time-to-botec

README.md (11052B)

---

 <!--
 
 @license Apache-2.0
 
 Copyright (c) 2020 The Stdlib Authors.
 
 Licensed under the Apache License, Version 2.0 (the "License");
 you may not use this file except in compliance with the License.
 You may obtain a copy of the License at
 
    http://www.apache.org/licenses/LICENSE-2.0
 
 Unless required by applicable law or agreed to in writing, software
 distributed under the License is distributed on an "AS IS" BASIS,
 WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 See the License for the specific language governing permissions and
 limitations under the License.
 
 -->
 
 # stdevch
 
 > Calculate the [standard deviation][standard-deviation] of a strided array using a one-pass trial mean algorithm.
 
 <section class="intro">
 
 The population [standard deviation][standard-deviation] of a finite size population of size `N` is given by
 
 <!-- <equation class="equation" label="eq:population_standard_deviation" align="center" raw="\sigma = \sqrt{\frac{1}{N} \sum_{i=0}^{N-1} (x_i - \mu)^2}" alt="Equation for the population standard deviation."> -->
 
 <div class="equation" align="center" data-raw-text="\sigma = \sqrt{\frac{1}{N} \sum_{i=0}^{N-1} (x_i - \mu)^2}" data-equation="eq:population_standard_deviation">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@10c5eae557369a7da1da27a5f12a45b56e592834/lib/node_modules/@stdlib/stats/base/stdevch/docs/img/equation_population_standard_deviation.svg" alt="Equation for the population standard deviation.">
     <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@10c5eae557369a7da1da27a5f12a45b56e592834/lib/node_modules/@stdlib/stats/base/stdevch/docs/img/equation_population_mean.svg" alt="Equation for the population mean.">
     <br>
 </div>
 
 <!-- </equation> -->
 
 Often in the analysis of data, the true population [standard deviation][standard-deviation] 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 [standard deviation][standard-deviation], the result is biased and yields an **uncorrected sample standard deviation**. To compute a **corrected sample standard deviation** for a sample of size `n`,
 
 <!-- <equation class="equation" label="eq:corrected_sample_standard_deviation" align="center" raw="s = \sqrt{\frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x})^2}" alt="Equation for computing a corrected sample standard deviation."> -->
 
 <div class="equation" align="center" data-raw-text="s = \sqrt{\frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x})^2}" data-equation="eq:corrected_sample_standard_deviation">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@10c5eae557369a7da1da27a5f12a45b56e592834/lib/node_modules/@stdlib/stats/base/stdevch/docs/img/equation_corrected_sample_standard_deviation.svg" alt="Equation for computing a corrected sample standard deviation.">
     <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@10c5eae557369a7da1da27a5f12a45b56e592834/lib/node_modules/@stdlib/stats/base/stdevch/docs/img/equation_sample_mean.svg" alt="Equation for the sample mean.">
     <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 standard deviation and population standard deviation. 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 stdevch = require( '@stdlib/stats/base/stdevch' );
 ```
 
 #### stdevch( N, correction, x, stride )
 
 Computes the [standard deviation][standard-deviation] of a strided array `x` using a one-pass trial mean algorithm.
 
 ```javascript
 var x = [ 1.0, -2.0, 2.0 ];
 
 var v = stdevch( x.length, 1, x, 1 );
 // returns ~2.0817
 ```
 
 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 [standard deviation][standard-deviation] according to `N-c` where `c` corresponds to the provided degrees of freedom adjustment. When computing the [standard deviation][standard-deviation] 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 corrected sample [standard deviation][standard-deviation], 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 [`Array`][mdn-array] or [`typed array`][mdn-typed-array].
 -   **stride**: index increment for `x`.
 
 The `N` and `stride` parameters determine which elements in `x` are accessed at runtime. For example, to compute the [standard deviation][standard-deviation] of every other element in `x`,
 
 ```javascript
 var floor = require( '@stdlib/math/base/special/floor' );
 
 var x = [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ];
 var N = floor( x.length / 2 );
 
 var v = stdevch( N, 1, x, 2 );
 // returns 2.5
 ```
 
 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 Float64Array = require( '@stdlib/array/float64' );
 var floor = require( '@stdlib/math/base/special/floor' );
 
 var x0 = new Float64Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
 var x1 = new Float64Array( x0.buffer, x0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
 
 var N = floor( x0.length / 2 );
 
 var v = stdevch( N, 1, x1, 2 );
 // returns 2.5
 ```
 
 #### stdevch.ndarray( N, correction, x, stride, offset )
 
 Computes the [standard deviation][standard-deviation] of a strided array using a one-pass trial mean algorithm and alternative indexing semantics.
 
 ```javascript
 var x = [ 1.0, -2.0, 2.0 ];
 
 var v = stdevch.ndarray( x.length, 1, x, 1, 0 );
 // returns ~2.0817
 ```
 
 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 [standard deviation][standard-deviation] for every other value in `x` starting from the second value
 
 ```javascript
 var floor = require( '@stdlib/math/base/special/floor' );
 
 var x = [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ];
 var N = floor( x.length / 2 );
 
 var v = stdevch.ndarray( N, 1, x, 2, 1 );
 // returns 2.5
 ```
 
 </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`.
 -   The underlying algorithm is a specialized case of Neely's two-pass algorithm. As the standard deviation is invariant with respect to changes in the location parameter, the underlying algorithm uses the first strided array element as a trial mean to shift subsequent data values and thus mitigate catastrophic cancellation. Accordingly, the algorithm's accuracy is best when data is **unordered** (i.e., the data is **not** sorted in either ascending or descending order such that the first value is an "extreme" value).
 -   Depending on the environment, the typed versions ([`dstdevch`][@stdlib/stats/base/dstdevch], [`sstdevch`][@stdlib/stats/base/sstdevch], etc.) are likely to be significantly more performant.
 
 </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 Float64Array = require( '@stdlib/array/float64' );
 var stdevch = require( '@stdlib/stats/base/stdevch' );
 
 var x;
 var i;
 
 x = new Float64Array( 10 );
 for ( i = 0; i < x.length; i++ ) {
     x[ i ] = round( (randu()*100.0) - 50.0 );
 }
 console.log( x );
 
 var v = stdevch( x.length, 1, x, 1 );
 console.log( v );
 ```
 
 </section>
 
 <!-- /.examples -->
 
 * * *
 
 <section class="references">
 
 ## References
 
 -   Neely, Peter M. 1966. "Comparison of Several Algorithms for Computation of Means, Standard Deviations and Correlation Coefficients." _Communications of the ACM_ 9 (7). Association for Computing Machinery: 496–99. doi:[10.1145/365719.365958][@neely:1966a].
 -   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].
 -   Chan, Tony F., Gene H. Golub, and Randall J. LeVeque. 1983. "Algorithms for Computing the Sample Variance: Analysis and Recommendations." _The American Statistician_ 37 (3). American Statistical Association, Taylor & Francis, Ltd.: 242–47. doi:[10.1080/00031305.1983.10483115][@chan:1983a].
 -   Schubert, Erich, and Michael Gertz. 2018. "Numerically Stable Parallel Computation of (Co-)Variance." In _Proceedings of the 30th International Conference on Scientific and Statistical Database Management_. New York, NY, USA: Association for Computing Machinery. doi:[10.1145/3221269.3223036][@schubert:2018a].
 
 </section>
 
 <!-- /.references -->
 
 <section class="links">
 
 [standard-deviation]: https://en.wikipedia.org/wiki/Standard_deviation
 
 [mdn-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array
 
 [mdn-typed-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/TypedArray
 
 [@stdlib/stats/base/dstdevch]: https://www.npmjs.com/package/@stdlib/stats/tree/main/base/dstdevch
 
 [@stdlib/stats/base/sstdevch]: https://www.npmjs.com/package/@stdlib/stats/tree/main/base/sstdevch
 
 [@neely:1966a]: https://doi.org/10.1145/365719.365958
 
 [@ling:1974a]: https://doi.org/10.2307/2286154
 
 [@chan:1983a]: https://doi.org/10.1080/00031305.1983.10483115
 
 [@schubert:2018a]: https://doi.org/10.1145/3221269.3223036
 
 </section>
 
 <!-- /.links -->