∑⟮xᵢ - μ⟯²
| public static sumOfSquaresDeviations ( array $numbers ) : number | ||
| $numbers | array | |
| return | number | |
/**
* SSreg - The Sum Squares of the regression (Explained sum of squares)
*
* The sum of the squares of the deviations of the predicted values from
* the mean value of a response variable, in a standard regression model.
* https://en.wikipedia.org/wiki/Explained_sum_of_squares
*
* SSreg = ∑(ŷᵢ - ȳ)²
* When a constant is fit to the regression, the average of y = average of ŷ.
*
* In the case where the constant is not fit, we use the sum of squares of the predicted value
* SSreg = ∑ŷᵢ²
*
* @return number
*/
public function sumOfSquaresRegression()
{
if ($this->fit_constant == 1) {
return RandomVariable::sumOfSquaresDeviations($this->Yhat());
}
return array_sum(Single::square($this->reg_Yhat));
}
/**
* One-way ANOVA
* Technique used to compare means of three or more samples
* (using the F distribution).
* https://en.wikipedia.org/wiki/One-way_analysis_of_variance
*
* Produces the following analysis of the data:
*
* ANOVA hypothesis test summary data
*
* | SS | df | MS | F | P |
* Treatment | | | | | |
* Error | | | |
* Total | | |
*
* where:
* Treament is between groups
* Error is within groups
* SS = Sum of squares
* df = Degrees of freedom
* MS = Mean squares
* F = F statistic
* P = P value
*
* Data summary table
*
* | N | Sum | Mean | SS | Variance | SD | SEM |
* 0 | | | | | | | |
* 1 | | | | | | | |
* ... | | | | | | | |
* Total | | | | | | | |
*
* where:
* Each row is the summary for a sample, numbered from 0 to m - 1
* m = Number of samples
* N = Sample size
* SS = Sum of squares
* SD = Standard deviation
* SEM = Standard error of the mean
*
* Calculations
*
* Sum of Squares
* SST (sum of squares total)
* ∑⟮xᵢ − μ⟯²
* where:
* xᵢ = each element of all samples
* μ = mean total of all elements of all samples
*
* SSB (sum of squares between - treatment)
* ∑n(x - μ)²
* where:
* n = sample size
* x = sample mean
* μ = mean total of all elements of all samples
*
* SSW (sum of squares within - error)
* ∑∑⟮xᵢ − μ⟯² Sum of sum of squared deviations of each sample
* where:
* xᵢ = each element of the sample
* μ = mean of the sample
*
* Degrees of Freedom
* dfT (degrees of freedom for the total)
* mn - 1
*
* dfB (degrees of freedom between - treatment)
* m - 1
*
* dfW (degrees of freedom within - error)
* m(n - 1)
*
* where:
* m = number of samples
* n = number of elements in each sample
*
* Mean Squares
* MSB (Mean squares between - treatment)
* SSB / dfB
*
* MSW (Mean squares within - error)
* SSW / dfW
*
* Test Statistics
* F = MSB / MSW
* P = F distribution CDF above F with degrees of freedom dfB and dfW
*
* @param array ...$samples Samples to analyze (at least 3 or more samples)
*
* @return array [
* ANOVA => [
* treatment => [SS, df, MS, F, P],
* error => [SS, df, MS],
* total => [SS, df],
* ],
* total_summary => [n, sum, mean, SS, variance, sd, sem],
* data_summary => [
* 0 => [n, sum, mean, SS, variance, sd, sem],
* 1 => [n, sum, mean, SS, variance, sd, sem],
* ...
* ]
* ]
*
* @throws BadDataException if less than three samples, or if all samples don't have the same number of values
*/
public static function oneWay(array ...$samples)
{
// Must have at least three samples
$m = count($samples);
if ($m < 3) {
throw new Exception\BadDataException('Must have at least three samples');
}
// All samples must have the same number of items
$n = count($samples[0]);
for ($i = 1; $i < $m; $i++) {
if (count($samples[$i]) !== $n) {
throw new Exception\BadDataException('All samples must have the same number of values');
}
}
// Summary data for each sample
$summary_data = [];
foreach ($samples as $i => $sample) {
$summary_data[$i] = [];
$summary_data[$i]['n'] = $n;
$summary_data[$i]['sum'] = array_sum($sample);
$summary_data[$i]['mean'] = Average::mean($sample);
$summary_data[$i]['SS'] = RandomVariable::sumOfSquares($sample);
$summary_data[$i]['variance'] = Descriptive::sampleVariance($sample);
$summary_data[$i]['sd'] = Descriptive::sd($sample);
$summary_data[$i]['sem'] = RandomVariable::standardErrorOfTheMean($sample);
}
// Totals summary
$all_elements = array_reduce($samples, function ($merged, $sample) {
return array_merge($merged, $sample);
}, array());
$μ = Average::mean($all_elements);
$total = ['n' => count($all_elements), 'sum' => array_sum($all_elements), 'mean' => $μ, 'SS' => RandomVariable::sumOfSquares($all_elements), 'variance' => Descriptive::sampleVariance($all_elements), 'sd' => Descriptive::sd($all_elements), 'sem' => RandomVariable::standardErrorOfTheMean($all_elements)];
// ANOVA sum of squares
$SST = RandomVariable::sumOfSquaresDeviations($all_elements);
$SSB = array_sum(array_map(function ($sample) use($n, $μ) {
return $n * (Average::mean($sample) - $μ) ** 2;
}, $samples));
$SSW = array_sum(array_map('MathPHP\\Statistics\\RandomVariable::sumOfSquaresDeviations', $samples));
// ANOVA degrees of freedom
$dfT = $m * $n - 1;
$dfB = $m - 1;
$dfW = $m * ($n - 1);
// ANOVA mean squares
$MSB = $SSB / $dfB;
$MSW = $SSW / $dfW;
// Test statistics
$F = $MSB / $MSW;
$P = F::above($F, $dfB, $dfW);
// Return ANOVA report
return ['ANOVA' => ['treatment' => ['SS' => $SSB, 'df' => $dfB, 'MS' => $MSB, 'F' => $F, 'P' => $P], 'error' => ['SS' => $SSW, 'df' => $dfW, 'MS' => $MSW], 'total' => ['SS' => $SST, 'df' => $dfT]], 'total_summary' => $total, 'data_summary' => $summary_data];
}
/**
* Variance
*
* Variance measures how far a set of numbers are spread out.
* A variance of zero indicates that all the values are identical.
* Variance is always non-negative: a small variance indicates that the data points
* tend to be very close to the mean (expected value) and hence to each other.
* A high variance indicates that the data points are very spread out around the mean
* and from each other.
* (https://en.wikipedia.org/wiki/Variance)
*
* ∑⟮xᵢ - μ⟯²
* σ² = ----------
* ν
*
* Generalized method that allows setting the degrees of freedom.
* For population variance, set d.f. (ν) to n
* For sample variance, set d.f (ν) to n - 1
* Or use popluationVariance or sampleVaraince covenience methods.
*
* μ is the population mean
* ν is the degrees of freedom, which usually is
* the number of numbers in the population set or n - 1 for sample set.
*
* @param array $numbers
* @param int $ν degrees of freedom
* @return numeric
*
* @throws OutOfBoundsException if degrees of freedom is ≤ 0
*/
public static function variance(array $numbers, int $ν)
{
if (empty($numbers)) {
return null;
}
if ($ν <= 0) {
throw new Exception\OutOfBoundsException('Degrees of freedom must be > 0');
}
$∑⟮xᵢ − μ⟯² = RandomVariable::sumOfSquaresDeviations($numbers);
return $∑⟮xᵢ − μ⟯² / $ν;
}
