time-to-botec

README.md (12132B)

---

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 # svariancetk
 
 > Calculate the [variance][variance] of a single-precision floating-point strided array using a one-pass textbook algorithm.
 
 <section class="intro">
 
 The population [variance][variance] of a finite size population of size `N` is given by
 
 <!-- <equation class="equation" label="eq:population_variance" align="center" raw="\sigma^2 = \frac{1}{N} \sum_{i=0}^{N-1} (x_i - \mu)^2" alt="Equation for the population variance."> -->
 
 <div class="equation" align="center" data-raw-text="\sigma^2 = \frac{1}{N} \sum_{i=0}^{N-1} (x_i - \mu)^2" data-equation="eq:population_variance">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@6da3e7388e483798f23a9ce30fcb35f454e7e3b4/lib/node_modules/@stdlib/stats/base/svariancetk/docs/img/equation_population_variance.svg" alt="Equation for the population variance.">
     <br>
 </div>
 
 <!-- </equation> -->
 
 where the population mean is given by
 
 <!-- <equation class="equation" label="eq:population_mean" align="center" raw="\mu = \frac{1}{N} \sum_{i=0}^{N-1} x_i" alt="Equation for the population mean."> -->
 
 <div class="equation" align="center" data-raw-text="\mu = \frac{1}{N} \sum_{i=0}^{N-1} x_i" data-equation="eq:population_mean">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@6da3e7388e483798f23a9ce30fcb35f454e7e3b4/lib/node_modules/@stdlib/stats/base/svariancetk/docs/img/equation_population_mean.svg" alt="Equation for the population mean.">
     <br>
 </div>
 
 <!-- </equation> -->
 
 After rearranging terms, the population [variance][variance] can be equivalently expressed as
 
 <!-- <equation class="equation" label="eq:population_variance_textbook" align="center" raw="\sigma^2 = \frac{1}{N}\biggl(\ \sum_{i=0}^{N-1} x_i^2 - \frac{1}{N}\biggl(\ \sum_{i=0}^{N-1} x_i \ \biggr)^2\ \biggr)" alt="Equation for the population variance (one-pass textbook formula)."> -->
 
 <div class="equation" align="center" data-raw-text="\sigma^2 = \frac{1}{N}\biggl(\ \sum_{i=0}^{N-1} x_i^2 - \frac{1}{N}\biggl(\ \sum_{i=0}^{N-1} x_i \ \biggr)^2\ \biggr)" data-equation="eq:population_variance_textbook">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@6da3e7388e483798f23a9ce30fcb35f454e7e3b4/lib/node_modules/@stdlib/stats/base/svariancetk/docs/img/equation_population_variance_textbook.svg" alt="Equation for the population variance (one-pass textbook formula).">
     <br>
 </div>
 
 <!-- </equation> -->
 
 Often in the analysis of data, the true population [variance][variance] is not known _a priori_ and must be estimated from a sample drawn from the population distribution. If one attempts to use the formula for the population [variance][variance], the result is biased and yields a **biased sample variance**. To compute an **unbiased sample variance** for a sample of size `n`,
 
 <!-- <equation class="equation" label="eq:unbiased_sample_variance" align="center" raw="s^2 = \frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x})^2" alt="Equation for computing an unbiased sample variance."> -->
 
 <div class="equation" align="center" data-raw-text="s^2 = \frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x})^2" data-equation="eq:unbiased_sample_variance">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@6da3e7388e483798f23a9ce30fcb35f454e7e3b4/lib/node_modules/@stdlib/stats/base/svariancetk/docs/img/equation_unbiased_sample_variance.svg" alt="Equation for computing an unbiased sample variance.">
     <br>
 </div>
 
 <!-- </equation> -->
 
 where the sample mean is given by
 
 <!-- <equation class="equation" label="eq:sample_mean" align="center" raw="\bar{x} = \frac{1}{n} \sum_{i=0}^{n-1} x_i" alt="Equation for the sample mean."> -->
 
 <div class="equation" align="center" data-raw-text="\bar{x} = \frac{1}{n} \sum_{i=0}^{n-1} x_i" data-equation="eq:sample_mean">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@6da3e7388e483798f23a9ce30fcb35f454e7e3b4/lib/node_modules/@stdlib/stats/base/svariancetk/docs/img/equation_sample_mean.svg" alt="Equation for the sample mean.">
     <br>
 </div>
 
 <!-- </equation> -->
 
 Similar to the population [variance][variance], after rearranging terms, the **unbiased sample variance** can be equivalently expressed as
 
 <!-- <equation class="equation" label="eq:unbiased_sample_variance_textbook" align="center" raw="s^2 = \frac{1}{n-1}\biggl(\ \sum_{i=0}^{n-1} x_i^2 - \frac{1}{n}\biggl(\ \sum_{i=0}^{n-1} x_i \ \biggr)^2\ \biggr)" alt="Equation for the unbiased sample variance (one-pass textbook formula)."> -->
 
 <div class="equation" align="center" data-raw-text="s^2 = \frac{1}{n-1}\biggl(\ \sum_{i=0}^{n-1} x_i^2 - \frac{1}{n}\biggl(\ \sum_{i=0}^{n-1} x_i \ \biggr)^2\ \biggr)" data-equation="eq:unbiased_sample_variance_textbook">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@6da3e7388e483798f23a9ce30fcb35f454e7e3b4/lib/node_modules/@stdlib/stats/base/svariancetk/docs/img/equation_unbiased_sample_variance_textbook.svg" alt="Equation for the unbiased sample variance (one-pass textbook formula).">
     <br>
 </div>
 
 <!-- </equation> -->
 
 The use of the term `n-1` is commonly referred to as Bessel's correction. Note, however, that applying Bessel's correction can increase the mean squared error between the sample variance and population variance. Depending on the characteristics of the population distribution, other correction factors (e.g., `n-1.5`, `n+1`, etc) can yield better estimators.
 
 </section>
 
 <!-- /.intro -->
 
 <section class="usage">
 
 ## Usage
 
 ```javascript
 var svariancetk = require( '@stdlib/stats/base/svariancetk' );
 ```
 
 #### svariancetk( N, correction, x, stride )
 
 Computes the [variance][variance] of a single-precision floating-point strided array `x` using a one-pass textbook algorithm.
 
 ```javascript
 var Float32Array = require( '@stdlib/array/float32' );
 
 var x = new Float32Array( [ 1.0, -2.0, 2.0 ] );
 var N = x.length;
 
 var v = svariancetk( N, 1, x, 1 );
 // returns ~4.3333
 ```
 
 The function has the following parameters:
 
 -   **N**: number of indexed elements.
 -   **correction**: degrees of freedom adjustment. Setting this parameter to a value other than `0` has the effect of adjusting the divisor during the calculation of the [variance][variance] according to `N-c` where `c` corresponds to the provided degrees of freedom adjustment. When computing the [variance][variance] of a population, setting this parameter to `0` is the standard choice (i.e., the provided array contains data constituting an entire population). When computing the unbiased sample [variance][variance], setting this parameter to `1` is the standard choice (i.e., the provided array contains data sampled from a larger population; this is commonly referred to as Bessel's correction).
 -   **x**: input [`Float32Array`][@stdlib/array/float32].
 -   **stride**: index increment for `x`.
 
 The `N` and `stride` parameters determine which elements in `x` are accessed at runtime. For example, to compute the [variance][variance] of every other element in `x`,
 
 ```javascript
 var Float32Array = require( '@stdlib/array/float32' );
 var floor = require( '@stdlib/math/base/special/floor' );
 
 var x = new Float32Array( [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ] );
 var N = floor( x.length / 2 );
 
 var v = svariancetk( N, 1, x, 2 );
 // returns 6.25
 ```
 
 Note that indexing is relative to the first index. To introduce an offset, use [`typed array`][mdn-typed-array] views.
 
 <!-- eslint-disable stdlib/capitalized-comments -->
 
 ```javascript
 var Float32Array = require( '@stdlib/array/float32' );
 var floor = require( '@stdlib/math/base/special/floor' );
 
 var x0 = new Float32Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
 var x1 = new Float32Array( x0.buffer, x0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
 
 var N = floor( x0.length / 2 );
 
 var v = svariancetk( N, 1, x1, 2 );
 // returns 6.25
 ```
 
 #### svariancetk.ndarray( N, correction, x, stride, offset )
 
 Computes the [variance][variance] of a single-precision floating-point strided array using a one-pass textbook algorithm and alternative indexing semantics.
 
 ```javascript
 var Float32Array = require( '@stdlib/array/float32' );
 
 var x = new Float32Array( [ 1.0, -2.0, 2.0 ] );
 var N = x.length;
 
 var v = svariancetk.ndarray( N, 1, x, 1, 0 );
 // returns ~4.33333
 ```
 
 The function has the following additional parameters:
 
 -   **offset**: starting index for `x`.
 
 While [`typed array`][mdn-typed-array] views mandate a view offset based on the underlying `buffer`, the `offset` parameter supports indexing semantics based on a starting index. For example, to calculate the [variance][variance] for every other value in `x` starting from the second value
 
 ```javascript
 var Float32Array = require( '@stdlib/array/float32' );
 var floor = require( '@stdlib/math/base/special/floor' );
 
 var x = new Float32Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
 var N = floor( x.length / 2 );
 
 var v = svariancetk.ndarray( N, 1, x, 2, 1 );
 // returns 6.25
 ```
 
 </section>
 
 <!-- /.usage -->
 
 <section class="notes">
 
 ## Notes
 
 -   If `N <= 0`, both functions return `NaN`.
 -   If `N - c` is less than or equal to `0` (where `c` corresponds to the provided degrees of freedom adjustment), both functions return `NaN`.
 -   Some caution should be exercised when using the one-pass textbook algorithm. Literature overwhelmingly discourages the algorithm's use for two reasons: 1) the lack of safeguards against underflow and overflow and 2) the risk of catastrophic cancellation when subtracting the two sums if the sums are large and the variance small. These concerns have merit; however, the one-pass textbook algorithm should not be dismissed outright. For data distributions with a moderately large standard deviation to mean ratio (i.e., **coefficient of variation**), the one-pass textbook algorithm may be acceptable, especially when performance is paramount and some precision loss is acceptable (including a risk of returning a negative variance due to floating-point rounding errors!). In short, no single "best" algorithm for computing the variance exists. The "best" algorithm depends on the underlying data distribution, your performance requirements, and your minimum precision requirements. When evaluating which algorithm to use, consider the relative pros and cons, and choose the algorithm which best serves your needs.
 
 </section>
 
 <!-- /.notes -->
 
 <section class="examples">
 
 ## Examples
 
 <!-- eslint no-undef: "error" -->
 
 ```javascript
 var randu = require( '@stdlib/random/base/randu' );
 var round = require( '@stdlib/math/base/special/round' );
 var Float32Array = require( '@stdlib/array/float32' );
 var svariancetk = require( '@stdlib/stats/base/svariancetk' );
 
 var x;
 var i;
 
 x = new Float32Array( 10 );
 for ( i = 0; i < x.length; i++ ) {
     x[ i ] = round( (randu()*100.0) - 50.0 );
 }
 console.log( x );
 
 var v = svariancetk( x.length, 1, x, 1 );
 console.log( v );
 ```
 
 </section>
 
 <!-- /.examples -->
 
 * * *
 
 <section class="references">
 
 ## References
 
 -   Ling, Robert F. 1974. "Comparison of Several Algorithms for Computing Sample Means and Variances." _Journal of the American Statistical Association_ 69 (348). American Statistical Association, Taylor & Francis, Ltd.: 859–66. doi:[10.2307/2286154][@ling:1974a].
 
 </section>
 
 <!-- /.references -->
 
 <section class="links">
 
 [variance]: https://en.wikipedia.org/wiki/Variance
 
 [@stdlib/array/float32]: https://www.npmjs.com/package/@stdlib/array-float32
 
 [mdn-typed-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/TypedArray
 
 [@ling:1974a]: https://doi.org/10.2307/2286154
 
 </section>
 
 <!-- /.links -->