning and extra code to support those great machine learning modules AND data visualization modules and tools. Your 2D polynomial (of arbitrary degree) can be defined using polyval2d from the numpy library. We have the Level column to represent the positions. The code below is stored in the repo for this post, and its name is LeastSquaresPractice_Using_SKLearn.py. In Python, there are many different ways to conduct the least square regression. We can see a gradual decrease in the error as we increase the order. We're also begin preparing a plot for the final section. To learn more, see our tips on writing great answers. The values of \hat y may not pass through many or any of the measured y values for each x. And that system has output data that can be measured. If you know a bit about NIR spectroscopy, you sure know very well that NIR is a secondary method and NIR data needs to be calibrated against primary reference data of the parameter one seeks to measure. Let us perform a few more iterations by increasing the order of the model and tabulate the root mean squared error. Thanks! Now, normalize the data. Our objective is to minimize the square errors. We have a real world system susceptible to noisy input data. In Sections 3 and 4, the fake data is prepared to be put into our desired polynomial format and then fit using our least squares regression tools using our pure python and scikit learn tools, respectively. Lets make a prediction using the model. When have an exact number of equations for the number of unknowns, we say that \footnotesize{\bold{Y_1}} is in the column space of \footnotesize{\bold{X_1}}. [795000. Nice! The simplest example of polynomial regression has a single independent variable, and the estimated regression function is a polynomial of degree two: () = + + . So to find that we've to first find the equation of such a line. Figure 1 shows our plot. Let's first convert our data to float as they are integer values now. These are your unknowns! We are still sort of finding a solution for \footnotesize{m} like we did above with the single input variable least squares derivation in the previous section. The data are as below. Well even throw in some visualizations finally. Each of the data in these lists, correspond to the same elements at the same index. Lets substitute \hat y with mx_i+b and use calculus to reduce this error. The actual data points are x and y, and measured values for y will likely have small errors. Understanding the derivation is still better than not seeking to understand it. Define the hypothesis function. With the tools created in the previous posts (chronologically speaking), were finally at a point to discuss our first serious machine learning tool starting from the foundational linear algebra all the way to complete python code. where the \footnotesize{x_i} are the rows of \footnotesize{\bold{X}} and \footnotesize{\bold{W}} is the column vector of coefficients that we want to find to minimize \footnotesize{E}. Section 4 is where the machine learning is performed. y1 = hypothesis(X, theta) We then fit the model using the training data and make predictions with our test data. X is the input feature and Y is the output variable. Fourth and final, solve for the least squares coefficients that will fit the data using the forms of both equations 2.7b and 3.9, and, to do that, we use our solve_equations function from the solve a system of equations post. First, get the transpose of the input data (system matrix). Computes the vector x that approximately solves the equation a @ x = b. It could find the relationship between input features and the output variable in a better way even if the relationship is not linear. Second step: substitude these initial guess in ODR as beta0 parameter. A simple and common real world example of linear regression would be Hookes law for coiled springs: If there were some other force in the mechanical circuit that was constant over time, we might instead have another term such as F_b that we could call the force bias. These substitutions are helpful in that they simplify all of our known quantities into single letters. The mathematical convenience of this will become more apparent as we progress. The Hello World of machine learning and computational neural networks usually start with a technique called regression that comes in statistics. Least Squares solution Sums of residuals (error) Rank of the matrix (X) Singular values of the matrix (X) np.linalg.lstsq (X, y) First, deducting the hypothesis from the original output variable. Section 1 prepares the fake data for usage. As opposed to linear regression, polynomial regression is used to model relationships between features and the dependent variable that are not linear. The system of equations are the following. By polynomial transformation, what we are doing is adding another variable from a higher degree. I am initializing an array of zero. 149.03108359133125, Now heres a spoiler alert. You can take any other random values. If not, I hope you will hang in there, because this approach of math theory all the way to code without relying on modules should help us to continue to grow our insights. In the data science jargon, the dependent variable is also known as y and the independent variables are known as x1, x2, xi. Second, multiply the transpose of the input data matrix onto the input data matrix. Let us now try to model the data using polynomial regression. If you run the 3a version of the file, you will see this. Visualizing the Polynomial Regression model X = df.drop(columns = 'Salary') For example, suppose x = 4. First step: find the initial guess by using ordinaty least squares method. It is apparent that the model produced by linear regression has not been able to accurately model the dataset by capturing the distinctive features of it. 1100, 1150, 1200, 1250, 1300, 1350, 1400, 1450], minutestoprocess = [1700.01, 1400.02, 1100.1, 800.01, 496.25, 734.0069349845201, It is easily discernible that the relationship is not linear. rev2022.11.7.43014. If we used the nth column, wed create a linear dependency (colinearity), and then our columns for the encoded variables would not be orthogonal as discussed in the previous post. In this tutorial video, we learned how to do Polynomial Regression in Python using Sklearn. ], It has grown to include our new least_squares function above and one other convenience function called insert_at_nth_column_of_matrix, which simply inserts a column into a matrix. Here is the step by step implementation of Polynomial regression. These are the a and b values we were looking for in the linear function formula. Houses of this city are mainly dependent on how many square feet it has. If you take the partial differential of the cost function on each theta, we can derive these formulas: Here, alpha is the learning rate. We can find this using the mean squared error between the true Y values and predicted Y values. 12. Our model says that the property is only worth about USD 733,902. In case the term column space is confusing to you, think of it as the established independent (orthogonal) dimensions in the space described by our system of equations. 222.15306501548002, Can lead-acid batteries be stored by removing the liquid from them? Finally, lets give names to our matrix and vectors. Now we want to find a solution for m and b that minimizes the error defined by equations 1.5 and 1.6. Please feel free to try it with a different number of epochs and different learning rates (alpha). This function uses least squares and the solution is to minimize the squared errors in the given polynomial. However, high variance models such as polynomial models of higher orders, KNN models of higher N values suggest that they are prone to quick changes trying to fit through all the data points. We can plot these data points in a scatter plot to see how they look like. With the pure tools, the coefficients with one of the collinear variables were 0.0. Applying Polynomial Features to Least Squares Regression using Pure Python without Numpy or Scipy, \tag{1.3} x=0, \,\,\,\,\, F = k \cdot 0 + F_b \\ x=1, \,\,\,\,\, F = k \cdot 1 + F_b \\ x=2, \,\,\,\,\, F = k \cdot 2 + F_b, \tag{1.5} E=\sum_{i=1}^N \lparen y_i - \hat y_i \rparen ^ 2, \tag{1.6} E=\sum_{i=1}^N \lparen y_i - \lparen mx_i+b \rparen \rparen ^ 2, \tag{1.7} a= \lparen y_i - \lparen mx_i+b \rparen \rparen ^ 2, \tag{1.8} \frac{\partial E}{\partial a} = 2 \sum_{i=1}^N \lparen y_i - \lparen mx_i+b \rparen \rparen, \tag{1.9} \frac{\partial a}{\partial m} = -x_i, \tag{1.10} \frac{\partial E}{\partial m} = \frac{\partial E}{\partial a} \frac{\partial a}{\partial m} = 2 \sum_{i=1}^N \lparen y_i - \lparen mx_i+b \rparen \rparen \lparen -x_i \rparen), \tag{1.11} \frac{\partial a}{\partial b} = -1, \tag{1.12} \frac{\partial E}{\partial b} = \frac{\partial E}{\partial a} \frac{\partial a}{\partial b} = 2 \sum_{i=1}^N \lparen y_i - \lparen mx_i+b \rparen \rparen \lparen -1 \rparen), 0 = 2 \sum_{i=1}^N \lparen y_i - \lparen mx_i+b \rparen \rparen \lparen -x_i \rparen), 0 = \sum_{i=1}^N \lparen -y_i x_i + m x_i^2 + b x_i \rparen), 0 = \sum_{i=1}^N -y_i x_i + \sum_{i=1}^N m x_i^2 + \sum_{i=1}^N b x_i, \tag{1.13} \sum_{i=1}^N y_i x_i = \sum_{i=1}^N m x_i^2 + \sum_{i=1}^N b x_i, 0 = 2 \sum_{i=1}^N \lparen -y_i + \lparen mx_i+b \rparen \rparen, 0 = \sum_{i=1}^N -y_i + m \sum_{i=1}^N x_i + b \sum_{i=1} 1, \tag{1.14} \sum_{i=1}^N y_i = m \sum_{i=1}^N x_i + N b, T = \sum_{i=1}^N x_i^2, \,\,\, U = \sum_{i=1}^N x_i, \,\,\, V = \sum_{i=1}^N y_i x_i, \,\,\, W = \sum_{i=1}^N y_i, \begin{alignedat} ~&mTU + bU^2 &= &~VU \\ -&mTU - bNT &= &-WT \\ \hline \\ &b \lparen U^2 - NT \rparen &= &~VU - WT \end{alignedat}, \begin{alignedat} ~&mNT + bUN &= &~VN \\ -&mU^2 - bUN &= &-WU \\ \hline \\ &m \lparen TN - U^2 \rparen &= &~VN - WU \end{alignedat}, \tag{1.18} m = \frac{-1}{-1} \frac {VN - WU} {TN - U^2} = \frac {WU - VN} {U^2 - TN}, \tag{1.19} m = \dfrac{\sum\limits_{i=1}^N x_i \sum\limits_{i=1}^N y_i - N \sum\limits_{i=1}^N x_i y_i}{ \lparen \sum\limits_{i=1}^N x_i \rparen ^2 - N \sum\limits_{i=1}^N x_i^2 }, \tag{1.20} b = \dfrac{\sum\limits_{i=1}^N x_i y_i \sum\limits_{i=1}^N x_i - N \sum\limits_{i=1}^N y_i \sum\limits_{i=1}^N x_i^2 }{ \lparen \sum\limits_{i=1}^N x_i \rparen ^2 - N \sum\limits_{i=1}^N x_i^2 }, \overline{x} = \frac{1}{N} \sum_{i=1}^N x_i, \,\,\,\,\,\,\, \overline{xy} = \frac{1}{N} \sum_{i=1}^N x_i y_i, \tag{1.21} m = \frac{N^2 \overline{x} ~ \overline{y} - N^2 \overline{xy} } {N^2 \overline{x}^2 - N^2 \overline{x^2} } = \frac{\overline{x} ~ \overline{y} - \overline{xy} } {\overline{x}^2 - \overline{x^2} }, \tag{1.22} b = \frac{\overline{xy} ~ \overline{x} - \overline{y} ~ \overline{x^2} } {\overline{x}^2 - \overline{x^2} }, \tag{Equations 2.1} f_1 = x_{11} ~ w_1 + x_{12} ~ w_2 + b \\ f_2 = x_{21} ~ w_1 + x_{22} ~ w_2 + b \\ f_3 = x_{31} ~ w_1 + x_{32} ~ w_2 + b \\ f_4 = x_{41} ~ w_1 + x_{42} ~ w_2 + b, \tag{Equations 2.2} f_1 = x_{10} ~ w_0 + x_{11} ~ w_1 + x_{12} ~ w_2 \\ f_2 = x_{20} ~ w_0 + x_{21} ~ w_1 + x_{22} ~ w_2 \\ f_3 = x_{30} ~ w_0 + x_{31} ~ w_1 + x_{32} ~ w_2 \\ f_4 = x_{40} ~ w_0 + x_{41} ~ w_1 + x_{42} ~ w_2, \tag{2.3} \bold{F = X W} \,\,\, or \,\,\, \bold{Y = X W}, \tag{2.4} E=\sum_{i=1}^N \lparen y_i - \hat y_i \rparen ^ 2 = \sum_{i=1}^N \lparen y_i - x_i ~ \bold{W} \rparen ^ 2, \tag{Equations 2.5} \frac{\partial E}{\partial w_j} = 2 \sum_{i=1}^N \lparen y_i - x_i \bold{W} \rparen \lparen -x_{ij} \rparen = 2 \sum_{i=1}^N \lparen f_i - x_i \bold{W} \rparen \lparen -x_{ij} \rparen \\ ~ \\ or~using~just~w_1~for~example \\ ~ \\ \begin{alignedat}{1} \frac{\partial E}{\partial w_1} &= 2 \lparen f_1 - \lparen x_{10} ~ w_0 + x_{11} ~ w_1 + x_{12} ~ w_2 \rparen \rparen x_{11} \\ &+ 2 \lparen f_2 - \lparen x_{20} ~ w_0 + x_{21} ~ w_1 + x_{22} ~ w_2 \rparen \rparen x_{21} \\ &+ 2 \lparen f_3 - \lparen x_{30} ~ w_0 + x_{31} ~ w_1 + x_{32} ~ w_2 \rparen \rparen x_{31} \\ &+ 2 \lparen f_4 - \lparen x_{40} ~ w_0 + x_{41} ~ w_1 + x_{42} ~ w_2 \rparen \rparen x_{41} \end{alignedat}, \tag{2.6} 0 = 2 \sum_{i=1}^N \lparen y_i - x_i \bold{W} \rparen \lparen -x_{ij} \rparen, \,\,\,\,\, \sum_{i=1}^N y_i x_{ij} = \sum_{i=1}^N x_i \bold{W} x_{ij} \\ ~ \\ or~using~just~w_1~for~example \\ ~ \\ f_1 x_{11} + f_2 x_{21} + f_3 x_{31} + f_4 x_{41} \\ = \left( x_{10} ~ w_0 + x_{11} ~ w_1 + x_{12} ~ w_2 \right) x_{11} \\ + \left( x_{20} ~ w_0 + x_{21} ~ w_1 + x_{22} ~ w_2 \right) x_{21} \\ + \left( x_{30} ~ w_0 + x_{31} ~ w_1 + x_{32} ~ w_2 \right) x_{31} \\ + \left( x_{40} ~ w_0 + x_{41} ~ w_1 + x_{42} ~ w_2 \right) x_{41} \\ ~ \\ the~above~in~matrix~form~is \\ ~ \\ \bold{ X_j^T Y = X_j^T F = X_j^T X W}, \tag{2.7b} \bold{ \left(X^T X \right) W = \left(X^T Y \right)}, \tag{3.1a}m_1 x_1 + b_1 = y_1\\m_1 x_2 + b_1 = y_2, \tag{3.1b} \begin{bmatrix}x_1 & 1 \\ x_2 & 1 \end{bmatrix} \begin{bmatrix}m_1 \\ b_1 \end{bmatrix} = \begin{bmatrix}y_1 \\ y_2 \end{bmatrix}, \tag{3.1c} \bold{X_1} = \begin{bmatrix}x_1 & 1 \\ x_2 & 1 \end{bmatrix}, \,\,\, \bold{W_1} = \begin{bmatrix}m_1 \\ b_1 \end{bmatrix}, \,\,\, \bold{Y_1} = \begin{bmatrix}y_1 \\ y_2 \end{bmatrix}, \tag{3.1d} \bold{X_1 W_1 = Y_1}, \,\,\, where~ \bold{Y_1} \isin \bold{X_{1~ column~space}}, \tag{3.2a}m_2 x_1 + b_2 = y_1 \\ m_2 x_2 + b_2 = y_2 \\ m_2 x_3 + b_2 = y_3 \\ m_2 x_4 + b_2 = y_4, \tag{3.1b} \begin{bmatrix}x_1 & 1 \\ x_2 & 1 \\ x_3 & 1 \\ x_4 & 1 \end{bmatrix} \begin{bmatrix}m_2 \\ b_2 \end{bmatrix} = \begin{bmatrix}y_1 \\ y_2 \\ y_3 \\ y_4 \end{bmatrix}, \tag{3.2c} \bold{X_2} = \begin{bmatrix}x_1 & 1 \\ x_2 & 1 \\ x_3 & 1 \\ x_4 & 1 \end{bmatrix}, \,\,\, \bold{W_2} = \begin{bmatrix}m_2 \\ b_2 \end{bmatrix}, \,\,\, \bold{Y_2} = \begin{bmatrix}y_1 \\ y_2 \\ y_3 \\ y_4 \end{bmatrix}, \tag{3.2d} \bold{X_2 W_2 = Y_2}, \,\,\, where~ \bold{Y_2} \notin \bold{X_{2~ column~space}}, \tag{3.4} \bold{X_2 W_2^* = proj_{C_s (X_2)}( Y_2 )}, \tag{3.5} \bold{X_2 W_2^* - Y_2 = proj_{C_s (X_2)} (Y_2) - Y_2}, \tag{3.6} \bold{X_2 W_2^* - Y_2 \isin C_s (X_2) ^{\perp} }, \tag{3.7} \bold{C_s (A) ^{\perp} = N(A^T) }, \tag{3.8} \bold{X_2 W_2^* - Y_2 \isin N (X_2^T) }, \tag{3.9} \bold{X_2^T X_2 W_2^* - X_2^T Y_2 = 0} \\ ~ \\ \bold{X_2^T X_2 W_2^* = X_2^T Y_2 }, BASIC Linear Algebra Tools in Pure Python without Numpy or Scipy, Find the Determinant of a Matrix with Pure Python without Numpy or Scipy, Simple Matrix Inversion in Pure Python without Numpy or Scipy, Solving a System of Equations in Pure Python without Numpy or Scipy, Gradient Descent Using Pure Python without Numpy or Scipy, Clustering using Pure Python without Numpy or Scipy, Least Squares with Polynomial Features Fit using Pure Python without Numpy or Scipy, Single Input Linear Regression Using Calculus, Multiple Input Linear Regression Using Calculus, Multiple Input Linear Regression Using Linear Algebraic Principles.
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