Extending the concept of multi-selfsimilar random field we study multi-scale invariant (MSI) fields which have component-wise discrete scale invariant property. Assuming scale parameters as $\lambda_i>1$, $i=1,\ldots,d$ and the parameter space as $(1, \infty)^d$, the first scale rectangle is referred to the rectangle $ (1, \lambda_1)\times \ldots \times (1, \lambda_d)$. Applying certain component-wise geometric sampling of MSI field, the harmonic-like representation and spectral density of the sampled MSI field are characterized. Furthermore, the covariance function and spectral density of the sampled Markov MSI field are presented by the variances and covariances of samples inside first scale rectangle. As an example of MSI field, a two-dimensional simple fractional Brownian sheet (sfBs) is demonstrated. Also real data of the precipitation in some area of Brisbane in Australia for two days (25 and 26 January 2013) are examined. We show that precipitation on this area has MSI property and estimate it as a simple MSI field with stationary increments inside scale intervals. This structure enables us to predict the precipitation in surface and time. We apply the mean absolute percentage error as a measure for the accuracy of the predictions.

中文翻译：

---

多尺度不变场：估计和预测

扩展多自相似随机场的概念，我们研究了具有分量离散尺度不变性的多尺度不变 (MSI) 场。假设尺度参数为$\lambda_i>1$, $i=1,\ldots,d$，参数空间为$(1, \infty)^d$，第一个尺度矩形称为矩形$(1, \lambda_1)\times \ldots \times (1, \lambda_d)$。应用MSI场的某些分量几何采样，表征了采样的MSI场的类谐波表示和谱密度。此外，采样马尔可夫MSI场的协方差函数和谱密度由第一尺度矩形内样本的方差和协方差表示。作为 MSI 字段的示例，演示了二维简单分数布朗表 (sfBs)。还检查了澳大利亚布里斯班某些地区两天（2013 年 1 月 25 日和 26 日）降水的真实数据。我们表明该地区的降水具有 MSI 属性，并将其估计为一个简单的 MSI 场，在尺度区间内具有固定增量。这种结构使我们能够预测地表和时间的降水。我们应用平均绝对百分比误差作为预测准确性的衡量标准。