∑⟮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; }