c# - 具有动态时间扭曲内核的 SVM 返回错误率大于 0

Question

我在研究中使用 Accord.net。我有一个可变大小的向量序列作为输入，所以我使用 DynamicTimeWarping 作为 MulticlassSupportVectorMachine 的内核。

IKernel kernel = new DynamicTimeWarping(dimension);
        var machine = new MulticlassSupportVectorMachine(0, kernel, 2);
        // Create the Multi-class learning algorithm for the machine
        var teacher = new MulticlassSupportVectorLearning(machine, inputs.ToArray(), outputs.ToArray());
        // Configure the learning algorithm to use SMO to train the
        //  underlying SVMs in each of the binary class subproblems.
        teacher.Algorithm = (svm, classInputs, classOutputs, i, j) =>
            new SequentialMinimalOptimization(svm, classInputs, classOutputs)
            {
                Complexity = 1.5
            };
        // Run the learning algorithm
        double error = teacher.Run();

输入和输出如下所示:

?inputs.ToArray()
{double[22][]}
    [0]: {double[10656]}
    [1]: {double[9360]}
    [2]: {double[9216]}
    [3]: {double[9864]}
    [4]: {double[10296]}
    [5]: {double[10152]}
    [6]: {double[9936]}
    [7]: {double[9216]}
    [8]: {double[10944]}
    [9]: {double[9504]}
    [10]: {double[11880]}
    [11]: {double[22752]}
    [12]: {double[23688]}
    [13]: {double[29880]}
    [14]: {double[32328]}
    [15]: {double[37224]}
    [16]: {double[30024]}
    [17]: {double[27288]}
    [18]: {double[26064]}
    [19]: {double[22032]}
    [20]: {double[21672]}
    [21]: {double[22680]}
?inputs[0]
{double[10656]}
    [0]: 7.6413027545068823
    [1]: -61.607203372756942
    [2]: 7.7375128997886513
    [3]: -25.704529598536471
    [4]: -0.4124927191531238
    [5]: 9.6820255661415011
    [6]: 3.0674374003781861
    [7]: 4.6364653722537668
    [8]: 3.3559314278499177
    [9]: 0.93969394152714925
    [10]: -6.3800159552064146
    [11]: 1.4239779356781062
    [12]: -2.25349154655782
    [13]: -1.5457194406236221
    [14]: -0.7612541874802764
    [15]: -3.3364791133985348
    [16]: 0.67816801816804861
    [17]: -3.4117217877592343
    [18]: 1.5785492543017225
    [19]: 0.31091690789261689
    [20]: -2.5526646739208712
    [21]: -1.0550268575680164
    [22]: -0.9598271201088191
    [23]: -1.1797916101998056
    [24]: 0.56157735657438412
    [25]: -0.16309890421998655
    [26]: 0.29765136770064271
    [27]: -0.35684735108472643
    [28]: -0.52382117896006564
    [29]: -0.052087258844925849
    [30]: -0.45363669419489172
    [31]: -0.16216259086709361
    [32]: -0.25958480481802632
    [33]: 0.081248839173330589
    [34]: -0.019783293216956807
    [35]: 0.14139773316964666
    [36]: 0.088466551256948273
    [37]: -0.019528343614348152
    [38]: 0.087073343332064762
    [39]: 0.048432068369313144
    [40]: -0.0069171431858626713
    [41]: -0.0095272766950126042
    [42]: 0.016639887499893875
    [43]: -0.009108847017642599
    [44]: 0.0017424263597747487
    [45]: 0.0042160613810267641
    [46]: -0.002793626734247919
    [47]: 0.00092130299196750763
    [48]: 0.0024488939699103319
    [49]: 0.0021684669072286468
    [50]: 0.000000000000010673294119695543
    [51]: -0.000000000000014072530108313123
    [52]: 0.000000000000000069063495074940116
    [53]: 8.73342598612937E-17
    [54]: 0.000000000000000030048643853749834
    [55]: -6.95380121971215E-17
    [56]: 0.00000000000000010093927767292201
    [57]: 0.000000000000000046158366228268829
    [58]: 0.000000000000000039070100378142324
    [59]: 0.00000000000000010492059540665321
    [60]: -0.000000000000000014254591247067773
    [61]: -0.0000000000000000015902697756329909
    [62]: 0.0000000000000000017024249964704589
    [63]: 0.0000000000000000010277956708903136
    [64]: 3.5875442986020568E-28
    [65]: -2.215158998843094E-31
    [66]: 1.041379591973569E-31
    [67]: -4.3897715186113276E-31
    [68]: 4.248432864156974E-34
    [69]: 4.3718530099471368E-47
    [70]: 1.4551856970655856E-50
    [71]: 0.0
    [72]: 11.031182384920639
    [73]: -63.434486026723626
    [74]: 1.7731679007864651
    [75]: -23.968196466652273
    [76]: 2.2753564408666507
    [77]: 9.5492641110324534
    [78]: 3.4465209481281054
    [79]: 4.7979691924966161
    [80]: 2.0147801482840508
    [81]: 1.1858337013571998
    [82]: -4.607944757859336
    [83]: 0.75637871318664485
    [84]: -3.8397810581420115
    [85]: -2.1276086210477514
    [86]: -0.4060620782117581
    [87]: -2.9313848427777227
    [88]: 0.052605148372525556
    [89]: -1.5948208186863277
    [90]: 0.36061926783486992
    [91]: -0.12623742266247567
    [92]: -1.1889713301479885
    [93]: -0.33299631607409635
    [94]: -0.00912650336180437
    [95]: -0.52707950657313729
    [96]: 0.52115933681848092
    [97]: 0.46870463636533816
    [98]: -0.18482093982467213
    [99]: -0.49350561475314514
    < More... (The first 100 of 10656 items were displayed.) >
?outputs
Count = 22
    [0]: 0
    [1]: 0
    [2]: 0
    [3]: 0
    [4]: 0
    [5]: 0
    [6]: 0
    [7]: 0
    [8]: 0
    [9]: 0
    [10]: 0
    [11]: 1
    [12]: 1
    [13]: 1
    [14]: 1
    [15]: 1
    [16]: 1
    [17]: 1
    [18]: 1
    [19]: 1
    [20]: 1
    [21]: 1

使用该代码，错误返回值为 0.5。

问题:

1. 这是否意味着我的训练数据有问题？

2. 是否有任何其他内核可用于我的可变大小序列？

谢谢。

Answer 1

我将举例说明如何结合使用 DynamicTimeWarping 内核和高斯内核来执行序列分类，这有望提供更好的结果。

序列分类问题的首要任务是正确组织序列以提供给学习算法。每个序列都可以由多元向量组成，因此，输入数据必须相应地组织(这是使用序列机器时最有可能的错误来源，所以请花一两分钟来理解下面的代码在做什么).

// Suppose you have sequences of multivariate observations, and that
// those sequences could be of arbitrary length. On the other hand, 
// each observation have a fixed, delimited number of dimensions.

// In this example, we have sequences of 3-dimensional observations. 
// Each sequence can have an arbitrary length, but each observation
// will always have length 3:

double[][][] sequences =
{
    new double[][] // first sequence
    {
        new double[] { 1, 1, 1 }, // first observation of the first sequence
        new double[] { 1, 2, 1 }, // second observation of the first sequence
        new double[] { 1, 4, 2 }, // third observation of the first sequence
        new double[] { 2, 2, 2 }, // fourth observation of the first sequence
    },

    new double[][] // second sequence (note that this sequence has a different length)
    {
        new double[] { 1, 1, 1 }, // first observation of the second sequence
        new double[] { 1, 5, 6 }, // second observation of the second sequence
        new double[] { 2, 7, 1 }, // third observation of the second sequence
    },

    new double[][] // third sequence 
    {
        new double[] { 8, 2, 1 }, // first observation of the third sequence
    },

    new double[][] // fourth sequence 
    {
        new double[] { 8, 2, 5 }, // first observation of the fourth sequence
        new double[] { 1, 5, 4 }, // second observation of the fourth sequence
    }
};

那些是我们的输入序列。现在，由于我们正在尝试执行分类问题，因此我们必须有一个与每个序列相关联的输出类标签。如果我们有 4 个序列，那么我们将需要 4 个类标签:

// Now, we will also have different class labels associated which each 
// sequence. We will assign -1 to sequences whose observations start 
// with { 1, 1, 1 } and +1 to those that do not:

int[] outputs =
{
    -1,-1,  // First two sequences are of class -1 (those start with {1,1,1})
        1, 1,  // Last two sequences are of class +1  (don't start with {1,1,1})
};

既然问题已经定义好了，就必须对其进行一些转换，以便 DTW-SVM 可以对其进行处理:

// At this point, we will have to "flat" out the input sequences from double[][][]
// to a double[][] so they can be properly understood by the SVMs. The problem is 
// that, normally, SVMs usually expect the data to be comprised of fixed-length 
// input vectors and associated class labels. But in this case, we will be feeding
// them arbitrary-length sequences of input vectors and class labels associated with
// each sequence, instead of each vector.

double[][] inputs = new double[sequences.Length][];
for (int i = 0; i < sequences.Length; i++)
    inputs[i] = Matrix.Concatenate(sequences[i]);


// Now we have to setup the Dynamic Time Warping kernel. We will have to
// inform the length of the fixed-length observations contained in each
// arbitrary-length sequence:
// 
var kernel = new Gaussian<DynamicTimeWarping>(new DynamicTimeWarping(length: 3));

// Now we can create the machine. When using variable-length
// kernels, we will need to pass zero as the input length:
var svm = new KernelSupportVectorMachine(kernel, inputs: 0);

// Create the Sequential Minimal Optimization learning algorithm
var smo = new SequentialMinimalOptimization(svm, inputs, outputs);

// And start learning it!
double error = smo.Run(); // error will be 0.0

此时，应该已经创建了机器。参数 C 应该是由机器自动猜测的，但您可以事后对其进行微调，以尝试提高机器的泛化性能。无论如何，

// At this point, we should have obtained an useful machine. Let's
// see if it can understand a few examples it hasn't seem before:

double[][] a = 
{ 
    new double[] { 1, 1, 1 },
    new double[] { 7, 2, 5 },
    new double[] { 2, 5, 1 },
};

double[][] b =
{
    new double[] { 7, 5, 2 },
    new double[] { 4, 2, 5 },
    new double[] { 1, 1, 1 },
};

// Following the aforementioned logic, sequence (a) should be
// classified as -1, and sequence (b) should be classified as +1.

int resultA = System.Math.Sign(svm.Compute(Matrix.Concatenate(a))); // -1
int resultB = System.Math.Sign(svm.Compute(Matrix.Concatenate(b))); // +1