LSQ2: An update to LSQ (Casio fx-9750GIII, TI-84 Plus CE)
Least Square Matrix and Correlation
The program LSQ2 fits the allows to fit data to a function with minimal error possible.
Multiple Linear Regression:
f(x1, x2, x3, ...) = b0 + b1 * x1 + b2 * x2 + b3 * x3 + ....
Polynomial Regression:
f(x) = b0 + b1 * x + b2 * x^2 + b3 * x^3 + ...
General:
f(x) = b0 + b1 * g1(x) + b2 * g2(x) + b3 * g3(x) + ...
f(x1, x2, x3, ...) = b0 * g0(x1, x2, x3, ...) + b1 * g1(x1, x2, x3, ...) + ...
The Matrices X, Y, and B
X is your data matrix and is set up as columns:
[ g0(x), g1(x), g2(x), ... ]
Where the function g(x) represents functions applied to every data point x_i.
Example 1: f(x) = b0 + b1 * x
The columns of the data matrix are set up as:
[ 1, x ]
A column of ones set up solving for a constant.
Example 2: f(x) = b0 + b1 * x + b2 * x^2
The columns of the data matrix are set up as:
[ 1, x, x^2 ]
Example 3: f(x0, x1) = b0 + b1 * x1 + b2 * x2
The columns of the data matrix are set up as:
[ 1, x1, x2 ] (note, not x squared in this case)
Y is the answer matrix, of size n rows and 1 column. There are n data points. We are fitting the function to y_i.
B is the coefficient matrix, consisting of values b0, b1, b2, ....
Simply put, to find B using the least squares method given the data points:
B = (X^T X)^-1 X^T Y
X^T is the transpose matrix of X
How well does the function fit?
We can predict y values by multiplying X by B.
P' = X B
Determining Coefficient of Correlation:
r^2 = SSreg ÷ SStot = [ B^T X^T Y - (O Y)^2 ÷ n ] ÷ [ Y^T Y - (O Y)^2 ÷ n ]
where O is a ones matrix [[ 1, 1, 1, 1, ... ]] of size 1 x n.
Casio fx-9750GIII Program: LSQ2
From the text file:
'ProgramMode:RUN
ClrText
"2022_-_07_-_19 EWS"
"LEAST SQUARES"
"_Mat _X"?->Mat X
"_Mat _Y"?->Mat Y
Dim Mat Y->List 26
List 26[1]->N
(Trn Mat X*Mat X)^-1*Trn Mat X*Mat Y->Mat B
"_Mat _B:"Disps
Mat BDisps
{1,N}->Dim Mat O
Fill(1,Mat O)
Mat O*Mat Y->Mat S
Mat S*Mat S/N->Mat S
(Trn Mat B*Trn Mat X*Mat Y)-Mat S->Mat R
Mat R*(Trn Mat Y*Mat Y-Mat S)^-1->Mat R
"R_^<2>_:"Disps
Mat R
Listing:
ClrText
"2022-07-19 EWS"
"LEAST SQUARES"
"Mat X"? → Mat X
"Mat Y"? → Mat Y
Dim Mat Y → List 26
List 26[1] → N
(Trn Mat X × Mat X)^-1 × Trn Mat X × Mat Y → Mat B
"Mat B:" ⊿
Mat B ⊿
{1, N} → Dim Mat O
Fill(1, Mat O)
Mat O × Mat Y → Mat S
Mat S × Mat S ÷ N → Mat S
(Trn Mat B × Trn Mat X × Mat Y) - Mat S → Mat R
Mat R × (Trn Mat Y × Mat Y - Mat S)^-1 → Mat R
"R^2:" ⊿
Mat R
Matrices:
Mat X: data matrix, X
Mat Y: answer matrix, Y
Mat B: coefficient matrix, B
Mat O: ones matrix
Mat S: used for calculation
Mat R: correlation
TI-84 Plus CE Program: LSQ2 (TI-Basic)
