从数据到预测：线性回归与残差图
几何画板及绘图内容加载中，请稍等……
① 本范例演示如何从一组散点数据出发，通过线性回归拟合直线，并利用残差图评估拟合效果，最终实现从数据到预测的完整过程。② 核心数学概念包括线性回归、最小二乘法、残差分析、统计特征（均值、方差等），以及直线方程与函数图像的关系。③ 作图要点：使用GeoGebra的ResidualPlot命令生成残差图，涉及多个数据点对象（如A1至B16等），通过直线拟合和残差可视化展示数据与预测值之间的偏差。④ 适合高中或大学统计学、数据分析课程，用于直观理解回归模型的构建与诊断，也可用于数学实验或项目式学习，帮助学生掌握数据建模的基本方法。
GeoGebra 命令及参数
ResidualPlot({(1880, -0.24), (1881, -0.3), (1882, -0.24), (1883, -0.23), (1884, -0.28), (1885, -0.2), (1886, -0.32), (1887, -0.33), (1888, 0.02), (1889, -0.27), (1890, -0.36), (1891, -0.3), (1892, -0.29), (1893, -0.25), (1894, -0.3), (1895, -0.12), (1896, 0), (1897, -0.17), (1898, -0.37), (1899, -0.1), (1900, 0.07), (1901, -0.34), (1902, -0.3), (1903, -0.52), (1904, -0.44), (1905, -0.27), (1906, -0.29), (1907, -0.3), (1908, -0.5), (1909, -0.37), (1910, -0.44), (1911, -0.32), (1912, -0.54), (1913, -0.31), (1914, -0.11), (1915, -0.24), (1916, -0.28), (1917, -0.33), (1918, -0.12), (1919, -0.24), (1920, -0.23), (1921, -0.06), (1922, -0.28), (1923, -0.19), (1924, -0.31), (1925, -0.17), (1926, -0.07), (1927, 0.01), (1928, -0.16), (1929, -0.14), (1930, -0.02), (1931, 0.03), (1932, -0.1), (1933, -0.2), (1934, -0.04), (1935, -0.03), (1936, -0.02), (1937, 0.1), (1938, 0.08), (1939, 0), (1940, 0.16), (1941, 0.43), (1942, 0.07), (1943, 0.26), (1944, 0.32), (1945, 0.27), (1946, 0), (1947, 0.05), (1948, -0.06), (1949, -0.06), (1950, -0.16), (1951, 0.13), (1952, 0.01), (1953, 0.12), (1954, -0.05), (1955, -0.12), (1956, -0.19), (1957, 0.03), (1958, 0.08), (1959, 0), (1960, 0.06), (1961, -0.02), (1962, 0.12), (1963, 0.23), (1964, -0.25), (1965, 0.02), (1966, -0.07), (1967, 0.15), (1968, 0.06), (1969, 0.09), (1970, -0.02), (1971, -0.08), (1972, 0.1), (1973, 0.07), (1974, -0.06), (1975, -0.12), (1976, -0.19), (1977, 0.12), (1978, 0.08), (1979, 0.3), (1980, 0.19), (1981, 0.18), (1982, 0.18), (1983, 0.24), (1984, 0.15), (1985, 0.15), (1986, 0.22), (1987, 0.35), (1988, 0.31), (1989, 0.29), (1990, 0.43), (1991, 0.29), (1992, 0.05), (1993, 0.23), (1994, 0.43), (1995, 0.44), (1996, 0.22), (1997, 0.64), (1998, 0.49), (1999, 0.35), (2000, 0.32), (2001, 0.52), (2002, 0.49), (2003, 0.74), (2004, 0.6), (2005, 0.68), (2006, 0.68), (2007, 0.56), (2008, 0.67), (2009, 0.64), (2010, 0.63), (2011, 0.61), (2012, 0.7), (2013, 0.67), (2014, 0.76), (2015, 1.03), (2016, 0.79), (2017, 0.81), (2018, 0.93), (2019, 0.95), (2020, 0.85)}, FitPoly[{(1880, -0.24), (1881, -0.3), (1882, -0.24), (1883, -0.23), (1884, -0.28), (1885, -0.2), (1886, -0.32), (1887, -0.33), (1888, 0.02), (1889, -0.27), (1890, -0.36), (1891, -0.3), (1892, -0.29), (1893, -0.25), (1894, -0.3), (1895, -0.12), (1896, 0), (1897, -0.17), (1898, -0.37), (1899, -0.1), (1900, 0.07), (1901, -0.34), (1902, -0.3), (1903, -0.52), (1904, -0.44), (1905, -0.27), (1906, -0.29), (1907, -0.3), (1908, -0.5), (1909, -0.37), (1910, -0.44), (1911, -0.32), (1912, -0.54), (1913, -0.31), (1914, -0.11), (1915, -0.24), (1916, -0.28), (1917, -0.33), (1918, -0.12), (1919, -0.24), (1920, -0.23), (1921, -0.06), (1922, -0.28), (1923, -0.19), (1924, -0.31), (1925, -0.17), (1926, -0.07), (1927, 0.01), (1928, -0.16), (1929, -0.14), (1930, -0.02), (1931, 0.03), (1932, -0.1), (1933, -0.2), (1934, -0.04), (1935, -0.03), (1936, -0.02), (1937, 0.1), (1938, 0.08), (1939, 0), (1940, 0.16), (1941, 0.43), (1942, 0.07), (1943, 0.26), (1944, 0.32), (1945, 0.27), (1946, 0), (1947, 0.05), (1948, -0.06), (1949, -0.06), (1950, -0.16), (1951, 0.13), (1952, 0.01), (1953, 0.12), (1954, -0.05), (1955, -0.12), (1956, -0.19), (1957, 0.03), (1958, 0.08), (1959, 0), (1960, 0.06), (1961, -0.02), (1962, 0.12), (1963, 0.23), (1964, -0.25), (1965, 0.02), (1966, -0.07), (1967, 0.15), (1968, 0.06), (1969, 0.09), (1970, -0.02), (1971, -0.08), (1972, 0.1), (1973, 0.07), (1974, -0.06), (1975, -0.12), (1976, -0.19), (1977, 0.12), (1978, 0.08), (1979, 0.3), (1980, 0.19), (1981, 0.18), (1982, 0.18), (1983, 0.24), (1984, 0.15), (1985, 0.15), (1986, 0.22), (1987, 0.35), (1988, 0.31), (1989, 0.29), (1990, 0.43), (1991, 0.29), (1992, 0.05), (1993, 0.23), (1994, 0.43), (1995, 0.44), (1996, 0.22), (1997, 0.64), (1998, 0.49), (1999, 0.35), (2000, 0.32), (2001, 0.52), (2002, 0.49), (2003, 0.74), (2004, 0.6), (2005, 0.68), (2006, 0.68), (2007, 0.56), (2008, 0.67), (2009, 0.64), (2010, 0.63), (2011, 0.61), (2012, 0.7), (2013, 0.67), (2014, 0.76), (2015, 1.03), (2016, 0.79), (2017, 0.81), (2018, 0.93), (2019, 0.95), (2020, 0.85)}, 1]) →ResidualPlot({(1880, -0.24), (1881, -0.3), (1882, -0.24), (1883, -0.23), (1884, -0.28), (1885, -0.2), (1886, -0.32), (1887, -0.33), (1888, 0.02), (1889, -0.27), (1890, -0.36), (1891, -0.3), (1892, -0.29), (1893, -0.25), (1894, -0.3), (1895, -0.12), (1896, 0), (1897, -0.17), (1898, -0.37), (1899, -0.1), (1900, 0.07), (1901, -0.34), (1902, -0.3), (1903, -0.52), (1904, -0.44), (1905, -0.27), (1906, -0.29), (1907, -0.3), (1908, -0.5), (1909, -0.37), (1910, -0.44), (1911, -0.32), (1912, -0.54), (1913, -0.31), (1914, -0.11), (1915, -0.24), (1916, -0.28), (1917, -0.33), (1918, -0.12), (1919, -0.24), (1920, -0.23), (1921, -0.06), (1922, -0.28), (1923, -0.19), (1924, -0.31), (1925, -0.17), (1926, -0.07), (1927, 0.01), (1928, -0.16), (1929, -0.14), (1930, -0.02), (1931, 0.03), (1932, -0.1), (1933, -0.2), (1934, -0.04), (1935, -0.03), (1936, -0.02), (1937, 0.1), (1938, 0.08), (1939, 0), (1940, 0.16), (1941, 0.43), (1942, 0.07), (1943, 0.26), (1944, 0.32), (1945, 0.27), (1946, 0), (1947, 0.05), (1948, -0.06), (1949, -0.06), (1950, -0.16), (1951, 0.13), (1952, 0.01), (1953, 0.12), (1954, -0.05), (1955, -0.12), (1956, -0.19), (1957, 0.03), (1958, 0.08), (1959, 0), (1960, 0.06), (1961, -0.02), (1962, 0.12), (1963, 0.23), (1964, -0.25), (1965, 0.02), (1966, -0.07), (1967, 0.15), (1968, 0.06), (1969, 0.09), (1970, -0.02), (1971, -0.08), (1972, 0.1), (1973, 0.07), (1974, -0.06), (1975, -0.12), (1976, -0.19), (1977, 0.12), (1978, 0.08), (1979, 0.3), (1980, 0.19), (1981, 0.18), (1982, 0.18), (1983, 0.24), (1984, 0.15), (1985, 0.15), (1986, 0.22), (1987, 0.35), (1988, 0.31), (1989, 0.29), (1990, 0.43), (1991, 0.29), (1992, 0.05), (1993, 0.23), (1994, 0.43), (1995, 0.44), (1996, 0.22), (1997, 0.64), (1998, 0.49), (1999, 0.35), (2000, 0.32), (2001, 0.52), (2002, 0.49), (2003, 0.74), (2004, 0.6), (2005, 0.68), (2006, 0.68), (2007, 0.56), (2008, 0.67), (2009, 0.64), (2010, 0.63), (2011, 0.61), (2012, 0.7), (2013, 0.67), (2014, 0.76), (2015, 1.03), (2016, 0.79), (2017, 0.81), (2018, 0.93), (2019, 0.95), (2020, 0.85)}, FitPoly[{(1880, -0.24), (1881, -0.3), (1882, -0.24), (1883, -0.23), (1884, -0.28), (1885, -0.2), (1886, -0.32), (1887, -0.33), (1888, 0.02), (1889, -0.27), (1890, -0.36), (1891, -0.3), (1892, -0.29), (1893, -0.25), (1894, -0.3), (1895, -0.12), (1896, 0), (1897, -0.17), (1898, -0.37), (1899, -0.1), (1900, 0.07), (1901, -0.34), (1902, -0.3), (1903, -0.52), (1904, -0.44), (1905, -0.27), (1906, -0.29), (1907, -0.3), (1908, -0.5), (1909, -0.37), (1910, -0.44), (1911, -0.32), (1912, -0.54), (1913, -0.31), (1914, -0.11), (1915, -0.24), (1916, -0.28), (1917, -0.33), (1918, -0.12), (1919, -0.24), (1920, -0.23), (1921, -0.06), (1922, -0.28), (1923, -0.19), (1924, -0.31), (1925, -0.17), (1926, -0.07), (1927, 0.01), (1928, -0.16), (1929, -0.14), (1930, -0.02), (1931, 0.03), (1932, -0.1), (1933, -0.2), (1934, -0.04), (1935, -0.03), (1936, -0.02), (1937, 0.1), (1938, 0.08), (1939, 0), (1940, 0.16), (1941, 0.43), (1942, 0.07), (1943, 0.26), (1944, 0.32), (1945, 0.27), (1946, 0), (1947, 0.05), (1948, -0.06), (1949, -0.06), (1950, -0.16), (1951, 0.13), (1952, 0.01), (1953, 0.12), (1954, -0.05), (1955, -0.12), (1956, -0.19), (1957, 0.03), (1958, 0.08), (1959, 0), (1960, 0.06), (1961, -0.02), (1962, 0.12), (1963, 0.23), (1964, -0.25), (1965, 0.02), (1966, -0.07), (1967, 0.15), (1968, 0.06), (1969, 0.09), (1970, -0.02), (1971, -0.08), (1972, 0.1), (1973, 0.07), (1974, -0.06), (1975, -0.12), (1976, -0.19), (1977, 0.12), (1978, 0.08), (1979, 0.3), (1980, 0.19), (1981, 0.18), (1982, 0.18), (1983, 0.24), (1984, 0.15), (1985, 0.15), (1986, 0.22), (1987, 0.35), (1988, 0.31), (1989, 0.29), (1990, 0.43), (1991, 0.29), (1992, 0.05), (1993, 0.23), (1994, 0.43), (1995, 0.44), (1996, 0.22), (1997, 0.64), (1998, 0.49), (1999, 0.35), (2000, 0.32), (2001, 0.52), (2002, 0.49), (2003, 0.74), (2004, 0.6), (2005, 0.68), (2006, 0.68), (2007, 0.56), (2008, 0.67), (2009, 0.64), (2010, 0.63), (2011, 0.61), (2012, 0.7), (2013, 0.67), (2014, 0.76), (2015, 1.03), (2016, 0.79), (2017, 0.81), (2018, 0.93), (2019, 0.95), (2020, 0.85)}, 1]) →
AI 绘图提示语参考
绘制一组从1880年到2020年的散点数据，横轴为年份，纵轴为数值。对这些点进行线性回归拟合，得到一条直线。同时绘制残差图，展示每个数据点与拟合直线之间的垂直偏差，即残差。散点图与残差图应并排或上下排列，以便对比观察拟合效果。
以上提示语为根据原始绘图命令整理而来，仅供参考。