Vector valued functions are often encountered in machine learning, computer graphics and computer vision algorithms. They are particularly useful for defining the parametric<\/span> equations of space curves. It is important to gain a basic understanding of vector valued functions to grasp more complex concepts.<\/p>\n In this tutorial, you will discover what vector valued functions are, how to define them and some examples.<\/p>\n After completing this tutorial, you will know:<\/p>\n Let\u2019s get started.<\/p>\n A gentle iIntroduction to vector valued functions. Photo by Noreen Saeed, some rights reserved<\/p>\n<\/div>\n This tutorial is divided into two parts; they are:<\/p>\n A vector valued function is also called a vector function. It is a function with the following two properties:<\/p>\n Vector functions are, therefore, simply an extension of scalar functions, where both the domain and the range are the set of real numbers.<\/p>\n In this tutorial we\u2019ll consider vector functions whose range is the set of two or three dimensional vectors. Hence, such functions can be used to define a set of points in space.<\/p>\n Given the unit vectors i,j,k parallel to the x,y,z-axis respectively, we can write a three dimensional vector valued function as:<\/p>\n r(t) = x(t)i + y(t)j + z(t)k<\/p>\n It can also be written as:<\/p>\n r(t) = <x(t), y(t), z(t)><\/p>\n Both the above notations are equivalent and often used in various textbooks.<\/p>\n We defined a vector function r(t) in the preceding section. For different values of t we get the corresponding (x,y,z) coordinates, defined by the functions x(t), y(t) and z(t). The set of generated points (x,y,z), therefore, define a curve called the space curve C. The equations for x(t), y(t) and z(t) are also called the parametric<\/span> equations of the curve C.<\/p>\n This section shows some examples of vector valued functions that define space curves. All the examples are also plotted in the figure shown after the examples.<\/p>\n Let\u2019s start with a simple example of a vector function in 2D space:<\/p>\n r_1(t) = cos(t)i + sin(t)j<\/p>\n Here the parametric equations are:<\/p>\n x(t) = cos(t)<\/p>\n y(t) = sin(t)<\/p>\n The space curve defined by the parametric<\/span> equations is a circle in 2D space as shown in the figure. If we vary t from -\ud835\udf0b to\u00a0\ud835\udf0b, we\u2019ll generate all the points that lie on the circle.<\/p>\n We can extend the r_1(t) function of example 1.1, to easily generate a helix in 3D space. We just need to add the value along the z axis\u00a0<\/span>that changes with t. Hence, we have the following function:<\/p>\n r_2(t) = cos(t)i + sin(t)j + tk<\/p>\n We can also define a curve called the twisted cubic with an interesting shape as:<\/p>\n r_3(t) = ti + t^2j + t^3k<\/p>\n Parametric curves<\/p>\n<\/div>\n We can easily extend the idea of the derivative of a scalar function to the derivative of a vector function. As the range of a vector function is a set of vectors, its derivative is also a vector.<\/p>\n If<\/p>\n r(t) = x(t)i + y(t)j + z(t)k<\/p>\n then the derivative of r(t) is given by r'(t) computed as:<\/p>\n r'(t) = x'(t)i + y'(t)i + z'(t)k<\/p>\n We can find the derivatives of the functions defined in the previous example as:<\/p>\n The parametric\u00a0equation of a circle in 2D is given by:<\/p>\n r_1(t) = cos(t)i + sin(t)j<\/p>\n Its derivative is therefore computed by computing the corresponding derivatives of x(t) and y(t) as shown below:<\/p>\n x'(t) = -sin(t)<\/p>\n y'(t) = cos(t)<\/p>\n This gives us:<\/p>\n r_1\u2032(t) = x'(t)i + y'(t)j<\/p>\n r_1\u2032(t) = -sin(t)i + cos(t)j<\/p>\n The space curve defined by the parametric equations is a circle in 2D space as shown in the figure. If we vary t from -\ud835\udf0b to \u03c0, we\u2019ll generate all the points that lie on the circle.<\/p>\n Similar to the previous example, we can compute the derivative of r_2(t) as:<\/p>\n r_2(t) = cos(t)i + sin(t)j + tk<\/p>\n r_2\u2032(t) = -sin(t)i + cos(t)j + k<\/p>\n The derivative of r_3(t) is given by:<\/p>\n r_3(t) = ti + t^2j + t^3k<\/p>\n r_3\u2032(t) = i + 2tj + 3t^2k<\/p>\n All the above examples are shown in the figure, where the derivatives are plotted in red. Note the circle\u2019s derivative also defines a circle in space.<\/p>\n Parametric functions and their derivatives<\/p>\n<\/div>\n Once you gain a basic understanding of these functions, you can have a lot of fun defining various shapes and curves in space. Other popular examples used by the mathematical community are defined below and illustrated in the figure.<\/p>\n The toroidal spira<\/strong>l:<\/p>\n r_4(t) = (4 + sin(20t))cos(t)i + (4 + sin(20t))sin(t)j + cos(20t)k<\/p>\n The trefoil knot<\/strong>:<\/p>\n r_5(t) = (2 + cos(1.5t)cos (t)i + (2 + cos(1.5t))sin(t)j + sin(1.5t)k<\/p>\n The cardioid:<\/strong><\/p>\n r_6(t) = cos(t)(1-cos(t))i + sin(t)(1-cos(t))j<\/p>\n More complex curves<\/p>\n<\/div>\n Vector valued functions play an important role in machine learning algorithms. Being an extension of scalar valued functions, \u00a0you would encounter them in tasks such as multi-class classification and multi-label problems. Kernel methods<\/a>, an important area of machine learning, can involve computing vector valued functions, which can be later used in multi-task learning or transfer learning.<\/p>\n This section lists some ideas for extending the tutorial that you may wish to explore.<\/p>\n If you explore any of these extensions, I\u2019d love to know. Post your findings in the comments below.<\/p>\n This section provides more resources on the topic if you are looking to go deeper.<\/p>\n In this tutorial, you discovered what vector functions are and how to differentiate them.<\/p>\n Specifically, you learned:<\/p>\n Ask your questions in the comments below and I will do my best to answer.<\/p>\n The post A Gentle Introduction To Vector Valued Functions<\/a> appeared first on Machine Learning Mastery<\/a>.<\/p>\n<\/div>\n Go to Source<\/a><\/p>\n","protected":false},"excerpt":{"rendered":" Author: Mehreen Saeed Vector valued functions are often encountered in machine learning, computer graphics and computer vision algorithms. They are particularly useful for defining the […] Read More<\/a><\/span><\/p>\n","protected":false},"author":1,"featured_media":4863,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_bbp_topic_count":0,"_bbp_reply_count":0,"_bbp_total_topic_count":0,"_bbp_total_reply_count":0,"_bbp_voice_count":0,"_bbp_anonymous_reply_count":0,"_bbp_topic_count_hidden":0,"_bbp_reply_count_hidden":0,"_bbp_forum_subforum_count":0,"footnotes":""},"categories":[24],"tags":[],"_links":{"self":[{"href":"https:\/\/www.aiproblog.com\/index.php\/wp-json\/wp\/v2\/posts\/4862"}],"collection":[{"href":"https:\/\/www.aiproblog.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.aiproblog.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.aiproblog.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.aiproblog.com\/index.php\/wp-json\/wp\/v2\/comments?post=4862"}],"version-history":[{"count":0,"href":"https:\/\/www.aiproblog.com\/index.php\/wp-json\/wp\/v2\/posts\/4862\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.aiproblog.com\/index.php\/wp-json\/wp\/v2\/media\/4863"}],"wp:attachment":[{"href":"https:\/\/www.aiproblog.com\/index.php\/wp-json\/wp\/v2\/media?parent=4862"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.aiproblog.com\/index.php\/wp-json\/wp\/v2\/categories?post=4862"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.aiproblog.com\/index.php\/wp-json\/wp\/v2\/tags?post=4862"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\n
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<\/a><\/p>\nTutorial Overview<\/h2>\n
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Definition of a Vector Valued Function<\/h2>\n
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Space Curves and Parametric Equations<\/h3>\n
Examples of Vector Functions<\/h2>\n
1.1 A Circle<\/h3>\n
1.2 A Helix<\/h3>\n
1.3 A Twisted Cubic<\/h3>\n
<\/a><\/p>\nDerivatives of Vector Functions<\/h2>\n
Examples of Derivatives of Vector Functions<\/h2>\n
2.1 A Circle<\/h3>\n
2.2 A Helix<\/h3>\n
2.3 A Twisted Cubic<\/h3>\n
<\/a><\/p>\nMore Complex Examples<\/h2>\n
<\/a><\/p>\nImportance of Vector Valued Functions in Machine Learning<\/h2>\n
Extensions<\/h2>\n
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Further Reading<\/h2>\n
Tutorials<\/h3>\n
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Resources<\/h3>\n
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Books<\/h3>\n
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Summary<\/h2>\n
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Do you have any questions?<\/h2>\n