GenLSFit

Advanced Analysis Library Only

AnalysisLibErrType GenLSFit (void *hMatrix, ssize_t numberOfRows, ssize_t numberOfColumns, double yArray[], double standardDeviation[], int algorithm, double zArray[], double coefficientArray[], void *covariance, double *meanSquaredError);

Purpose

Finds the best fit k-dimensional plane and the set of linear coefficients using the least chi-squares method for observation data sets.

You can use GenLSFit to solve multiple linear regression problems and to solve for the linear coefficients in a multiple-function equation.

The general least squares linear fit problem can be described as follows. Given a set of observation data, find a set of coefficients that fit the linear model:

[Equation 1]

You can write Equation 1 as Y = HB.

The previous discussion leads to a multiple linear regression model, which uses several variables:

x_i0, x_i1, ..., x_{ik – 1}

to predict one variable y_i. In contrast, LinearFitEx, ExpFitEx, and PolyFitEx are all based on a single predictor variable, which uses one variable to predict another variable.

In most cases, we have more observation data than coefficients. The formulas in Equation 1 might not produce the solution. The fit problem becomes to find the coefficients B that minimize the difference between the observed data, y_i, and the predicted value:

GenLSFit uses the least chi-squares plane method to obtain the coefficients in Equation 1, that is, finding the solution, B, which minimizes the following quantity:

[Equation 2]

where

In the previous equation, σ_i is the standard deviation, stdDeviation. If the measurement errors are independent and normally distributed with constant standard deviation σ_i = σ, the previous equation is also the least squares estimation.

There are different ways to minimize χ². One way to minimize χ² is to set the partial derivatives of χ² to zero with respect to b₀, b₁, ..., b_{k – 1}:

The previous equations can be written as

is the transposition of H₀.

The previous equation and Equation 2 are also called normal equations of the least squares problems. You can solve them using LU or Cholesky factorization algorithms, but the solution from the normal equations is susceptible to round-off error.

The preferred way to minimize χ² is to find the least squares solution of the equations:

H₀B = Y₀

You can use QR or Singular Value Decomposition factorization to find the solution, B. For QR factorization, you can choose Householder, Givens, or Givens2, also called fast Givens.

Different algorithms can give you different precision. In some cases, if one algorithm cannot solve the equation, perhaps another algorithm can. You can try different algorithms to find the one best suited to your data.

GenLSFit calculates the covariance matrix covariance as follows:

The best fitted curve z is given by the following formula:

GenLSFit obtains the mean squared error using the following formula:

You can think of the polynomial fit that has a single predictor variable as a special case of multiple regression. If the observation data sets are (x_i, y_i) where i = 0, 1, . . ., n – 1, the model for polynomial fit is as follows:

[Equation 3]

where i = 0, 1, 2, . . ., n – 1

Comparing Equation 1 and the Equation 3 shows that . In other words:

In this case, you can build HMatrix as follows:

Instead of using , you can choose another function formula to fit the data sets (x_i, y_i). In general, you can select x_ij = f_j(x_i). Here, f_j(x_i) is the function model that you choose to fit your observation data. In polynomial fit, .

In general, you can build HMatrix as follows:

Your fit model is

y_i = b₀f₀(x) + b₁f₁(x) + ... + b_{k – 1}f_{k – 1}(x)

The following two examples show how to use GenLSFit. The first example uses the function to perform multiple regression analysis based entirely on tabulated observation data. The second solves for the linear coefficients in a multiple-function equation.

Predicting Cost

Suppose you want to estimate the total cost, in dollars, of a production of baked scones using the quantity produced, X₁, and the price of one pound of flour, X₂. To keep things simple, the following five data points form the sample data table shown in the following table.

You want to estimate the coefficients to the following formula:

Y = b₀ + b₁X₁ + b₂X₂

The only parameters you must build are the H (observation matrix) and y arrays. Each column of H is the observed data for each independent variable: The first column is one because the coefficient b₀ is not associated with any independent variable. Fill in H as follows:

The following code is based on this example.

// Example of predicting cost using GenLSFit
int k, n, algorithm, status;
double H[5][3], y[5], z[5], b[3], X1[5], X2[5], mse;
double *stdDev=0, *covar=0; /* Define empty arrays; the function will
ignore these parameters. */
n = 5;
k = 3;
// Read in data for X1, X2, and y.

.
.
.

// Construct matrix H.
for(i=0;i<n;i++) {

H[i][0] = 1; // Fill in the first column of H.
H[i][1] = X1[i]; // Fill in the second column of H.
H[i][2] = X2[i]; // Fill in the third column of H.

}
algorithm = 0; // Use SVD algorithm.
status = GenLSFit (H, n, k, y, stdDev, algorithm, z, b, covar, &mse);

Linear Combinations

Suppose that you have samples from a transducer, y values, and you want to solve for the coefficients of the model:

y = b₀ + b₁sin(ωx) + b₂cos(ωx) + b₃x³

To build H, set each column to the independent functions evaluated at each x value. Assuming there are 100 x values, H would be the following array:

The following code is based on this example.

// Example of linear combinations using GenLSFit
int i, k, n, algorithm, status;
double H[100][4], y[100], z[100], b[4], x[100], mse, w;
double *stdDev=0, *covar=0; /* Define empty arrays, the function will
ignore these parameters.*/
n = 100;
k = 4;
w = 0.2;
// Read in data for x and y.

.
.
.

// Construct matrix H.
for(i=0;i<n;i++) {

H[i][0] = 1; // Fill in the first column of H.
H[i][1] = sin(w*x[i]); // Fill in the second column of H.
H[i][2] = cos(w*x[i]); // Fill in the third column of H.
H[i][3] = pow(x[i],3); // Fill in the fourth column of H.

}
algorithm = 0; // Use SVD algorithm.
status = GenLSFit (H, n, k, y, stdDev, algorithm, z, b, covar, &mse);

Parameters

Return Value

Additional Information

Library: Advanced Analysis Library

Include file: analysis.h

LabWindows/CVI compatibility: LabWindows/CVI 4.0 and later