document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178892>Question 1178892</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the correct way to write the following function:<BR>\n" );
document.write( "K(x, xi) = Square of alpha multiplied with exp power minus of summation of k of (x - xi)² if we have following condition:<BR>\n" );
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document.write( "Vectors x1, x2,   x3, ... , xn belongs to d dimensional Real space (R with superscript d) and d is from 1 to D dimensional. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850285 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850285\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down how to write this function in a clear and standard mathematical notation, along with explanations.\r<BR>\n" );
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document.write( "**Understanding the Components:**\r<BR>\n" );
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document.write( "* **K(x, xi):** This indicates that the function K takes two inputs:<BR>\n" );
document.write( "    * `x`: A vector in d-dimensional real space (ℝᵈ).<BR>\n" );
document.write( "    * `xi`: Another vector in d-dimensional real space (ℝᵈ).<BR>\n" );
document.write( "* **α²:** \"Square of alpha\" means α raised to the power of 2 (α²).<BR>\n" );
document.write( "* **exp(...):** This represents the exponential function (e raised to the power of...).<BR>\n" );
document.write( "* **∑k (...):** This is a summation over an index k. We need to clarify what k is summing over.<BR>\n" );
document.write( "* **(x - xi)²:** This represents the squared Euclidean distance between vectors x and xi.\r<BR>\n" );
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document.write( "**Clarifying the Summation:**\r<BR>\n" );
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document.write( "Since x and xi are d-dimensional vectors, the summation is likely over the components of these vectors. So, if:\r<BR>\n" );
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document.write( "* x = (x₁, x₂, ..., xᵈ)<BR>\n" );
document.write( "* xi = (xi₁, xi₂, ..., xiᵈ)\r<BR>\n" );
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document.write( "Then:\r<BR>\n" );
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document.write( "* (x - xi)² = ∑(k=1 to d) (xk - xik)²\r<BR>\n" );
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document.write( "**Putting it All Together:**\r<BR>\n" );
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document.write( "The function K(x, xi) can be written as:\r<BR>\n" );
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document.write( "**K(x, xi) = α² * exp(-∑(k=1 to d) (xk - xik)²)**\r<BR>\n" );
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document.write( "**More Detailed Notation:**\r<BR>\n" );
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document.write( "If you want to be even more explicit, you can write:\r<BR>\n" );
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document.write( "**K(x, xi) = α² * exp(-∑(k=1)ᵈ (x<0xE2><0x82><0x96> - xi<0xE2><0x82><0x96>)²)**\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* x<0xE2><0x82><0x96> represents the k-th component of vector x.<BR>\n" );
document.write( "* xi<0xE2><0x82><0x96> represents the k-th component of vector xi.\r<BR>\n" );
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document.write( "**Python Example (for clarity):**\r<BR>\n" );
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document.write( "```python<BR>\n" );
document.write( "import numpy as np\r<BR>\n" );
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document.write( "def K(x, xi, alpha):<BR>\n" );
document.write( "    \"\"\"<BR>\n" );
document.write( "    Calculates the function K(x, xi).\r<BR>\n" );
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document.write( "    Args:<BR>\n" );
document.write( "        x: A numpy array representing vector x.<BR>\n" );
document.write( "        xi: A numpy array representing vector xi.<BR>\n" );
document.write( "        alpha: A scalar value.\r<BR>\n" );
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document.write( "    Returns:<BR>\n" );
document.write( "        The value of K(x, xi).<BR>\n" );
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document.write( "    diff = x - xi<BR>\n" );
document.write( "    squared_distance = np.sum(diff**2)<BR>\n" );
document.write( "    return alpha**2 * np.exp(-squared_distance)\r<BR>\n" );
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document.write( "# Example usage:<BR>\n" );
document.write( "x = np.array([1, 2, 3])<BR>\n" );
document.write( "xi = np.array([4, 5, 6])<BR>\n" );
document.write( "alpha = 2.0\r<BR>\n" );
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document.write( "result = K(x, xi, alpha)<BR>\n" );
document.write( "print(result)<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**Key Points:**\r<BR>\n" );
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document.write( "* The summation is crucial. It's summing the squared differences of the corresponding components of the vectors.<BR>\n" );
document.write( "* The exponential function applies to the negative of the squared Euclidean distance.<BR>\n" );
document.write( "* Alpha is a scaling factor.<BR>\n" );
document.write( "* The function is a radial basis function (RBF) kernel, and is used in machine learning.<BR>\n" );
document.write( " </SPAN>\n" );
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