Consider a temperature-sensing network consisting of N sensors, deployed on an interval X in R. Let x₁ € X and T₁ € R represent, respectively, the location and temperature reading of sensor i. Suppose, upon deployment, the sensors transmit their locations ri's and temperature readings Ti's to a base station. Suppose the base station wishes to construct an approximate temperature-versus-location profile by fitting a polynomial Î(x) =ão +â₁x +â₂x² + ... + âmxm of degree m≥ 0 to the data (x₁, T₁), (x2, T₂),..., (xn, Tn) from the first n = {1, 2, ..., N} sensors. Show that this may be accomplished using the least-squares method. For the least-squares solution to be well-defined, what conditions must m, n, x1, x2,..., n, T1, T2,..., Tn satisfy?
Consider a temperature-sensing network consisting of N sensors, deployed on an interval X in R. Let x₁ € X and T₁ € R represent, respectively, the location and temperature reading of sensor i. Suppose, upon deployment, the sensors transmit their locations ri's and temperature readings Ti's to a base station. Suppose the base station wishes to construct an approximate temperature-versus-location profile by fitting a polynomial Î(x) =ão +â₁x +â₂x² + ... + âmxm of degree m≥ 0 to the data (x₁, T₁), (x2, T₂),..., (xn, Tn) from the first n = {1, 2, ..., N} sensors. Show that this may be accomplished using the least-squares method. For the least-squares solution to be well-defined, what conditions must m, n, x1, x2,..., n, T1, T2,..., Tn satisfy?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Consider a temperature-sensing network consisting of N sensors, deployed on an interval X in R. Let
x₁ € X and T₁ € R represent, respectively, the location and temperature reading of sensor i. Suppose,
upon deployment, the sensors transmit their locations ri's and temperature readings Ti's to a base station.
Suppose the base station wishes to construct an approximate temperature-versus-location profile
by fitting a polynomial
Î(x) =ão +â₁x +â₂x² + ... + âmxm
of degree m≥ 0 to the data (x₁, T₁), (x2, T₂),..., (xn, Tn) from the first n = {1, 2,..., N} sensors.
Show that this may be accomplished using the least-squares method. For the least-squares
solution to be well-defined, what conditions must m, n, x1, x2,..., n, T1, T2,..., Tn satisfy?
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