add 统计

src/algos/math/statistics.md

@@ -1 +1,176 @@
 # 统计
+
+### 测量中心趋势
+
+下面的一些例子为 Rust 数组中的数据计算它们的中心趋势。
+
+#### 平均值
+首先计算的是平均值。
+
+```rust,editable
+fn main() {
+    let data = [3, 1, 6, 1, 5, 8, 1, 8, 10, 11];
+
+    let sum = data.iter().sum::<i32>() as f32;
+    let count = data.len();
+
+    let mean = match count {
+       positive if positive > 0 => Some(sum  / count as f32),
+       _ => None
+    };
+
+    println!("Mean of the data is {:?}", mean);
+}
+```
+
+#### 中位数
+下面使用快速选择算法来计算中位数。该算法只会对可能包含中位数的数据分区进行排序，从而避免了对所有数据进行全排序。
+
+```rust,editable
+use std::cmp::Ordering;
+
+fn partition(data: &[i32]) -> Option<(Vec<i32>, i32, Vec<i32>)> {
+    match data.len() {
+        0 => None,
+        _ => {
+            let (pivot_slice, tail) = data.split_at(1);
+            let pivot = pivot_slice[0];
+            let (left, right) = tail.iter()
+                .fold((vec![], vec![]), |mut splits, next| {
+                    {
+                        let (ref mut left, ref mut right) = &mut splits;
+                        if next < &pivot {
+                            left.push(*next);
+                        } else {
+                            right.push(*next);
+                        }
+                    }
+                    splits
+                });
+
+            Some((left, pivot, right))
+        }
+    }
+}
+
+fn select(data: &[i32], k: usize) -> Option<i32> {
+    let part = partition(data);
+
+    match part {
+        None => None,
+        Some((left, pivot, right)) => {
+            let pivot_idx = left.len();
+
+            match pivot_idx.cmp(&k) {
+                Ordering::Equal => Some(pivot),
+                Ordering::Greater => select(&left, k),
+                Ordering::Less => select(&right, k - (pivot_idx + 1)),
+            }
+        },
+    }
+}
+
+fn median(data: &[i32]) -> Option<f32> {
+    let size = data.len();
+
+    match size {
+        even if even % 2 == 0 => {
+            let fst_med = select(data, (even / 2) - 1);
+            let snd_med = select(data, even / 2);
+
+            match (fst_med, snd_med) {
+                (Some(fst), Some(snd)) => Some((fst + snd) as f32 / 2.0),
+                _ => None
+            }
+        },
+        odd => select(data, odd / 2).map(|x| x as f32)
+    }
+}
+
+fn main() {
+    let data = [3, 1, 6, 1, 5, 8, 1, 8, 10, 11];
+
+    let part = partition(&data);
+    println!("Partition is {:?}", part);
+
+    let sel = select(&data, 5);
+    println!("Selection at ordered index {} is {:?}", 5, sel);
+
+    let med = median(&data);
+    println!("Median is {:?}", med);
+}
+```
+
+#### 众数( mode )
+下面使用了 `HashMap` 对不同数字出现的次数进行了分别统计。
+
+```rust,editable
+use std::collections::HashMap;
+
+fn main() {
+    let data = [3, 1, 6, 1, 5, 8, 1, 8, 10, 11];
+
+    let frequencies = data.iter().fold(HashMap::new(), |mut freqs, value| {
+        *freqs.entry(value).or_insert(0) += 1;
+        freqs
+    });
+
+    let mode = frequencies
+        .into_iter()
+        .max_by_key(|&(_, count)| count)
+        .map(|(value, _)| *value);
+
+    println!("Mode of the data is {:?}", mode);
+}
+```
+
+### 标准偏差
+
+下面一起来看看该如何计算一组测量值的标准偏差和 z-score。
+
+```rust,editable
+fn mean(data: &[i32]) -> Option<f32> {
+    let sum = data.iter().sum::<i32>() as f32;
+    let count = data.len();
+
+    match count {
+        positive if positive > 0 => Some(sum / count as f32),
+        _ => None,
+    }
+}
+
+fn std_deviation(data: &[i32]) -> Option<f32> {
+    match (mean(data), data.len()) {
+        (Some(data_mean), count) if count > 0 => {
+            let variance = data.iter().map(|value| {
+                let diff = data_mean - (*value as f32);
+
+                diff * diff
+            }).sum::<f32>() / count as f32;
+
+            Some(variance.sqrt())
+        },
+        _ => None
+    }
+}
+
+fn main() {
+    let data = [3, 1, 6, 1, 5, 8, 1, 8, 10, 11];
+
+    let data_mean = mean(&data);
+    println!("Mean is {:?}", data_mean);
+
+    let data_std_deviation = std_deviation(&data);
+    println!("Standard deviation is {:?}", data_std_deviation);
+
+    let zscore = match (data_mean, data_std_deviation) {
+        (Some(mean), Some(std_deviation)) => {
+            let diff = data[4] as f32 - mean;
+
+            Some(diff / std_deviation)
+        },
+        _ => None
+    };
+    println!("Z-score of data at index 4 (with value {}) is {:?}", data[4], zscore);
+}
+```