Alink漫谈(二十一) ：回归评估之源码分析

Alink漫谈(二十一) ：回归评估之源码分析

0x00 摘要

Alink 是阿里巴巴基于实时计算引擎 Flink 研发的新一代机器学习算法平台，是业界首个同时支持批式算法、流式算法的机器学习平台。本文和将带领大家来分析Alink中 回归评估 的实现。

这是剖析Alink以来，最轻松的一次了。因为这里的概念和实现逻辑都非常清晰。

0x01 背景概念

1.1 功能介绍

回归评估是对回归算法的预测结果进行效果评估，支持下列评估指标。这些指标基本都是统计领域概念。

1.2 具体指标

Alink 提供如下指标：

count 行数

SST 总平方和(Sum of Squared for Total)，度量了Y在样本中的分散程度。

\[SST=\sum_{i=1}^{N}(y_i-\bar{y})^2 \]

SSE 误差平方和(Sum of Squares for Error)，度量了总样本变异。

\[SSE=\sum_{i=1}^{N}(y_i-f_i)^2" \]

SSR 回归平方和(Sum of Squares for Regression)，度量了残差的样本变异。

\[SSR=\sum_{i=1}^{N}(f_i-\bar{y})^2 \]

R^2 判定系数(Coefficient of Determination)，用于估计回归方程是否很好的拟合了样本的数据，判定系数为估计的回归方程提供了一个拟合优度的度量。

\[R^2=1-\dfrac{SSE}{SST} \]

R 多重相关系数(Multiple Correlation Coeffient)，指一个随机变量与某一组随机变量间线性相依性的度量。

\[R=\sqrt{R^2} \]

MSE 均方误差(Mean Squared Error)，均方差（标准差）、方差都是用来描述数据集的离散程度。

均方误差是衡量“平均误差”的一种较方便的方法，可以评价数据的变化程度。从类别来看属于预测评价与预测组合；从字面上看来，“均”指的是平均，即求其平均值，“方差”即是在概率论中用来衡量随机变量和其估计值（其平均值）之间的偏离程度的度量值，“误”可以理解为测定值与真实值之间的误差。

\[MSE=\dfrac{1}{N}\sum_{i=1}^{N}(f_i-y_i)^2 \]

RMSE 均方根误差(Root Mean Squared Error)

\[RMSE=\sqrt{MSE} \]

SAE/SAD 绝对误差(Sum of Absolute Error/Difference)

\[SAE=\sum_{i=1}^{N}|f_i-y_i| \]

MAE/MAD 平均绝对误差(Mean Absolute Error/Difference)

\[MAE=\dfrac{1}{N}\sum_{i=1}^{N}|f_i-y_i| \]

MAPE 平均绝对百分误差(Mean Absolute Percentage Error)

\[MAPE=\dfrac{100}{N}\sum_{i=1}^{N}|\dfrac{f_i-y_i}{y_i}| \]

explained variance 解释方差

\[explained Variance=\dfrac{SSR}{N} \]

0x02 示例代码

直接拿出来Alink的示例代码。

public class EvalRegressionBatchOpExp {
    
    public static void main(String[] args) throws Exception {
        Row[] data =
                new Row[] {
                        Row.of(0.4, 0.5),
                        Row.of(0.3, 0.5),
                        Row.of(0.2, 0.6),
                        Row.of(0.6, 0.7),
                        Row.of(0.1, 0.5)
                };

        MemSourceBatchOp input = new MemSourceBatchOp(data, new String[] {"label", "pred"});

        RegressionMetrics metrics = new EvalRegressionBatchOp()
                .setLabelCol("label")
                .setPredictionCol("pred")
                .linkFrom(input)
                .collectMetrics();

        System.out.println(metrics.getRmse());
        System.out.println(metrics.getR2());
        System.out.println(metrics.getSse());
        System.out.println(metrics.getMape());
        System.out.println(metrics.getMae());
        System.out.println(metrics.getSsr());
        System.out.println(metrics.getSst());
    }
}

输出为：

0.27568097504180444
-1.5675675675675653
0.38
141.66666666666669
0.24
0.31999999999999973
0.14800000000000013

0x03 总体逻辑

总体逻辑是：

• 调用 CalcLocal 进行分区计算各种统计数值；
• reduce 调用 ReduceBaseMetrics 进行归并各种统计数值；
• 调用 SaveDataAsParams 存储；

getLabelCol 就是 y，getPredictionCol 就是 y_hat。

public EvalRegressionBatchOp linkFrom(BatchOperator<?>... inputs) {
    BatchOperator in = checkAndGetFirst(inputs);

    // 这里就是找到y, y_hat
    TableUtil.findColIndexWithAssertAndHint(in.getColNames(), this.getLabelCol());
    TableUtil.findColIndexWithAssertAndHint(in.getColNames(), this.getPredictionCol());
	
  	// 利用y, y_hat来构建Metrics
    TableUtil.assertNumericalCols(in.getSchema(), this.getLabelCol(), this.getPredictionCol());
    DataSet<Row> out = in.select(new String[] {this.getLabelCol(), this.getPredictionCol()})
        .getDataSet()
        .rebalance()
        .mapPartition(new CalcLocal())
        .reduce(new EvaluationUtil.ReduceBaseMetrics())
        .flatMap(new EvaluationUtil.SaveDataAsParams());

    this.setOutputTable(DataSetConversionUtil.toTable(getMLEnvironmentId(),
        out, new TableSchema(new String[] {"regression_eval_result"}, new TypeInformation[] {Types.STRING})
    ));
    return this;
}

0x04 分区计算统计数值

调用 CalcLocal 进行分区计算各种统计数值，间接调用getRegressionStatistics。

/**
 * Get the label sum, predResult sum, SSE, MAE, MAPE of one partition.
 */
public static class CalcLocal implements MapPartitionFunction<Row, BaseMetricsSummary> {
    @Override
    public void mapPartition(Iterable<Row> rows, Collector<BaseMetricsSummary> collector)
        throws Exception {
        collector.collect(getRegressionStatistics(rows));
    }
}

getRegressionStatistics作用是遍历输入数据，在本Partition内部计算各种累积数值，为后续做准备。

/**
 * Calculate the RegressionMetrics from local data.
 *
 * @param rows Input rows, the first field is label value, the second field is prediction value.
 * @return RegressionMetricsSummary.
 */
public static RegressionMetricsSummary getRegressionStatistics(Iterable<Row> rows) {
    RegressionMetricsSummary regressionSummary = new RegressionMetricsSummary();
    for (Row row : rows) {
        if (checkRowFieldNotNull(row)) {
            double yVal = ((Number)row.getField(0)).doubleValue();
            double predictVal = ((Number)row.getField(1)).doubleValue();
            double diff = Math.abs(yVal - predictVal);
            regressionSummary.ySumLocal += yVal;
            regressionSummary.ySum2Local += yVal * yVal;
            regressionSummary.predSumLocal += predictVal;
            regressionSummary.predSum2Local += predictVal * predictVal;
            regressionSummary.maeLocal += diff;
            regressionSummary.sseLocal += diff * diff;
            regressionSummary.mapeLocal += Math.abs(diff / yVal);
            regressionSummary.total++;
        }
    }
    return regressionSummary.total == 0 ? null : regressionSummary;
}

0x05 归并统计数值

reduce 调用 ReduceBaseMetrics 进行归并各种统计数值：

/**
 * Merge the BaseMetrics calculated locally.
 */
public static class ReduceBaseMetrics implements ReduceFunction<BaseMetricsSummary> {
    @Override
    public BaseMetricsSummary reduce(BaseMetricsSummary t1, BaseMetricsSummary t2) throws Exception {
        return null == t1 ? t2 : t1.merge(t2);
    }
}

0x06 存储模型

这里调用SaveDataAsParams来存储模型。

/**
 * After merging all the BaseMetrics, we get the total BaseMetrics. Calculate the indexes and save them into params.
 */
public static class SaveDataAsParams implements FlatMapFunction<BaseMetricsSummary, Row> {
    @Override
    public void flatMap(BaseMetricsSummary t, Collector<Row> collector) throws Exception {
        collector.collect(t.toMetrics().serialize());
    }
}

0x07 toMetrics

最后呈现出统计指标。

public RegressionMetrics toMetrics() {
    Params params = new Params();
    params.set(RegressionMetrics.SST, ySum2Local - ySumLocal * ySumLocal / total);
    params.set(RegressionMetrics.SSE, sseLocal);
    params.set(RegressionMetrics.SSR,
        predSum2Local - 2 * ySumLocal * predSumLocal / total + ySumLocal * ySumLocal / total);
    params.set(RegressionMetrics.R2, 1 - params.get(RegressionMetrics.SSE) / params.get(RegressionMetrics.SST));
    params.set(RegressionMetrics.R, Math.sqrt(params.get(RegressionMetrics.R2)));
    params.set(RegressionMetrics.MSE, params.get(RegressionMetrics.SSE) / total);
    params.set(RegressionMetrics.RMSE, Math.sqrt(params.get(RegressionMetrics.MSE)));
    params.set(RegressionMetrics.SAE, maeLocal);
    params.set(RegressionMetrics.MAE, params.get(RegressionMetrics.SAE) / total);
    params.set(RegressionMetrics.COUNT, (double)total);
    params.set(RegressionMetrics.MAPE, mapeLocal * 100 / total);
    params.set(RegressionMetrics.Y_MEAN, ySumLocal / total);
    params.set(RegressionMetrics.PREDICTION_MEAN, predSumLocal / total);
    params.set(RegressionMetrics.EXPLAINED_VARIANCE, params.get(RegressionMetrics.SSR) / total);

    return new RegressionMetrics(params);
}

最后得到结果

params = {Params@9098} "Params {R2=-1.5675675675675693, predictionMean=0.5599999999999999, SSE=0.38, count=5.0, MAPE=141.66666666666666, RMSE=0.27568097504180444, MAE=0.24, R=NaN, SSR=0.3200000000000002, yMean=0.32, SST=0.1479999999999999, SAE=1.2, Explained Variance=0.06400000000000003, MSE=0.076}"
 params = {HashMap@9101}  size = 14
  "R2" -> "-1.5675675675675693"
  "predictionMean" -> "0.5599999999999999"
  "SSE" -> "0.38"
  "count" -> "5.0"
  "MAPE" -> "141.66666666666666"
  "RMSE" -> "0.27568097504180444"
  "MAE" -> "0.24"
  "R" -> "NaN"
  "SSR" -> "0.3200000000000002"
  "yMean" -> "0.32"
  "SST" -> "0.1479999999999999"
  "SAE" -> "1.2"
  "Explained Variance" -> "0.06400000000000003"
  "MSE" -> "0.076"

0xFF 参考

均方误差