Questions about (linear or nonlinear) least-squares, an estimation method used in statistics, signal processing and elsewhere.
Questions tagged [least-squares]
1853 questions
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Least Squares Optimization Converging on wrong solution
I'm trying to calculate the position of a multi-constellation GNSS receiver using GPS and GLONASS satellites using least-squares optimization. Sparing the details, I have 5 equations to solve for 5 unknowns: (x, y, z, r_gps, r_gal).
4 equations are…
Shawn Lim
- 101
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How is QR factorization used to solve Least Squares problem for ill conditioned matrix
Is there a method to find QR factorization for ill conditioned matrix... here the matrix (eye($3$) and geographical $xyz$ coordinate) $n*7$ matrix
1 0 0 0 -4.2495e+06 1.0366e+06 4.6289e+06
0 1 0 4.2495e+06 0.0 …
claudio
- 11
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Need help understanding least squares solution to overdetermined system
(Sorry I had to post the images as links. I don't have enough cred to post pictures directly yet)
I'm trying to understand what the least squares solution to an overdetermined system means geometrically in the case of the following system:
$$
y =…
vyb
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Is this understanding of the derivation of the Gauss-Newton algorithm correct?
Given a loss function $S$, with some data and some function which we want to approximate the data with, etc., the Gauss Newton algorithm for finding parameters (packed in a vector) $\vec\beta$ of the function $f$ that best minimise the loss, $S$, as…
FShrike
- 40,125
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Ordinary Least Squares: Why we need mean independent errors?
This is from my lecture on classic linear regression model:
$$
\text { Assumption 1: } E\left(\varepsilon \mid x_{1}, \ldots, x_{K-1}\right)=0
$$
Q: I am able to follow this fine until "Assumption 1 applies ... so that upon substitution". Where is…
aisync
- 365
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1 answer
When does this least squares analytical solution based on zeros of partial derivatives start providing more than one solution?
If I want to fit a quadratic function of two variables to some data, I can use
$$f(x, y) = c_1 x^2 + c_2 xy + c_3 y^2 + c_4 x + c_5 y + c_6$$
$$\frac{\partial}{\partial c_i} \sum_j\left( z_j - f(x_j, y_j) \right)^2 = 0$$
to obtain six equations, and…
uhoh
- 1,864
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derive the least squares estimate b1 from the normal equations
I am stuck on the following question*:
derive the least squares estimate:
$$ b1 = \frac{\sum X_iY_i - \frac{\sum X_i \sum Y_i}{n}}
{\sum X_i^2 - \frac{( \sum X_i)^2}{n}} $$
from the normal equations:
$$ (i) \sum Y_i = nb_0 + b_1 \sum…
Joseph
- 361
- 1
- 12
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Least squares estimation
I have the following linear regression model: $y_t=\beta_0+\beta_1x_t+\sigma \epsilon_t$, where $\epsilon_t$ is iid $N(0,1)$.
I am trying to estimate the parameters $\beta_0, \beta_1, \sigma$ using Least-Squares estimation. I am struggling about how…
user608881
- 77
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- 6
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Linear Least-Squares Frequency Domain
I am doing an implementation of the Poly-Reference Least Squares Complex Frequency Domain algorithm for modal analysis as described in various papers like: "A poly-reference implementation of the least-squares complex frequencydomain estimator" by…
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1 answer
Fitting a line through intercept 0
I need to code a least squares routine to fit a line
$$y = m*x$$
into a 2d set of points
$$(x_i,y_i)$$
How can I find the regression line without an interceptor?
Jan Hackenberg
- 105
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1 answer
Show that Var$(\beta_0)$ $\leq$ Var$(\beta'_0)$
The least squares estimator of $\beta_0$ $=$ $(Y\bar)$ $-$ $\beta_1$$(X\bar)$ can be expressed as a linear function of $Y_i$. Let $(\beta'_0)$ be another unbiased estimator of $\beta_0$, say $(\beta'_0)$ $=$ $$\sum_{i=1}^n c_iY_i $$ where $c_i$ $=$…
theshah
- 13
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Matrix Calculus in Least-Square method (Why setting first order derivatives to be zero guarantees it is minimum)
Assume V + Ax = b is the equation where V is the vectors of residuals, A is the matrix for coefficients, x is the vector for unknowns, and b is the vector for observation.
It is common to read something like "The least squares estimator is obtained…
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ln(N * e^st) to a Matrix A * x = b
given is this function $y = N \cdot e^st$.
I have to transform it to a "linear"(least square approx.)
$\implies y = \ln(N)+ st$
How do I put this to a Matrix form $A \cdot x = b$
$N$ and $s$ are unknown.
The only problem that I have is I don't know…
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2 answers
Average of a set of values using least squares formula
To get the equation of a line $y = ax+b$ passing through a set of $n$ points $(x_i, y_i)$ using least squares formula, we have to solve the following system of linear equations to determine the coefficients, $a$ and…
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Househbolder transformation identity matrix dimensions
When performing a householder transformation and generating an elementary reflector matrix of the form:
$$H = I - 2\dfrac{vv^T}{v^Tv}$$
How do we know the dimensions of the identity matrix?
Paradox
- 285