1140513 meeting

前言

以下與上週訓練方式相同，惟訓練方式改為 forecast 方法。

各組合也顯示資料集中 CO 、 NOx 、 O3 、 PM2.5 、 SO2 的 MSPE 。

原程式 (test 無缺值)

200

rm

MSPE: 0.2670296334605754

MSPE (per polluant): [0.24853024 0.2530521 0.26650984 0.29022381 0.27683218]

bm

MSPE: 0.028377005098096042

MSPE (per polluant): [0.05570757 0.00931698 0.00982079 0.01584567 0.05119401]

mnr

MSPE: 0.031035673202506362

MSPE (per polluant): [0.04832559 0.01763471 0.01713394 0.05337523 0.01870889]

forecast

MSPE: 0.01983729877681366

MSPE (per polluant): [0.02048197 0.01991615 0.02261555 0.01805628 0.01811654]

24

rm

MSPE: 0.00619986746721049

MSPE (per polluant): [0.00633583 0.00567932 0.00643217 0.00653991 0.0060121 ]

bm

MSPE: 0.014311867730900778

MSPE (per polluant): [0.05506475 0.00345052 0.00333255 0.00523976 0.00447176]

mnr

MSPE: 0.018981540433120517

MSPE (per polluant): [0.00333461 0.03810891 0.00481243 0.00331675 0.04533501]

forecast

MSPE: 0.0070626912237513545

MSPE (per polluant): [0.00720427 0.00756547 0.00656687 0.00792716 0.00604968]

12

rm

MSPE: 0.002616900708467343

MSPE (per polluant): [0.00300699 0.0035922 0.00263202 0.00171722 0.00213607]

bm

MSPE: 0.038285284933024764

MSPE (per polluant): [0.02292427 0.06918261 0.0394378 0.02133293 0.03854882]

mnr

MSPE: 0.04627111620176548

MSPE (per polluant): [0.0442022 0.04389888 0.04129601 0.04729015 0.05466834]

forecast

MSPE: 0.027468193901030315

MSPE (per polluant): [0.03450684 0.02567697 0.02923344 0.01771871 0.030205 ]

預測 (test 可缺值)

200

rm

MSPE for all: 0.26359314172785486

MSPE for all (per polluant): [0.26905376 0.25574947 0.28052578 0.25100442 0.26163227]

MSPE only for missing: 0.266999942757196

MSPE for all (per polluant): [0.28228508 0.27145088 0.29145757 0.23416063 0.25510918]

bm

MSPE for all: 0.03627450973249119

MSPE for all (per polluant): [0.03418577 0.03615802 0.03825413 0.03674731 0.03602732]

MSPE only for missing: 0.09063353972406932

MSPE for all (per polluant): [0.08541148 0.09034198 0.09558279 0.09181558 0.09001587]

rbm

MSPE for all: 0.18872160069272334

MSPE for all (per polluant): [0.1245216 0.23241761 0.27848055 0.17466932 0.13351893]

MSPE only for missing: 0.1529691304212167

MSPE for all (per polluant): [0.31117033 0.05161114 0.09800657 0.07323075 0.23082686]

forecast

MSPE for all: 0.021272910649979906

MSPE for all (per polluant): [0.02169129 0.0203196 0.02180517 0.02046255 0.02208595]

MSPE only for missing: 0.05313452497446276

MSPE for all (per polluant): [0.05418063 0.05075108 0.05446552 0.05110837 0.05516703]

24

rm

MSPE for all: 0.006075528992989291

MSPE for all (per polluant): [0.00671391 0.00600151 0.00490925 0.00661012 0.00614285]

MSPE only for missing: 0.02425226355592921

MSPE for all (per polluant): [0.00531567 0.00777571 0.10396352 0.00475068 0.00439044]

bm

MSPE for all: 0.002255499816708605

MSPE for all (per polluant): [0.00236381 0.00218572 0.00243353 0.00219861 0.00209583]

MSPE only for missing: 0.0458826803307174

MSPE for all (per polluant): [0.04814393 0.04443191 0.04959418 0.04469903 0.04254436]

rbm

MSPE for all: 0.0075089527580064555

MSPE for all (per polluant): [0.00872188 0.00656201 0.00420567 0.01047179 0.00758342]

MSPE only for missing: 0.01891713509648322

MSPE for all (per polluant): [0.03741054 0.00052717 0.00024734 0.05448549 0.00191513]

forecast

MSPE for all: 0.007943497468487118

MSPE for all (per polluant): [0.00652371 0.00969486 0.00874908 0.0077423 0.00700754]

MSPE only for missing: 0.16438706211646267

MSPE for all (per polluant): [0.13480245 0.20087991 0.18117356 0.16019934 0.14488005]

12

rm

MSPE for all: 0.0024877418248716965

MSPE for all (per polluant): [0.00212427 0.00400996 0.00201409 0.00214242 0.00214798]

MSPE only for missing: 0.002110897565069707

MSPE for all (per polluant): [0.00154594 0.0008708 0.00047257 0.00196384 0.00552956]

bm

MSPE for all: 0.023524741195117407

MSPE for all (per polluant): [0.01595404 0.02913753 0.02277857 0.02257204 0.02718153]

MSPE only for missing: 0.9778849302630097

MSPE for all (per polluant): [0.66240434 1.21178666 0.94678264 0.93818589 1.13026512]

rbm

MSPE for all: 0.0028292502494105744

MSPE for all (per polluant): [0.0027832 0.0033312 0.00269622 0.00275 0.00258563]

MSPE only for missing: 0.004499003182155347

MSPE for all (per polluant): [1.84505694e-02 3.92772396e-04 5.71754454e-05 2.03783543e-03 1.55666327e-03]

forecast

MSPE for all: 0.028298511239477558

MSPE for all (per polluant): [0.02675121 0.02386053 0.03320219 0.02575034 0.03192828]

MSPE only for missing: 1.1767571661660086

MSPE for all (per polluant): [1.11228154 0.99181863 1.38109155 1.07055341 1.3280407 ]

結果

Test 無缺值

Test 可缺值

MSPE for All

MSPE only for missing

autoFRK 嵌入 SSSDS4

在嵌入 autoFRK 的 Basis function 前，先將 Basis function 於 Python 中實現。

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class FRKBasis(nn.Module):
    def __init__(self, locs: torch.Tensor, num_centers: int = 64):
        """
        Multi-resolution spline basis function.
        Args:
            locs: [N, d] tensor of spatial locations
            num_centers: number of TPS centers to use
        """
        super().__init__()
        self.register_buffer("locs", locs)  # [N, d]
        self.N, self.d = locs.shape

        # Select centers from locs
        idx = torch.randperm(self.N)[:num_centers]
        self.register_buffer("centers", locs[idx])  # [K, d]

    def tps_phi(self, r: torch.Tensor) -> torch.Tensor:
        """
        Apply TPS radial basis based on input distance r
        """
        eps = 1e-10  # avoid log(0)
        if self.d == 1:
            return (r ** 3) / 12
        elif self.d == 2:
            return (r ** 2) * torch.log(r + eps) / (8 * torch.pi)
        elif self.d == 3:
            return -r / 8
        else:
            raise NotImplementedError(f"TPS not implemented for d={self.d}")

    def forward(self) -> torch.Tensor:
        # Step 1: Construct Φ (TPS basis)
        dists = torch.cdist(self.locs, self.centers)  # [N, K] (Euclidean distances)
        Phi = self.tps_phi(dists)  # [N, K]

        # Step 2: Construct X = [1, x1, x2, ..., xd]
        ones = torch.ones(self.N, 1, device=self.locs.device)
        X = torch.cat([ones, self.locs], dim=1)  # [N, d+1]

        # Step 3: Compute projection matrix Q = I - X(XᵗX)⁻¹Xᵗ
        XtX_inv = torch.inverse(X.T @ X)  # [(d+1), (d+1)]
        Q = torch.eye(self.N, device=self.locs.device) - X @ XtX_inv @ X.T  # [N, N]

        # Step 4: Compute QΦ
        QPhi = Q @ Phi  # [N, K]

        # Step 5: Eigen-decomposition of QΦ(QΦ)ᵗ
        #G = QPhi @ QPhi.T  # [N, N]
        G = Q @ Phi @ Q
        eigenvalues, eigenvectors = torch.linalg.eigh(G)  # ascending order

        # Step 6: Filter out near-zero eigenvalues and sort descending
        valid = eigenvalues > 1e-6
        eigenvalues = eigenvalues[valid]
        eigenvectors = eigenvectors[:, valid]
        idx_desc = torch.argsort(eigenvalues, descending=True)
        eigenvalues = eigenvalues[idx_desc]
        eigenvectors = eigenvectors[:, idx_desc]

        # Step 7: Construct basis functions
        sqrt_inv_lambda = torch.diag(1.0 / torch.sqrt(eigenvalues))  # [r, r]
        B_high = eigenvectors @ sqrt_inv_lambda  # [N, r]

        # Step 8: Concatenate [1, x1, ..., xd] and B_high
        B_low = X  # [N, d+1]
        F = torch.cat([B_low, B_high], dim=1)  # [N, d+1 + r]
        return F