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Introduction of Matlab polyfit()

MATLAB function polyfit() is defined to fit a specific set of data points to a polynomialquickly and easily computing polynomial with the least squares for the given set of data. It generates the coefficients for the elements of the polynomial, which are used for modeling a curve to fit to the given data.

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Polyfit() function uses input vector (x) to form Vandermonde matrix (V ) having n+1 columns and r = length(x) rows, which is nothing but results in a linear system.

Syntax of Matlab polyfit()

Syntax of Matlab polyfit() are given below:

Syntax

poly = polyfit(x,y,n) It generates the coefficients of the resultant polynomial p(x) with a degree of ‘n’, for the data set in yas the best fit in the view of a least-square. The coefficients in p are assigned to power in descending order and matching length of p to n+1.

[poly,Struct] = polyfit(x,y,n) It results in a structure S which can be used as input to the function polyval() in order to obtain error estimation.

[poly,Struct,mu] = polyfit(x,y,n) It results in a two-element vector having values-centered and scaled.

mu(1) holds a value of the mean of (x), and

mu(2) ) holds the value of standard of (x).

Using these two values, function polyfit()makes x centered at zero and scaledx to have a unit standard deviation,

Input Arguments

Query Points: Query points are specified as an input of vector type. If x is non-vector element, then this function polyfit() converts x into a column chúng tôi data points in x and their corresponding fitted function values contained in the vector y are formed.

Note

If the vector x has recurring data points or if it needs centering and scaling, warning messages may result out.

Fitted values at query points: Fitted values as inputs are available at query points being specified with the vector data type. The data points in x and their corresponding fitted function values contained in the vector y are formed. If y is the non-vector element, then this function polyfit() converts y into a column vector.

Degree of polynomial fit: Degree of polynomial fit as inputs, are available being specified as any positive integer scalar. In the respective syntax, ‘n’refers to the polynomial power to that of the left-most coefficient in the polynomial ‘p’.

Example of Matlab polyfit()

The below code is designed to generate data points placed equally spaced across a sine curve drawn in a specific interval.

Code:

hold off

Output:

Use Cases for polyfit() Function

Use cases for polyfit() function are given below:

Fitting  Polynomial to Set of data Points: The below code snippet carry out the fitting process on the polynomial poly of degree 4 towards 5  points.

Code:

legend(‘ydata’,’ydata1′,’fig1′)

Output:

Fitting the Polynomial function to Error Function: The below code generate a vector having x data points being placed equally in the interval of [0,3.5] and co-efficient are assigned to the polynomial assuming the degree as 6.

Code:

hold off

Output:

Improving Numerical Properties using Centering and Scaling: While solving the equation p = Vy, the condition number for V is usually large for higher-order fits and results in a matrix with singular coefficient, as the columns of V (Vandermonde matrix) are powers of the x vector.

In such cases, functions like scaling and centering are helpful to improve the numerical properties associated with the system in order to find a fit that is more reliable. In the below example polyfit() is called on three outputs to fit a polynomial of degree 5 along with the process of centering and scaling. The data is centered for the quarter, at 0, and scaled to have a unit standard deviation.

Code:

hold off

Output:

Simple Linear Regression: A simple linear regression model can be used to apply a fitting to a set of discrete two-dimensional data points.

legend(‘data’,’func’)

Output:

Combining Linear Regression and Error Estimation: A linear model can be set fit to a set of specified data points and the results can be plotted including an estimation of a prediction interval of 95%.

A few vectors can be created containing sample data points. The function polyfit can be called to fit a polynomial of degree 1 to the given set of data. Dual outputs can be specified to hold the values of coefficients supporting a linear fit as well as a structure containing error estimation.

Code:

[poly,Samp] = polyfit(xdata,ydata,1);

/*The error estimation structure is specified as the third input so that the function polyval()computes an estimated standard error. The estimated standard error estimate is stored in the second output variable delta. */

title(‘Usage of polyfit and polyval’)

Output:

For n number of data points, a polynomial can be fit to that of degree n-1 to passing exactly through the points.

With the increase in the degree of the polynomial, a fitting process using polyfit()loses the accuracy resulting in to a poorer fit for the data. In such cases, a low-order polynomial is preferable to use that tends to be smoother between the data points or apply a different technique, based on the requirement.

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This is a guide to Matlab polyfit(). Here we also discuss the introduction and use cases for polyfit() function along with examples and its code implementation. you may also have a look at the following articles to learn more –

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## Learn The Example Of Opencv Puttext

Introduction to OpenCV putText()

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It is also needed that the starting point for the text has to be defined within the matrix. there is also a requirement for defining the font color for the text, the font style for the text, and the weight that the text has two be specified with. The function is present in the OpenCV library of Python programming language, which is a one-stop solution that has been designed in order to solve problems related to computer vision.

Syntax for OpenCV putText()

cv2.put Text (* image *, text *, org *, font *, fontScale *, color [*, thickness [,line Type [*,*  bottom Left Origin *] *] *] *) * Parameters for OpenCV put text function

The following parameters are accepted by the OpenCV putText() function:

Parameters Description of the parameters:

image this parameter represents the original image that the user has selected to add text by the system

text This parameter represents the text that has to be drawn by the system as specified by the user

org This parameter is used to represent the coordinate with respect to the text present on the bottom left corner of the image. These coordinates are represented with the help of 2 double values that are passed, which represent the X coordinate and the Y coordinate values, respectively

font This parameter is used to represent the type or style of font, which would be denoted for the string text that the user specifies. some instances for the kinds of font types that can be used are FONT * _ * HERSHEY * _ * PLAIN OR FONT * _ * HERSHEY * _ * SIMPLEX

font * scale This parameter represents the best size for the specified font size, which is relevant to the font scale factor, which acts as a multiplying factor further font size of the text that has to be entered

thickness this parameter represents the thickness that has to be given for the line of text that has to be entered by the user. it is measured in terms of pixel size

color this parameter represents the specific color that has to be given to the text string that is being entered into the image that is being drawn on the screen. The color is extracted from the BGR tuple, which is passed to it. for instance, for a text of blue color the tuple to be passed would be * (* 255 *, * 0 *, * 0) *

Line * Type This parameter is used to define the type of line used for the text, which has to be entered into the image. This parameter is an optional parameter.

Bottom * Left * Origin this parameter is used for defining the position for the image data origin with respect to the directional position in the image. this parameter is an optional parameter. If the parameter is taken as true, the image data origin is found to be placed at the bottom left corner of the image. If it is not true, the image data origin is placed at the top left corner of the image.

Return * Value This method is responsible for returning an output image that is to be loaded from the file which has been specified.

Example of OpenCV putText()

Following is the example which is used in order to demonstrate how the OpenCV putText() command is utilized in the Python programming language

# command used to import the OpenCV library to utilize OpenCV read image function import cv2 # path being defined from where the system will read the image path = r'C:Users Priyanka Desktop educba OpenCV edu cba logo.png' # command used for reading an image from the disk disk, cv2.imread function is used image1 = cv2.imread(path) # Window name being specified where the image will be displayed window_name1 = 'image' # font for the text being specified font1 = cv2.FONT_HERSHEY_SIMPLEX # org for the text being specified org1 = (50, 50) # font scale for the text being specified fontScale1 = 1 # Blue color for the text being specified from BGR color1 = ( 255 , 0 , 0 ) # Line thickness for the text being specified at 2 px thickness1 = 2 # Using the cv2.putText() method for inserting text in the image of the specified path image_1 = cv2.putText(image1, 'EDU CBA', org1, font1, fontScale1, color1, thickness1, cv2.LINE_AA) # Displaying the output image cv2.imshow(window_name, image_1) cv2.waitKey(0) cv2.destroyAllWindows()

Conclusion

The OpenCV putText() method is a very useful function present in the OpenCV library, which allows the system to add text to an image that the user has provided. There are several image processing areas where text needs to be associated with the images that are being processed, and there needs to be a variety in the color, font style, width, and orientation in terms of the position whether the text has to be placed on the image which can easily be utilized by using the put text method. It also reduces the verbosity of the program that is being written and increases the overall processing speed for the program to be executed.

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## Different Function Of Linspace In Matlab With Examples

Introduction to Linspace MATLAB

MATLAB is a technical computing language. MATLAB gets its popularity from providing an easy environment for performing and integrating computing tasks, visualizing & programming.

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Uses of MATLAB include (but not limited to)

Computation

Simulation

Modeling

Data analytics (Analysing and Visualizing data)

Prototyping

Application development

Engineering & Scientific graphics

Linspace Function in MATLAB

In this article, we will understand a very useful function of MATLAB called ‘linspace’. This function will generate a vector of values linearly spaced between two endpoints. It will need two inputs for the endpoints and an optional input to specify the number of points to include in the two endpoints.

X = linspace(a1, a2)

Now let us understand this one by one

1. X=linspace(a1,a2)

This function will return a row of a vector of 100(default) linearly spaced points between a1 and a2

a1 and a2 can be real or complex

a2 can be either larger or smaller than a1

If a2 is smaller than a1 then the vector contains descending values

Here is an example to understand this:

Example #1

X = linspace(-1, 1)

It will generate a vector of 100 evenly spaced vectors for the interval [-1, 1]

Output:

Example #2

X = linspace(2, 3)

It will generate a vector of 100 evenly spaced vectors for the interval [2,3]

Output:

Example #3

X = linspace(2, 1)

Here a2 is smaller than a1, it will generate a vector of 100 evenly spaced vectors for the interval [2,1] in descending order

Output:

2. X=linspace(a1,a2,n)

This function will return a row of a vector of “n” points as specified in input for linearly spaced points between a1 and a2. This function gives control of the number of points and will always include the endpoints specified in the input as well.

If n is 1, the function will return a2 as output

If n is zero or negative, the function will return 1by0 empty matrix

Here is an example to understand this:

Example #1

X = linspace(-1, 1, 7 )

It will generate a vector of 7 evenly spaced vectors for the interval [-1, 1]

Output:

Example #2

X = linspace(2,3,5)

It will generate a vector of 5 evenly spaced vectors for the interval [2,3]

Output:

Example #3

X = linspace(2, 3, 1)

Here n = 1, so the function will return a2 input parameter

Output:

Example #4

Here n = 0, so function will return 1X0 empty double row vector

Output:

Vector of evenly spaced Complex numbers

X = linspace(2+2i, 3+3i)

Here a1 and a2 are complex numbers, it will generate a vector of complex numbers for 100 evenly spaced points for the interval [2+21, 3+3i]

Output:

X= linspace(1+1i, 5+5i, 4)

It will generate a vector of complex numbers with 4 evenly spaced point for the interval [1+1i, 5+5i]

Output:

The linspace function in MATLAB provides us with an array/matrix comprising the desired number of values starting from and ending at a declared value. The produced array will have exactly the desired number of terms which will be evenly spaced. The values will be in the range of start and end values passed. So, the linspace function will help us in creating an instantiated matrix or array.

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## Example And Graphical Representation Of Tail Risk

Definition of Tail Risk

Explanation

Usually, the financial market follows a normal distribution but with the introduction of tail risk, the idea that the distribution of returns is not normal but has fatter tails crosses the mind. A tail risk or fat tail risk is an event of a risk for the portfolios of assets. It occurs when there is a possibility of fluctuation in the investment from its mean point to three standard deviations. Although the probability of such events to occur is quite low but still, it can cause great negative consequences for the market as well as for the portfolios.

How does It Work?

A tail risk occurs when there is a possibility of three standard deviations to move further than the mean point. Usually, in financial markets, it is believed that returns that are generated from the securities and portfolios have a normal distribution. Such distribution is also called bell-shaped distribution. But tail risk suggests that these distributions are not normal but has fatter tails, the left side of the tail represents the lowest return an investor can get on the investment while the right side of the tail represents the highest return an investor can get. It can also be named as left tail risk and right tail risk depending upon the sides; left tail risk shows negative returns while right tail risk shows positive returns of a portfolio.

Example of Tail Risk

A tail risk can be understood better by taking a real-life example. In 2007, the health of 30 public companies of the United States of America was contained in the Dow Index, these companies were also a part of the S&P 500 Index. The index showed great performance and went above the 24k mark in December 2023 and continued to show an upward movement since then.

In January 2023, the index which had been continuing to show an upward movement hit the mark of 26k and investors were expecting that this growth will continue to flourish but due to trade wars and economic slowdown, the whole market of the US suffered. It also created an impact on the Dow Index and it reached fall back to the 24k mark in October 2023. It was a massive fall for the market and the Dow index fell to 21k in December 2023. It was a 19℅ downward move in the year. The tail risk in the Dow Index was when it went below the 24k mark in October 2023, as the downfall from 26k to 24k was a behavioral movement but when it went below that, the condition got worse.

Graphical Representation of Tail Risk

A tail risk can be better understood with the help of graphical representation. Here,

The bluebell curve indicates the normal distribution that is generated from the securities or the portfolios.

The green bell curve indicates the tail risk fluctuations that could happen in the investment from the mean point to the three normal deviations.

The left tail of the green curve shows how that much loss an investor can suffer while the right tail of the green curve shows how much the investor can gain on investments.

Uses of Tail Risk

It shows investors all the possible risks that could happen which makes them able to gauge the risks.

Tail risk enables investors to take quick and accurate decisions in the investment strategy.

Tail risk is used to support hedging which results in better and increased cash flow in the market.

Tail risk is used to get knowledge about the negative activity that could happen in the market.

Enhanced decision making: Tail risk enables the investor in measuring the unforeseen risks which enable the investor to take accurate decisions.

Encouragement of hedging: Tail risk encourages hedging which results in the better and increased flow of funds in the market.

Create awareness: Tail risk creates awareness about all the possible risks that could happen in the future which could create a great negative impact thus creating awareness.

Possibility of not occurring: There is a high possibility for the tail risk that it wouldn’t occur even onetime as there is a quite low chance of such occurring.

The fear among the investors: Tail risk creates a sense of fear among the investors which can result in fewer investments and can create a negative outlook.

Conclusion

The possibility of losses that could happen shortly due to any unforeseen event is known as tail risk. It can be on both the sides of the curve as the right tail and the left tail, the right side indicates gains while the left side indicates losses. It is beneficial for the businesses and the investors as they can study tail risk which will help them to take accurate decisions for investment.

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## How To Create Matlab Object? (Examples)

Introduction to Matlab Object

The MATLAB language uses many specialized objects. For example, Exception objects, timer objects, the serial object, and so on. MATLAB toolboxes are used to define objects to manage data and analyses performed by the toolbox. Objects provide specific functionality that is not easily available from general-purpose language components. Objects are used to inform when errors occur, to execute code at a certain time interval, to enables you to communicate with devices connected to your computer’s serial port, etc.

Syntax:

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Object_name = class_name;

How to Create Matlab Object?

To create an object, first, we need to create a class, using ‘ classdef ’ we create a class, in class we take some properties and end the class and then we take methods some methods using function statements after all these lastly we end the class with an end statement. First, we save the class using the .m extension.

Now, take a new Matlab script and create an object using the same class name which we used to create a class. For creating the object we write syntax like:

Object_name = class_name;

Let’s consider a1 is an object name and BasicClass1 is a class name. In class, we create 1 property and 3 methods. After creating an object of class we can perform the several operations on a class by using that class object:-

For accessing the properties to assign the data.

To perform operations on data calling of the method.

We saw all the properties and their current values available in the class.

Examples to Implement Matlab Object

Let see the first example, in this example first, we create one class, that class name is ‘BasicClass1 ’. In this class we take a property like value, the must be only the numbers. Then we take a method, basically, methods are nothing but operations that defined by the class. We create some operations and then we end the method also with an end statement. And then finally we end the BasicClass1 class with an end statement. In methods for operations, we use some functions. This code saves using the .m extension.

Code:

end

Example #1

Let us see to create an object and how it’s used. In this example we can create an object with the respected class name in our example it’s ‘ BasicClass1 ’, we create objects namely ‘a1’. Initially the value of the property it’s empty. Then using objects we can access the properties of the class that’s nothing but the property is value. So we assign some number to the value property using a class object, and simply it can display it.

Code:

Output:

As we have seen in the command window the output was value = [ ] empty. Because the initial value of property is empty. Then we assign the number to property value and the number is pi (3.142) / 3 and displays it using the class object.

Example #2

Let us see an example, for this example also we use the same class that’s ‘BasicClass1’. First, we simply create an object for the same class name that’s BasicClass1. Then we can assign the number to property value and display it. Then after we call the methods of that class by using the help of class object ‘a1’. We called the method name, the method name is ‘multiplyBy (obj1 , n1)’ multiplying operation by the given number in parenthesis we simply pass the argument and then display the result.

Code:

multiplyBy(a1,3)

Example #3

Let us see an example, for this example also we use the same class that’s ‘BasicClass1’. First, we simply create an object for the same class name that’s BasicClass1. Then we can assign the number to property value and display it. Then after we call the methods of that class by using the help of class object ‘a1’. We called the method name, the method is ‘divideBy (obj1 , n1)’ divide operation by the given number in parenthesis we simply pass the argument and then display the result.

Code:

a1.divideBy( 2 )

Output:

Conclusion

In this article, we saw the concept of Matlab object. We understood the basic concept and different ways to use Matlab objects and what exactly is a Matlab object.  And also we saw the syntax. Also first we saw how to declare or create a class on Matlab and then how to create a class object. And what operations we perform using that class object.

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## How Autocorrelation Function Works In Matlab?

Definition of Matlab Autocorrelation

In Matlab, Autocorrelation function means a correlation between numbers in a set or series with other numbers in the same set or series separated by provided time interval. Autocorrelation is also called a serial correlation because it correlates numbers with a delayed copy of itself set or series. Autocorrelation is used in signal processing for analyzing a series of values like time-domain signals. Autocorrelation means the correlation between the observation at the current time spot and the observation at previous time spots. Autocorrelation is used to determine the terms used in the MA model. Autocorrelation is used to measure the relation between the elements’ current value and past values of the same element.

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How does Autocorrelation Function work in Matlab?

Autocorrelation measures the relation between elements’ current value and past values of the same element. There are the following steps of autocorrelation function to work in Matlab: –

Step 2: Assign all data to a variable.

Step 3: Then, use the appropriate syntax of the ‘Matlab Autocorrelation’ function.

Step 4: then execute the code.

Examples of Matlab Autocorrelation

Lets us discuss the examples of Matlab Autocorrelation.

Example #1

In this example, we calculate the autocorrelation of random Gaussian noise in Matlab. We know that autocorrelation means matching signals with the delayed version itself. Now for random Gaussian noise, only when shift= 0 is there some autocorrelation value, and for all other cases, the autocorrelation result will be zero. Now first, we will generate random Gaussian noise in Matlab. For generating random Gaussian noise, we will use randn function in Matlab. “x= randn(1, length(t))” generates length t Gaussian sequence with mean 0 and variance 1. After that, we use the subplot and plot function to plot the random Gaussian noise signal. Here we will use the Matlab autocorrelation function to calculate the autocorrelation of random Gaussian noise in Matlab.“autocorr(x)” this syntax is used for calculating the autocorrelation of random Gaussian noise. Then subplot and plot function is used for plotting the autocorrelation of random Gaussian noise. To calculate the autocorrelation of a random Gaussian signal, execute the Matlab code.

Code :

Output:

Example #2

In this example, we can see how we can find the periodicity of the signal using the function. So let’s first load the data. Here we use office temperature for data. This is, by default, available in Matlab. Once we load the data, we plot the data. We use a plot function to plot the data. After the plotting data, we will find the temperature oscillates. So we take the normal temperature by using mean temperature. “normal_temp= temp -mean(temp)” ones we subtract mean temperature from temperature, we get the normal temperature. After that, we will plot the normal temperature using the plot function. so we get normal temperatures varying around zero. Now we will set sampling ‘fs’ as 24. Then we are going to create a time vector t. The t will start from 0 and go up to the length of the normal temperature. Then we use Matlab autocorrelation to find the periodicity of the signal. Then we use above syntax “[autocor, lags]= xcorr (normal_temp,3*7*fs,’coeff’)”. Here ‘autocor’ variable stores the autocorrelation matrix, and ‘lags’ this variable stores the lags between the data. ‘xcorr’ correlates between normal temperature and sampling frequency. Then we plot the data that lag/fs, and autocor plot the autocorrelation of the signal.

Code:

plot(lags/fs,autocor);

Output:

Example #3

In this example, we calculate the autocorrelation of the input sine signal. Now we load the signal in variable ‘x.’ “x= sin(2*t)” is used to get the sine signal in Matlab. After that, we use the subplot and plot function to plot the sine signal. Here we will use the function to calculate the autocorrelation of random Gaussian noise in Matlab.“autocorr(x)” this syntax is used for calculating the autocorrelation of sine signal. Then subplot and plot function is used for plotting the autocorrelation of the sine signal.

Code:

autocorr(x)

Output:

Conclusion

In this article, we saw the concept of Matlab autocorrelation. Basically, this function is used to measure the relation between elements’ current values and past values of the same element. Then we saw how we could find the periodicity of the signal using this function and the efficient work of the Matlab autocorrelation function.

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