org.apache.commons.math.optimization.fitting

## Class HarmonicCoefficientsGuesser

• java.lang.Object
• org.apache.commons.math.optimization.fitting.HarmonicCoefficientsGuesser

• ```public class HarmonicCoefficientsGuesser
extends java.lang.Object```
This class guesses harmonic coefficients from a sample.

The algorithm used to guess the coefficients is as follows:

We know f (t) at some sampling points ti and want to find a, ω and φ such that f (t) = a cos (ω t + φ).

From the analytical expression, we can compute two primitives :

```     If2  (t) = ∫ f2  = a2 × [t + S (t)] / 2
If'2 (t) = ∫ f'2 = a2 ω2 × [t - S (t)] / 2
where S (t) = sin (2 (ω t + φ)) / (2 ω)
```

We can remove S between these expressions :

```     If'2 (t) = a2 ω2 t - ω2 If2 (t)
```

The preceding expression shows that If'2 (t) is a linear combination of both t and If2 (t): If'2 (t) = A × t + B × If2 (t)

From the primitive, we can deduce the same form for definite integrals between t1 and ti for each ti :

```   If2 (ti) - If2 (t1) = A × (ti - t1) + B × (If2 (ti) - If2 (t1))
```

We can find the coefficients A and B that best fit the sample to this linear expression by computing the definite integrals for each sample points.

For a bilinear expression z (xi, yi) = A × xi + B × yi, the coefficients A and B that minimize a least square criterion ∑ (zi - z (xi, yi))2 are given by these expressions:

```
∑yiyi ∑xizi - ∑xiyi ∑yizi
A = ------------------------
∑xixi ∑yiyi - ∑xiyi ∑xiyi

∑xixi ∑yizi - ∑xiyi ∑xizi
B = ------------------------
∑xixi ∑yiyi - ∑xiyi ∑xiyi
```

In fact, we can assume both a and ω are positive and compute them directly, knowing that A = a2 ω2 and that B = - ω2. The complete algorithm is therefore:

```
for each ti from t1 to tn-1, compute:
f  (ti)
f' (ti) = (f (ti+1) - f(ti-1)) / (ti+1 - ti-1)
xi = ti - t1
yi = ∫ f2 from t1 to ti
zi = ∫ f'2 from t1 to ti
update the sums ∑xixi, ∑yiyi, ∑xiyi, ∑xizi and ∑yizi
end for

|--------------------------
\  | ∑yiyi ∑xizi - ∑xiyi ∑yizi
a     =  \ | ------------------------
\| ∑xiyi ∑xizi - ∑xixi ∑yizi

|--------------------------
\  | ∑xiyi ∑xizi - ∑xixi ∑yizi
ω     =  \ | ------------------------
\| ∑xixi ∑yiyi - ∑xiyi ∑xiyi

```

Once we know ω, we can compute:

```    fc = ω f (t) cos (ω t) - f' (t) sin (ω t)
fs = ω f (t) sin (ω t) + f' (t) cos (ω t)
```

It appears that `fc = a ω cos (φ)` and `fs = -a ω sin (φ)`, so we can use these expressions to compute φ. The best estimate over the sample is given by averaging these expressions.

Since integrals and means are involved in the preceding estimations, these operations run in O(n) time, where n is the number of measurements.

Since:
2.0
• ### Constructor Summary

Constructors
Constructor and Description
`HarmonicCoefficientsGuesser(WeightedObservedPoint[] observations)`
Simple constructor.
• ### Method Summary

All Methods
Modifier and Type Method and Description
`double` `getGuessedAmplitude()`
Get the guessed amplitude a.
`double` `getGuessedPhase()`
Get the guessed phase φ.
`double` `getGuessedPulsation()`
Get the guessed pulsation ω.
`void` `guess()`
Estimate a first guess of the coefficients.
• ### Methods inherited from class java.lang.Object

`equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`
• ### Constructor Detail

• #### HarmonicCoefficientsGuesser

`public HarmonicCoefficientsGuesser(WeightedObservedPoint[] observations)`
Simple constructor.
Parameters:
`observations` - sampled observations
• ### Method Detail

• #### guess

```public void guess()
throws OptimizationException```
Estimate a first guess of the coefficients.
Throws:
`OptimizationException` - if the sample is too short or if the first guess cannot be computed (when the elements under the square roots are negative).
• #### getGuessedAmplitude

`public double getGuessedAmplitude()`
Get the guessed amplitude a.
Returns:
guessed amplitude a;
• #### getGuessedPulsation

`public double getGuessedPulsation()`
Get the guessed pulsation ω.
Returns:
guessed pulsation ω
• #### getGuessedPhase

`public double getGuessedPhase()`
Get the guessed phase φ.
Returns:
guessed phase φ