Simple linear regression#
In case of one feature (\(d=1\)) linear regression is written as
(5)#\[ y = a x + b.\]
The parameters of this model \(\boldsymbol \theta = (a, b)\), where \(b\) is intercept (or bias), \(a\) is slope.
The feature matrix here has only one column, denote it \(\boldsymbol x\) and let \(\boldsymbol y\) be the vector of corresponding labels. Also denote
\(\overline {\boldsymbol x} = \frac 1n \sum\limits_{i=1}^n x_i\) — sample mean of predictors;
\(\overline {\boldsymbol y} = \frac 1n \sum\limits_{i=1}^n y_i\) — sample mean of targets.
MSE fit#
Use MSE to fit parameters \(\boldsymbol \theta = (a, b)\):
(6)#\[\mathcal L(a, b) = \frac 1n\sum\limits_{i=1}^n (y_i - ax_i - b)^2 \to \min\limits_{a, b}.\]
What about some calculus?
We have
\[
\frac{\partial \mathcal L}{\partial a} = -\frac 2n\sum\limits_{i=1}^n x_i(y_i - ax_i - b) = 0,
\]
\[
\frac{\partial \mathcal L}{\partial b} = -\frac 2n\sum\limits_{i=1}^n (y_i - ax_i - b) = 0.
\]
From the last equality it follows that
\[
b = \overline {\boldsymbol y} - a \overline {\boldsymbol x}
\]
TODO: Finish the proof
The optimal parameters are
(7)#\[ \widehat a = \frac{\sum\limits_{i=1}^n (x_i - \overline {\boldsymbol x})(y_i - \overline {\boldsymbol y})}{\sum\limits_{i=1}^n (x_i - \overline {\boldsymbol x})^2}, \quad
\widehat b = \overline {\boldsymbol y} - \widehat a \overline {\boldsymbol x}.\]
Note that the slope is equal to the ratio of sample correlation between \(\boldsymbol x\) and \(\boldsymbol y\) to the sample variance of \(\boldsymbol x\).
Question
Does (7) work for all possible values of \(\boldsymbol x\) and \(\boldsymbol y\)?
Dummy model#
The model (5) is simple but we can do even simpler! Let \(a=0\) and predict labels by a constant (or dummy) model \(\widehat y = b\). In fact, for constant predictions you don’t need any features, only labels do the job.
Question
Which value of \(b\) minimizes the MSE (6)?
Linear regression can be used with different loss functions. For example, we can choose mean absolute error (MAE) instead of MSE:
(8)#\[\mathcal L(a, b) = \frac 1n\sum\limits_{i=1}^n \vert y_i - ax_i - b\vert \to \min\limits_{a, b}.\]
This time it is unlikely that we can find the analytical solution. But maybe it can be done for the dummy model?
Question
For which value of \(b\) the value of MAE
\[
\frac 1n\sum\limits_{i=1}^n \vert y_i - b\vert
\]
is minimal?
Answer
\(\widehat b = \mathrm{med}(\boldsymbol y)\) (see this discussion for details)
RSS and \(R^2\)-score#
Putting the optimal weights \(\widehat a\) and \(\widehat b\) into the loss function (6), we obtain residual square error (RSE). Multiplying by \(n\) we get residual sum of squares
\[
RSS = \sum\limits_{i=1}^n(y_i - \widehat a x_i - \widehat b)^2.
\]
Also, total sum of squares is defined as
\[
TSS = \sum\limits_{i=1}^n(y_i - \overline {\boldsymbol y})^2.
\]
A popular metric called coefficient of determination (or \(R^2\)-score) is defined as
(9)#\[R^2 = 1 - \frac{RSS}{TSS} = 1 - \frac{\sum\limits_{i=1}^n(y_i - \widehat y_i)^2}{\sum\limits_{i=1}^n(y_i - \overline {\boldsymbol y})^2}.\]
The coefficient of determination shows proportion of variance explained. \(R^2\)-score does not exceed \(1\) (the greater — the better).
\(R^2\)-score measures the amount of variability that is left unexplained after performing the regression. It shows how better the model works in comparison with dummy prediction.
Example: Boston dataset#
import pandas as pd
boston = pd.read_csv("../ISLP_datsets/Boston.csv").drop("Unnamed: 0", axis=1)
boston.head()
| crim | zn | indus | chas | nox | rm | age | dis | rad | tax | ptratio | lstat | medv | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0.00632 | 18.0 | 2.31 | 0 | 0.538 | 6.575 | 65.2 | 4.0900 | 1 | 296 | 15.3 | 4.98 | 24.0 |
| 1 | 0.02731 | 0.0 | 7.07 | 0 | 0.469 | 6.421 | 78.9 | 4.9671 | 2 | 242 | 17.8 | 9.14 | 21.6 |
| 2 | 0.02729 | 0.0 | 7.07 | 0 | 0.469 | 7.185 | 61.1 | 4.9671 | 2 | 242 | 17.8 | 4.03 | 34.7 |
| 3 | 0.03237 | 0.0 | 2.18 | 0 | 0.458 | 6.998 | 45.8 | 6.0622 | 3 | 222 | 18.7 | 2.94 | 33.4 |
| 4 | 0.06905 | 0.0 | 2.18 | 0 | 0.458 | 7.147 | 54.2 | 6.0622 | 3 | 222 | 18.7 | 5.33 | 36.2 |
Let predictors be lstat, target — medv. Let’s calculate the regression coefficients using (7):
import numpy as np
x = boston['lstat']
y = boston['medv']
a = np.sum((x -x.mean()) * (y - y.mean())) / np.sum((x -x.mean()) ** 2)
b = y.mean() - a*x.mean()
a, b
(-0.9500493537579907, 34.5538408793831)
Now plot the data and the regression line:
import matplotlib.pyplot as plt
%config InlineBackend.figure_format = 'svg'
plt.scatter(x, y, s=10, c='b', alpha=0.7)
xs = np.linspace(x.min(), x.max(), num=10)
plt.plot(xs, a*xs + b, c='r', lw=2, label=r"$y = \widehat a x + \widehat b$")
plt.plot([xs.min(), xs.max()], [y.mean(), y.mean()], c='orange', lw=2, label=r"$y = \overline y$")
plt.xlabel("lstat")
plt.ylabel("medv")
plt.title("Simple linear regression vs dummy model")
plt.legend()
plt.grid(ls=":");
Error in callback <function _draw_all_if_interactive at 0x7fa5c6881260> (for post_execute), with arguments args (),kwargs {}:
---------------------------------------------------------------------------
FileNotFoundError Traceback (most recent call last)
File ~/.local/lib/python3.12/site-packages/matplotlib/texmanager.py:250, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
249 try:
--> 250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
File /usr/local/lib/python3.12/subprocess.py:466, in check_output(timeout, *popenargs, **kwargs)
464 kwargs['input'] = empty
--> 466 return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,
467 **kwargs).stdout
File /usr/local/lib/python3.12/subprocess.py:548, in run(input, capture_output, timeout, check, *popenargs, **kwargs)
546 kwargs['stderr'] = PIPE
--> 548 with Popen(*popenargs, **kwargs) as process:
549 try:
File /usr/local/lib/python3.12/subprocess.py:1026, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask, pipesize, process_group)
1023 self.stderr = io.TextIOWrapper(self.stderr,
1024 encoding=encoding, errors=errors)
-> 1026 self._execute_child(args, executable, preexec_fn, close_fds,
1027 pass_fds, cwd, env,
1028 startupinfo, creationflags, shell,
1029 p2cread, p2cwrite,
1030 c2pread, c2pwrite,
1031 errread, errwrite,
1032 restore_signals,
1033 gid, gids, uid, umask,
1034 start_new_session, process_group)
1035 except:
1036 # Cleanup if the child failed starting.
File /usr/local/lib/python3.12/subprocess.py:1950, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session, process_group)
1949 err_msg = os.strerror(errno_num)
-> 1950 raise child_exception_type(errno_num, err_msg, err_filename)
1951 raise child_exception_type(err_msg)
FileNotFoundError: [Errno 2] No such file or directory: 'latex'
The above exception was the direct cause of the following exception:
RuntimeError Traceback (most recent call last)
File ~/.local/lib/python3.12/site-packages/matplotlib/pyplot.py:197, in _draw_all_if_interactive()
195 def _draw_all_if_interactive() -> None:
196 if matplotlib.is_interactive():
--> 197 draw_all()
File ~/.local/lib/python3.12/site-packages/matplotlib/_pylab_helpers.py:132, in Gcf.draw_all(cls, force)
130 for manager in cls.get_all_fig_managers():
131 if force or manager.canvas.figure.stale:
--> 132 manager.canvas.draw_idle()
File ~/.local/lib/python3.12/site-packages/matplotlib/backend_bases.py:1893, in FigureCanvasBase.draw_idle(self, *args, **kwargs)
1891 if not self._is_idle_drawing:
1892 with self._idle_draw_cntx():
-> 1893 self.draw(*args, **kwargs)
File ~/.local/lib/python3.12/site-packages/matplotlib/backends/backend_agg.py:388, in FigureCanvasAgg.draw(self)
385 # Acquire a lock on the shared font cache.
386 with (self.toolbar._wait_cursor_for_draw_cm() if self.toolbar
387 else nullcontext()):
--> 388 self.figure.draw(self.renderer)
389 # A GUI class may be need to update a window using this draw, so
390 # don't forget to call the superclass.
391 super().draw()
File ~/.local/lib/python3.12/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File ~/.local/lib/python3.12/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/.local/lib/python3.12/site-packages/matplotlib/figure.py:3154, in Figure.draw(self, renderer)
3151 # ValueError can occur when resizing a window.
3153 self.patch.draw(renderer)
-> 3154 mimage._draw_list_compositing_images(
3155 renderer, self, artists, self.suppressComposite)
3157 for sfig in self.subfigs:
3158 sfig.draw(renderer)
File ~/.local/lib/python3.12/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/.local/lib/python3.12/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/.local/lib/python3.12/site-packages/matplotlib/axes/_base.py:3034, in _AxesBase.draw(self, renderer)
3031 for spine in self.spines.values():
3032 artists.remove(spine)
-> 3034 self._update_title_position(renderer)
3036 if not self.axison:
3037 for _axis in self._axis_map.values():
File ~/.local/lib/python3.12/site-packages/matplotlib/axes/_base.py:2978, in _AxesBase._update_title_position(self, renderer)
2976 top = max(top, bb.ymax)
2977 if title.get_text():
-> 2978 ax.yaxis.get_tightbbox(renderer) # update offsetText
2979 if ax.yaxis.offsetText.get_text():
2980 bb = ax.yaxis.offsetText.get_tightbbox(renderer)
File ~/.local/lib/python3.12/site-packages/matplotlib/axis.py:1336, in Axis.get_tightbbox(self, renderer, for_layout_only)
1333 renderer = self.figure._get_renderer()
1334 ticks_to_draw = self._update_ticks()
-> 1336 self._update_label_position(renderer)
1338 # go back to just this axis's tick labels
1339 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
File ~/.local/lib/python3.12/site-packages/matplotlib/axis.py:2609, in YAxis._update_label_position(self, renderer)
2605 return
2607 # get bounding boxes for this axis and any siblings
2608 # that have been set by `fig.align_ylabels()`
-> 2609 bboxes, bboxes2 = self._get_tick_boxes_siblings(renderer=renderer)
2610 x, y = self.label.get_position()
2611 if self.label_position == 'left':
File ~/.local/lib/python3.12/site-packages/matplotlib/axis.py:2161, in Axis._get_tick_boxes_siblings(self, renderer)
2159 axis = ax._axis_map[name]
2160 ticks_to_draw = axis._update_ticks()
-> 2161 tlb, tlb2 = axis._get_ticklabel_bboxes(ticks_to_draw, renderer)
2162 bboxes.extend(tlb)
2163 bboxes2.extend(tlb2)
File ~/.local/lib/python3.12/site-packages/matplotlib/axis.py:1315, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/.local/lib/python3.12/site-packages/matplotlib/text.py:956, in Text.get_window_extent(self, renderer, dpi)
951 raise RuntimeError(
952 "Cannot get window extent of text w/o renderer. You likely "
953 "want to call 'figure.draw_without_rendering()' first.")
955 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 956 bbox, info, descent = self._get_layout(self._renderer)
957 x, y = self.get_unitless_position()
958 x, y = self.get_transform().transform((x, y))
File ~/.local/lib/python3.12/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File ~/.local/lib/python3.12/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File ~/.local/lib/python3.12/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File ~/.local/lib/python3.12/site-packages/matplotlib/backends/backend_agg.py:213, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
211 _api.check_in_list(["TeX", True, False], ismath=ismath)
212 if ismath == "TeX":
--> 213 return super().get_text_width_height_descent(s, prop, ismath)
215 if ismath:
216 ox, oy, width, height, descent, font_image = \
217 self.mathtext_parser.parse(s, self.dpi, prop)
File ~/.local/lib/python3.12/site-packages/matplotlib/backend_bases.py:652, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
648 fontsize = prop.get_size_in_points()
650 if ismath == 'TeX':
651 # todo: handle properties
--> 652 return self.get_texmanager().get_text_width_height_descent(
653 s, fontsize, renderer=self)
655 dpi = self.points_to_pixels(72)
656 if ismath:
File ~/.local/lib/python3.12/site-packages/matplotlib/texmanager.py:363, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
361 if tex.strip() == '':
362 return 0, 0, 0
--> 363 dvifile = cls.make_dvi(tex, fontsize)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
File ~/.local/lib/python3.12/site-packages/matplotlib/texmanager.py:295, in TexManager.make_dvi(cls, tex, fontsize)
293 with TemporaryDirectory(dir=cwd) as tmpdir:
294 tmppath = Path(tmpdir)
--> 295 cls._run_checked_subprocess(
296 ["latex", "-interaction=nonstopmode", "--halt-on-error",
297 f"--output-directory={tmppath.name}",
298 f"{texfile.name}"], tex, cwd=cwd)
299 (tmppath / Path(dvifile).name).replace(dvifile)
300 return dvifile
File ~/.local/lib/python3.12/site-packages/matplotlib/texmanager.py:254, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
--> 254 raise RuntimeError(
255 f'Failed to process string with tex because {command[0]} '
256 'could not be found') from exc
257 except subprocess.CalledProcessError as exc:
258 raise RuntimeError(
259 '{prog} was not able to process the following string:\n'
260 '{tex!r}\n\n'
(...)
267 exc=exc.output.decode('utf-8', 'backslashreplace'))
268 ) from None
RuntimeError: Failed to process string with tex because latex could not be found
---------------------------------------------------------------------------
FileNotFoundError Traceback (most recent call last)
File ~/.local/lib/python3.12/site-packages/matplotlib/texmanager.py:250, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
249 try:
--> 250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
File /usr/local/lib/python3.12/subprocess.py:466, in check_output(timeout, *popenargs, **kwargs)
464 kwargs['input'] = empty
--> 466 return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,
467 **kwargs).stdout
File /usr/local/lib/python3.12/subprocess.py:548, in run(input, capture_output, timeout, check, *popenargs, **kwargs)
546 kwargs['stderr'] = PIPE
--> 548 with Popen(*popenargs, **kwargs) as process:
549 try:
File /usr/local/lib/python3.12/subprocess.py:1026, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask, pipesize, process_group)
1023 self.stderr = io.TextIOWrapper(self.stderr,
1024 encoding=encoding, errors=errors)
-> 1026 self._execute_child(args, executable, preexec_fn, close_fds,
1027 pass_fds, cwd, env,
1028 startupinfo, creationflags, shell,
1029 p2cread, p2cwrite,
1030 c2pread, c2pwrite,
1031 errread, errwrite,
1032 restore_signals,
1033 gid, gids, uid, umask,
1034 start_new_session, process_group)
1035 except:
1036 # Cleanup if the child failed starting.
File /usr/local/lib/python3.12/subprocess.py:1950, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session, process_group)
1949 err_msg = os.strerror(errno_num)
-> 1950 raise child_exception_type(errno_num, err_msg, err_filename)
1951 raise child_exception_type(err_msg)
FileNotFoundError: [Errno 2] No such file or directory: 'latex'
The above exception was the direct cause of the following exception:
RuntimeError Traceback (most recent call last)
File ~/.local/lib/python3.12/site-packages/IPython/core/formatters.py:340, in BaseFormatter.__call__(self, obj)
338 pass
339 else:
--> 340 return printer(obj)
341 # Finally look for special method names
342 method = get_real_method(obj, self.print_method)
File ~/.local/lib/python3.12/site-packages/IPython/core/pylabtools.py:152, in print_figure(fig, fmt, bbox_inches, base64, **kwargs)
149 from matplotlib.backend_bases import FigureCanvasBase
150 FigureCanvasBase(fig)
--> 152 fig.canvas.print_figure(bytes_io, **kw)
153 data = bytes_io.getvalue()
154 if fmt == 'svg':
File ~/.local/lib/python3.12/site-packages/matplotlib/backend_bases.py:2164, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
2161 # we do this instead of `self.figure.draw_without_rendering`
2162 # so that we can inject the orientation
2163 with getattr(renderer, "_draw_disabled", nullcontext)():
-> 2164 self.figure.draw(renderer)
2165 if bbox_inches:
2166 if bbox_inches == "tight":
File ~/.local/lib/python3.12/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File ~/.local/lib/python3.12/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/.local/lib/python3.12/site-packages/matplotlib/figure.py:3154, in Figure.draw(self, renderer)
3151 # ValueError can occur when resizing a window.
3153 self.patch.draw(renderer)
-> 3154 mimage._draw_list_compositing_images(
3155 renderer, self, artists, self.suppressComposite)
3157 for sfig in self.subfigs:
3158 sfig.draw(renderer)
File ~/.local/lib/python3.12/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/.local/lib/python3.12/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/.local/lib/python3.12/site-packages/matplotlib/axes/_base.py:3034, in _AxesBase.draw(self, renderer)
3031 for spine in self.spines.values():
3032 artists.remove(spine)
-> 3034 self._update_title_position(renderer)
3036 if not self.axison:
3037 for _axis in self._axis_map.values():
File ~/.local/lib/python3.12/site-packages/matplotlib/axes/_base.py:2978, in _AxesBase._update_title_position(self, renderer)
2976 top = max(top, bb.ymax)
2977 if title.get_text():
-> 2978 ax.yaxis.get_tightbbox(renderer) # update offsetText
2979 if ax.yaxis.offsetText.get_text():
2980 bb = ax.yaxis.offsetText.get_tightbbox(renderer)
File ~/.local/lib/python3.12/site-packages/matplotlib/axis.py:1336, in Axis.get_tightbbox(self, renderer, for_layout_only)
1333 renderer = self.figure._get_renderer()
1334 ticks_to_draw = self._update_ticks()
-> 1336 self._update_label_position(renderer)
1338 # go back to just this axis's tick labels
1339 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
File ~/.local/lib/python3.12/site-packages/matplotlib/axis.py:2609, in YAxis._update_label_position(self, renderer)
2605 return
2607 # get bounding boxes for this axis and any siblings
2608 # that have been set by `fig.align_ylabels()`
-> 2609 bboxes, bboxes2 = self._get_tick_boxes_siblings(renderer=renderer)
2610 x, y = self.label.get_position()
2611 if self.label_position == 'left':
File ~/.local/lib/python3.12/site-packages/matplotlib/axis.py:2161, in Axis._get_tick_boxes_siblings(self, renderer)
2159 axis = ax._axis_map[name]
2160 ticks_to_draw = axis._update_ticks()
-> 2161 tlb, tlb2 = axis._get_ticklabel_bboxes(ticks_to_draw, renderer)
2162 bboxes.extend(tlb)
2163 bboxes2.extend(tlb2)
File ~/.local/lib/python3.12/site-packages/matplotlib/axis.py:1315, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/.local/lib/python3.12/site-packages/matplotlib/text.py:956, in Text.get_window_extent(self, renderer, dpi)
951 raise RuntimeError(
952 "Cannot get window extent of text w/o renderer. You likely "
953 "want to call 'figure.draw_without_rendering()' first.")
955 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 956 bbox, info, descent = self._get_layout(self._renderer)
957 x, y = self.get_unitless_position()
958 x, y = self.get_transform().transform((x, y))
File ~/.local/lib/python3.12/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File ~/.local/lib/python3.12/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File ~/.local/lib/python3.12/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File ~/.local/lib/python3.12/site-packages/matplotlib/backends/backend_svg.py:1287, in RendererSVG.get_text_width_height_descent(self, s, prop, ismath)
1285 def get_text_width_height_descent(self, s, prop, ismath):
1286 # docstring inherited
-> 1287 return self._text2path.get_text_width_height_descent(s, prop, ismath)
File ~/.local/lib/python3.12/site-packages/matplotlib/textpath.py:52, in TextToPath.get_text_width_height_descent(self, s, prop, ismath)
49 fontsize = prop.get_size_in_points()
51 if ismath == "TeX":
---> 52 return TexManager().get_text_width_height_descent(s, fontsize)
54 scale = fontsize / self.FONT_SCALE
56 if ismath:
File ~/.local/lib/python3.12/site-packages/matplotlib/texmanager.py:363, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
361 if tex.strip() == '':
362 return 0, 0, 0
--> 363 dvifile = cls.make_dvi(tex, fontsize)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
File ~/.local/lib/python3.12/site-packages/matplotlib/texmanager.py:295, in TexManager.make_dvi(cls, tex, fontsize)
293 with TemporaryDirectory(dir=cwd) as tmpdir:
294 tmppath = Path(tmpdir)
--> 295 cls._run_checked_subprocess(
296 ["latex", "-interaction=nonstopmode", "--halt-on-error",
297 f"--output-directory={tmppath.name}",
298 f"{texfile.name}"], tex, cwd=cwd)
299 (tmppath / Path(dvifile).name).replace(dvifile)
300 return dvifile
File ~/.local/lib/python3.12/site-packages/matplotlib/texmanager.py:254, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
--> 254 raise RuntimeError(
255 f'Failed to process string with tex because {command[0]} '
256 'could not be found') from exc
257 except subprocess.CalledProcessError as exc:
258 raise RuntimeError(
259 '{prog} was not able to process the following string:\n'
260 '{tex!r}\n\n'
(...)
267 exc=exc.output.decode('utf-8', 'backslashreplace'))
268 ) from None
RuntimeError: Failed to process string with tex because latex could not be found
<Figure size 640x480 with 1 Axes>
Calculate MSE:
mse_lin_reg = np.mean((y - a*x - b)**2)
mse_dummy = np.mean((y - y.mean())**2)
print("Linear regression MSE:", mse_lin_reg)
print("Dummy model MSE:", mse_dummy)
Linear regression MSE: 38.48296722989415
Dummy model MSE: 84.41955615616556
Coefficient of determination:
print("R2-score:", 1 - mse_lin_reg / np.mean((y - y.mean())**2))
print("Dummy R2-score:", 1 - mse_dummy / np.mean((y - y.mean())**2))
R2-score: 0.5441462975864798
Dummy R2-score: 0.0
Of course, the linear regression line can be found automatically:
import seaborn as sb
sb.regplot(boston, x="lstat", y="medv",
scatter_kws={"color": "blue", "s": 10}, line_kws={"color": "red"},
).set_title('Boston linear regression');
Error in callback <function _draw_all_if_interactive at 0x7fa5c6881260> (for post_execute), with arguments args (),kwargs {}:
---------------------------------------------------------------------------
FileNotFoundError Traceback (most recent call last)
File ~/.local/lib/python3.12/site-packages/matplotlib/texmanager.py:250, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
249 try:
--> 250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
File /usr/local/lib/python3.12/subprocess.py:466, in check_output(timeout, *popenargs, **kwargs)
464 kwargs['input'] = empty
--> 466 return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,
467 **kwargs).stdout
File /usr/local/lib/python3.12/subprocess.py:548, in run(input, capture_output, timeout, check, *popenargs, **kwargs)
546 kwargs['stderr'] = PIPE
--> 548 with Popen(*popenargs, **kwargs) as process:
549 try:
File /usr/local/lib/python3.12/subprocess.py:1026, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask, pipesize, process_group)
1023 self.stderr = io.TextIOWrapper(self.stderr,
1024 encoding=encoding, errors=errors)
-> 1026 self._execute_child(args, executable, preexec_fn, close_fds,
1027 pass_fds, cwd, env,
1028 startupinfo, creationflags, shell,
1029 p2cread, p2cwrite,
1030 c2pread, c2pwrite,
1031 errread, errwrite,
1032 restore_signals,
1033 gid, gids, uid, umask,
1034 start_new_session, process_group)
1035 except:
1036 # Cleanup if the child failed starting.
File /usr/local/lib/python3.12/subprocess.py:1950, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session, process_group)
1949 err_msg = os.strerror(errno_num)
-> 1950 raise child_exception_type(errno_num, err_msg, err_filename)
1951 raise child_exception_type(err_msg)
FileNotFoundError: [Errno 2] No such file or directory: 'latex'
The above exception was the direct cause of the following exception:
RuntimeError Traceback (most recent call last)
File ~/.local/lib/python3.12/site-packages/matplotlib/pyplot.py:197, in _draw_all_if_interactive()
195 def _draw_all_if_interactive() -> None:
196 if matplotlib.is_interactive():
--> 197 draw_all()
File ~/.local/lib/python3.12/site-packages/matplotlib/_pylab_helpers.py:132, in Gcf.draw_all(cls, force)
130 for manager in cls.get_all_fig_managers():
131 if force or manager.canvas.figure.stale:
--> 132 manager.canvas.draw_idle()
File ~/.local/lib/python3.12/site-packages/matplotlib/backend_bases.py:1893, in FigureCanvasBase.draw_idle(self, *args, **kwargs)
1891 if not self._is_idle_drawing:
1892 with self._idle_draw_cntx():
-> 1893 self.draw(*args, **kwargs)
File ~/.local/lib/python3.12/site-packages/matplotlib/backends/backend_agg.py:388, in FigureCanvasAgg.draw(self)
385 # Acquire a lock on the shared font cache.
386 with (self.toolbar._wait_cursor_for_draw_cm() if self.toolbar
387 else nullcontext()):
--> 388 self.figure.draw(self.renderer)
389 # A GUI class may be need to update a window using this draw, so
390 # don't forget to call the superclass.
391 super().draw()
File ~/.local/lib/python3.12/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File ~/.local/lib/python3.12/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/.local/lib/python3.12/site-packages/matplotlib/figure.py:3154, in Figure.draw(self, renderer)
3151 # ValueError can occur when resizing a window.
3153 self.patch.draw(renderer)
-> 3154 mimage._draw_list_compositing_images(
3155 renderer, self, artists, self.suppressComposite)
3157 for sfig in self.subfigs:
3158 sfig.draw(renderer)
File ~/.local/lib/python3.12/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/.local/lib/python3.12/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/.local/lib/python3.12/site-packages/matplotlib/axes/_base.py:3034, in _AxesBase.draw(self, renderer)
3031 for spine in self.spines.values():
3032 artists.remove(spine)
-> 3034 self._update_title_position(renderer)
3036 if not self.axison:
3037 for _axis in self._axis_map.values():
File ~/.local/lib/python3.12/site-packages/matplotlib/axes/_base.py:2978, in _AxesBase._update_title_position(self, renderer)
2976 top = max(top, bb.ymax)
2977 if title.get_text():
-> 2978 ax.yaxis.get_tightbbox(renderer) # update offsetText
2979 if ax.yaxis.offsetText.get_text():
2980 bb = ax.yaxis.offsetText.get_tightbbox(renderer)
File ~/.local/lib/python3.12/site-packages/matplotlib/axis.py:1336, in Axis.get_tightbbox(self, renderer, for_layout_only)
1333 renderer = self.figure._get_renderer()
1334 ticks_to_draw = self._update_ticks()
-> 1336 self._update_label_position(renderer)
1338 # go back to just this axis's tick labels
1339 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
File ~/.local/lib/python3.12/site-packages/matplotlib/axis.py:2609, in YAxis._update_label_position(self, renderer)
2605 return
2607 # get bounding boxes for this axis and any siblings
2608 # that have been set by `fig.align_ylabels()`
-> 2609 bboxes, bboxes2 = self._get_tick_boxes_siblings(renderer=renderer)
2610 x, y = self.label.get_position()
2611 if self.label_position == 'left':
File ~/.local/lib/python3.12/site-packages/matplotlib/axis.py:2161, in Axis._get_tick_boxes_siblings(self, renderer)
2159 axis = ax._axis_map[name]
2160 ticks_to_draw = axis._update_ticks()
-> 2161 tlb, tlb2 = axis._get_ticklabel_bboxes(ticks_to_draw, renderer)
2162 bboxes.extend(tlb)
2163 bboxes2.extend(tlb2)
File ~/.local/lib/python3.12/site-packages/matplotlib/axis.py:1315, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/.local/lib/python3.12/site-packages/matplotlib/text.py:956, in Text.get_window_extent(self, renderer, dpi)
951 raise RuntimeError(
952 "Cannot get window extent of text w/o renderer. You likely "
953 "want to call 'figure.draw_without_rendering()' first.")
955 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 956 bbox, info, descent = self._get_layout(self._renderer)
957 x, y = self.get_unitless_position()
958 x, y = self.get_transform().transform((x, y))
File ~/.local/lib/python3.12/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File ~/.local/lib/python3.12/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File ~/.local/lib/python3.12/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File ~/.local/lib/python3.12/site-packages/matplotlib/backends/backend_agg.py:213, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
211 _api.check_in_list(["TeX", True, False], ismath=ismath)
212 if ismath == "TeX":
--> 213 return super().get_text_width_height_descent(s, prop, ismath)
215 if ismath:
216 ox, oy, width, height, descent, font_image = \
217 self.mathtext_parser.parse(s, self.dpi, prop)
File ~/.local/lib/python3.12/site-packages/matplotlib/backend_bases.py:652, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
648 fontsize = prop.get_size_in_points()
650 if ismath == 'TeX':
651 # todo: handle properties
--> 652 return self.get_texmanager().get_text_width_height_descent(
653 s, fontsize, renderer=self)
655 dpi = self.points_to_pixels(72)
656 if ismath:
File ~/.local/lib/python3.12/site-packages/matplotlib/texmanager.py:363, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
361 if tex.strip() == '':
362 return 0, 0, 0
--> 363 dvifile = cls.make_dvi(tex, fontsize)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
File ~/.local/lib/python3.12/site-packages/matplotlib/texmanager.py:295, in TexManager.make_dvi(cls, tex, fontsize)
293 with TemporaryDirectory(dir=cwd) as tmpdir:
294 tmppath = Path(tmpdir)
--> 295 cls._run_checked_subprocess(
296 ["latex", "-interaction=nonstopmode", "--halt-on-error",
297 f"--output-directory={tmppath.name}",
298 f"{texfile.name}"], tex, cwd=cwd)
299 (tmppath / Path(dvifile).name).replace(dvifile)
300 return dvifile
File ~/.local/lib/python3.12/site-packages/matplotlib/texmanager.py:254, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
--> 254 raise RuntimeError(
255 f'Failed to process string with tex because {command[0]} '
256 'could not be found') from exc
257 except subprocess.CalledProcessError as exc:
258 raise RuntimeError(
259 '{prog} was not able to process the following string:\n'
260 '{tex!r}\n\n'
(...)
267 exc=exc.output.decode('utf-8', 'backslashreplace'))
268 ) from None
RuntimeError: Failed to process string with tex because latex could not be found
---------------------------------------------------------------------------
FileNotFoundError Traceback (most recent call last)
File ~/.local/lib/python3.12/site-packages/matplotlib/texmanager.py:250, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
249 try:
--> 250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
File /usr/local/lib/python3.12/subprocess.py:466, in check_output(timeout, *popenargs, **kwargs)
464 kwargs['input'] = empty
--> 466 return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,
467 **kwargs).stdout
File /usr/local/lib/python3.12/subprocess.py:548, in run(input, capture_output, timeout, check, *popenargs, **kwargs)
546 kwargs['stderr'] = PIPE
--> 548 with Popen(*popenargs, **kwargs) as process:
549 try:
File /usr/local/lib/python3.12/subprocess.py:1026, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask, pipesize, process_group)
1023 self.stderr = io.TextIOWrapper(self.stderr,
1024 encoding=encoding, errors=errors)
-> 1026 self._execute_child(args, executable, preexec_fn, close_fds,
1027 pass_fds, cwd, env,
1028 startupinfo, creationflags, shell,
1029 p2cread, p2cwrite,
1030 c2pread, c2pwrite,
1031 errread, errwrite,
1032 restore_signals,
1033 gid, gids, uid, umask,
1034 start_new_session, process_group)
1035 except:
1036 # Cleanup if the child failed starting.
File /usr/local/lib/python3.12/subprocess.py:1950, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session, process_group)
1949 err_msg = os.strerror(errno_num)
-> 1950 raise child_exception_type(errno_num, err_msg, err_filename)
1951 raise child_exception_type(err_msg)
FileNotFoundError: [Errno 2] No such file or directory: 'latex'
The above exception was the direct cause of the following exception:
RuntimeError Traceback (most recent call last)
File ~/.local/lib/python3.12/site-packages/IPython/core/formatters.py:340, in BaseFormatter.__call__(self, obj)
338 pass
339 else:
--> 340 return printer(obj)
341 # Finally look for special method names
342 method = get_real_method(obj, self.print_method)
File ~/.local/lib/python3.12/site-packages/IPython/core/pylabtools.py:152, in print_figure(fig, fmt, bbox_inches, base64, **kwargs)
149 from matplotlib.backend_bases import FigureCanvasBase
150 FigureCanvasBase(fig)
--> 152 fig.canvas.print_figure(bytes_io, **kw)
153 data = bytes_io.getvalue()
154 if fmt == 'svg':
File ~/.local/lib/python3.12/site-packages/matplotlib/backend_bases.py:2164, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
2161 # we do this instead of `self.figure.draw_without_rendering`
2162 # so that we can inject the orientation
2163 with getattr(renderer, "_draw_disabled", nullcontext)():
-> 2164 self.figure.draw(renderer)
2165 if bbox_inches:
2166 if bbox_inches == "tight":
File ~/.local/lib/python3.12/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File ~/.local/lib/python3.12/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/.local/lib/python3.12/site-packages/matplotlib/figure.py:3154, in Figure.draw(self, renderer)
3151 # ValueError can occur when resizing a window.
3153 self.patch.draw(renderer)
-> 3154 mimage._draw_list_compositing_images(
3155 renderer, self, artists, self.suppressComposite)
3157 for sfig in self.subfigs:
3158 sfig.draw(renderer)
File ~/.local/lib/python3.12/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File ~/.local/lib/python3.12/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File ~/.local/lib/python3.12/site-packages/matplotlib/axes/_base.py:3034, in _AxesBase.draw(self, renderer)
3031 for spine in self.spines.values():
3032 artists.remove(spine)
-> 3034 self._update_title_position(renderer)
3036 if not self.axison:
3037 for _axis in self._axis_map.values():
File ~/.local/lib/python3.12/site-packages/matplotlib/axes/_base.py:2978, in _AxesBase._update_title_position(self, renderer)
2976 top = max(top, bb.ymax)
2977 if title.get_text():
-> 2978 ax.yaxis.get_tightbbox(renderer) # update offsetText
2979 if ax.yaxis.offsetText.get_text():
2980 bb = ax.yaxis.offsetText.get_tightbbox(renderer)
File ~/.local/lib/python3.12/site-packages/matplotlib/axis.py:1336, in Axis.get_tightbbox(self, renderer, for_layout_only)
1333 renderer = self.figure._get_renderer()
1334 ticks_to_draw = self._update_ticks()
-> 1336 self._update_label_position(renderer)
1338 # go back to just this axis's tick labels
1339 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
File ~/.local/lib/python3.12/site-packages/matplotlib/axis.py:2609, in YAxis._update_label_position(self, renderer)
2605 return
2607 # get bounding boxes for this axis and any siblings
2608 # that have been set by `fig.align_ylabels()`
-> 2609 bboxes, bboxes2 = self._get_tick_boxes_siblings(renderer=renderer)
2610 x, y = self.label.get_position()
2611 if self.label_position == 'left':
File ~/.local/lib/python3.12/site-packages/matplotlib/axis.py:2161, in Axis._get_tick_boxes_siblings(self, renderer)
2159 axis = ax._axis_map[name]
2160 ticks_to_draw = axis._update_ticks()
-> 2161 tlb, tlb2 = axis._get_ticklabel_bboxes(ticks_to_draw, renderer)
2162 bboxes.extend(tlb)
2163 bboxes2.extend(tlb2)
File ~/.local/lib/python3.12/site-packages/matplotlib/axis.py:1315, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1313 if renderer is None:
1314 renderer = self.figure._get_renderer()
-> 1315 return ([tick.label1.get_window_extent(renderer)
1316 for tick in ticks if tick.label1.get_visible()],
1317 [tick.label2.get_window_extent(renderer)
1318 for tick in ticks if tick.label2.get_visible()])
File ~/.local/lib/python3.12/site-packages/matplotlib/text.py:956, in Text.get_window_extent(self, renderer, dpi)
951 raise RuntimeError(
952 "Cannot get window extent of text w/o renderer. You likely "
953 "want to call 'figure.draw_without_rendering()' first.")
955 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 956 bbox, info, descent = self._get_layout(self._renderer)
957 x, y = self.get_unitless_position()
958 x, y = self.get_transform().transform((x, y))
File ~/.local/lib/python3.12/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File ~/.local/lib/python3.12/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File ~/.local/lib/python3.12/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File ~/.local/lib/python3.12/site-packages/matplotlib/backends/backend_svg.py:1287, in RendererSVG.get_text_width_height_descent(self, s, prop, ismath)
1285 def get_text_width_height_descent(self, s, prop, ismath):
1286 # docstring inherited
-> 1287 return self._text2path.get_text_width_height_descent(s, prop, ismath)
File ~/.local/lib/python3.12/site-packages/matplotlib/textpath.py:52, in TextToPath.get_text_width_height_descent(self, s, prop, ismath)
49 fontsize = prop.get_size_in_points()
51 if ismath == "TeX":
---> 52 return TexManager().get_text_width_height_descent(s, fontsize)
54 scale = fontsize / self.FONT_SCALE
56 if ismath:
File ~/.local/lib/python3.12/site-packages/matplotlib/texmanager.py:363, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
361 if tex.strip() == '':
362 return 0, 0, 0
--> 363 dvifile = cls.make_dvi(tex, fontsize)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
File ~/.local/lib/python3.12/site-packages/matplotlib/texmanager.py:295, in TexManager.make_dvi(cls, tex, fontsize)
293 with TemporaryDirectory(dir=cwd) as tmpdir:
294 tmppath = Path(tmpdir)
--> 295 cls._run_checked_subprocess(
296 ["latex", "-interaction=nonstopmode", "--halt-on-error",
297 f"--output-directory={tmppath.name}",
298 f"{texfile.name}"], tex, cwd=cwd)
299 (tmppath / Path(dvifile).name).replace(dvifile)
300 return dvifile
File ~/.local/lib/python3.12/site-packages/matplotlib/texmanager.py:254, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
--> 254 raise RuntimeError(
255 f'Failed to process string with tex because {command[0]} '
256 'could not be found') from exc
257 except subprocess.CalledProcessError as exc:
258 raise RuntimeError(
259 '{prog} was not able to process the following string:\n'
260 '{tex!r}\n\n'
(...)
267 exc=exc.output.decode('utf-8', 'backslashreplace'))
268 ) from None
RuntimeError: Failed to process string with tex because latex could not be found
<Figure size 640x480 with 1 Axes>
Linear regression from sklearn:
from sklearn.linear_model import LinearRegression
LR = LinearRegression()
x_reshaped = x.values.reshape(-1, 1)
LR.fit(x_reshaped, y)
print("intercept:", LR.intercept_)
print("slope:", LR.coef_[0])
print("r-score:", LR.score(x_reshaped, y))
print("MSE:", np.mean((LR.predict(x_reshaped) - y) ** 2))
intercept: 34.5538408793831
slope: -0.9500493537579906
r-score: 0.5441462975864797
MSE: 38.48296722989415
Compare this with dummy model:
dummy_mse = np.mean((y - y.mean())**2)
print(dummy_mse)
84.41955615616556
Exercises#
Prove that \(\frac 1n \sum\limits_{i=1}^n (x_i - \overline {\boldsymbol x})^2 = \overline {\boldsymbol x^2} - (\overline {\boldsymbol x})^2\) where \(\overline {\boldsymbol x^2} = \frac 1n\sum\limits_{i=1}^n x_i^2\).
TODO
Finish analytical derivation of \(a\) and \(b\)
Add some quizzes
Add more datasets (may me even simulated)
Think about cases where the performance of linear model is poor