Least squares method calculator: exponential approximation
Calculator finds out coefficient of exponent function y=a×exp(bx) that fits best into series of (x, y) points.

Beta version#

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Calculation data - measurement points#

Format of input data
x-values
y-values

Results - approximation of your dataset#

Regression typeApproximation formulaCoefficient of determination R2
Exponential regressionShow sourcey=0.8308190769 e1.0503191452 xy=0.8308190769~{ e}^{1.0503191452~ x}0.999554898

Summary - function best fitting to your data#

Measurement points
Number of points4
Points you entered(1, 2), (2, 7), (3, 20), (4, 55)
Approximation
Regression typeExponential regression
Function formulaShow sourcey=0.8308190769 e1.0503191452 xy=0.8308190769~{ e}^{1.0503191452~ x}
Coefficient of determination R20.999554898

Some facts#

  • ⓘ Hint: If you're not sure what type of regression this is, let us do the hard work for you and visit another calculator: Regression types.
  • Approximation of a function consists in finding a function formula that best matches to a set of points e.g. obtained as measurement data.
  • The least squares method is one of the methods for finding such a function.
  • The least squares method is the optimization method. As a result we get function that the sum of squares of deviations from the measured data is the smallest. Mathematically, we can write it as follows:
    i=1n[yif(xi)]2=min.\sum_{i=1}^{n} \left[y_i - f(x_i)\right]^2 = min.
    where:
    • (xi,yi)(x_i, y_i) - coordinations of the i-th measurement point, these are points that we know,
    • f(x)f(x) - the function we are searching for, we want this function to best match to the measurement points,
    • nn - number of measurement points.
  • If we limit the search to exponential function only, then we say about exponential regression or approximation.
    f(x)=a×ebxf(x) = a \times e^{bx}
    • f(x) - function that best approximates the input data in the best way,
    • a,b - unknown function parameters, which we want to find,
    • ln - the base of natural logarithm.
  • Exponential approximation is an example of non-linear regression i.e. estimation with function other than linear function.
  • Using the method of least squares we can find a and b parameters of the above function, at which the sum of squares of deviations from the measured data is the smallest, so the final function fits best to the the input data.
  • Example phenomena, which may be estimated by exponential function are dynamics of infection spread in epidemics or execution time of some computer algorithms.

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